EPA/600/8-89/102a
January 1990
MOUSE
A COMPUTERIZED UNCERTAINTY ANALYSIS SYSTEM
OPERATIONAL MANUAL
by
Albert J. Klee
Risk Reduction Engineering Laboratory
U.S. Environmental Protection Agency
Cincinnati, Ohio 45268
RISK REDUCTION ENGINEERING LABORATOR\
OFFICE OF RESEARCH AND DEVELOPMENT
U.S. ENVIRONMENTAL PROTECTION AGENCY
CINCINNATI, OHIO 45268
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TECHNICAL REPORT DATA
(PUasr rt»d Immienom on tht rrrent btfo>t complerini)
1 REPORT NO. , 2.
EPA/600/S--89/102a
3. RECIPIENTS ACCESSION NO.
PB9Q~l7?^n
*. title and subtitle
MOUSE - A Computer-ized Uncertainty Analysis System
5. REPORT DATE
January 1990
6. PERFORMING ORGANIZATION COSE
?. AUTMORtS)
Albert J. Klee
i. PERFORMING ORGANIZATION REPORT NO.
EPA/SW/DK-90/O19a
9. PERFORMING ORGANIZATION NAME. AND ADDRESS
Risk Reduction Engineering Laboratory--Cincinnati OH ¦
Office of Research and Devel opment
U.S. Environmental Protection Agency
Cincinnati, OH
10. program element NO.
D109N32G2
11. CONTRAC+Z&fUNT NO.
InHouse
12. SPONSORING AGENCY NAME AND ADDRESS
Risk Reduction Engineering Laboratory—Cincinnati, OH
Office of Research and Development
U.S. Environmental Protection Agency
Cincinnati, OH
13. TYPE Of REPORT *NO PERIOD COVERED
Complete
1«. SPONSORING AGENCV CODE
EPA/600/14
15. SUPPLEMENTARY NOTES
P.O. Albert J, Klee Phone: (513} 569-7493 FTS 684-7493
For diskette see: PB90-50f?7n
16. ABSTRACT
Environmental engineering calculations involving uncertain-
ties are far beyond the capabilities of hand analysis for any but the
simplest of models. There exist a number of computer simulation lan-
guages that certainly can do the job but learning such languages and
implementing them quickly and error-free on personal computers is not
the quickest or easiest of tasks. HOUSE {an acronym for Modular Orient-
ed Uncertainty System) deals with the problem of uncertainties in
models that consist of one or more algebraic equations. It was espe-
cially designed for use by those with little or no knowledge of comput-
er languages or programming. It is compact (and thus can be run on
almost any personal computer), easy and fast to learn, and has most of
the features for substantive uncertainty analysis (built-in probabili-
ty distributions, plotting and graphing capabilities, sensitivity
analysis, interest functions for cost analyses, etc.). Moreover, a
series of unique companion utility programs not only write much of the
necessary computer code for the user and help in analyzing sample data
to determine the probability distributions that best suit that data,
but they check each program for errors in syntax, and assist in find-
ing logical errors in the models under study.
1?, KEY WQRDS AND DOCUMENT ANALYSIS
a DESCRIPTORS
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c. COSaTi Field/Gjoup
is, distribution statement
RELEASE TO PUBLIC
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UNCLASSIFTFn
SI. NO. OF PAGES
233
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UNCLASSIFIED
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fl/j
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DISCLAIMER
This report has been reviewed by the U.S. Environmental
Protection Agency and approved for publication. Approval does
not signify that the contents necessarily reflect the views and
policies of the U.S. Environmental Protection Agency, nor does
mention of trade names or commercial products constitute endorse-
ment or recommendation of use.
ii
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FOREWORD
Today's rapidly developing and changing technologies and
industrial products and practices frequently carry with them the
increased generation of materials that, if improperly dealt with,
can threaten both public health and the environment. The U.S.
Environmental Protection Agency is charged by Congress with
protecting the Nation's land, air, and water resources. Under a
mandate of national environmental laws, the agency strives to
formulate and implement actions leading to a compatible balance
between human activities and the ability of natural systems to
support and nurture life. These laws direct the EPA to perform
research to define our environmental problems, measure the im-
pacts, and search for solutions.
The Risk Reduction Engineering Laboratory is responsible for
planning, implementing, and managing research, development, and
demonstration programs to provide an authoritative, defensible
A Jl fit U£¦)uXT Li O JL wll6 Q JL JL w JL t2q f XT Oy J. gLHib j u JT1Q
regulations of the EPA with respect to drinking water, waste-
water, pesticides, toxic substances, solid and hazardous wastes,
and Superfund-related activities. This publication is one of the
products of that research and provides a vital communication link
between the researcher and the user community.
This manual describes a system, called MOUSE (for Modular
Oriented Sncertainty £yst£m), for dealing with the computational
problems of uncertainty, specifically in models that consist of a
set of one or more algebraic equations. Since such models are
frequently encountered in the fields of environmental science,
risk analysis, economics, and engineering, the system has broad
application throughout these fields. No prior programming knowl-
edge is needed to learn MOUSE and, because of its transportabili-
ty and compactness, the system can be run on a wide variety of
personal computers available to the U.S. Environmental Protection
Agency and/or its contractors and grantees.
For further information, please contact the Waste Minimiza-
tion, Destruction and Disposal Research Division of the Risk
Reduction Engineering Laboratory.
E. Timothy Oppelt, Director
Risk Reduction Engineering Laboratory
iii
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ABSTRACT
Environmental engineering calculations involving uncertain-
ties, either in the model itself or in the data, are far beyond
the capabilities of conventional analysis for any but the sim-
plest of models. There exist a number of general-purpose computer
simulation languages, using Monte Carlo methods, that are capable
of such analysis, but these languages are difficult to learn and
to implement quickly.
MOUSE (an acronym for Modular £riented uncertainty £yst£m)
deals with the problem of uncertainties in models that consist of
one or more algebraic equations. It was especially designed for
use by those with little or no knowledge of computer languages,
programming, or simulation. It is designed to be run on almost
any personal computer, easy and fast to learn, and has all of the
features needed for substantive uncertainty analysis (built-in
probability distributions, plotting and graphing capabilities,
sensitivity analysis, interest functions for cost analyses,
etc.). Moreover, a series of unique companion utility programs
write much of the necessary computer code for the user, help in
analyzing sample data to determine the probability distributions
that best fit those data, check each program for errors in syn-
tax, and assist in finding logical errors in the model that is
subject to uncertainty.
Some typical examples of the use of MOUSE within the U.S.
Environmental Protection Agency include; studying the migration
of pollution plumes in streams, establishing regulations for
hazardous wastes in landfills, and estimating pollution control
costs.
iv
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PREFACE
This manual describes, using a new personal computer-based
programming language called MOUSE (for Modular firiented Uncer-
tainty SvstEm), how to deal with the problem of uncertainty,
specifically in models that consist of a set of one or more
algebraic equations. Such models are frequently encountered in
the fields of environmental science, risk analysis, economics,
and engineering. No prior programming knowledge is needed to
learn MOUSE and, because of its transportability and compactness,
the system can be run on a wide variety of PC-DOS or MS-DOS
personal computers available to the U.S. Environmental Protection
Agency and/or its contractors and grantees. (The basic require-
ment is that the system have a FORTRAN compiler.)
The manual introduces the concept of uncertainty and its
effects, discuss the implications of traditional solutions to
the problem, and compares the alternative solutions available. A
step-by-step approach, via examples, is taken to introduce the
various features of MOUSE. The reader's understanding of these
features is tested by numerous exercises, to which the solutions
are given. Writing computer programs is often a tedious process,
and one that is seldom error-free. To this end, several utilities
are provided that make life easier for the user. One utility,
TRAP, actually writes a great deal of the program while another,
CHECKER, examines it for errors. Two additional utilities, TRACER
and MSHELP, provide additional assistance in writing and debug-
ging MOUSE programs.
The determination of probability distributions is perhaps
the most difficult concept for newcomers to the field to
master. The manual explores the methods by which the user can
determine the probability distributions to be used in his model,
whether they be subjective or based upon empirical data.
In addition, an additional computer-based system, called IMP
(for interactive Modeler for Probabilities), is introduced as an
aid in performing the data analysis required in the process.
The concept of simulation is not a new one, of course, but
any attempt to apply simulation usually entails utilizing profes-
sionals who are familiar with simulation and simulation languages
or else training in-house or contractor staff to do the same.
Limited resources often preclude the former; the latter is not
cost-effective unless a professional is expected to devote a
major portion of his or her time to such matters. The concepts
of simulation, coupled with the necessity to learn one of the
v
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standard simulation languages such as SIMSCRIPT or SLAM, are not
skills transferable within a short period of time, however.
Furthermore, without constant reinforcement through frequent use,
such skills are quickly forgotten. MOUSE, on the other hand, was
designed to make the -writing of computer simulation models,
based upon algebraic equations, as simple as possible.
Although some previous exposure to probability and statis-
tics would be helpful, readers without formal training in these
areas will not find the material difficult. The exposition has
deliberately been kept on a tutorial level to facilitate imple-
mentation.
NOTE: The author would appreciate it if any errors found in this
manual and/or bugs in any of the programs are reported to;
"i -r rrt
Ai&erx J, Kiee
Risk Reduction Engineering Laboratory
US Environmental Protection Agency
26 Martin Luther King Drive
Cincinnati, Ohio 45268
513/569-7493
vi
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CONTENTS
Page
Chapter 1 EXPLORING UNCERTAINTY
1.1 Defining the Problem l
1.2 Consequences of Failure to Deal With Uncertainty.. 1
1.3 Traditional Approaches to Uncertainty. 2
1.4 Problems with Traditional Approaches...... 3
1.5 Alternative Solutions to the Uncertainty Problem.. 5
1.6 Traditional Sensitivity and Uncertainty
Sensitivity Analysis Compared. 9
1.7 Languages For A Computerized Uncertainty
Analysis System. 9
Addendum. 15
References 17
1 V71UDT WO fi *1? TT'MII'V "AVUT VflTfl
V»JUflLCr til"VUG mi ISi«iJfl JLiCiD VJ<1T URWaAI A**l
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4.9 Notes On The Generation Of Stochastic Variates
in MOUSE and Checks on MOUSE Probability
Call Syntax. — 55
4.10 Fitting Classical Probability Distributions
With IMP. 55
4.11 Fitting Empirical Continuous Probability
Distributions With IMP 63
1 £! "7
References 69
Chapter 5 MOUSE LIBRARY INTEREST FUNCTIONS
5.1 Introduction 71
5.2 Compound Interest 71
5.3 Gradient Series.... 74
5.4 Nominal Interest Rates. 74
5.5 Continuous Compounding. 75
5.6 Nominal Interest Rates, Continuous Compounding.... 76
5.7 Example of Probability Function Calls and MOUSE
Interest Call Syntax 77
Exercises " 7a
erei^ce78
Chapter 6 THE MOUSE TABLE
6.1 Introduction............. 79
6.2 The MOUSE-Dimensioned Table. 80
6.3 The User—Dimensioned Table 81
6.4 The Short Table 82
6.5 The Dummy Table 82
6.6 User Table Scaling...... 83
6.7 Literal Names in MOUSE.. 83
6.8 Mixing Table Types84
Exercises84
References 84
Chapter 7 USER CONTROL DECISIONS
7.1 Introduction.. 85
'7.2 The Iteration Number Line......................... 85
7.3 The Snap Interval Line. 85
7.4 The Seed Reset Line 86
7.5 The Output Destination and Column Output Lines.... 88
7.6 Summary of the Specification Block........ 89
7.7 Multiple Runs: Matrices and Vectors 89
7.8 Multiple Runs: The MOUSE Pass 92
viii
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7.9 Program Output Titles: The Name Line . 93
7.10 The Program Listing Line 95
7.11 Preparing and Checking Specification Block Lines.. 95
iff"'' i C5L4S ts E
CiAtSi.vJ.0C0t 7 v
Reference..........a................................... 96
Chapter 8 ADDITIONAL MOUSE FUNCTIONS
8.1 Introduction 97
8.2 The Plot Output Line.. 97
8.3 MOUSE Automated Sensitivity: Introduction 101
8.4 The Sensitivity Control Line... 102
8.5 Sensitivity Level Line 104
Exercises105
Chapter 9 CORRELATED STOCHASTIC INPUT VARIABLES
9.1 Introduction......... 106
9.2 Linear Correlation 106
9.3 Eivariate Normal Distribution.. 108
9\4 Bivariate Lognormal Distribution..... 112
9.5 Using IMP to Analyze Bivariate Normal And
Bivariate Lognormal Distributions 114
9.6 Time Series and Autocorrelation. 116
Exercises. .»»..« .*• 1*. ••«••«».#• 119
- References ................ 119
Chapter 10 NON-UNIFORM CASH FLOWS, VECTORS, AND THE
INTERNAL-RATE-OF-RETURN FUNCTION
10.1 Introduction 121
10.2 The Present Worth Sum Function 123
10.3 Calculating The Elements of a Vector 124
10.4 The Use of the IF Statement in Vector Loops 125
10.5 Vector Loops and Order of Statements 126
10.6 The Vector Table. 126
10.7 Vectors, The MOUSE Pass, And Plots ... 131
Exercises. 135
Chapter XI Troubleshooting# Error Messages, An4 Help
11.1 Introduction..... 137
11.2 CHECKER, A Utility for Catching MOUSE
Syntactical Errors. 137
IX
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11.3 A Case History. 141
11.4. MOUSE Run-Tine Error Checks 143
11.5 The MOUSE Trace. 147
11.6 TRAP, A utility For Preparing MOUSE
Source Programs .,.., 151
11.7 Using TRAP To Prepare MOUSE Data Files........... 156
11.8 MSHELP - Another MOUSE Help Utility 158
Exercises .. 159
Chapter 12 FITTING PROBABILITY DISTRIBUTIONS
TO SAMPLE DATA
12.1 Introduction........... .. 160
12.2 ' Goodness-of-fit Tests. . 162
12.3 Subjective Determination Of Probability
Distributions.. 163
12.4 The Variable Interval Method 163
12.5 The Fixed Interval Method. 165
12.6 Other Methods 166
12.7 Improving Elicitation Methods. 168
12.8 Some Useful Distributions for Subjective
Methods. 172
^ References. 174
Appendix A: RUNNING MOUSE, IMP, AND TRAP PROGRAMS
A. 1 Introduction 176
A.2 Running CHECKER, IMP, TRACER, and TRAP on
PC-DOS or MS-DOS Computers. 177
A.3 Running MOUSE On PC-DOS or MS-DOS Computers....... 177
Appendix B; INSTALLATION AND CUSTOMIZATION NOTES FOR MOUSE
B.l Installation. 180
B.2 Compilation and Linking With Libraries...... 181
B.3 Adding Modules to MOUSE.. 183
References 184
Appendix C: SUMMARY TABLES
Table C.l: MOUSE Control Lines 185
Table C.2: Alphabetical List of MOUSE Interest
Functions 186
Table C.3: Alphabetical List of MOUSE Probability
Functions..; 187
Table C.4: MOUSE Line Types 188
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Appendix D:
MOOSE Error Messages. 189
Appendix E: ANSWERS TO EXERCISES
Chapter 3 .. 192
Chapter 4 . 192
Chapter 5 193
Chapter 6... 194
Chapter 7 194
Chapter 8 195
Chapter 9 195
Chapter 10... 195
Chapter 11.......... 198
Appe&dix FJ MOUSE RANDOM NUMBER TESTER
F.l Introduction....... 199
F.2 Gap Test.. 200
F.3 Two-dimensional serial test....... 201
F.4 Kolmogorov-smirnov test 201
F.5 Poker test 201
F.6 Permutation test 201
F. 7 Runs test 202
F.8 Maximum-of-T test... 202
F.9 D-square test.. ,.. 202
F.10 Serial correlations and correlograms. 202
F.11 Lag test203
I&dex»........ ................................... .<*...... 204
XI
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LIST OF FIGURES
Kuober Page
1.1: Examples of Uncertainty 4
1.2: Direct (Complete) Enumeration 6
1.3: Probability Calculus Approach..... 8
1.4: Monte Carlo Simulation 10
1.5: Sensitivity Analysis: Traditional Method..... 10
1.6: Sensitivity Analysis: Uncertainty Method. 11
2.1: MOUSE Program for the Waste Oil Relamation
Problem .............a............................... 23
2.2: Partial Echoed Listing of Empirical Input
Data Variables, Waste Oil Relamation
Problem 24
2.3: Statistics, Histogram, and Graphs Produced
by MOUSE for the Waste Oil Relamation
Problem — 25
2.4: Schematic Diagram of the Model For The
Regulatory Problem....... . 27
2.5: MOUSE Program for the Regulatory Problem...... 29
2.6: Statistics, Histogram And Graphs Produced
By MOUSE For The Regulatory Problem — ......... 30
3.1: General Overview of a MOUSE Program................... 32
3.2: Commented MOUSE Program For The Waste Oil
Reclamation Problem. 33
3.3: Commented MOUSE Program For The
Regulatory Problem 34
4.1: The M0USE.DAT Data File Containing the
Empirical Variable Data Lines for the
Waste Oil Reclamation Problem. 42
4.2: Example Of A Step Rectangular Probability
Distribution.. 44
4.3: Partial Data file for Fitting a Classical
Distribution Using IMP 57
4.4: Imp Plots Of Fitted Distibutions To
Original Data - 59
4.5: Sample IMP Output, Continuous Distribution. 60
4.6: Sample IMP Output, Discrete Distribution 62
4.7: Integrating An Empirical Continuous
Probability Distribution 64
4.8: Input Data for an Empirical Continuous
Probability Distribution........ 65
4.9: IMP Output, Integrated Continuous Empirical
Distrlbution, Data Of Fxyure 4.S.66
4.10: Empirical Continuous Probability
Distribution For Exercise Number 17. 70
xii
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7.1; Waste Oil Reclamation Problem,
Modification #1. 87
7.2; Snap Output For Waste Oil Reclamation
Problem, Modification #1 88
7.3: An Outline Of The MOUSE Pass 93
7.4: Partial PASS Output For Waste Oil Reclamation
Problem, Modification #2 94
8.1: Regulatory Problem Modification #1 98
8.2: Plot output For Regulatory Problem,
Modification #1 100
8.3: Sensitivity Output For The Regulatory Problem,
Modification f 1 103
9.1: Portion Of Example IMP Data File
For Analysis Of A Bivariate Distribution 108
9.2: IMP Output, Bivariate Distribution,
Data Of Figure 9.1 109
9.3; Portion Of Example IMP Data File For Analysis
Of A Bivariate Normal Distribution 115
9.4: IMP Output, Bivariate Normal Distribution
Analysis, Data of Figure 9.3........... 116
9.5: Portion Of Example IMP Data File For
Analysis Of An Autocorrelation (Exponential)
Time Series. 120
9.6: IMP Output, Autocorrelated (Exponential
Time Series, Data of Figure 9.5............ 120
10.1: MOUSE Program For The Waste Oil Reclamation
Problem, Modification #2 122
10.2; Excerpt From Waste Oil Reclamation Problem,
Modification #2, Showing Lines Relevant To
Vectors, Vector Tables, And Present
Value Sum Function.... 127
10.3: Vector Table Output For Waste Oil
Reclamation Problem, Modification %2 127
10.4: Examples Of Slcewness And Kurtosis..................... 130
10.5: Waste Oil Reclamation Problem,
Modification #3 132
10.6: Output Of The Waste Oil Reclamation Problem,
o^^ i f ica t ion 3 133
10.7: Plot Output For Waste Oil Reclamation
Problem, Modification #2 134
11.1: Original Case History 142
11.2: Corrected Case History Program..... 142
11.3; Table Output of Original Case History Program 144
11.4; Table Output of Corrected Case
History Program 145
11.5: Placement of Trace Lines in Modification #2
of the Waste Oil Reclamation Problem. 149
xiii
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11.6: Output of The Traced Program of Figure 11.5 150
11.7: Output Prom Sample TRAP Run For Waste Oil
Reclamation Problem, Modification #2 157
12.1: Shapes of Some Beta Distributions 173
xiv
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LIST OF TABLES
number Page
1.1: A Comparison of Computer Languages for
Uncertainty Analysis. 16
2.1: Distributions of the Input Stochastic
Variables for the Irrigation Project Problem 20
2.2: Regulatory Levels As A Function of Delmax.. 28
2.3: Distributions of the Input Stochastic Variables
for the Regulatory Problem 28
3.1: Summary of Rules for Use of Decimal Points
in MOUSE. . 36
12.1: Summary of Consensus Techniques 171
C.l: MOUSE Control Lines 185
C.2: Alphabetical List of MOUSE Interest Functions 186
C.3: Alphabetical List of MOUSE Probability
Functions 187
C.4: MOUSE Line Types. 188
xv
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Mouse Manual Page 1
CHAPTER 1
EXPLORING UNCERTAINTY
l.X Defining the Problea
If we define a model as a physical or symbolic representa-
tion of reality, we find among the set of all models one called
the mathematical model of which one particular type consists of a
series of one or more algebraic equations. Mathematical models
of this type are extremely important for they are found almost
everywhere, including the environment, business, economics,
engineering, and science. The use of an equation is understood
by almost everyone,* in a somewhat "inelegant" sense, numbers "go
into" the equation and an answer is obtained. For example,
consider the following very simple equation,
Y = AB [1.1]
where ¥ might stand for an equipment purchase cost, given that A
is the^unit price and B is the number of units required. Alter-
natively, equation 1.1 might represent an engineering calculation
where Y is the cross-sectional area of a heating duct, given that
A is its height and B is its width. In any event, to "solve"
equation 1.1 one has to know the values of A and B. If A is
equal to 2, for example, and B is equal to 15, then Y = 2 x 15 =
30. Often, however, we are not sure of the values of A or of B.
A might be 3 and B might be 20; in such event Y would be equal
to 60, not 30. The greater our uncertainty about the input
variables A and B, clearly the greater our uncertainty about the
output variable, Y.
1.2 Consequences of Failure to Deal vith uncertainty
The "best" estimate of a input variable is what we state if
we were restricted to single numbers for our estimate. (This is
one of the single value approaches to be discussed in Section
1.3.) The use of single values (sometimes known as "point esti-
mates") for input variables, of course, leads to single values or
point estimates for output values. If we ignore uncertainty in
our calculations, the point estimates obtained will almost
assuredly be either too high or too low. Either may have unde-
sirable consequences. In our equipment purchase example, an
overestimate of requirements may result in overstocking the item,
leading to increased inventory carrying costs. If the require-
ments are underestimated, downtime may occur and production
schedules may not be met. In our engineering duct problem, if Y
is underestimated then our heating duct will be inadequate to
carry the airflow required,- if Y is overestimated we will have
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Mouse Manual Page 2
wasted money by building it too large and the reduced air veloci-
ty night encourage the unwanted deposition of particulate matter
along the duct before the air reaches the filter.
It should be pointed out that, depending upon the circum-
stances, overestimation may be more harmful than underestimation
and vice versa. For example, it is probably better to oversize a
duct rather than to undersize it since tearing out existing
ductwork is very expensive. If the inventory carrying charges
for an item stocked is high (an expensive, seldom-used special-
ized pump, for example), it may be better to understock the item
and incur some occasional out-of-stock cost rather than to pay
the high interest charges on unmoved inventory.
1.3 Traditional Approaches to Uncertainty
The most often encountered approaches to uncertainty in
mathematical models are. (1) the best value approach, (2) the
conservative approach, and (3) sensitivity analysis. The first
two are single-value approaches. The "best" in "best value" is
not precisely defined; generally it refers to some measure of
centra! tendency such as an average or a mode. In a probabi-
listic sense, the mode is the "most likely" value in that it has
the highest probability of occurring. If a probability distribu-
tion is not symmetric then the mode differs from the mean; conse-
quently it is not always clear what another person's "best value"
means. In the equipment purchase problem, we might suppose that
the value of A of 2 and of B of 15 are average values. Hope-
fully, the answer of 3 0 is also some sort of average value. As
we shall see in the next section, however, this is not always the
case. The best value approach is an admittedly simple technique
since single values are used for the input numbers to the equa-
tions. In reality, however, the best value approach is more a
matter of ignoring uncertainty that it is of any conscious
effort to come to grips with it.
The conservative approach does make an attempt to consider
the consequences of uncertainty. As the name implies, the input
values selected are not the average or most likely ones but
rather those that produce conservative results with regard to the
consequences of over- or underestimating. For example, in the
conservative approach for the duct example, the values of A and B
selected to go into equation 1.1 would be greater than their
average values of 2 and 15 since, as we have suggested in the
previous section, overestimation is better than underestimation
in this case. If "best" values were used, there would be too
great a chance that the duct area would be underestimated.
A popular traditional approach to the problem of uncertainty
is sensitivity analysis. Traditional sensitivity analysis
usually starts with a best value estimate, followed by a pertur-
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Mouse Manual Page 3
bation or change in one of the input variables (holding all
other input variables at their previous values). The perturba-
tion can be either an increase or a decrease in the value of the
variable and hence can be either of a "conservative" nature or a
"liberal" one. The perturbation is generally within the known
or believed uncertainty range of the variable. The process is
repeated for as many variables and for as great a change as is
desired. For example, to examine the effects of a modest change
in A in equation 1.1, we might increase the value of A by 10%
over its "best" estimate value of 2. A 10% increase in A (to
2.2) results in an estimated Y-value of 33. If a value of A of
2.2 is "reasonably likely" to occur, then the traditional sensi-
tivity analysis suggests that a value of Y of 33 is also "reason-
ably likely" to occur.
1.4 Problems with Traditional Approaches
As has been mentioned before, the best value approach really
does not address the problem of uncertainty at all. Consider the
three hypothetical outcomes for Y shown in Figure 1.1. In all
cases the average value of Y is 30, In Case A, where there is
relatively little variation in Y, there is a high probability
that the actual value will be in the vicinity of the estimated
(by the best value method) value of 30; in Case B, however, this
likelihood is very much smaller. Estimates made by using best
input values, therefore, do not necessarily have high probabili-
ties of occurring. Note that in Case D the most likely value
of Y is not the mean value of 30 but a value considerably higher.
Thus, "uncertainty" is not just variance; it refers to the entire
distribution, including its shape.
There is another difficulty with the best value approach
which is not generally recognized. The difficulty arises when
the algebraic model contains non-linear elements such as multi-
plications or divisions and the variables are correlated. if
variables A and B of equation 1.1 were correlated, for example,
the average of Y would not be equal to the average of A times the
average of B. In point of fact, if A and B were positively
correlated, then the average of Y would be greater than the
product of the averages; conversely, if A and B were negatively
correlated, the average of Y would be less than the product of
the averages (see Addendum to this chapter).
The conservative approach also has its deficiencies. First,
in a complex calculation involving many equations and many input
variables (some of which may be correlated) it may not be obvious
what values of the input variables constitute "conservative" ones
with respect to the output. Second, because conservative input
values generally are those with a low probability of occurring,
the estimates obtained by using such values perforce will not
have a high probability of occurring.
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House Manual Page 4
A
B
C
D
FIGURE 1.1: EXAMPLES OF UNCERTAINTY
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Mouse Manual Page 5
It should be recognized that the point estimates involved m
using either the best value or the conservative approach do not
utilize all of the information that is usually available. One
usually has at least some Idea of the uncertainties in the input
figures. Since a greater loss may occur in a decision from an
underestimate than from an overestimate {or vice versa), there
are other problems with single value approaches as well. For
example, since people like success better than failure, the
activities they undertake are more likely to be the ones for
which they have overestimated their ability to succeed.
Traditional sensitivity analysis, being largely an amalgama-
tion of elements of both the best value and conservative ap-
proaches, suffers the defects of both methods. An arbitrary
change in the value of an input variable, even though the change
falls within the expected range of the variable, tells us little
about the likelihood of occurrence of the new estimate obtained.
In other words, if we know little about the likelihood of such a
change occurring, it follows that we know little about the like-
lihood- of the calculated output occurring. Furthermore, in tradi-
tional sensitivity analysis all other variables are held at their
previous values, the so-called "all other things being equal"
view of the world. The problem is that "all other things" are
seldom equal. In actuality, the change we introduce in a varia-
ble under traditional sensitivity analysis may well be either
mitigated or intensified by what is happening to the other varia-
bles. In short, traditional sensitivity analysis does not show
the combined net effect of changes in all variables or the like-
lihood of various changes occurring together. Viewed in this
manner, the traditional sensitivity analysis can be misleading.
More will be said about sensitivity analysis in Section 1.6.
1.5 Alternative Solutions to the Uncertainty Problem
Let us now examine three alternative solutions to the prob-
lem of uncertainty. The first, illustrated in Figure 1,2, is
Direct or Complete Enumeration. The model of equation 1.1 is
employed, and we assume the uncertainties of A and B as given in
the two probability distributions for these variables shown in
the upper left-hand corner of the figure. In other words, we
suppose that there is a 25% chance that A is equal to 1, 50% that
it is equal to 2, and 25% that it is equal to 3. For B, there is
a 50-50 chance that it is equal to either 10 or 20. (For this
simple example, we assume no correlation between the two input
variables.) In complete enumeration we list all of the possible
combinations of the input variables and then calculate the proba-
bilities of these combinations occurring. In this example there
are 3 choices for A and 2 for B, resulting in 6 possible outcomes
for Y. The probabilities of these combinations are shown in the
middle top of the figure. Since some of the combinations are
duplications, the table of combinations of A and B may be simpli-
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Mouse Manual Page 6
MODEL; Y =AB
A p(A) B p(B)
1 .25 .10 .50
2 .50 20 30
3 ' .25
.401
nv>
.30
.20
.10
.00
£ 1 AB p(AB)
AB p(AB)
1
2
3
1
2
3
10 10
10 20
10 30
20 20
20 40
20 60
.125
.250
.125
.125
.250
.125
10
20
30
40
60
.125
.375
.125
.250
.125
Note: Average = Y = AB = 2(15) = 30
0 10 20 30 40 50 60 Y
PROBABILITY DISTRIBUTION OF Y <= AB)
FIGURE 1.2: DIRECT (COMPLETE) ENUMERATION
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fied to the 5 entries shown at the upper right of the figure.
The average value of Y is shown to be 30. At the bottom of
Figure 1.2 is a graph of the frequency or probability distribu-
tion of Y. Note that the most likely value is not the average
but rather values to either side of it. Furthermore, one of the
extreme values (Y = 60} has a higher probability of occurrence
than has the average value. As can be seen, the complete enumera-
tion method tells us everything about the distribution of Y,
including its mean, standard deviation, minimum, maximum, and the
probability of occurrence of any value of Y.
The second method we shall consider is the Probability
Calculus method. The method, as the name implies, requires some
knowledge of the calculus of probabxlxti.es (sometimes known xn
engineering as the "propagation of error".... see Deming, 1950).
Using the model of equation 1.1 as before, the method is illus-
trated in Figure 1.3. The error formula is given at the top of
the figure, and involves three terms and knowledge of the vari-
ances of A and B, The latter are calculated, as is shown in the
figure, from the probability distributions of A and B given
previously in Figure 1.2. The error formula shows the variance
of Y to be 225. The Probability Calculus method produces no more
than the mean and the variance of the output (i.e., Y) distribu-
tion. That the variance alone is not sufficient to determine the
nature of uncertainty is clear from Figure 1.1. It is entirely
possible that Cases B and C of Figure 1.1 have the same
variance; they may have, however, quite different consequences to
the decision maker.
The third approach to the problem of uncertainty is a form
of Monte Carlo simulation known as Model Samplinct. The term
"Monte Carlo" originated during World War II where it referred to
a method, utilizing random numbers, used for solving certain
problems where the passage of time played no substantive role.
The idea of Model Sampling is relatively simple*
a. A value for each of the input variables is
drawn at random from their respective
probability distributions, and the model
is computed using this particular set of
values.
b. The above process is repeated many times.
Since the results vary with each itera-
tion, the outputs themselves (i.e., the
Y's) are gathered in the form of a proba-
bility distribution. Thus the uncertain-
ties of the model's inputs are transferred
to the output which can then be studied
and subsequently utilized in decision
processes.
-------
MODEL; Y - AB
ERROR FORMULA IS var (AB) = *ar (Y) = A 2 var (BJ + § 2 var (A) + var (A) var (B)
A ~ 2 and B = 15
p(A) A (A - A)2 p(A) (A - A)2
.25 1 1 .25
.50 2 0 .00
.25 3 1 ^
var(A) = .50
p (B) B (B - B)2 p(B) (B - I)2
.50 10 25 12.5
.50 20 25 12.5
var (B)= 25.0
therefore, w (Y) = 22 (25.0) ~ 152 (0.50) ~ (0.50)(25.0) = 225
FIGURE 1.3: PROBABILITY CALCULUS APPROACH
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The procedure is shown schematically in Figure 1.4. The output
(and the accuracy) of the Monte Carlo simulation method becomes
almost equal to that of complete enumeration as the number of
iterations becomes large. The Monte Carlo simulation method is
the one used in this manual and forms the basis for mouse, a
computerized uncertainty analysis system which is the subject of
this manual.
1.6 Traditional Sensitivity and Uncertainty
Sensitivity Analysis compared
A pictorial comparison of traditional sensitivity analysis
with uncertainty sensitivity analysis is shown in Figures 1.5 and
1.6. In traditional sensitivity analysis we alter the location
(i.e., value) of one variable (holding all others at their previ-
ous values) and examine the effect on the output variable (Figure
1.5). This is clearly a two-step procedure (T1 and T2). In
uncertainty sensitivity analysis (Figure 1.6) this is a one-step
procedure (Ul) since all perturbations in X are automatically
reflected in the distribution of Y. The perturbations in the
X-values, unlike those in traditional sensitivity analysis, are
not arbitrary, however? they are performed in accordance with
their likelihoods of occurring.
In uncertainty sensitivity analysis we can not only alter
the value of an input variable, but we can alter its location (U2
in Figure 6), the amount of its uncertainty (U3) and its nature
(U4) . The U3 pair of distributions in Figure 1.6 shows one
possible result. By decreasing the variability of the input
variable by the amount shown, we decrease our uncertainty about
the output by a fairly large amount. The implication is that it
would pay us to reduce the uncertainty in this input variable,
perhaps by obtaining better data. The decrease in the uncertainty
of the output variable could have been much less. If this were
so, we would have concluded that it would not have paid to
invest additional money in reducing the uncertainty of the input
variable. The possibilities for sensitivity analysis using the
uncertainty approach are therefore, far greater than those avail-
able using any single value, traditional approach.
1.7 languages For A Computerized Uncertainty Analysis System
Monte Carlo simulation requires the use of a digital comput-
er for any substantive problem. It is important, therefore, to
consider the desirable characteristics of any computerized uncer-
tainty analysis system:
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Mouse Manual Page 10
METHOD 3: MONTE CARLO SIMULATION
random
START: I = 1
¥
< random
cample
Y| = AB
1
sample
record Yj
repeat n times
FINISH
FROM COLLECTION OF Y'« OBTAIN:
1. MEAN
2. STANDARD DEVIATION
3. COEFFICIENT OF VARIATION
4. MINIMUM
5. MAXIMUM
S. GRAPH OF FREQUENCY
DISTRIBUTION
7. GRAPH OF CUMULATIVE
FREQUENCY DISTRIBUTION
FIGURE 1.4: MONTE CARLO SIMULATION
T1
THIS ORIGINAL VALUE OF X PRODUCES
THIS VALUE OF Y
T2
THIS ALTERED VALUE OF X
PRODUCES
THIS VALUE OF Y
FIGURE 1.5: SENSITIVITY ANALYSIS: TRADITIONAL METHOD
-------
U1
THIS ORIGINAL DISTRIBUTION OF X
PRODUCES
THIS DISTRIBUTION OF Y
U2
THIS ALTERED DISTRIBUTION OF X
(CHANGE IN LOCATION)
PRODUCES
THIS DISTRIBUTION OF Y
U3
THIS ALTERED DISTRIBUTION OF X
(CHANGE IN DISPERSION)
PRODUCES
THIS DISTRIBUTION OF Y
U4
THIS ALTERED DISTRIBUTION OF X
(CHANGE IN SHAPE)
PRODUCES
THIS DISTRIBUTION OF Y
FIGURE 1.6: SENSITIVITY ANALYSIS: UNCERTAINTY METHOD
X
o
c
in
n>
X
(V
3
C
0»
•d
0
iQ
rt>
i
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Mouse Manual Page 12
a. Simplicity
In order to conduct Monte Carlo simulation upon an
environmental, business, economic, engineering or scientific
model, you must communicate the model and any input information
to the computer, and to instruct it as to what is required with
regard to the nature and the output of the analysis. Naturally
enough, this requires a large number of tasks to be accomplished
by the computer. Obviously, it would be highly desirable if you
need not have prior Knowledge of computer programming or even of
computers in general to use such a system. Furthermore, the
programming language used should be simple so that you can spend
his time improving his model and his probability estimates
rather than worrying about the details of programming.
b. Library
An important part of uncertainty analysis is the probability
distributions involved arid, if economic matters form part of the
model,sthe interest functions used. A desirable feature of any
computerized uncertainty analysis system, then, would be a large
library of probability and interest functions.
c. Automatic Output Distributions ¦
The major end product of any uncertainty analysis consists
of the probability distributions of the quantities we are trying
to estimate as well as various statistics concerning these dis-
tributions. It is essential, therefore, that any uncertainty
analysis system provide automatic output of these critical ele-
ments.
Intermediate Output Distributions
In any complex model, the analyst is often interested in
intermediate outputs as well. In an equipment expansion problem,
for example, in addition to the total cost both capital and
operating costs are also of great concern. The uncertainty
analysis system should be able to describe the uncertainties in
any of these intermediate outputs.
e. Generalized Sensitivity Analysis
In Section 1.6 the value of more generalized sensitivity
analysis was discussed. The uncertainty analysis system should
facilitate such types of investigations.
f. De-buqaing Tools
It is easy to lose one's way in a complex model. Errors in
such models are often not easy to find. An answer may look
reasonable but the model or its implementation may still be
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Mouse Manual Page 13
faulty. For small models a hand calculation, line-by-line, is
one way to uncover such errors but for large models this approach
is slow and prone to error. A desirable feature of an
uncertainty analysis system, then, is some painless way to "de-
bug" the program.
g. Correlated Variables
It is true that calculations are much simpler when one
assumes that all of the input variables are uncorrelated. Unfor-
tunately, this rarely occurs in the real world. Supply varies
with price, yield varies with pressure, and sales vary with the
season. Any uncertainty analysis system should be able to accom-
modate correlated variables, i.e., correlation of more than just
the "A = 0.7B" kind.
h. Derivation M Probability Distributions
A critical aspect of uncertainty analysis is the determina-
tion o"f the probability distributions of the input variables.
These may be determined subjectively, by fitting empirical data
to classical probability distributions or by a combination of
these methods. To newcomers to uncertainty analysis this is
often a more formidable task than the construction of the model
itself. A truly useful uncertainty analysis system should pro-
vide some means for helping you derive the probability distribu-
tions required.
i. Modular Expansion
At times you may wish to modify an uncertainty analysis
system to meet special requirements of his own particular model,
e.g., the addition of a probability distribution not included in
the original system. Modular construction of such systems facil-
itates such "tailoring".
It is possible to use general purpose computer languages
such as BASIC, FORTRAN or PASCAL for uncertainty analysis but
the problems are formidable. For one thing, specialized program-
ming knowledge is required and these languages are not mastered
without some effort. There is nothing particularly simple about
either FORTRAN or PASCAL as those who have had to deal with the
mysteries of the FORMAT statement in the former can attest.
There is no such thing as "automatic output" or provision for
sensitivity analysis. If you want these features you must pains-
takingly program them himself. They are, however, transportable
in that compilers or translators are available for these lan-
guages for almost every type of computer.
Another alternative is the specialized computer simulation
programs, foremost among them being GPSS, SIMSCRIPT, and SLAM.
(For references to these simulation languages see Maryanski, I960
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Mouse Manual Page 14
and Pritsker & Pegden, 1979.) One major difficulty with GPSS is
that, although easy to learn, it is based upon integer arith-
metic, a decidedly unhealthy environment for the kind of calcula-
tions needed for uncertainty analysis. GPSS was primarily de-
signed for job-shop problems and is not appropriate to uncertain-
ty analysis. SIMSCRIPT, on the other hand, although perfectly
capable of performing uncertainty analysis, is a complex language
usually taught as a hierarchal set of languages: first, as a
scientific language similar to FORTRAN,* then as a general purpose
language comparable to ALGOL; next as a list processing language
to aid in the creation and handling of data structures? and
finally as a simulator and statistics gathering language. SLAM is
a collection of FORTRAN functions and subroutines that you must
assemble having first a knowledge of FORTRAN.
There are three general approaches taken by general purpose
simulation languages: they are either event-oriented, process-
oriented, equation-oriented or some combination of the three. In
event-orientation, a system is modeled by identifying its charac-
terizing events and then writing a set of event routines that
describe how the state of the system changes whenever an event
occurs. In a simple queuing system, for example, the two major
events are the arrival of a unit to be serviced and the comple-
tion of service of a unit. In process-orientation, the model is
constructed about entities that are "processed" as they move
through the system being modeled. In the simple queuing system,
for example, a unit enters and leaves a queue, enters a service
station, receives service, leaves the service station, etc. Most
general purpose simulation languages are event- or process-ori-
ented and, accordingly, contain features such as control of time,
set membership, and queuing routines not needed for dealing with
sets of equations.
Table l.l presents a comparison among general purpose com-
puter languages, general purpose simulation languages, and MOUSE
(the system introduced in this manual) for uncertainty problems
that deal in sets of algebraic equations. Table 1.1 is not
designed to devalue the capabilities of either general purpose
computer languages or general purpose simulation languages. MOUSE
is a restricted special purpose simulation language and the table
makes its comparison assuming that the problem at hand is of this
restricted form. (However, see Exercise No. 8, Chapter 10, for
an example of MOUSE used as a more general purpose simulation
language.) For such restricted models, MOUSE is clearly superior
to the other two groups of languages. It is concise, powerful,
and convenient to use. Whether you are a scientist, engineer,
businessman or economist, MOUSE can solve uncertainty problems
faster and easier than can other languages. With MOUSE, your
attention is on problem-solving, rather than on the details of
coding a program to compute a solution. Further, MOUSE programs
are easier to understand, easier to explain to others, and easier
to modify than are general purpose languages.
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Mouse Manual Page 15
ADDENDUM
In Section 1.4 it was mentioned that, for non-linear models,
the use of average values as inputs does not guarantee that the
output is an average value. For example, consider the quantity
A2, with A having the probability distribution shown in Figure
1.2. The average or "expectation" of A, E[A], is shown in Figure
1.2 to be 2. It might be thought that, since A2 = A times A,
the average of A2 (i.e., E[A2]) is E[A]E[A] or 2(2) =4. This is
not the case, however, since E[A2] is easily seen to be:
1(0.25) + 4(0.50) + 9(0.25) = 4.50.
The expectation of a product in qeneral is (Kane, 1968):
E[AB] = E[A] E[B] + COV(A,B) , [1.2]
where COV(A,B) is the covariance between A and B. Thus,
, . E[A2] = E[A] E[A] + COV(A,A). [1.3]
Since COV (A, A) = VAJR(A) (i.e., the variance of A) and noting from
Figure 1.3 that VAR(A) = 0.50, using equation 1.3 we obtain:
E[A2] =2(2) + 0.50 = 4.50
i.e., the answer previously obtained.
If A and B are independent, then COV(A,B) = 0 and E[AB] =
E[A]E[B]. In equation [l.l], A and B were selected independently
of one another? accordingly, E [¥] «= E[AB] = E[A]E[B] or 2(15) =
30. Note, however, that
COV(A,B) ¦ rho[VAR(A) VAR(B)]0'5 [1-4]
where rho is the correlation coefficient between A and B. From
Figure 1.3, VAR(B) = 25.0. If A and B were correlated to the
extent that rho ¦ 0.5, then from equations 1.2 and 1.4:
E[AB] = 2(15) + 0.5[(0.50)(25.0)]0*5 = 31.77
If the correlation were negative, i.e., rho ¦ -0.50, then
E[AB] - 2(15) - 0.5[(0.50)(25.0)]0*5 = 28.23
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TABU 1.1
& COMPARISON OP COMPUTER LANGUAGES FOR OHCERTAIMTY ANALYSIS
\ Language
\
\
Element \
General Purpose
• Languages
(e.g., FORTRAN,
PASCAL, BASIC)
General Purpose
Simulation Languages
GPSS
SIMSCRIPT
SLAM
MOUSE
Simplicity
complex
simple ¦
complex
complex
simple
Built-in Proba-
bility functions
no-
a few
yes
yes
yes
Built-in Inter-
est functions
no
no
no
no
yes
Automatic Output
no
yes
, yes.
yes
yes
Decimal Numbers
yes •
no
yes
yes
¦ yes '
Automatic Sens-
itivity Analysis
no-
no
no
no
yes
Correlation
Among Variables
must be
programmed
no
must be
p r ogrammed
must be -
programmed
yes
De-bugging
simple
no
no
no
no
yes
Aid To Deriving
Probability
Distributions
no
no
no
no
yes*
Ability to
Expand by
Modules
yes"
no
no
yes
yes
*With its companion system, IMP. See, Chapters 12 and 13.
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Mouse Manual Page 17
»H PHBRHmfi
Denting, W.E., "Some Variances in Random Sampling", in Some
Theories of Sampling. John Wiley & Sons, New York, N.Y., p. 127-
134,1950.
Kane, E. J., Economic Statistics and Econometrics, Harper & Row,
New York, N.Y., 1968.
Maryanski, F. J., Digital Computer Simulation, Hayden Book Co.,
Rochelle Park, N. J., 1980.
Pritsker, A., and Fegden, C. D., Introduction To Simulation and
Slam. John Wiley & Sons, New York, N.Y., 1979.
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Mouse Manual Page 18
CHAPTER 2
ror»MPT.Tt«; of DlfCERTAINTY ANALYSIS
2.1 A Cost Exaaple
This is a simple cost example that involves a waste recla-
mation project where the present value of the returns is a func-
tion of the, cost of establishing the system, which involves the
re-refining of spent motor oil, and of a number of annual equal
net returns after the project is completed. The annual net re-
turns for this project consist of revenues obtained from the sale
of three reclaimed petroleum products, and can be calculated from
a series of algebraic equations as follows:
a. The throughput in production of motor oil grade 1, THRU1,
is a function of the contaminants of the feed, CONTAM, i.e.,
THRUl = 10 - CONTAM
In other words, as the level of contaminants increases, less of
motor oil grade 1 will be produced.
b. The yield of motor oil grade 1, YIELDl, is also a func-
tion of this same contaminant level, i.e.,
YIELDl = 10 - 2*CONTAM
c. The production of motor oil grade 1, PRODI, is
throughput, THRUl, times yield, YIELDl, plus a random effect,
RAND1, i.e.,
PRODI = THRUl*YIELDl + RAND1
This random effect can either be positive or negative, and repre-
sents uncertainties of the demand for motor oil grade 1.
d. The gross revenue from motor oil grade 1, GREV1, is its
price, PRICE1, times the quantity produced, PRODI, i.e.,
GREV1 = PRICE1*PR0D1
e. The price of motor oil grade 2, another product of the
reclaiming facility, is designated by FRICE2. This price is
dependent upon both the production of motor oil grade 2, PROD2,
and another random effect, RAND2, i.e.,
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Mouse Manual Page 19
PRICE2 - 10 - SL0PE*PR0D2 + RAND2
SLOPE 3»s part of the functional relationship between price and
production in this equation, and RAND2 represents the uncertain-
ties of the marketplace.
f. The gross revenue from motor oil grade 2, GREV2, is
price, PRICE2, times production, PR0D2, i.e.,
GREV2 « PRICE2 *PROD2
g. The gross revenue from motor oil grade 3, the third and
last product produced at the facility, is price, PRICE3, times
production, PR0D3, i.e.,
GREV3 = PRICES*PR0D3
h. The annual net benefit, ANETB, is the sum of the gross
revenues, GREV1, GREV2, and GREV3, less the annual cost (assumed
to be fixed in this problem and equal to 70), i.e.,
ANETB = GREV1 + GREV2 + GREV3 - 70
i. The present value, PVALUE, of the returns is a function
of the life of the investment, XLIFE, the interest rate (which is
assumed in this problem to be a constant and equal to 0.08), the
annual benefits, ANETB, and the initial investment, XININV, i.e.,
XLIFE
PVALUE - ANETB £ (1 + 0.08)-t - XININV
t=l '
These nine equations comprise the mathematical model.
2.2 The Input. Variables
Ten of the input variables in this model are assumed to
represent uncertain quantities. Two of them, RAND1 and RAND2,
have already been mentioned. However, our analyst is also uncer-
tain about the values of CONTAM, PRICE1, the SLOPE coefficient in
the PRICE2 equation, PROD2, PRICE3, PROD3, XLIFE, and XININV.
tCa ** i Sk K1 a i e sk <5 n Vi sua st s ^ \r H 1 c1^ t* 1 K11+1 i
LaCn Variayic 15 a5g>Ulu6u wv ilcIVS a prODayJll Ity Uia wl iDUuiun § XJ1
the language of probability they are termed stochastic input
variables. The problem of determining the probability distribu-
tion of stochastic input variables is left to Chapters 12 and 13.
For the time being, however, let us assume that the analyst has
determined these distributions to be those shown in Table 2.1.
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TABLE 2.1
DISTRIBUTIONS OF TtUS INFUT STOCHASTIC VAHTftBTiKS
FOR THE WASTE Oil. RECLAMATION PROBLEM
Variable
CONTAM
RAND1
PRICE1
SLOPE
Tvi>e
Discrete
Uniform
Continuous
Uniform
Discrete
Empirical
Continuous
Empirical
Outcome Probability
Cumulative
Probability
PR0D2 Step
Rectangular
1
2
3
¦ 4
-10
10
7.5
8.0
10.5
.05
.10
.15
.20
30 to 35
35 to 60
60 to 65
.25
.25
.25
.25
, 35
.40
.25
,15
.50
35
.00
1.00
.35
.75
1.00
.00
.30
.70
1.00
.15
.65
1.00
Variable
RAND2
PRICE3
PR0D3
XLIFE
XININV
Type
Normal
Empirical
Beta
Exponential
Discrete
Triangular
Trapezoidal
Parameters
0.00 {= mean)
0.10 (= standard deviation)
3.5 (= lowest value)
5.0 (= most likely value)
6.5 (» highest value)
18 (= mean)
5 (= lowest value)
10 (= most likely value)
15 (= highest value)
1600 (= lowest value)
2000 (= most likely value,
2200 (= most likely value,
2400 (= highest value)
lower boundary)
upper boundary)
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Mouse Manual Page 21
Table 2.1 shows, for example, that variable CONTAM has four
possible outcomes (representing four different levels of contami-
nants); 1, 2, 3, and 4, each with probabilities 0.25. Detailed
descriptions of all of the probability distributions used can be
found in Chapter 4.
2-3 The "Best Value" Approach
In the traditional best value approach, the average values
of the input variables are entered into the mathematical model as
follows;
CONTAM
RAND1
PRICE1
SLOPE
PR0D2
RAND2
PRICE3
PR0D3
XLIFE
XININV
2.50000
.00000
8.45000
.12500
52.50000
.00000
8.00000
18.00000
10.00000
2100.00000
The intermediate and final equations evaluated at these values
are: .
THRU1
YIELD1
PRODI
GREV1
PRICE2
GREV2
GREV3
ANETB
PVALUE
7.50000
5.00000
37.50000
316.87500
3.43750
180.46880
144.00000
571.34380
1733.76300
As we shall see shortly, 1733.76 is not the correct value for the
present worth of this project.
2.4 The MOOSE Oil Reclamation Progra* And Results
Figure 2.1 shows the HOUSE program required for the Waste
Oil Reclamation problem. There are 43 lines of computer code but
only the following nine lines were written by the user directly:
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House Manual Page 22
THRU1=10.O-CONTAM
YIELD1=10.0-2.0*C0NTAM
PR0D1=THRU1*YIELD1+RAND1
GREV1=FRICE1*PR0D1
PRICE2=10.O-SLOPE*PROD2+RAN02
GREV2=PRICE2*PROD2
GREV3=PRICE3*PROD3
ANETB=GREV1+GREV2 +GREV 3-70.0
PVALUE=ANETB*UPW(0.08,XLIFE,0.0)-XININV
since a MOUSE utility, called "TRAP" (see Chapter 11), wrote
(with some user input) the remaining lines. A full explanation
for the MOUSE program of Figure 2.1 will be given in the chap-
ters that follow. The output for this program is shown in Fig-
ures 2.2 and 2.3. Figure 2.2 shows a portion of the automatical-
ly echoed listing of the empirical input variables data. Figure
2.3 shows the statistics, histogram and graph produced. The
quantity tabled in Figure 2.3 is PVALUE, to which the table name
"PRES VAL" has been given. The number of iterations (as de-
scribed in Chapter 1, Section 1.5), mean, standard' deviation,
coefficient of variation, average of values less than zero,
minimum, and maximum of PVALUE is part of this standard output,
as is s the histogram table containing the number of entries,
percent entries, cumulative percent entries, and complement of
the cumulative percent entries for each interval of the histo-
gram. Graphical output is also automatically provided in the
form of (a) the frequency distribution of PVALUE (asterisks), and
(b) the cumulative frequency distribution of PVALUE (circles) .
Note that the true mean is estimated as 581.57. The "best" value
approach as calculated in the previous section resulted in a
value approximately 50% greater. This is due to the non-linear
functional relationships reflected in the uncertainty equations.
The exact mean present value, obtained by applying mathematical
expectations, is 574.
2.5 A Regulatory Example
This example concerns the development of a means to estab-
lish the regulatory level of those wastes disposed of in a land-
fill that pose a hazard due to their toxic organic constituents.
The specific environment modeled was that of a non-secure sani-
tary landfill that receives a small amount of industrial waste,
the source of the toxic materials. The specific features of the
model are as follows:
(1) The landfill receives predominately domestic
refuse, i.e., only 5% of the waste is industri-
al in nature;
-------
Mouse Manual Page 23
C
C THE WASTE OIL RECLAMATION PROBLEM
C
DIMENSION D1(3,2),D2(4,2),D3(4,2)
DIMENSION TAB(46)
COMMON/Q1/CSO/Q2/JUMP/Q3/LOOP/Q4/SENS/Q5/IDATA/Q6/IOUT/Q7/IPASS/Q8
&/IRAN/Q9/IREAD/Q10/ISCR1/Q11/ISCR2/Q12/ISNAP/Q13/ITER/Q14/ITOG/Q15
&/IWRIT/Q16/IXXX/Q17/OK1/Q18/OK2/Q19/OK3/Q20/OK4/Q21/OK5/Q2 2/OK6
CALL NAME('(WASTE OIL RECLAMATION PROBLEM)«)
CALL READ(D1,3)
CALL READ(D2,4)
CALL READ(D3,4)
ITER=5000
5 DO 10 IXXX—1,ITER
CONTAM=DUNI(1.0,4.0)
RAND1-CUNI(-10.0,10.0)
PRICE1«EDIS(D1,3)
SLOPE=ECTS(D2,4)
PRQD2=STEP(D3,4)
RAND2=GAU(0.00,0.10)
PRICE3=EBETA(3.5,5.0,6.5)
PROD3=EXPON(18.0)
XLLFE-DTRI(5.0,10.0,15.0)
XININV=TRA(1600.0,2000.0,2200.0,2400.0)
THRU1=10.O-CONTAM
¥IELD1=10.0-2.0*CONTAM
PR0D1=THRU1*¥IELD1+RAND1
. GREV1=PRICE1*PR0D1
PRICE2=10.0-SLOPE*PROD2+RAND2
GREV2=PRICE2*PROD2
GREV3=PRICE3 *PROD3
ANETB=GREVl+GREV2+GREV3-70.0
PVALUE=ANETB*UPW(0.08,XLIFE,0.0)-XININV
CALL TABLE(PVALUE,•(PRES VAL)',DUMO,DUM1,40,TAB)
10 CONTINUE
GOTO(5,15), JUMP
15 CONTINUE
END
FIGURE 2.1
MOUSE PROGRAM FOR THE WASTE OIL RECLAMATION PROBLEM
-------
Mouse Manual Page
********************
* INPUT DATA *
********************
INPUT DATA DISTRIBUTION FOR
PRICE1 (PRICE OF MOTOR OIL GRADE 1)
VALUE OF VARIABLE
CUMULATIVE PROBABILITY
7.50000
8.00000
10.50000
.35000
.75000
1.00000
INPUT DATA DISTRIBUTION FOR
SLOPE (SLOPE OF MOTOR OIL GRADE 2 PRICE EQUATION)
VALUE OF VARIABLE
CUMULATIVE PROBABILITY
.05000
,10000
,15000
,20000
.00000
.30000
.70000
1.00000
INPUT DATA DISTRIBUTION FOR
PR0D2 (PRODUCTION OF MOTOR OIL GRADE 2)
VALUE OF VARIABLE
CUMULATIVE PROBABILITY
30.00000
35.00000
60.00000
65.00000
.00000
.15000
.65000
1.00000
FIGURE 2.2:
PARTIAL ECHOED LISTING OF EMPIRICAL INPUT DATA VARIABLES,
HASTE OIL RECLAMATION PROBLEM
-------
Mouse Manual Page 25
****** nnw***wt*t*wtmmHwn*wti»
* DISTRIBUTION FOR QUANTITY PRES VAL •
**•****•***»**•*»*«*«•«•**•••*««••»••*»•**
NUMBER OF ITERATIONS * 5000
MEAN
MINIMUM
MAXIMUM
1841.81500
-3190.54000
nan.57000
STANDARD DEVIATION
COEFFICIENT OF VARIATION, %
1942.75900
105.48070
AVERAGE OF VALUES LESS THAM ZERO -
•782.08850
LOWER
LIMIT
-3380.0000
-3040.0000
-2700.0000
-2360.0000
•2020.0000
•16BO.OOQO
•1340.0000
-1000.0000
-660,0000
-320.0000
20.0000
360,0000
700.0000
1040.0000
1380.0000
1720.0000
2060.0000
2400.0000
2740.0000
3080.0000
3420.0000
3760.0000
4100.0000
4440.0000
4780.0000
5120.0000
5460.0000
5800.0000
6140.0000
6480.0000
6820.0000
7160.0000
7500.0000
7840.0000
8180.0000
OVERFLOW
MUHBER OF
ENTRIES
1.
4.
12.
27.
44.
70.
91.
176.
203.
243.
310.
325.
369.
339.
327.
346.
285.
308.
248.
251.
222.
167.
152.
114,
81.
75.
60.
41.
25.
22.
21.
10.
8.
6.
7.
10.
PERCENT
ENTRIES
.02
.08
.24
.54
.88
1.40
1.82
3.52
4.06
4.86
6.20
6.50
7.38
6.78
6.54
6.92
5.70
6.16
4.96
5.02
4.44
3.34
3.04
2.28
1.62
1.50
1.20
.82
.50
.44
.42
.20
.16
.12
.14
.20
CUMULATIVE
X ENTRIES
.02
.10
.34
.88
1.76
3.16 '
4.98
8.50
12.56
17.42
23.62
30.12
37,50
44.28
50.82
57.74
63.44
69.60
74.56
79.58
84.02
87.36
90.40
92.68
94.30
95.80
97.00
97.82
98.32
98.76
99.18
99.38
99.54
99.66
99.80
100.00
CUMULATIVE
COMPLEMENT
99.98
99.90
99.66
99.12
98,24
96.84
95.02
91.50
87.44
82.58
76.38
69.88
62.50
55.72
49.18
42.26
36.56
30.40
25.44
20.42
15.98
12.64
9.60
7.32
5.70
4.20
3.00
2.18
1.68
1.24
.82
.62
.46
.34
.20
.00
DISTRIBUTIONS
* I FREQUENCY DISTRIBUTION 0 = CUMULATIVE DISTRIBUTION
0
0*
0"*
»o**«
*Q******
**Q**********
***Q************
*•***0*•*••*#•***•**••**»*•***
••••••••••••••OH***#*##**###**#**«**#********#******
»**********#******Q*******«i****************************
***********************0****** •****••********••• *************.
***************************g*****************************
•••••••********#*******************q***********•***»•»*•*«
•******************************* *******Q********
********************************************
****************************************** Q
************************************** Q
**************************
********************
**************
*************
***********
********
*****
*****
****
***
**
**
**
***
0
0
0
0
o
0
0
CUMULATIVE
X ENTRIES
CUMULATIVE
COMPLEMENT
VALUE Of
PRES VAL
.1
99.9
-3040.0000
1,0
99.0
-2313.6360
5.0
95.0
-1338.0680
10.0
90.0
-874.3841
25.0
75.0
92.1046
50,0
50.0
1337.3700
75,0
25.0
2769.8010
90.0
10.0
4055.2630
95,0
5.0
4938.6660
99.0
1.0
6674.2850
FIGURE 2.3
STATISTICS, HISTOGRAM AND GRAPHS PRCDUCS
BY HOUSE KK THE tttSTf OIL RECLAMATION PROBLEM
-------
Mouse Manual Page 26
(2) The character of the leaching fluid that the
waste will be exposed to is predominately a
function of the non-industrial material in the
.landfill?
(3) The landfill is situated over an aquifer that
is a source of drinking water;
(4) The soil below the site is composed of material
with limited attenuation capacity;
(5) Persons using the aquifer as a source of drink-
ing water are supplied from wells situated 150
meters down-gradient from the landfill;
(6) As constituents migrate from the landfill
through the saturated and unsaturated zones to
the source . of the drinking water, they are
attenuated by a factor of 100.
A schematic of the model is shown in Figure 2.4.
The regulatory level (REGLEV) of a toxic constituent is
obtained by simply multiplying the maximum permitted daily expo-
sure level (DELMAX) for the toxic constituent in question by a
suitable factor, i.e.,
REGLEV(ppm) = DELMAX(mg/day) x FACTOR(days/kg)
since mg/kg is equivalent to ppm. The factor can be explained by
the following dimensional analysis:
FACTOR = [(attenuation factor) x (liters leached/year) x years]/
[(liters consumed/day) x (industrial waste quantity, kg)].
If the (dimensionless) attenuation factor is assumed to be 100,
and if we assume 2 liters of water consumption per day, a 70-year
exposure period, and that 5% of the waste is industrial, the
factor becomes:
FACTOR « 3500 * [liters leached/year]/
[industrial waste quantity, kg]. =
[annual precipitation, liters/sq. meters per year x
rate of leaching (as a fraction of the annual precipitation)]/
[landfill depth, meters x waste density, kg/cubic meters x 0.05].
-------
© _ tfc»
CO-DISPOSAL
(A)
MUNICIPAL WASTE 95%
INDUSTRIAL WASTE 5%
WELL
(500 ft FROM
LANDFILL)
100X
ATTENUATION FACTOR
UNSATURATED
ZONE
GROUND WATER
FIGURE 2.4
SCHEMATIC DIAGRAM OF THE MODEL FOR THE REGULATORY PROBLEM
3
O
e
in
-------
Mouse Manual Page 28
Letting annual precipitation = PRECIP, rate of leaching = RLEACH,
landfill depth = DEPTHL, and waste density = WDENS, then
FACTOR = 70,000 * [PRECIP * RLEACH]/[DEPTHL * WDENS]
Thus, the regulatory level of a toxic constituent in this model
is a function of four factors: annual precipitation, rate of
leaching, landfill depth, and waste density. Suppose the annual
precipitation was 1000 liters/meter per year, the rate of leach-
ing was 0,8, the landfill depth was 8 meters, and the waste
density was 500 kg/cub. meter. Then FACTOR would equal 14000. At
this value of FACTOR, Table 2.2 shows the regulatory levels for
compounds with varying DELMAX1s, and Table 2.3 shows the
probability distributions assumed for the stochastic variables in
this problem.
TABLE 2.2: REGULATORY LEVELS AS * FUNCTION OF DELMAX
Compound DELHWnw/day) Regulatory Level(pom)
CoRfomd A
Compound B
Compound C
CoBpouid D
Conpoind E
Compound F
Compound S
.000001
.00001
.0001
.001
.01
..1 ¦
1.0
.014
.14
1.4
14
140
1400
14000
TABLE 2.2
DISTRIBUTIONS OF THE INPUT STOCHASTIC VARIABLES
FOR THE REGULATORY PROBLEM
Variable
RLEACH
IffiS
Continuous
Uniform
Outcome
0.5
1.0
Cumulative
Probability
0.00
1.00
Variable Type Parameters
PRECIP Trapezoids I 762.0 (¦ lowest value}
1016.0 <=¦ most likely value, loner boundary)
1270.0 <= Bost likely value, nsper bouidary)
1016.0 (¦ highest value)
DEPTHL Trupaioidal 3.0 (* lowest value)
6.1 £¦ most likely value, lower boundary)'
9.1 (a Most likely value, upper boundary)
1S.2 (¦ highest value)
WDENS Triangular 297.0 {= lowest value)
415.0 (* most likely value)
534.0 <¦ highest value)
-------
Mouse Manual Page 29
2.6 The MOUSE Regulatory Program And Results
Figure 2.5 shows the MOUSE program required for the
Regulatory problem. There are 18 lines of computer code but only
the following line was written by the user directly (TRAP - see
Chapter 11 - with user input, wrote the remainder of the
program):
FACTOR= 7 0000*(PRECIP*RLEACH)/(DEPTHL*WDENS)
A full explanation for the MOUSE program of Figure 2.5 will be
given in the Chapters that follow. The output for this program
is shown in Figure 2.6. Note that the mean value of FACTOR is
estimated as 20361. Since 5000 iterations were used, this is
within a fraction of a percent of the true value. The "best"
value approach (using the mean values for the four input distri-
butions) produces an average value equal to 17303, i.e., approxi-
mately 15% smaller than the true mean. This is due to the non-
linear functional relationships reflected in the uncertainty
equations.
C
C REGULATORY PROBLEM
C
DIMENSION TAB(46)
COMMON/Q1/C80/Q2/JUMP/Q3/LOOP/Q4/SENS/Q5/IDATA/Q6/IOUT/Q7/IPASS/Q8
&/IRAN/Q9/IREAD/Q10/ISCR1/Q11/ISCR2/Q12/ISNAP/Q13/ITER/Q14/ITOG/Q15
&/IWRIT/Q16/IXXX/Q17/QK1/Q18/OK2/Q19/OK3/Q2Q/OK4/Q21/OK5/Q22/OK6
CALL NAME < »(REGULATORY PROBLEM)*)
ITIR=5000
5 DO 10 IXXX=1,ITER
PRECIP=»TRA (762 .0,1016.0,1270.0,15 24.0)
RLEACH-CUNI(.5,1.0)
DEPTHL=TRA(3.0,6.1,9.1,15.2)
WDENS=TRI(297.0,415.0,534.0)
FACTOR=70000*(PR1CIP*RLEACH)/(DEPTHL*WDENS)
CALL TABLE(FACTOR,«(FACTOR)*,DUM0,DUK1,40,TAB)
10 CONTINUE
GOTO(5,15), JUMP
15 CONTINUE
END
FIGURE 2.5
MOUSE PROGRAM FOR THE REGULATORY PROBLEM
-------
Mouse Manual Page 30
»< >***** **********~*~****»**¦**~*~*
* DISTRIBUTION FOR OUAMTITT FACTOR *
WrtmtlHKWWWMWWWWIHWWtWWWW
NUMBER Of ITERATIONS = 5000
MEAN
MINIMUM
MAXIMUM
20361.21000
4219,64400
82200.32000
STANDARD DEVIATION
COEFFICIENT OF VARIATION, %
12154.16000
59.69275
LOWER NUMBER OF PERCENT CUMULATIVE CUMULATIVE
LIMIT
ENTRIES
ENTRIES
X ENTRIES
COMPLEX!
4200.0000
60.
1.20
1.20
98.80
6100.0000
245.
4.90
6.10
93.90
8000.0000
443.
8.86
14.96
85.04
9900.0000
585.
11.70
26.66
73.34
11800.0000
486.
9.72
, 36.38
63.62
13700.0000
, 467.
9.34
45.72
54.28
15600.0000
381.
7.62
53.34
46.66
17500.0000
374.
7.48
60.82
39.18
19400.0000
248.
4.96
65.78
34.22
21300.0000
239.
4.78
70.56
29.44
23200.0000
194.
3.88
74.44
25.56
25100.0000
158.
3.16
77.60
22.40
27000.0000
127..
2.54
80.14
19.86
28900.0000
121.
2.42
82.56
17.44
30800.0000
107.
2.14
84.70
15.30
32700.0000
94.
1.88
86.58
13.42
34600.0000
95.
1.90
88.48
11.52
36500.0000
69.
1.38
89.86
10.14
38400.0000
75.
1.50
91.36
B.64
40300.0000 v
70.
1.40
92.76
7.24
42200.0000
56.
1.12
93.88
6.12
44100.0000
56.
1.12
95.00
5.00
46000.0000
55.
1.10
96.10
3.90
47900.0000
35.
.70
96.80
3.20
49800.0000
26.
.52
97.32
2.68
51700.0000
24. ,
.48
97.80
• 2.20
53600.0000
28.
.56
98.36
1.64
55500.0000
'¦ 17.
- .34
98.70
1.30
17400.0000
11.
.22
98.92
1.08
59300.0000
15.
.30
99.22
.78
61200.0000
5.
.10
99.32
.68
63100.0000
6.
.12
99.44
.56
65000.0000
9.
.18
99.62
.38
OVERFLOW
19.
.38
100.00
.00
DISTRIBUTIONS
3 FREQUENCY DISTRIBUTION
*§»*•**
**¦**0******** **~~**•*~****~
CUWJLATIVE DISTRIBUTION
*******
******************** ********
**************************
••a******************
*****************
**************
**************
************
***********
***********
********
*********
********
*******
*******
*******'
*****
****
****'
¦ **-»
**
***
**
** •
- **
***
CUMULATIVE CUMULATIVE
X ENTRIES COMPLEMENT
VALUE OF
FACTO?
5.0
95.0
5673.4690
10.0
90.0
6936.3430
25.0
75.0
9630.4280
50.0
50.0
14767.1900
75.0
25.0
23536.7100
90.0
10.0
36677.3300
95.0
5.0
44100.0000
99.0
1.0
57906.6700
FIGURE 2.6: STATISTICS, HISTOGRAM MB «APHS PITCDUCB)
BT MOUSE FOR TIE REGULATOR PROBLEM
-------
Mouse Manual Page 31
COMTEK 3
THE MOOSE LANGUAGE
3.1 Overview of MOOSE Programs
Although MOUSE is written and based upon FORTRAN (and there-
fore requires a FORTRAN compiler as part of its system), it is
designed to be implemented by those with little or no knowledge
of computer languages or programming. MOUSE programs consist of:
(a) Eight Control lines (see Figure 3.1).
(b) A Name line, and
(c) Four Program Blocks,
The four blocks are:
(1) Dimension Block,
(2) Specification Block,
(3) Read Block, and
(4) Program Block.
Because we shall have need to refer to actual programs in order
to locate these elements of a MOUSE model, the two example
problems of Chapter 2 are redisplayed in Figur e s 3*2 and 3.3
along with Comment lines for clarity. Comment lines are those
that start with a "C" in the first column of a program line.The
computer ignores such lines but does print them if the program is
listed on a printer or on a terminal. You can (and should) use
as many comment lines as will help in following or clarifying a
given program.
The Dimension Block consists of all lines occurring before
Mouse Control line # 1 and contains lines that size or dimension
the various tables, vectors, and matrices used by a program for
storing data and results. The lines in the Dimension Block are
defined by a set of specific rules, but you need not be concerned
with their exact meaning. Examples of the lines in the Dimension
Blocks of Figures 3.2 and 3.3 include those starting with the
label DIMENSION.
The Specification Block informs the computer of certain
decisions under your control, such as the number of iterations in
the Monte Carlo method, the number of intervals in the table
histograms, and the number of columns in the MOUSE output. Exam-
ples of the lines in the Specification Blocks of Figures 3.2 and
3.3 include those starting with the label ITER.
-------
*******************
* DIMENSION BLOCK *
*******************
Mouse Manual Page 32
\
MOUSE CONTROL LINES 1, 2, AND 3
\
/
/
NAME LINE
\
/
+ READ BLOCK (Optional) +
++++ + ++++ + +++ + -T++++++ +++ +
\ /
***********************
uk CT>T?nTJ?rnhrnTnU RTArV *
* oyLLlrlLAilUJi dliULA *
***********************
\ /
MOUSE CONTROL LINE 4
\ /
******************* it-it ********************
* PROGRAM BLOCK *
* 1. Random Sampling of Input Variables *
* 2. Uncertainty Equations (Model) *
* 3. Output Requests *
* 4, Optional Control lines *
*****************************************
\
MOUSE CONTROL LIN!
/
IS 5, 6, 7, and 8
FIGURE 3.1
GENERAL OVERVIEW OF A. HOUSE PROGRAM
-------
C
C
C MOUSE CONTROL LINES 1, 2, AND 3
C
C0MHON/Q1/C80/Q2/JUMP/Q3/LOOP/Q4/SENS/Q5/IDATA/Q6/IOUT/07/IPASS/OS
t/lRAN/09/IREAD/Q10/ISCR1/a11/ISCR2/Q12/ISNAP/O13/ITER/Q14/ITOG/al5
&/IURIT/016/IXXX/Ql7/OK1/Q18/OK2/Q19/0K3/Q20/0K4/Q21/OK5/Q22/OK6
C NAME LINE
CALL NAME('(WASTE OIL RECLAMATION PROBLEM)1)
C
C EMPIRICAL VARIABLE READ LINES
C
CALL READ(D1,3)
CALL READ(D2,4)
CALL READ(D3,4)
C
C
-------
Mouse Manual Page 34
C - 'Start of DIMENSION BLOCK*
C
C TABLE DIMENSION LINE
C
DIMENSION TAB<46)
C
C -, JUMP
10 CONTINUE
END
FIGURE 3.3: COMMENTED MOUSE PROGRAM FOR TEE REGULATORY PROBLEM
-------
Mouse Manual Page 35
The Read Block enters input data, such as empirical proba-
bility distributions, and parameters (i.e., temporary constants)
of the model. Since not all MOOSE programs involve empirical
probability distributions or parameters, the Read Block may not be
needed in all programs,* hence the use of the term "optional" in
Figure 3.1. The Regulatory problem, for example, does not use a
Read Block. Examples of the lines in the Read Block of Figure 3.2
include those starting with the labels CALL READ.
The Program Block is the heart of any MOUSE program and
contains four sections, of which the first three are vital:
(1) A section dealing with the random sampling
of the input variables,
(2) Your mathematical model (i.e., the set of
uncertainty equations),
(3) Your requests for specific kinds of outputs
from the MOUSE program, and
(4) Optional Control lines.
The first two sections are clearly identified in Figures 3.2
and 3.3. Example of lines dealing with output requests in the
Program Blocks of Figures 3.2 and 3.3 are the ones starting with
the label CALL TABLE. In more complex MOUSE problems, there may
be (and hence ns6 of till© tsritt "optional" in Figure 3.1) a
requirement for additional, specialized control lines such as a
"Pass Control" line. No such lines are shown in Figures 3.2 or
3.3? these will be introduced in later chapters. It should be
noted that only the Program Block need be prepared by you; the
TRAP utility (Chapter 11) will take care of the rest. Actually,
TRAP will also prepare part of the Program Block, i.e., Output
Requests and Optional Control lines. Therefore, what appears to
be complicated coding is, in reality, quite simple and straight-
forward.
3.2 Arithmetic
Constants in MOUSE are merely numbers,* variables, on the
other hand, may be thought of as storage locations in the comput-
er. Variable names are denoted by one to six alphanumeric char-
acters (i.e., any combination of letters and numbers, starting
with a letter). Both constants and variables may be either
integer or decimal. An integer constant is any number without a
decimal point; an integer variable is any whose name starts with
the letters I, J, K, L, M, or N, This is an important distinc-
tion since all arithmetic in MOUSE is done in decimal numbers.
This is why XLIF1 and XININV were used instead of LIFE and ininv
-------
Mouse Manual Page 36
in the coding of the Waste Oil Reclamation problem and why deci-
mal points were used in the constants in the uncertainty equa-
tions. On the other hand, certain control functions require the
use of integer constants or variables. A summary of the rules
for the,use of decimal points in MOUSE appears in Table 3.1.
TABLE 3.1 RULES FOR USE OF DECIMAL POINTS IK MOUSE
. Decimal Points Used Decimal Points Wot Used
Classical Probability Functions Column Output line
Dimension Arguments (all)
Empirical Probability
Functions
GOTO Statement Number
Iteration line
Output Destination line
Pass Control line
Plot Output lines (with
exception of scaling factor)
Read lines (all)
Seed Reset lines
Table Output Lines (with
exception of number of Sensitivity Curtailment line
histogram intervals argument)
Snap line
Trace lines
Under certain circumstances, it is necessary to change
decimal variables to integer variables and vice versa. This is
done by simple renaming, e.g.,
XLIFE=LIFE
LIFE=XLIFE
In the first case a new variable, the decimal variable XLIFE, is
produced from the integer variable, LIFE (without changing LIFE)?
in the second case, the reverse occurs. To repeat, neither
variable on the right-hand side of the equal sign is altered;
instead, a new variable is created.
Correlation Function (with
exception of dimension
argument)
Data lines (all)
IF Statement Operators and
Numerical Arguments
Interest Functions
Numbers Used in Uncertainty
Equations
Scaling Factors in Table and
Plot Output Lines
lines
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The valid arithmetic operations available in MOUSE are;
Operation Symbol BxiTOlg
Addition + A+B
Subtraction - A-B
. Multiplication * A*B
Division / A/1
Exponentiation ** A**B (= AB)
Remark; MOUSE does not allow -two arithmetic operations to appear
next to each other. Thus, A*-B must be written as A*(-B).
The conventional use of parentheses in arithmetic expres-
sions is acceptable in MOUSE and, indeed, the use of parentheses
is encouraged whenever it contributes to the clarity of an ex-
pression - There is a hierarchal sequence of arithmetic opera-
tions in MOUSE, Exponentiation is done first, then multiplica-
tion or division, then addition or subtraction. Parentheses
override this hierarchal sequence, however.
Along with these arithmetic operations, there are also a
number" of standard functions available as follows:
Function
square root
exponential
natural logarithm
common logarithm
absolute value
sine
cosine
tangent
Definition
(a)0,5
ln(a)
log(a)
|a|
sin(a)
cos(a)
tan(a)
Function Name
With Argument
SQRT(A)
EXP(A)
ALOG(A)
ALOGIO(A)
ABS(A)
SIN(A)
COS(A)
TAN(A)
There are other functions available but they are not likely to be
encountered in uncertainty equations. The reader is referred to
a FORTRAN handbook should such a need arise (see Lipschutz & Poe,
1978),
As an example, consider how the following uncertainty equa-
tion would be written in MOUSE:
Y4 = (3X3)0-5 + X20*8
In MOUSE this equation would appear as
Y4 « SQRT(3.0*X3) + X2**0.8
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Note that there is a multiplication sign between the 3 and the
X3, and that a decimal point (and an optional zero for clarity)
has been added to the 3 in accordance with the decimal arithmetic
rule stated earlier. Note also that the left-hand side of any
equation must be a single variable name. One cannot write, for
example,
4 . Q*X5 «= Y8
In general, program lines start in or after column 7 and no
entry is permitted after column 72. (Data lines — see Chapter 4
— follow a different protocol, however. Statement numbers used
in conjunction with GOTO statements are also exceptions. See
Chapter 4, Sections 4.6 and 4.7, and Table C.l, Appendix C.) if
a particularly long equation is encountered, it can be continued
on the next line but the continuation line (or lines) must have
a some character (the author recommends a •'&") placed in column
6. As many continuation lines as are necessary generally may be
used; there are limits, of course, that depend upon the FORTRAN
compiler being used but as these limits are fairly generous this
usually poses no serious problems. Your local system FORTRAN
manuals should be consulted for details. It should also be noted
that spaces (blanks) between arithmetic operations are ignored by
the computer, e.g., the following are identical:
¥=1.0+X3
¥ - 1.0 + X3
3.3 Restricted Names
The following are not allowed as variable names in your
uncertainty equations because they are reserved for special MOUSE
program codes;
Q1
Q6
Qll
Q16
Q21
C80
IDATA
ISCR1
IWRIT
OK4
Q2
Q7
Q12
Q17
Q22
JUMP
I OUT
ISCR2
IXXX
0K5
Q3
Q8
Q13
Q18
LOOP
IPASS '
ISNAP
OKI
0K6
Q4
Q9
Q14
Q19
RAN
IRAN
• ITER
OK2
Q5
Q10
015
Q20
SENS
IREAD
ITOG
0K3
There is, furthermore, a prohibition on the use of names used by
MOUSE as names of functions and subroutines, and names used in
the various MOUSE line types. Alphabetized lists of these names
can be found in the tables in Appendix B found at the end of this
book.
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3.4 stochastic Input Variables
The method by which stochastic input variables are usually
introduced into MOUSE uncertainty equations is indirect, i.e.,
the probability distributions are first sampled and then the
random values produced are assigned to a surrogate variable:
Referring to Figure 3.2, although probability distributions will
be explained in detail in Chapter 4, it is sufficient to say at
this point that DUNI(1.0,4.0) returns a random value from the
probability distribution for the variable CONTAM. (In the Waste
Oil Reclamation problem, therefore, it will return a value of l,
25% of the time; a value of 2, 25% of the time,* a value of 3, 25%
of the time, and a value of 4# 25% of the time.) In other words,
one usually samples first and then assigns the random value pro-
duced to a surrogate variable, such as CONTAM.
The indirect method is used for two reasons: (a) it minimizes
coding errors, and (b) it is the method that is usually the most
appropriate for any stochastic input variable that appears more
than once in the set of uncertainty equations. For the Waste Oil
Reclamation problem, this includes the variables CONTAM and PR0D2
since they appear twj.ce in the equations for THRU1 and YIELD1, and
PRICE2 and GRREV2, respectively. In model sampling, for any
iteration the set of values for the stochastic variables is usual-
ly assumed to be constant. It would not make sense, for example,
to use one value of CONTAM to calculate the acres of crop 1 and
yet another to calculate yield per acre of crop 1. Each time the
computer refers to DUNI(1.0,4.0), however, a new value of CONTAM
is produced. The indirect method results in only one value being
produced since DUNI(1,0,4.0) is used only once per iteration.
However, under certain circumstances (see Chapter 10, Section
10.5), the direct method may be safely used.
EXERCISES
Which of the following are valid MOUSE equations?
1. 3RDSHF»HRS**2-SQRT(Y4) 6.
2fYE,f*=TTMT? * 1 *> "7
• Lr£lVr * * *} * £t t «
3. ¥=EXF(-COUNT)+ALQG(X26) 8.
4. X/2.=A6.*PRICE+SEND 9,
5. Ql=RATE+5.*TIME
COST=BASE+SHIP-DISC
CONTAM=3.*SQRT(X6)-ABS(X7)
DISCOUNT=ALOG((X1-X2)/(Y1-Y2)
SALES=ALOG(Y1+Y2)*PRICE-25.4
REFERENCE
Lipsehutz, S., and Foe, A., Programming With Fortran. Schaum's
Outline Series in Computers, McGraw-Hill Book Co., New York, p.
1-314 (1978).
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' CHAPTER 4
MOUSE LIBRARY PROBABILITY FUNCTIONS
4.1 Introduction
In this chapter, the methods used in MOUSE to obtain random
samples from various probability distributions are described.
(Techniques to determine which distributions are appropriate are
discussed in Chapters 12 and 13.) A probability distribution is
any rule which assigns a probability to each possible value of a
random variable. Discrete random variables in MOUSE are those
that are allowed to take on only selected values, while continu-
ous random variables can take on the value of any real number.
MOUSE provides nineteen probability functions that may be called
by you when desired. Two empirical and fifteen classical distri-
butions are provided, with two of the classical distributions
appearing in alternative forms. Furthermore, any of these proba-
bility distributions can be truncated and/or composited. (In
addition, two correlated probability functions are available; see
Chapter 9.) The general form of a probability function in MOUSE
is:
function(argument1,argument2, etc.)
The number of arguments varies from one to four, according to the
distribution required. The empirical distributions involve the
most input from you and will be discussed first.
4.2 Empirical Probability Functions
A continuous empirical probability random variable is called
by:
ECTS(name,n)
and a discrete empirical probability random variable is called
by:
EDIS(name,n)
where "name" is the name of a cumulative probability distribution
array of size n x 2, and 15n" is the number of points in the
distribution. Note that the n in both the ECTS and EDIS
functions are of integer form., i.e»f no decidual points are
used. Also note that the name must be a decimal name. (A decimal
name is one that starts with any letter except I, J, K, L, K, or
N. See Chapter 3, Section 3.2).
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As an example, in the waste Oil Reclamation problem a random
value of the variable PRICE1 was called by:
PRICE1=EDIS(Dl, 3)
In this example, we assigned the name Dl to the cumulative proba-
bility distribution for PRICE1, the distribution having 3
observations.
Whenever an empirical probability function is part of a
MOUSE program, the cumulative probability function must be dimen-
sioned in a Empirical Variable Dimension line and a corresponding
Empirical Variable Read line must be supplied. The Empirical
Variable Dimension line has the following form:
DIMENSION namel(nl,2),name2(n2,2),name3(n3,2), etc.
where "namei" is the name of the ith cumulative probability
distribution, and "ni" is the number of points in the distribu-
tion. These two arguments must correspond to those used in the
calling function. An example is:
DIMENSION Dl(3,2),D2(3,2),D3(3,2),D4(3,2),D5(3,2),D6(3,2),
&D7(3,2),D8(3,2},D9(3,2),D10(3,2)
This example dimensions ten probability distributions. The D of
DIMENSION starts in column 7; note that an & has been placed in
column 6 of the second line since this is a continuation line.
Note: See Chapter 11, Section 11.6, for a discussion of TRAP, a
MOUSE utility that greatly facilitates the preparation of the
Dimension Block.
The Empirical Variable Read line (one is needed for each
probability distribution read in) has the following form:
CALL READ(name,n)
where "name" and "n" have the same definitions as described for
the Empirical Variable Dimension line. An example from Figure
3.1 is:
CALL READ(D1,3)
Empirical Variable Read lines, if there are any in the
program, must be placed immediately after the Name Line (see
Chapter 7, Section 7.9).
The data for empirical probability distributions are en-
tered, in a free-form format, into Empirical Variable Data lines
which form the MOUSE Data File, MOUSE.DAT. Figure 4.1 shows
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these lines for the Waste Oil Reclamation problem. PRICE1 in
this figure is distributed according to a Discrete Empirical
probability distribution, and SLOPE is distributed according to
the continuous Empirical probability distribution, (PR0D2 is
distributed according to the Step Rectangular distribution, which
will be discussed shortly.) The value of the variable is entered,
starting in column 1, and is followed by a comma and the proba-
bility data in its cumulative form. Decimal points are used in
all numbers, including zeros. Each probability distribution is
preceded by a name line, i.e., a line with information to identi-
fy the probability distribution when it is read out via the echo
print feature of MOUSE. The name line, which is required (it is
not optional), is also shown in Figure 4.1. Data in the name
line can start in any column.
It is important that the first Empirical Variable Data line
contains the left-most observation of the probability distribu-
jl on u s ng ^»lie c^oxi 3*nxxous 6uipr .x ca 1 J3rol^a 13 1 jlty funct jL^^n ^
ECTS. The cumulative probability in this line must perforce be
PRICE!
7.5,0.35
8.0,0.75
10.5,1.00
SLOPE
0.05,0.00
0.10,0.30
0.15,0.70
0.20,1.00
PROD2
30.0,0.00
35.0,0•15
60.0,0.65
65.0,1.00
(PRICE OF MOTOR OIL GRADE 1)
(SLOPE OF MOTOR OIL GRADE 2 PRICE EQUATION)
(PRODUCTION OF MOTOR OIL GRADE 2)
FXGORE 4.1
THE MOUSE.DAT DATA FILE CONTAINING THE EMPIRICAL VARIABLE DATA
LINES FOR THE WASTE OIL RECLAMATION PROBLEM
zero since there are no values to the left of this point. For
example, if we wish to consider the PRICE1 variable as continuous
over the interval 7 to 10.5 rather than discrete for the values
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7.5, 8, and 10.5, then the Empirical Variable Data lines for
PRICE1 would appear as:
PRICE1 (PRICE OF MOTOR OIL GRADE 1)
7.0,0.00
7.5,0.35
8.0,0.75
10.5,1.00
and the appropriate Empirical Variable Dimension and Read lines
would have to be changed to include arguments (4,2) and (Dl,4),
respectively. TRAP (see Chapter 11) can be used to prepare MOUSE
data files. This program not only performs the probability cumu-
lation for you, but also inserts the zero probability point in
continuous empirical probability functions in its proper place in
the data set.
4.3 ft semi-lapirical Probability Function:
The Step Rectangular Function
The step rectangular distribution is a combination of empir-
ical and classical probability functions (the classical function
embedded in this distribution is that of the uniform continuous
kind), but it is treated in MOUSE more like the empirical distri-
butions just discussed than the classical distributions to be
described in Section 4.4. An example of a step rectangular
probability distribution is shown in Figure 4.2. In this distri-
bution, discrete probabilities are assigned to ranges of the
variable, with the assumption that within each range the variable
is distributed uniformly.
The step rectangular probability function is called by:
STEP(name,n)
where "name" and "n" have definitions identical to those given in
Section 4.2. The data for step rectangular distributions are
handled exactly as are those for continuous empirical probability
functions, i.e., the distribution must be dimensioned in an
Empirical Variable Dimension line, an Empirical Variable Read
line must be used, and the data must be entered via Empirical
Variable Data lines. For the diagram above, these first two
lines might appear (in the appropriate places in the program) as:
DIMENSION D1(4,2}
CALL READ(D1,4)
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p2 =
0.40
. •
P3 =
0.35
Pl ¦ 0.25
0 3 4 .6
FIGURE 4.2:
EXAMPLE OF A STEP RECTANGULAR
PROBABILITY DISTRIBUTION
The Empirical Variable Read lines would appear as follows:
XI STEP RECTANGULAR DISTRIBUTION
0.0,0.00
3.0,0.25
4.0,0,65
6.0,1.00
As with continuous empirical probability distribution data lines,
the first cumulative probability must be set equal to zero. (TRAP
- see Chapter 11 - can do the probability cumulation for you, and
also insert the zero probability point in the step rectangular
probability function in its proper place in the data set.)
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4.4 Classical Probability Functions: Continuous
Classical probability functions (discrete or continuous)
supplied by MOUSE do not require Dimension, Read or Data lines.
You supply the parameters of these distributions in the arguments
of the callxng functxons dxrectly xa ca 11 xng arguments • Ar gu™
ments to classical probability distribution calls, however, can
be simple or complex. For example, a simple call to a normal
probability distribution with mean 50 and standard deviation 10
might look like the following:
VAK=GAU(50.0,10.0).
However, the arguments need not be direct; they can be functions
of other variables, e.g.,
VAR=GAU(PRICE/5.0,30.0*COST).
In any case, decimal points are used in all numbers that appear
in the arguments.
4^4.1 Normal Distribution:
The normal distribution is called by:
GAU(mean,std)
where "mean" is the mean of the distribution and
"std" is its standard deviation.
4.4.2 Exponential Distribution:
The exponential distribution is called by:
EXPON(mean)
4.4.3 Continuous Uniform Distribution:
The continuous uniform distribution is called by:
CUNI(lower,upper)
where "lower11 is the lower bound and "upper" is the
upper bound of the distribution. The mean and
variance of the continuous uniform distribution are:
mean = (upper + lower)/2
var = (upper - lower) */12
[4.1]
[4.2]
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4.4.4 Lognormal Distribution; .
The lognormal distribution is called by:
GAUL(mean,std)
4.4,.5 Lognormal Distribution. Alternate form? .
An alternate form of the lognormal distribution is
called by:
AGAUL(mode, median)
where "mode" and "median" are the mode and median,
respectively, of the distribution. Note that the
median must be greater than the mode or else an error
message will occur and the MOUSE program will be
terminated automatically.
4v4.6 Beta Distribution. Standard Form:
The general mathematical form of the beta
distribution is:
xa_1(l-x)b-l
f (x) =» : [4.3]
B(a,b)
where a and b are parameters greater than zero and
where B(a,b) is the beta function. The standard form
of the beta distribution is called by:
BETA(a,b)
The mean and variance of the standard beta
distribution are;
mean ¦ a/(a + b) [4.4]
var ¦ ab(a + b)"2(a + b - 1)[4.5]
4.4.7 Beta Distribution. Empirical Form:
The empirical form of the beta distribution is called
by:
EBETA(l,m,u)
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where 1 is the lowest value the variable can reason-
ably assume, m is the most likely value, and u is the
highest value the variable can reasonably assume.
MOUSE will automatically scale and fit an appropriate
beta distribution when presented with these three
parameters. The method works, however, only when the
parameters of the underlying standard beta distribu-
tion both are greater than zero. If MOUSE finds that
either of these parameters is less than zero, it will
so inform you and terminate the run. The mean and
standard deviation of the empirical beta distribution
are:
mean = (4m + 1 + u)/6 [4.6]
std - (u - l)/6 14.7]
4.4.8 Gamma Distribution:
The general mathematical form of the gamma
distribution is:
xa-le-x/b
f(x) [4#8]
baG(a)
where a .and b are parameters greater than zero and
where G(a) is the gamma function. The gamma
distribution is called by:
GAM(a,b)
The moari anH anco 4~ fho Hi ct"T'ihnt"i ftTi
•Lilts uicaii a.iiu vaj. idilvc uj. wilts uidti jluut'i.uii qic«
mean = ab [4.9]
var = ab2 [4.10]
4.4.9 Triangular Distribution:
The triangular distribution has the following shape:
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4.4.10
Mouse Manual Page 48
b
a
c
The triangular distribution is called by:
TRI(a,b,c)
The mean and variance of the triangular distribution
are:
mean = (a + b + c)/3 [4.11]
var = [a(a - b) + c(c - a) + b(b - c) ]/18 [4-1.2]
Trapezoidal Distribution:
The trapezoidal distribution has the following shape:
The
trapezoidal distribution is
TRA(a,b,c,d)
called by:
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4.5 Classical Probability Functions:-Discrete
The discrete classical probability functions supplied by
MOUSE also do not require Dimension lines, Read lines or Data
lines. ¥ou supply the parameters of these distributions in the
arguments of the calling functions directly. Decimal points are
used in all discrete classical probability functions.
4.5.1 Erlana Distribution:
The general mathematical form of the erlang
distribution is:
xk-le-x/a
f(X) = [4.13]
ak(k-l)I
where a and k are greater than zero and k is an
integer (but, as in all MOUSE probability functions,
must be written with a decimal point). The erlang
distribution is called by:
ERL(a,k)
The mean and variance of the erlang distribution are:
mean = ka [4.14]
var = ka2 [4.15]
4.5.2 Poisson Distribution:
The Poisson distribution is called by:
POI(mean)
4.5.3 Binomial Distribution:
The binomial distribution is called by:
BIN(p,n)
where n and p are the usual parameters of the
binomial distribution (i.e., p is the probability of
a success and n is the number of trials). The mean
and variance of the binomial distribution are:
mean = np
var = np(l - p)
[4.16]
[4.17]
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4.5.4 Pascal/Geometric Distribution;
Pascal
[4.18]
and describes the number of failures fx) that
occur before k successes have occurred (where p is
the probability of a success). When k ¦ 1, the
Pascal distribution becomes the Geometric ,
distribution. The pascal/geometric distribution is
called by:
PAS(k,p)
Once again, k and p are greater than zero and k is an
integer (written with a decimal point). The mean and
variance of the pascal/geometric distribution are;
mean = k(l - p)/p
var - k(l - p)/p
4.5.6 Hvnergeometric Distribution:
The hypergeometric distribution is called by:
HYP(N,n,p)
where N is the total population size, n is the sample
size, and p is the probability of success. N, n, and
p must be greater than zero, and N and n are integers
(written with a decimal point).
4.5.7 Discrete Uniform Distribution •
The discrete uniform distribution is called by:
DUNI(lower,upper)
where "lower" is the lower bound and "upper" is the
upper bound of the distribution.
4.5.8 pj.s
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where a, b, and c are the parameters shown as
follows;
/IN
/
/
/
~
4
/
\
N
N
N
N
I \
c
4.5.9 Discrete Trapezoidal Distribution
The discrete trapezoidal distribution is called by:
DTRA(a,b,c,d)
where a, b, c, and d are the parameters shown as
follows:
r
/
/
a • •
c
' \
. d
4.6 Truncated Distributions
Any distribution can be truncated easily in MOUSE by using
an IF statement. The rationale is simply to discard any values
generated by the probability function that fall outside of the
truncation limits. There are three forms of truncation:
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IF(variable .LT. lower) GOTO n
IF(variable .GT. upper) GOTO n
IF(variable .LT, lower .OR. variable .GT. upper) GOTO n
where "variable" is the name of the random variable to be trun-
cated; "lower" and "upper" are the lower and upper truncation
limits, respectively; and NnM is a statement number consisting of
an integer between 1 and 9999 (except 5, 10, 15, and 100 which
are reserved for certain MOUSE control lines; see Table B.l in
Appendix B for a summary of the use of these reserved statement
numbers). The integer n is also placed, starting in column l, in
the line initially generating the random variable. The condi-
tional operators, .LT., .GT., and .OR. (which signify "less
than", "greater than", and "or", respectively), are written with
the dots or decimal points; the lower and upper limits must
contain decimal points, also.
For example, to generate random values of price (PRICE) from
.a truncated normal distribution with mean 100 and standard devia-
tion 20 such that no values less than 80 will be produced, we
write:
column 1— —column 7 (or beyond)
\ / \ /
200 PRICE=*GAU(100. ,20. )
IF(PRICE .LT. 80.) GOTO 200
In other words, if PRICE is less than 80, control of the program
reverts to the line with statement number 200, i.e., the price-
generation line.
If, in addition, we wanted to generate random values of
demand (DEMAND) from an exponential distribution with mean 5 such
that no values greater than 10 will be produced, we write:
200 PRICE=GAU(100.,80.)
IF(PRICE .LT. 80.) GOTO 200
201 DEMAND=EXP(5.)
IF(DEMAND .GT. 10.) GOTO 201
4.7 Composite Distributions
Composite distributions are mixtures of two or moire distri-
butions, the most commonly encountered being the bimodal type.
For example, in many production facilities the quantity produced
per unit of time is bimodally distributed with the first peak
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occurring at quantity equal to zero. This occurs when the pro-
duction line closes down for repair or preventive maintenance.
For composites consisting of a mixture of just two probabil-
ity distributions, another form of the IF statement can easily be
used to generate the appropriate random values:
variable=functionl
IF(RAN(IRAN) . LE, w) variable=function2
where "functionl"and "function2" refers to the two functions
involved and "w" is the fraction of the time that function2 is to
be used. [RAN(IRAN) is the MOUSE random number call? see Appen-
dix B, Section B.3.] The conditional operator, .LI. (which
signifies "less than or equal to"), as well as the w require the
decimal point. For example, a 50-50 mixture of normal distribu-
tions with means/standard deviations of 100/20 and 200/45, re-
spectively, could be generated by:
PRICE«GAU(100.,2 0.)
IF(RAN(IRAN) .LE. 0.5) PRICE=GAU(200.f45.)
In the production example cited above suppose that, when the line
was functioning, the quantity produced followed a normal distri-
bution with mean 50 and standard deviation of 15, and that the
line was down 10% of the time. The quantity would be generated
by:
QUANTITY=GAU(50.,15.)
IF(RAN(IRAN) .LE. 0.10) QUANTITY=0.0
More complex composites can be generated using this method; the
following are the conditional operator codes that are available
to you for the construction of such composites:
Operator Meaning
•EQ. equal to
.NE. not equal to
.GT. greater than
.GE. greater than or equal to
.LT. less than
.LE. less than or equal to
.OR. or
.AND. and
The compositing and truncating functions illustrate the use
of two powerful instructions in MOUSE: the IF statement and the
GOTO statement. The computer normally executes a program line by
line, but both the order of execution and the execution itself
can be altered by these transfer functions. The GOTO statement
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illustrated in Section 4,5 imposes an unconditional transfer of
control? the IF statement imposes a condition on the operation.
In Chapter 10, Section 10,4, an additional use of the IF state-
ment is described. Further uses of the IF and GOTO statements
are found in Exercises Nos. 8 and 9 of Chapter 10. The use of IF
and GOTO statements can easily transform MOUSE into a more gener-
al purpose simulation language.
4.8 Examples of Probability Function calls
The following stochastic variable sampling lines, along with
their appropriate Dimension and Read lines, were taken from the
Waste Oil Reclamation Problem shown in Figure 3.2:
C
C EMPIRICAL VARIABLE DIMENSION LINES
C
DIMENSION D1(3,2),D2{4,2),03(4,2)
C .
C EMPIRICAL VARIABLE READ LINES
C
CALL READ(D1,3) .
CALL READ(D2,4)
CALL READ(D3,4)
C.
C STOCHASTIC VARIABLE SAMPLING LINES
C
CONTAM=DUNI(1.0,4.0)
RAND1«CUNI(-10.0,10.0)
PRICE1=EDIS(D1,3)
SLOPE=ECTS(D2,4)
FROD2=STEP(D3,4)
RAND2=GAU(0.00,0.10)
PRICE3=EBETA(6.5,8.0,9.5)
PROD3=EXPON(18.0)
XLIFE=DTRI(5.0,10.0,15.0)
XININV=TRA(1600.0,2000.0,2200.0,2400.0)
In this case Empirical Variable Dimension and Read lines were
necessary since three probability distributions were of the
empirical or semi-empirical type. On the other hand, the follow-
ing probability function calls were taken from the Regulatory
Problem shown in Figure 3.3:
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Mouse Manual Page 55
C
C STOCHASTIC VARIABLE SAMPLING LINES
C
PRECIP=TRA(762.0,1016.0,1270.0,1524.0)
RLEACH=CUNI(.5,1.0)
DEPTHL=TRA(3.0,6.1,9.1,15.2)
WDENS=TRI(297.0,415.0,534.0)
No Stochastic Variable Dimension or Read lines were necessary
here because none of the distributions (Trapezoidal, Continuous
Uniform, and Triangular) used required them.
4.9 Rotes On The Generation Of Stochastic Variates In MOOSE,
and Checks On NOOSE Probability Call Syntax
The method of generation of the normal distribution is the
modified Box-Mueller algorithm of Marsaglia and Bray (1964); fpr
the gamma distribution the combination method described by Prit-
sker and Pegden (1979) is used; and for the exponential distribu-
tion the method is that of Von Neumann (1951). Those for the
empirical beta, discrete uniform, triangul arf ax^di t3^ajpezoidal
were developed by the author (the first is based oh re-scaling
the standard beta distribution to an arbitrary interval, and the
last three are based on the inverse transformation method). The
methods used for the other distributions can be found in Naylor,
et al. (1966).
A utility program, CHECKER (see Chapter 11, Section 11.2),
can be used to examine any MOUSE source program to insure that:
(1) the proper number of arguments are made to MOUSE probability
calls, (2) the arguments are reasonable (e.g., no negative stand-
ard deviations, "upper" is greater than "lower", etc.), the
proper form of argument is used, i.e., decimal versus integer
number or variable), and (4) the dimensions for empirical proba-
bility distributions agree with those specified in the Dimension
Block.
4.10 Fitting Classical Probability Distributions With IMP
IMP (Interactive Modeler for £robabilities) is a companion
system to MOUSE that serves a variety of functions that deal with
classical probability distributions. Unlike MOUSE, the user does
not write programs for IMP? it is an interactive system that
queries the user for information and data.
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Mouse Manual Page 56
The user has a choice between placing the observations into
a data file (named IMP.DAT) accessible to the program or entering
the data directly from the keyboard. The first line of the data
file contains descriptive information (up to 60 characters) used
to identify the file. (This line is mandatory.) The remainder of
the file contains the sample data, one observation per line
starting in column 1, with the use of the decimal point mandato-
ry. Figure 4.3 shows such a file for part of a 100-observation
i'9'if'Si c nnne i e+' i rtiT #"% f m i -*n K a i**ss 4* a Van at* va Hp ^Ain a vi ym a 1
del tq S6 v COil5 JL151Xily OX> li«*lDia>6Jl 15 v»Ci J*. €5#i Gi t"
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FITTING A DATA SET TO A CLASSICAL DISTRIBUTION: NORMAL (100,25)
89.48
97.07
106.23
90.26
107.13
125.30
83.86
81.47
112.08
103.82
121.52
104.38
84.66
108.69
131.95
100.78
123.32
107.21
87.83
105.28
81.05
104.12
86.37"
117.23
109.65
66.40
77.83
86.15
65.72
82.34
95.85
FIGURE
PARTIAL DATA FILE FOR
DISTRIBUTION
4.3
FITTING A CLASSICAL
USING IMP
4. IMP then asks whether you wish to review or edit the
data just entered via keyboard or file. This provides
an opportunity to correct erroneous entries and to add
or delete a data line. The editor is self-prompting, so
no further details will be given here. IMP then prints
out the following statistics (taken from the example
data set):
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ft*******************'***'***'*****************************************
* INPUT DATA STATISTICS , *
* *
* NUMBER OF OBSERVATIONS = 50 *
* MEAN = 101.13080 *
* MEDIAN OF INPUT DISTRIBUTION = 100.66500 *
* STANDARD DEVIATION = 21.35185 *
* MINIMUM - 57.36000 *
* MAXIMUM = 151.59000 *
*******************************************************************
5. IMP will now ask whether you wish the data plotted. If
the data are continuous, it will ask for the number of
intervals desired in the histogram; if the data are
discrete it will simply ask whether you wish the data
plotted or not. In the case of continuous data, the user
may have the data re-plotted, using a different number
of intervals in the histogram.
6. IMP then presents you with a list of options as follows:
*¦*"* #*•**#****##****## A *#*#*********** ik** ik-ie it *
* OPTIONS
* TEST FOR CONTINUOUS UNIFORM DISTRIBUTION
* TEST FOR NORMAL DISTRIBUTION
* TEST FOR L0GN01MAL DISTRIBUTION
* TEST FOR EXPONENTIAL DISTRIBUTION
* TEST FOR EMPIRICAL BETA DISTRIBUTION
* TEST FOR GAMMA DISTRIBUTION
* TEST FOR ERLANG DISTRIBUTION
* TEST FOR ALL OF ABOVE DISTRIBUTIONS
* QUIT
*******************************************
The options enable to user to:
a. Quit or exit from IMP;
b. Test for any of seven continuous distributions
(uniform, normal, lognormal, exponential,
empirical beta, gamma, and erlang);
c. Test for all seven continuous distributions.
7. If any of the testing options are selected, IMP will
query the user as to whether he wishes a plot of the'
original data along with the fitted distribution. If
plotting is selected, you will be asked to specify the
number of intervals desired in the histogram that will
be plotted. IMP will then keep the user informed as to
the distribution it is fitting if it is not printing
other information at the time. Figure 4.4 shows three
such plots; a fit of the uniform distribution to the
data, a fit of the normal distribution, and a fit of the
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HISTOGRAM OF ORIGINAL DATA AND FITTED UNiF/CTS DISTRIBUTION.
(CIRCLES • FITTED, ASTERISKS - ORIGINAL.)
60.50100
66.78300
73.06500
79.34700
85.62900
91.91100
98.19300
104.47500
110.75700
117.03900
123.32100
129.60300
135.8S500
142.16700
148.44900
*•#***• 0
************ 0
************ 0
************* ******0**«*******
*******************0****
*******************0*********************
****#************* 0
****ftft*1r*****»**«**Q****
0
0
************ 0
******* 0
************
HISTOGRAM OF ORIOINAL DATA AND FITTED NORMAL DISTRIBUTION.
(CIRCLES - FITTED, ASTERISKS « ORIGINAL.)
60.50100
66.78300
73. 06500
79.34700
85.62900
91.91100
98.19300
104.47500
110.75700
117.03900
123.32100
129.60300
135,88500
142,16700
148.44900
******0
*********0**
************ 0
¦**•»*•*•*•*********0«**#*****
************** *******11 ****0********
************************ 0
*********************************0*
************************
****************** 0
*******************0****
************ 0
******* 0
*****0******
***0***
HISTOGRAM OF ORIGINAL DATA AND FITTED BETA DISTRIBUTION.
(CIRCLES - FITTED, ASTERISKS - ORIGINAL.)
60.50100
66.78300
73.06500
79.34700
85.62900
91.91100
98.19300
104.47500
110.75700
117.03900
123.32100
129.60300
135.68500
142.16700
148.44900
*«.****.***» Q
•*.#*** Q
l»pMi
FIGURE 4.4: IMP PLOTS OF FITTED DISTRIBUTIONS TO ORIGINAL DATA
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empirical beta distribution. The normal distribution is
obviously better than the uniform distribution; whether
it xs better than the beta xs not clear at thxs poxnt•
After each distribution (or set of distributions under
the ALL option) have been plotted (assuming the plot
option was selected), IMP prints out the table shown in
Figure 4.5. For each distribution fitted, a
Goodness-of-Fit statistic known as the "Foutz Value" is
shown as well as the parameters of the fitted distribu-
tion. (Goodness-of-Fit statistics are discussed in
Section 12.2 of Chapter 12.) The parameters are given in
the same order of the probability function arguments
given in Chapter 4., For example, for the Uniform Con-
tinuous distribution, Parameter 1 is "lower" and Parame-
ter 2 is "upper" as described for that distribution in
Chapter 4. For the Normal distribution, Parameters 1
and 2 are "mean" and "std", respectively; for the empir-
ical beta distribution, Parameters 1, 2, and 3 are "1",
"m" # "u", respectively. The numbers in parentheses
under the Foutz value is the significance level of this
statistic. (A single asterisk signifies a level between
1% and 5%; a double asterisk signifies a level of 1% or
lower.
SUMMARY OF FITS OF CLASSICAL DISTRIBUTIONS TO DATA SET
DISTRIBUTION FOUTZ VALUE
PARAMETER 1
PARAMETER 2
UNIF/CTS
3,3865
57.3600
151.5900
( .9999)
NORMAL
.3071
101.1308
21.3518
( .0380)*
BETA
.3250
98.8574
41.6219
( .1059)
EXPONEN
.6610
101.1308
( .9999)
LOGNORM
• 3168
101.1308
21.3518
( .0691)
GAMMA
.3307
22.4334
4.5080
( .1398)
ERLANG
.3365
4.5080
22.0000
( .1806)
PARAMETER 3
169.7330
FIGURE 4.5:
SAMPLE 'HP OUTPUT, COHTOfPODS DISTRIBUTION
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The Foutz statistic was chosen for use with continuous
distributions because the author has found that it performs
better under many circumstances than the more usual Kolmogorov-
Sairnov statistic. In Figure 4.5 the minimum Foutz value is that
for the normal distribution so that based upon the Foutz value
alone, the best fit is that of the normal distribution (although
the beta distribution is very close). The numbers in parentheses
under the Foutz statistic show the probability of rejecting the
null hypothesis (i.e., that there is no difference between the
distribution from which the sample data was obtained and the
postulated distribution) incorrectly for the Foutz values shown.
Since the point to a goodness-of-fit test is not to reject the
null hypothesis, these significance levels generally should be
set fairly high (say at the .10 level or higher) for decision
purposes, based on the Foutz values in Figure 4.5 we most likely
would reject all hypotheses except those of the normal and empir-
ical beta distributions. However, goodness-of-fit tests are not
all that reliable for small data sets- It is suggested that the
fit of a classical distribution to sample data not be made solely
upon the basis of a classical test of hypothesis; the user should
examine the plots of the original data versus the fitted distri-
bution, the sample statistics calculated from the data, and
anything else the user knows or believes about the distribution
in question before a final decision is made.
The IMP example of this Section involved a continuous
distribution. For discrete distributions only two differences
should be noted;
1. IMP does not query the user for the number of intervals
to be used when plotting discrete data;
2. IMP test for four discrete distributions (pascal/geomet-
ric, poisson, binomial, and uniform);
3. In the Summary Of Fits Of Classical Distributions To
Data Set (Figure 4.6), the Chi-Square value is printed
out rather than the Foutz value, and the degrees of
freedom (d.f.) for the test are also indicated. The
principle is the same? that distribution with the
smallest Chi-Square value is given the greatest cre-
dence. As with continuous data, the numbers in paren-
theses under the Chi-Square value is the significance
level of this statistic. (A single asterisk signifies a
level between 1% and 5% r a double asterisk signifies a
level of 1% or lower.
Figure 4.6 is an example of IMP output for a discrete data set of
100 numbers, based on the binomial distribution with parameters
p=0.3 and n=l0:
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INPUT DATA STATISTICS
NUMBER OF OBSERVATIONS
MEAN
MEDIAN OF INPUT DISTRIBUTION*
STANDARD DEVIATION
MINIMUM
MAXIMUM
100
3.05000
3.00000
1.45904
.00000
6.00000
CTTMMLDV AP PTTC A1P f*T ECCTriT nTCTTDTRTTr7,TnMC frt nSTS CTTT*
i9UI?Jrl/u\X W/Jr C JL X 2) wf vlinwu X vnij I#jLw4K>Ji>uU 4 XviiO X V/ wJwJL
DISTRIBUTION CHI-SQUARE PARAMETER 1 PARAMETER 2
. f >
GEO/PAS DISTRIBUTION CANNOT BE FITTED TO THESE DATA.
POISSON 4.6158 3.0500 5.0000
( .5355)
BINOMIAL 2.1074 .3020 10.0000 4.0000
C .2840)
UNIF/DIS 34.5400 .0000 6.0000 4.0000
(1.0000)
FIGURE 4.6;
SAMPLE IMP OUTPUT, DISCRETE DISTRIBUTION
The lowest Chi-Square value is for the binomial distribution.
Note that the parameters (p-0.3 and n^lO) have been estimated
very well. Note that the pascal/geometric distribution could not
be estimated because the mean of the distribution was 3.05 and
the variance was (1.45)2=2.12. For a pascal/geometric distribu-
tion, the variance must be greater than the mean.
The following restrictions in using IMP should be noted:
1. The number of histogram intervals that can be processed
by IMP is between 2 and 100. If 1 is entered, it will
automatically be changed to 2} is a number greater than
100 is entered, it will automatically be changed to
100.
2. The number of histogram intervals must be less than or
equal to one-third the number of observations or else
an error message will be returned.
3. IMP will not accept fewer than 10 or more than 5000
observations. If fewer are indicated, an error mes-
sage will be returned, and if more are indicated only
the first 5000 will be read.
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Mouse Manual Page 63
Finally, before fitting a distribution to a data set, the data
should be checked for autocorrelation using the option, "ANALYZE
A DATA SET FOR AUTOCORRELATION" (see Chapter 13, Section 13.9),
Note: The binomial, pascal/geometric, beta, poisson, and gamma
probability generators used for grap>hing the fitted distributions
in IMP are from Peizer and Pratt (1968)j the generator for the
normal (and, indirectly, the lognormal) distribution is taken
from Burr (1967).
If it desired to subject output data to IMP analysis,
data can be collected using an Accumulation Line. which has the
following form:
CALL ACCUM(varl,var2,1(fname)1,1(info) 1)
where "varl" and Hvar2" are the names of the variables whose
values are to be collected, "fname" is the name of the file which
is to contain the data, and "info" is a descriptive information
string (up to 60 characters) to be placed in the first line of
the file. Note that both "fname" and "info" are literals so the
single quotes and parentheses pust accompany these items exactly
in the order shown.
The, following example of the use of the Accumulation Line is
taken from the Waste Oil Reclamation Problem of Figure 2.1 where
it is assumed that the variable PVALUE is to be sujected to IMP
analysis:
PVALUE=«ET8*UPW<0.08,XU?f,0.Q}-XIIIIilV
CALL ACajH
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House Manual Page
W
l/i
3
8
t«
r*
*
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Mouse Manual Page 65
INTEGRATE A CONTINUOUS DISTRIBUTION
9,0
9.5,14
10,26
10.5,34
11.39
11.5,41
12.40
12.5,3?
13,32
13.5,24.5
14,17.7
14.5,12.1
15,8
15.5,5
16,2.7
16.5,.8
17 , 0
FIGURE 4.8;
INPUT DATA FOR AM EMPIRICAL CONTINUOUS
PROBABILITY DISTRIBUTION
(i.e., the "portrait" approach of Pouliquen referred to in Sec-
tion 13.6). The human eye is capable of absorbing a great deal
of information, and if the subjective curve so drawn truly re-
flects the analyst's beliefs, then the only remaining task is to
transform the distribution to a form usable by MOUSE.
For illustrative purposes, we assume that the analyst has
drawn the pseudo-continuous probability distribution shown in
Figure 4.7. The term "pseudo" is used since the Y-axis is of
arbitrary scale, under such circumstances the area under the
curve will not necessarily add up to 1 and therefore the curve
will not be a true probability density function. Scaling can be
accomplished by several methods. Since MOUSE requires the cumula-
tive distribution, one method that can be used is to cut the
figure into vertical segments and weight them on a sensitive
balance. The weight of each individual segment, divided by the
total weight, gives the probability of each segment. Subsequent-
ly, these probabilities may be cumulated for entry into MOUSE
empirical continuous probability distribution data cards. To
spare the user from such tedium, option 2 on the IMP Main Menu is
used. First, however, the analyst draws a number of ecmi-spaced
vertical lines on his curve (except perhaps for the last interval
which, of course, is terminated when the height of the distribu-
tion is again zero), as shown in Figure 4.7. (The greater the
number of lines drawn, the more accurately the distribution will
be represented n*! 5s ) The hi© ti"i© 1 xn6s ar© re»adi off
on the arbitrary Y-axis, and the set of (X,¥) values are placed
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HISTOGRAM OF INTEGRATED EMPIRICAL CONTINUOUS DISTRIBUTION
9.00000
9.50000
10.00000
10.50000
11.00000
11.50000
12.00000
12.50000
13.00000
13.50000
14.00000
14.50000
15.00000
15.50000
16.00000
16.50000
17.00000
********
**************
**************
**************
**************
**************
**************
**************
**************
**************
**************
***********
*******
*****
***
*
*******
*****************
***********************
***************************
***************************
*************************
*********************
***************
********
**
INTEGRATED CONTINUOUS EMPIRICAL DISTRIBUTION
VALUE OF VARIABLE
CUMULATIVE PROBABILITY
9.00000
9.50000
10.00000
10.50000
11.00000
11.50000
12.00000
12,50000
13.00000
13.50000
14.00000
14.50000
15.00000
15.50000
16.00000
16.50000
17.00000
.00000
.02097
.08089
.17076
.28011
.39994
.52127
.63661
.73996
.82460
.88781
.93244
.96255
.98203
.99356
.99880
1.00000
FIGURE 4.9!
Tiff's f^rfTfiTJTTF|« TlWPfilll'T'tPn PnVTTlITTnnfl UWtiTPTMkT,
JLflJf vUlrUX | IfllfivKAlwU Wvlli IXiuwUD SAr lAlwUi
DISTRIBUTION, DATA OF FIGURE 4.8
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Mouse Manual Page 67
either (a) into an IMP.DAT data file (see Figure 4.8) somewhat
similar to that discussed in Section 4.9 or (b) entered directly
via the keyboard. The first line of the data file contains
descriptive information (up to 60 characters) used to identify
the file. (This line is mandatory.1 The pairs of (X, Y) values are
placed in subsequent lines of the file with the X-value starting
in column 1 (decimal point required) and the Y-value following
but separated by a comma (decimal point required). The end-
points of the distribution (i.e.. those with values of Y equal to
zerol must be included in this data set. Since 15 vertical lines
were drawn on Figure 4.7, the data file contains 17 entries after
the first line. Data entries from the keyboard follow the same
rules.
The results of selecting IMP option 2 on the Main Menu (and
using the data file shown in Figure 4.8) are shown in Figure
4.9. IMP integrates the discrete data entered, and then prints a
histogram of the integrated distribution. Finally, it prepares
the cumulative probability distribution required for the MOUSE
ECTS function.
EXIRCIS1S
Which of the following are valid probability function
statements and calls?
1.
ECTS(Rl,17)
2.
EBETA(200.,100.,400.)
3.
ECTS(D1,3.)
4.
TRA(5.,7.,10.,20.)
5.
BIN(0.3,10.)
6.
GAU(-15.,5.)
7.
GAUL(-15.,5.)
8.
AGAUL(22.,30.)
9.
TRI(5.,10.,20.)
10.
READ(D7,3 0)
Find the errors, if any, in the following Empirical Data
line sets.
(a) The function is SDIS(D1,3)
11.
PRICE... PRICE OF ITEM
10.,0.3
20.,0.7
30.,1.0
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Mouse Manual Page 68
12.
CONTAM FEED CONTAMINANTS
1.,0.0
2.,0.5
3.,1.0
(b) The function is ECTS(D1,3)
13
PRICE
PRICE OF,ITEM
10,,0.3
20.,0.7
30..1,0
14. Construct a truncated triangular distribution with
modal value 20, and lower and upper limits 10 and 70, respective-
ly. The truncation limits are 15 and 60.
15. Production on three shifts is normally distributed with
mean/standard deviations of 100/10, 75/5, and 55/5, respectively.
Construct a composite probability function for daily production.
16. Fit the following two data sets to a classical proba-
bility distribution.
Data Set 1
(continuous data)
Data Set 2
(discrete data)
48.87
60.63
88.99
143.06
19.31
145.66
165.16
34.17
78.40
42.53
123.09
47.72
76.65
81.56
99.67
28.77
33.81
109.39
290.52
169.18
68.63
303.63
33.27
306.19
20.15
71.20
75.99
37.68
80.46
135.46
93.72
60.97
253.71
219.31
121.76
144.76
165.64
223.06
78.71
21.99
25.83
55.57
292.71
16.82
137.94
372.50
44.01
48.09
160.71
82.71
6
2
6
8
10
¦5
, 3
7
3
6
5
5
3
4
8
7
3
5
4
: 7
1
14
6
7
1
5
5
4
5
5
3
2
6
8
3
7
5
2
6
4
7
7
2
5
5
4
6
4
1
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Mouse Manual Page 69
17. Using IMP, integrate the empirical continuous distribu-
tion shown in Figure 4.10 using (a) 13 points and (b) using 25
points. Does the 25-point integration provide a substantially
better fit than the 13-point integration?
KBFBRXHCES
Burr, I.W., "A Useful Approximation to the Normal Distribution
Function, With Application to Simulation", Technometrics. Vol.. 9.
NO. 4, p. 647-651 (1967).
Foutz, R.V., MA Test For Goodness-Of-Fit Based Upon An Empirical
Probability Measure", The Annual of Statistics. Vol. 8, No. 5,
p. 989-1001 (1980).
Marsaglia, G. and Bray, T.A., "A Convenient Method for Generating
Normal Variates*', SI AM Review, 6, p. 260-264 (1980).
Naylor, T. J., Balinfty, J.L., Burdick, D.S., and Chu, K., Com-
puter Simulation Techniques, John Wiley & Sons, New York, p. 1-
352 (1966)•
Peizer, D.B., and Pratt, J.W., "A Normal Approximation for
Binomial, F, Beta, and Other Common, Related Tail Probabilities,
I"i ii. Aroer. Statistical Association. Vol. 63, No. 324, p. 1416-
1456 (1968).
Pritsker, A.B., and Pegden, C.D., Introduction to Simulation and
SLAM. John Wiley & Sons, New York, p. 567 (1979).
Von Neumann, J., "Various Techniques Used in Connection with
Random Digits", Monte Carlo Method, Nat. Bur. of Standards,
Applied Math. Series 12, p. 36-38 (1951) .
-------
Mouse Manual Page 70
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Mouse Manual Page 71
CHAPTER 5
MOUSE UBRARY INTEREST FUNCTIONS
5.1 Introduction
Economic analyses invariably involve interest formulas of
various types, some of them unfortunately rather complex. For
this reason, MOUSE provides eleven interest functions that may be
called by you when desired. These functions cover almost all of
the interest situations encountered in practice. Unlike stochas-
tic input variables, the method of introduction of interest
functions is direct, i.e., they are used directly in the risk
equations. The general form of compound interest and gradient
interest functions in MOUSE is:
function(interest rate,term,period)
Explanation of the period will be deferred until Section 5.4.
For the time being, the period will be set to zero in the exam-
ples. .. - :
Decimal points are used in all interest functions. The
functions are divided into three groups; (a) compound interest,
(b) gradient series, and (c) continuous compounding. In the
following formulas, the symbol P generally refers to a present
worth, F refers to a future worth. and A refers to an annual
payment or annuity. The basic payment or receipt involved in the
formulas is the unit amount,* for other quantities the function
is merely multiplied by the actual amount involved.
5.2 Compound Interest
a- Single Payment Compound Amount Factor {SCAS
The formula for the compound amount, Fj_, obtainable in n
years at an interest rate of i from a single initial deposit of
$1 is
Fi = (1 +. i)n
[5.1]
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Mouse Manual Page 72
This is called by the function, SCAf i,n» 0.). where i represents
the interest rate per interest term and n represents the number
of interest terms. This convention is followed for all MOUSE
interest functions. Example: "If $1 is invested at 6% compounded
annually for 10 years, how much will be accumulated at the end of
the term? Using SCA(0.06,10.0,0.), MOOSE will return the value
of 1.79085.
b. Single Payment Present Worth Factor ISMl
The formula for the present worth, of $1 n years hence
at an interest rate of i is
. . PX « 1/(1 + i)" ' . . [5.21
This is called by the function, SPWfi.n.O.1. Example: What is
the present worth of $1, 5 years hence, if the interest rate is
10%? Using SPW(0.10,5.0,0 .) , MOUSE will return the value of
0.62092.
The two preceding formulas involved single payments. The
next four involve uniform end-of-year payments.
C- qnifola ifElgg'present worth Factor (RPffI
The formula for the present worth, ?2, of $1 deposited at
the end of every year for n years at an interest rate i is
P2 = [(1 + i)n -l]/i(l + i)n .¦ £5.3]
This is called by the function UPWTi.n.0.). Example: What is
the present worth of $1 deposited at the end of each of 9 years
at an interest rate of 20%? Using UPW(0.20,9.0,0.), MOUSE will
return the value of 4.03097.
d. Uniform Series Present Worth Factor With Constant
. 'Ferc?er?tage C^apge (UPWPt
The previous factor involved $1 deposited at the end of
every year. If each year after the first this amount is
increased or decreased by 100j percent of the previous year's
amount, the present worth, P3becomes
P3 = [i/U " j)][l " <1 + j)nd + D"n] [5.4]
This is called by the function UPWPfi.n.i.0.K Example: If $1
is deposited at the end of the first year,•and subsequent year-
end deposits are increased 12% per year, what is the present
worth at the end of 3 years at an interest rate of 8%. Using
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UPWP(0.08,3.0,0.12,0.)/ MOUSE will return the value of 2.8 8194.
Compare this result to that obtained when the change is a yearly
12% decrease: UPWP returns a value of 2.29512.
e. Uniform Series Compound Amount Factor (UCA)
The formula for the compound amount, F2, obtainable in n
years from a series of end-of-year payments of $1 at an interest
rate of i is
F2 - [(1 + i)n " l]/i [5.5]
This is called by the function UCAfi,n.0.i. Example: How much
will be accumulated in a fund, earning 6% interest, at the end of
10 years if Si is deposited at the end of each year? Using
UCA(0.06,10.0,0.), MOUSE will return the value of 13.18084.
f. Sinking Fund Factor fSINtO
The formula for the uniform annual series of deposits, A\,
to produce $1 at the end of n years at an interest rate of i is
Al = i/[(1 + i)n - 1] [5.6]
This is called by the function, SINKfi.n.0.>. Example: How much
must be deposited each year for 7 years at 8% interest in order
to accumulate $1? Using SINK(0.08,7.0,0.), MOUSE will return a
value of 0.11207.
g. Capital Recovery Factor (CAPREC)
The formula for the deposit, A2, necessary to draw out $1 at
the end of each year for n years, leaving nothing in the fund at
the end, is
A2 « [i(1 + i)n]/[(1 + i)n - l] [5.7]
The Capital Recovery Factor is also the uniform end-of-year
payment necessary to repay a debt of $1 in n years at an interest
rate i, ¦ The function is called by CAPREC(i,n.0.). Example;
What equal year-end withdrawals can be made each year for 10
years, leaving nothing in the fund after the tenth withdrawal, at
an interest rate of 6% and an initial deposit of $1? Using
CAPREC(0.06,10.0,0.), MOUSE will return the value of 0.13587.
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5.3 Gradient Series
Sometimes receipts or expenses increase or decrease each
year by varying amounts. If the change is the same each year, it
is known as a uniform arithmetic gradient. Calculations
involving gradients are most easily handled by first converting
them to equivalent uniform series.
a. Gradient Series Equivalent Uniform
Annual Amount Factor fGCA)
Using end-of-the-year payments, we assume that the payment
the second year is greater than the first year by $1, the third
is $1 greater than the second year, etc. The formula for the
equivalent uniform annual figure, A3, for n years at an interest
rate of i is
A3 = l/i - (n/i)[i/(1 + i)n - 1] [5.8]
This is called by the function GCAf i, n, 0.). Example: There is
an increase in cost of an item of $1 per year over a term of¦6
years; the interest rate is 12%. What is the equivalent uniform
annual cost? Using GCA(0.12,6.0,0.), MOUSE will return the value
of 2.17207.
b.• Gradient .Series Present Worth Factor fGFW)
The formula for the present worth of a gradient series, P4,
is merely the previous formula, A3, multiplied by the uniform
series present worth factor, P2, i.e.,
P4 = A3P2 [5.9]
This is called by the function GFW(i.n.0.)« Example: What is
the present worth of the gradient series in the last example?
Using GPW(0.12,6.0,0.), MOUSE will return the value of 8.93027.
5.4 Nominal Interest Sates
Although the interest period used in illustrating the above
nine compound interest formulas was the year, any interest
period could have been used. However, when other than annual
compounding is used, a real difference exists between nominal and
true (sometimes called " effective") interest rates. The MOUSE
functions so far described are assumed to be provided with the
true or effective rates. For example, to compute the compound
amount of $1 (single payment) for 8 years at a true annual inter-
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est rate of 6% compounded semiannually, we note that the number
of periods is 8(2) = 16 and the rate per interest period is 6/2 =
3%. Thus, using SCA(0.03,16.0,0.), MOUSE will return the value
of 1.60471.
If, on the other hand, a nominal annual interest rate, r, of
6% was given we would first have to convert it to an effective
rate per annum, i, by the following formula;
i = (1 + r/m)m - l [5.10]
where m is the number of times per year the amount is compounded.
Thus, for an r of 6* in the above example,
i = (1 + 0.06/2)^ - 1 = 0.0609.
Using SCA(0.0609,8.0,o.), MOUSE will return the value of 1.61596.
However, to spare you from having to make this sort of
calculation, a provision has been made to use the compound inter-
est and gradient series functions directly even when nominal
annual interest rates are specified. To do so, the nominal
interest rate is used and the "period" argument is set equal to
the number of yearly compounding periods. For example, for the
above problem in which the compounding is semiannual for 8 years
at a nominal rate of 6%, the interest function would appear in
the risk equations as SCA(0.06,8.0,2.0). The value of SCA would
be returned as 1.61596. Note that for true interest rates, the
value of the period is always zero.
Thus, MOUSE interest functions can be used readily for any
compounding period, regardless of whether true or nominal
interest rates are specified.
5.5 Continuous Compounding
All of the interest functions described so far have assumed
that cash flow occurring throughout the year was concentrated at
the end of the interest periods. An alternative convention is to
assume that cash flow occurs uniformly throughout the year. To
do so, however, it is necessary to use continuous compounding
interest formulas. The general form of a continuous compounded
interest function in MOUSE is:
function([-]interest rate,term)
The minus sign before the interest rate is optional and will be
explained later.
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.*• ' Swig- JZpyamnt E«fiAt SBtKfth Factor Igjjfltt
The formula for the present worth', P5, of $1 for the period *
of tine n-l to n at an interest rate of i, is
p5 - ((ei - lj/e^n ~ !> jie* [5,11)
This is called by the function CSPWfj.n). Example: _ what is the
present worth of $1 compounded continuously * from the period year
3 to year 4 at an interest rate of 25*? Using CSPW(0.25,4.0) ,
MOUSE will return the value of 0.45890.
b. Vnifprm ger|g§ Wqrt.h facfror ¦ 191,Pfl ¦'
The formula for the present worth, Pg, of Si deposited
continuously throughout the year for n years at an interest rate
of i is merely the sum of the P5 factors from year l to year n,
i.e.,
n
P6 - , I -Psi (5.12]
j=l
This is called by the function CUPWfi.nl. Example: What is the
present worth of $1 per year flowing continuously for 4 years at
an interest rate of 25.%? Using CUPW(0.25,4 .0) , MOUSE will return
the value of 2.64583.
5.6 Seminal Interest Sates, Continuous Compounding
In the above two continuous interest functions, the interest
rate is assumed to be the true or effective rate. As with the
compound interest and gradient series functions, however, it may
happen that the nominal interest rate, r, is specified instead
of i. (In continuous compounding, the nominal annual rate is
also known as the - force of interest.) To convert a nominal rate
to an effective rate, the following formula can be used:
i, - er -.1.' [5.13]
For -example, a nominal, rate of 25% is converted to an effective
•or true rate as follows;
i * e0.25 - 1 « 0.28403.
Using CSPW(o.28403,4.0), MOUSE will return the value of 0.41755.
Again, however,, in order to spare you from having to make this
sort of calculation, a minus sign is used before the interest
rate and the nominal interest rate is used. As an example, for
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the above problem in which the compounding is continuous with a
nominal interest rate of 25% from the year 3 to year 4, the
value of CSPW(~0,25,4.0) will be returned as 0.41795, the same
value as shown with the additional hand calculation. Thus, the
MOUSE continuous interest functions can also be used readily
regardless of whether true or nominal interest rates have been
specified.
For additional detail on the genesis and use of most of the
interest formulas described here (the Uniform Series Present
Worth Factor With Constant Percentage Change was developed by the
author), it is suggested that the reader consult Grant and Ireson
(1970).
5.7 Exaaple of Probability Function Calls and
MOUSE Interest Call Syntax
The following line was taken from the Waste Oil Reclamation
Problem shown in Figure 3.2:
PVALUE=ANETB*UPW(0.08,XLIFE,0.0)-XININV
The function, UPW(0.08,XLIFE,0.0), is the Uniform Series Present
Worth Factor with a nominal interest rate of 8%. The period
varies from iteration to iteration with the variable XLIFE,
i.e., project life is a stochastic variable. Note that even the
simplest of problems involving uncertain project periods are
almost impossible to calculate by hand, MOUSE removes this diffi-
culty entirely.
A utility program, CHECKER (see Chapter 11, Section 11.2),
can be used to examine a MOUSE source program to insure that: (1)
the proper number of arguments are made to MOUSE interest calls,
Qiiu 4L/ viic qj:gunignvs »rs rcdiSunaDic ^ i § no negative inwcj.cs w
rates, initial investments, or interest periods), the proper form
of argument is used, i.e., decimal versus integer number or
variable).
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EXERCISES
Which of the following are valid interest function
statements and calls?
1. SPW(.09,19,0.)
2. CAPREC(.12,3.,0.)
3• GCA(.20,10.0,0.)
4. SINK(10.,.10,0.)
5. UPWP(. 09,.10,20. ,0. )
6. CUPW(0.08,-15., 0. )
7. CSPW(0.08,15.,1.)
8. GPW(-.08,15.,0.)
9. SCA(. 18 , 3 . 5, 0.)
REFERENCE
Grant, E.L., and Ireson, W.G., Principles of Engineering Economy,
Fifth Edition, The Ronald Press Co., New York, p. 1-640 (1970).
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CHAPTER 6
THE MOOSE TABLE
€.1 Intr ochict ion
The MOUSE Table is the most important utility for any MOUSE
program since it provides all of the vital statistics needed for
the final output stochastic variables and for intermediate output
stochastic variables as well. The MOUSE Table not only includes
traditional statistics such as mean, standard deviation, minimum,
maximum, and coefficient of variation (the coefficient of varia-
tion is equal to the standard deviation divided by the mean, and
is a measure of relative variability of the variable), but it
also calculates the average of values less than zero. If the
random variable tabulated is in dollars, the average of values
less than zero is the familiar "average opportunity loss" of
economists or "expected value of perfect information" of deci-
sion scientists (see Winkler, 1972).
The MOUSE Table produces much more than just these statis-
tics, however. It also produces the complete probability distri-
bution and cumulative probability distribution of the variable.
Thus you can find the probability of the random variable equal to
or falling below a stated value, or the probability of the random
variable equal to or falling above a stated value. The distribu-
tions are given in both histogram and graphical form, providing
the advantages of both numeric and graphical presentation. Fur-
thermore, it is not necessary to interpolate within MOUSE tables
for the value of the variable at rounded fractiles of cumulative
probability. A short table is printed out after the histogram
giving the variable values at the following cumulative probabili-
ties (in percentages):
0,1,1,5,10,25,50,75,90,95,99/99.9
Although the request for table output is made in the Program
Block of a MOUSE program, every table (except for Dummy Tables -
see Section 6.5) must be dimensioned; accordingly, a table in-
volves two of the four basic MOUSE blocks; the Dimension Block
and the Program Block.
Requests for MOUSE tables are of two basic forms: MOUSE-
dimensioned and User-dimensioned. To structure a histogram
properly, you must have a good estimate of the minimum and maxi-
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mum values of the distribution. In complex problems, however,
this information is usually not available unless a MOUSE run has
already been made. Thus, if you dimension the histogram, two
MOUSE runs must be made to obtain a usable histogram. An alter-
native is to take advantage of this option whereby MOUSE does the
dimensioning automatically. Since it is simpler, the MQUSE-
Dimensioned Table will be discussed first.
6.2 The MOUSE-Di*ensioned Table
A MOUSE-dimensioned Table output line has the following
form:
CALL TABLE(variable,'(lname)',dl,d2,ni,ti)
where "variable" is the name of the variable to be tabulated,
"lname" is an alphanumeric name (up to eight characters) for the
table (note that this is a literal name and the single quotes and
the parentheses must accompany the name exactly in the order
shown) and can contain up to but not more than eight characters),
dl and d2 are dummy floating point variable names chosen arbi-
ter a r x 1 ^ iz s t' r1 e m e Bib e 3^ t h a t cl 1 a J^cl ci 2 a n o t a e 13 e e n u s e d
elsewhere in the program), "ni" (an integer) is the number of
histogram intervals desired for the table output (including a
possible underflow and/or overflow interval (in grouping data
into frequency distributions, a rough working rule used by stat-
isticians is to aim to have about 20 intervals), and "ti" is the
name of an array used for storing the histogram table between
iterations of the program. Also note that the variable name must
be a decimal name. (A decimal name is one that starts with any
letter except I, J, K, L, M, or N. See Chapter 3, Section 3.2).
To use this option, a minimum of 100 iterations (ITER"=100)
must be used. As an example, the following is the Table Output
line of Figure 3.2: .
CALL TABLE(PVALUE,'(PRES VAL)',DUMO,DUM1,40,TAB)
With one exception (the Program Listing line; see Chapter 7,
Section 7.10 the last executable statement in any MOUSE program
(i.e., the last user-written line before MOUSE Control line
Number 5) must be a Table Output line, although additional
Table Output lines many also be used at any appropriate place in
the Program Block. Each program must also contain one or more
Table Dimension lines that dimension the arrays used in the Table
Output lines (an exception is when a Dummy Table is used; see
Section 6.5). These have the following form:
DIMENSION tl(nl),t2(n2),etc.
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where the "ti" arguments are identical to those specified in
their respective Table Output lines and the "ni" arguments are
equal to the respective ni plus six. These additional six ele-
ments of the dimension number are used to accumulate the mean,
standard deviation, minimum, maximum, and percentage of entries
less than zero. For this reason, the minimum dimension of any
table is six. In the Waste Oil Reclamation Problem shown in
Figure 3.2, the Table Dimension line was:
DIMENSION TAB(46)
The Table Output line is placed after (usually immediately after)
the uncertainty equation that calculates the quantity to be
tabulated; all Table Dimension lines are placed in the Dimension
Block of the MOUSE program. (See Chapter 11, Section 11.6, for a
discussion of TRAP, a MOUSE utility that greatly facilitates the
preparation of the Dimension Block and even of the Table Output
lines themselves. Also, CHECKER, Chapter 11, Section 11.2, will
examine a MOUSE source file to insure that arguments of standard
and dummy tables are correct, and that the dimension of the array
used for storing the standard histogram table agrees with those
specified in the Dimension Block.)
The algorithm used in MOUSE for the MOUSE-dimensioned histogram
guarantees (see Grant and Leavenworth, 1972) that:
a. If the variable tabulated is distributed normally,
approximately 0.05% of the values will fall into the
underflow/overflow portions of the histogram?
b. If the variable is distributed unimodally and
symmetrically about the mean, less than 3.6% of the
values will fall into the underflow/overflow portions
(Camp-Meidell inequality); and
c. Regardless of the distribution of the variable, less
than 8.21 of the values will fall into the
underflow/overflow portions (Tchebycheff Inequality).
6.3 The User-Dimensioned Table
The user-dimensioned Table Output line has the following
form:
CALL TABLE(variable,'(lname)lower,interval,-ni,ti)
where "variable", "lname", "ni", and "Ti" are defined as for
MOUSE-dimensioned tables. Note. however, that ni is preceded by
a minus sign. The dummy floating point variable names are re-
placed in the User-dimensioned table by "lower", which is the
lower bound of the histogram to be produced by MOUSE, and "inter-
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val" (a decimal number greater than zero), which is the width of
the intervals desired. We could have, for example, replaced the
Table Output line of Figure 3.2 with!
CALL TABLE(FVALUE,'(PRES VAL)',-3500.,250.,-40,TAB)
This would start the lower interval at FVALUE equal to -3500, and
make the width of each histogram interval equal to 250. Like
MOUSE-dimensioned tables, the appropriate Table Dimension lines
must be used with User-dimensioned lines, also.
6.4 The Short Table
A "short" version of the Table Output line can be used that
produces only the six basic MOUSE table statistics, i.e., mean,
standard deviation, minimum, maximum, coefficient of variation,
and average of values less than zero, skipping the histogram and
graphs. The short version is of the form, ^
CALL TABLE(variable,'(lname)',0.,0.,0,Ti)
In this Output line, the number of histogram intervals is given
as zero, and "lower" and "interval" are given as zeros with
decimal points. A Table Output line with dimension equal to 6
will, of course, still have to be supplied.
6.5 The Diray Table
It has previously been mentioned that, with the exception of
the Program Listing line (see Chapter 7, Section 7.10), the last
executable statement in any MOUSE program must be a Table
Output line. However, when you do not want any table output
(e.g., when generating random data for testing purposes - see
Appendix E - or when you just desire Plot output - see Chapter
8), a Dummy Table can be substituted. The Dummy Table Output Line
is of the form:
CALL DTABLE(switch)
where "switch" is either "1" or "2". The usual default for the
"switch" is M1H in which case the standard MOUSE output file,
MOUSE.OUT, will be constructed. With "switch" set to "2", the
MOUSE.OUT file will deleted tM end of the run. Note that
Table Dimension lines are not used with Dummy Tables.
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6.6 User Table Scaling
The table subroutine in MOUSE involves the calculation of
the standard deviation of the variable involved, a m31©i" of
squaring and summing the values of the variable over the number
of iterations used. If these values are large, the combination
of squaring and adding over many iterations could easily exceed
the capacity of the computer. When this happens, most computers
will print a string of asterisks in lieu of the result or else
print some form of error message such as "FLOATING point
ARITHMETIC OVERFLOW". Although some automatic scaling is built
into MOUSE, it may occur that the computer capacity is exceeded.
In such a case, you may try another run, scaling the variable
right in the Table Output line. For example, to scale down the
value of PVALUE by 1000 in Figure 3.1, the Table Output line
would be written as:
CALL TABLE(PVALUE/1000.,'(PRES VAL) ',DUM0,DUM1,40,Tl)
You must remember, however, that all of the numbers printed in
the table so produced will also be scaled by this factor of 1000
(except the Coefficient of Variation which has no absolute
scale). User-scaling may be applied with any kind of MOUSE table
(user- or MOUSE-dimensioned and short forms). Scaling in a table
does not affect the value of the variable for use in subsequent
calculations in a model. Note that the scaling factor must have
a decimal point.
6.7 Literal liases in NOOSE
Literal names are names that ultimately are reported in
MOUSE output reports. Such names are always first enclosed in
parentheses, then enclosed in single quotes, e.g.
'(COST)'
This may appear to be somewhat awkward, but it necessary in order
to accommodate different FORTRAN compilers that may be used to
implement MOUSE. Literal names are found in the following MOUSE
output lines:
Name line
Plot Output Line
Program Listing line
Table Output line
Trace line
Vector Table Output line
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6.8 Mixing Table Types
Tables of different types (i.e., user-dimensioned, MQUSE-
dimensioned, and short tables) can appear in a single program.
EXERCISES
Which of the following table statements are valid?
1. CALL TABLE(COST,'(COST)',1000.,20,r-40,Tl)
2. CALL TABLE(REVENUE,'(REVENUE)',A1,A2,40.,T1)
3. CALL TABLE(WAGECOST,'WAGECOST',5.5,0.6,-40,Tl)
4. CALL TABLE(X5,'(COST FAM)',-450.,20.,-60,Tl)
5. DIMENSION Tl(40)
6. CALL TABLE(CUSTOMER,'(CUSTOMER)',ATI,BT1,-40,Tl)
7. CALL TABLE(SPEED/1000,'(SPEED)',Z1,Z2,100,Tl)
8. CALL TABLE(X5,'(X5)',100.,0.,-20,Tl)
9. DIMENSION (Tl,40)
i ft t T&fiT p fw&ft? 9 fw&.r'V \ 9 a a its tm \
XU« v/iJyJy InOJyL \ WnVjJy f ( nnuL ) | U i ^ U » ^ &U| 11J
11. DIMENSION R2(83)
REFERENCES
Grant, E.L., and Leavenworth, R.S., - Statistical Quality Control.
McGraw-Hill Book Co., New York, p. 74-75 (1972).
Winkler, R.L., Introduction to Bavesian Inference and Decision.
Holt, Rinehart and Winston, New York, p. 1-563 (1972).
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USSR CONTROL DECISIONS
7.1 Introduction
As was mentioned in Chapter 3, the Specification Block
{i.e., those lines placed immediately after MOOSE Control Line 3)
informs the computer of certain of your control decisions. Five
line types to be discussed in this Chapter and one in the next
Chapter will comprise the types to be found in the Specification
Block. This Chapter will also explain the use of the Name Line
and how multiple runs (i.e., runs in which parameters are altered
from run to run) are made in MOUSE.
7.2 Hue Iteration Nuaber line
The number of iterations within a MOUSE program is
determined by the Iteration Number Line. This line is of the
following form; ...
ITER«n
where n is the number of iterations desired (an integer).
Usually this is a fairly substantial number <1000 or greater),
and normally never lower than 100 (it can not be less than 2).
If you do not supply an Iteration Number line, the default number
of iterations is 1000.
7.3 The Snap Interval Mm
There are two types of Monte Carlo simulations with respect
to the number of iterations in a run (Law, 1980). A steady state
simulation is one for which the output is conceived as a limit as
the number of iterations becomes infinite. From a practical
standpoint, the actual number of iterations in a steady state
simulation is made large enough to get "good" estimates of the
output quantities of interest. The other type is the terminating
simulation where the outputs are defined relative to an interval
of simulated time, [ 0,Te], where Te is the instant in the
simulation when a specified event, E, occurs. (Te may be a
stochastic variable.) The event E is specified before the
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simulation begins. Since the outputs for terminating simulations
explicitly depend on the state of the simulated system at time 0,
care must be taken in choosing the initial conditions. (An
example of a terminating simulation is that of simulating a frame
in bowling. The frame is over when either two balls are thrown
or when all the pins go down, whichever comes first. The end of
the frame is E, and the time at the end of the frame is Te. The
initial conditions, as far as score is concerned, are represented
by the history of the prior frame. If, for example, a spare was
achieved in the prior frame, then the number of pins downed on
the first ball are also added to the score for the preceding
frame.)
The Monte Carlo uncertainty analysis simulations dealt with
in this book are of the steady state type? initial conditions,
therefore, are of no concern. As it is sometimes difficult to
decide how many iterations should be employed to achieve steady
state conditions in a, Monte Carlo simulation, MOUSE provides an
option called a "snap" which is used in conjunction with MOOSE
tables. The Snap Interval Line is of the form:
ISNAP=s
where s is some integer smaller than the n of the Iteration Line.
In Figure 7.1, which is Modification #1 of the Waste Oil Recla-
mation Problem, the use of the Snap line is illustrated. The
snap interval is set at 1000, and the iteration line is set at
5000. This will produce a total of four snap outputs, i.e., at
number of iterations = 1000, 2000, 3000, and 4000. At 5000, the
usual MOUSE output will be delivered. The snap output for the
program of Figure 7.1 is shown in Figure 7.2. (The snap output
follows the echo print of any input data you may have supplied,*
otherwise, the snap output is the first thing printed by MOUSE.)
At each snap interval the mean and standard deviation is printed
out for each table defined in the program. You can follow the
snap output to see whether the output has reached steady state.
At number of intervals = 4000, it does not appear from Figure 7.2
that steady state has been reached.
7.4 Hie Seed Reset Lim
For those who like to rerun programs using different random
number seeds in order to see what effect a different set of
random numbers has on the output, a Seed Reset Line is available.
This line is of the following form;
IRAN=seed
where "seed" is any odd integer not exceeding the integer
capability of the computer being used.
-------
c
C UASTE OIL RECLAMATION PROBLEM MODIFICATION #1
C
DIMENSION D1(3,2),D2(4,2),D3(4,2)
C MATRIX DIMENSION LINE
C
DIMENSION COEF<3,1)
DIMENSION TAB(46)
COMMON/O1/CB0/02/JUMP/03/L00P/Q4/SE NS/Q5/1DATA/Q6/1 OUT/Q7/1PASS/08 C
I/IRAN/O9/IREAD/O10/ISCR1/Q11/ISCR2/Q12/ISNAP/Q13/ITER/Q14/ITOG/Q15
l/IURIT/O16/IXXX/al7/0K1/Ql8/OK2/Q19/OK3/Q20/OK4/Q21/OK:5/Q22/OK6
C
C NAME LINE
C
CALL NAMEC(HASTE OIL RECLAMATION PROBLEM - MODIFICATION #1)')
C
CALL READ(D1,3)
CALL READ(D2,4)
CALL READ(D3,4) C
C
C MATRIX READ LINE C
C 10
CALL MREAD(C0EF,3,1)
C 15
C SNAP INTERVAL LINE
C
ISNAP-1000
C
C ITERATION NUMBER LINE
C
1TER-5000
C
C PASS CONTROL LINE
C
DO 15 lASS-1,3
C
5 DO 10 IXXX«1,ITER
C
FIGURE 7.1: UASTE OIL RECLAMATION PROBLEM, MODIFICATION fl
COHTAM=OUNI(1.0.4.0)
RAND1=CUNI<-10.6,10.0)
PRICE1=EDIS<01,3)
SL0PE=ECTS(D2,4)
PROD2=STEP(D3,4)
RAND2=GAU(0.0fi,0.10)
PRICE3=EBETA(6.5,8.0,9.5)
PR003=EXPON(1B.O)
XLIFE=DTRI(5.0.10.0,15.0)
XININV=TRA<1600.0,2000.0,2200.0,2400.0)
THRU1«10.0-CONTAM
YIELD1»10.0-COEF(IPASS.1)*CONTAM
PRG01°THRU1*TIELD1+RAND1
GREV1=PRICE1*PR0D1
PRICE2=10.0'SLOPE*PROD2+RAND2
GREV2=PR1CE2*PR002
GREV3=PRICE3*PR0D3
ANETB=GREVl+GREV2+GREV3-70.0
PVALUESANETB*UPW(0.08,XLIFE,0.0)-XININV
CALL TABLE(PVALUE,'(PRES VAL)•,DUM0,DUM1,40,TAB)
CONTINUE
GOTO(5,15), JUMP
CONTINUE
CALL LIST(1(UOl.FOR)1)
END
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Mouse Manual Page
* SUMMARY OF SNAPS •
* PASS NUMBER 1 *
SNAP AT NUMBER Of ITERATIONS = 1000 FOR QUANTITY PRES VAL
MIAN s 2855,27000 STANDARD DEVIATION
1874.10900
SNAP AT NUMBER OF ITERATIONS = 2000 FOR QUANTITY PRES VAL
MEAN » 2851.14600 STANDARD DEVIATION
1829.24000
SNAP AT NUMBER OF ITERATIONS
MIAN > 2865.37400
3000 FOR OUANTIT* PRES VAL
STANDARD DEVIATION
1817.95600
SNAP AT NUMBER OF ITERATIONS
MEAN = 2829.80000
4000 FOR QUANTITY PRES VAL
STANDARD DEVIATION
1820.22100
FIGURE 7.2
SNAP OUTPUT FOR WASTE OIL RECLAMATION PROBLEM, MODIFICATION #1
7.5 The Output Destination and Column Output Lines
The destination of MOUSE output depends upon the logical
unit or device number selected when modifying MOUSE to run on a
specific computer system (the distribution version of MOUSE
directs output to the file, MOUSE.OUT) For most purposes, it is
probably best that output be directed to a file; such files are
permanent (until you delete them) and can be read and printed at
your leisure.
If it is desired that output for a particular run be
directed to your terminal, however, this can be accomplished by
use of an Output Destination Line (which is placed in the Speci-
fication Block) of the following form:
IOUT=IWRIT
Normally, all MOUSE output is dimensioned for 132 columns,*
if the device used cannot be set this wide, such as an 80-column
video display terminal, then the output will "wrap around" and be
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Mouse Manual Page 89
difficult to read. You can, however, instruct MOUSE to re-
dimension output to 80 columns by way of a Column Output Line of
the following form:
C80=80
The 80-column output is not quite as elegant as the 132-column
output, but the identical information is presented in either
format. This line may be placed anywhere in the Specification
Block.
7.6 Summary of the Specification Block
Except for the Sensitivity Curtailment line to be discussed
in the next Chapter, all of the Specification Block lines have
now been introduced and can be summarized as follows:
Line
Type
C80=80
SENS=c
IOUT=IWRIT
ISNAP=S
ITER=n
IRAN=seed
Column Output
Sensitivity Curtailment
Output Destination
Snap Interval
Iteration Number
Seed Reset
(see chapter 8)
All of these lines are optional (but note the default to 1000 if
an ITER Line is not used). Note that TRAP (see Chapter 11,
Section 11.6) can be used to prepare the Specification Block.
7.7 Multiple Rubs: Matrices and Vectors
You may wish to vary one or more parameters of a model
without having to make separate MOUSE runs. By "parameter" is
meant a temporary constant as, for example, the coefficient of 2
in the YIELD1 equation shown in Figure 7.1, i.e.,
YIELD1»1G.-2.0*CONTAM
The investigator might wish, say, to study the effects of
changing this parameter from 2 to 1 and from 2 to 3. This is
accomplished by placing the values 1, 2, and 3 into a matrix. A
matrix is identified by the usual rules for naming variables in
MOUSE, followed by a parenthetical expression containing two
indices in integer form. The first index is a special one called
"IPASSW, e.g., C0EF(IPASS,n), and the reason for this special
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Mouse Manual Page 90
index will be explained in the next Section. The second index is
the number of parameters to be changed. In this example we are
changing only one parameter, i.e., the coefficient in the YIELD1
equation noted above? hence, the second index is set equal to 1,
i.e., C02F(IPASS,1).
Matrices must be dimensioned, a procedure similar to but
slightly different from that used for the dimensioning of
empirical variables where the second index is always 2. In ma-
trices, the second index can be any integer. The Matrix Dimen-
sion Line has the form:
DIMENSION name(n,m)
where "name" is the name of the matrix, n is the number of rows,
and m is the number of columns in the matrix. (See Chapter 11,
Section 11.6, for a discussion of TRAP, a MOUSE utility that
greatly facilitates the preparation of the Dimension Block.) Also
note that the name must be a decimal name. (A decimal name is one
that starts with any letter except I, J, K, L, M, or N. See Chap-
ter 3, Section 3.2).
The Waste Oil Reclamation Problem shown in Figure 7.1 is, as
has been mentioned, a modification of the original problem shown
in Figure 2.1. In Figure 7.1 the slope coefficient of the YIELDl
equation has been modified as discussed above. Accordingly, a
Matrix Dimension line was added to the program as follows:
DIMENSION C0EF(3,1)
the 3 signifying that you are going to make three runs or "pass-
es", each pass with a different value for the coefficient param-
eter, COEF. The YIELDl equation was also modified in Figure 3.1
to:
YIELD1-10.-COEF(IPASS,1)+C0NTAM
Note that, when used in an uncertainty equation, a matrix must
use IPASS as its first index.
To read in the value of any matrix a Matrix Read Line of the
following form is used:
CALL MREAD(name,n,m)
where the arguments are the same as discussed for the
Matrix Dimension line. Decimal points are not used in Matrix
Read lines.
The matrix data is treated much like empirical variable data
in that it is entered free-form (in a MOUSE.DAT Data File), i.e.,
each row of the matrix is entered, starting at column 1, with the
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Mouse Manual Page 91
numbers scpsrdtsd by commas • Decimals aire required in the nun~
bers. If large matrices are involved, do not split a number at the
end of a line but merely go to the next line and continue. Start
new rows of the matrix on new data lines. One should not combine
data from two separate rows in the matrix on one line.
The Matrix Data in the MOUSI.DAT file for this problem would
appear as follows;
COEF... COEFFICIENT IN YIELD1 EQUATION
1.
2.
3.
The first line is a name line (and is required) and may be
punched in any column. The numbered order of the MOUSE passes
corresponds to the rows of the matrix. Pass 3, for example, sets
the value of COEF(IPASS,l) equal to 3.0. As in all data lines,
decimal points are used.
Alternatively, we can bypass the Matrix Read Line by
defining the matrix data directly in the Specification Block-,
e.g.,
COEF(1,1)*1.0
COIF(2,1)=2.0
COEF(3,1)=3.0
However, a 3 x 1 matrix (as is COEF) is identical to a 3-
dimensional vector, which is a bit simpler. Vectors can be dimen-
sioned using a Vector Dimension Line which has the following
general form:
DIMENSION namel(nl)
where "namei" is the name of the ith vector and "ni" is its
dimension. Also note that the name must be a decimal name. (A
decimal name is one that starts with any letter except I, J, K, L,
M, or N. See Chapter 3, Section 3.2). In this case, our informa-
tion would be read in the Specification Block as follows:
COEF(1)=1.0
COEF(2)=2.0
COEF(3)=3.0
and the YIILD1 equation would be changed to:
YIELD1=10.-COEF(IPASS)*CONTAM
Note that again, the index used is "IPASS".
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Mouse Manual Page 92
7.8 Multiple Rubs: The HOUSE Pass
A Pass Control Line is required when implementing multiple
runs. The line is of the form:
DO 15 IPASS=l,n
where "n" is defined as for the Matrix or Vector Dimension lms>
This line must be placed immediately before MOUSE Control Line 4.
If a matrix is used, by adding additional columns to the
matrix you can alter more than one parameter during a pass. In
the THRUl and YIELD1 equations, for example, the parameter "10" is
used. We could alter this parameter on different MOUSE passes by
redefining it as COEF(IPASS,2), i.e.,
THRU1=C0EF(IPASS,2)-CONTAM
YIELD1=C0EF(IPASS,2)-COEF(IPASS,1)*CONTAM
Suppose we place the following Matrix Data in the MOUSE.DAT file:
COEF... COEFFICIENTS IN THRUl AND YIELD1 EQUATIONS
1.,10.
1.,20.
2.,10.
2.,20.
This provides four MOUSE passes. Pass 3, for example, would set
the intercepts in both equations equal to 10 and the slope in the
second equation equal to 2. The Matrix Dimension line in this
case would appear as:
DIMENSION COEF(4,2)
and the Matrix Read line would appear as:
CALL MREAD(COEF,4,2)
As can be seen, the use of the MOUSE pass permits you to make
multiple runs, altering any combination of parameters of the
model desired. A schematic outline of the lines involved in a
typical MOUSE pass is shown in Figure 7.3, and a small portion of
the table output from the program of Figure 7.1 is shown in
Figure 7.4 where the effect of the COEF variable is very clear.
In addition to altering parameters, the MOUSE pass can also
be used with IF statements to completely change the probability
distributions of the model in multiple runs (see Chapter 14,
Section 14.4). This adds very powerful sensitivity analysis
capabilities to MOUSE.
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Mouse Manual Page 93
[Dimension Block] DIMENSION PAR(n,m)
a
[Read Block] CALL MREAD(PAR,n,m)
[PCC] DO 15 IPASS=1,n
[MCC 4] 5 DO 10 IXXX=1, ITER
[Program Block] =...PAR(IPASS, i) . . .
[MCC 5] 10 CALL END
[MCC 6] GOTO(5,15), JUMP
[MCC 7] 15 CONTINUE
[MCC 8] END
Key: PCC = Pass Control Line
MCC = MOUSE Control Line
i = ith Parameter (i is a value between 1 and m)
FIGURE 7.3:
AN OUTLINE OF THE MOUSE PASS
7.9 Program output Titles: The Kama Line
Program titles, up to 60 characters in length, are placed in
MOUSE output by use of a Name Line of the following form:
CALL NAME(1(lname)1)
where "lname" is a literal name descriptor of no more than 60
characters in length. Note that the descriptor must be enclosed
in the single quotes and parentheses exactly as shown. The Name
Line is placed immediately after Mouse Control Line # 3 and it is
mandatory. The TRAP utility (see Chapter 11, Section 11.6)
facilitates the preparation of this line.
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• distribution for ouantitt pres val *
* PASS NUMBER 1 *
NUMBER OF ITERATIONS
5000
MEAN
MINIMUM
MAXIHUM
2815.72500
-1834.04600
11838.67000
STANDARD DEVIATION a. 1818,11100
COEFFICIENT OF VARIATION, % = 64.56989
AVERAGE OF VALUES LESS THAN ZERO
-493.14260
a***********#********#******#*******••••••
* DISTRIBUTION FOR QUANTITY PRES VAL *
* PASS NUMBER • 2 *
*«*****#*#*******•*****•****#*************
NUMBER OF ITERATIONS = 5000. MEAN = 1876.57200 STANDARD DEVIATION = 1922.17400
MINIMUM = -5250.27700 COEFFICIENT OF VARIATION, % ,= 102.43010
¦ MAXIMUM - , 10609.75000
AVERAGE OF VALUES LESS THAN ZERO = -766.87190 . .
* DISTRIBUTION FOR QUANTITY PRES VAL *
* PASS NUMBER 3 ¦
NUMBER Of ITERATIONS = 5000
MEAN -
MINIMUM s
MAXIMUM =
860.50080
-4596.54200
9859.02800
STANDARD DEVIATION = 2151.11500
COEFFICIENT OF VARIATION, % = 249.98410
AVERAGE Of VALUES LESS THAN ZERO
-1212.52300
FIGURE 7.4: Partial PASS Output For Waste Oil Reclamation Problem,
Modification #2
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Mouse Manual Page 95
7.10 The Program Listing Line
It is sometimes desirable to append a listing of your
source program file (see Chapter 7, Section 7.10) to the MOUSE
output file, MOUSE.OUT. This is accomplished by use of a Program
Listing Line of the following form:
CALL LIST(1(lname)')
where "lname" (a literal name enclosed in single quotes and
parentheses as shown) is the name of the source file to be list-
ed. An example is:
CALL LIST('(PROGRAM.FOR)')
The Program Listing line must be placed between MOUSE Control
Lines numbers 7 and 8, e.g.,
15 CONTINUE
CALL LIST(•(PROGRAM.FOR)1)
END
If a non-existent file is referenced, MOUSE will report the fact
and, of course, no listing will be produced.
7.11 Preparing and checking Specification slock Lines
See Chapter 11, Section 11.6, for a discussion of TRAP, a
MOUSE utility that greatly facilitates the preparation of the
Specification Block lines discussed in this chapter. Also, CHECKER
(Chapter 11, Section 11.2) will examine a MOUSE source file to
insure that these lines are correct both m form and argument.
EXERCISES
Which of the following control lines are valid ?
1. ITIR=27S6
2. IRAN=37654
3. DIMENSION COEF(3,15)
4. ¥ISLD=TAXES+SL0PE(2,1)-500.
5. CALL MREAD(REV,4)
What, if anything, is wrong with the following Pass control
line?
6.
DO 15 1=1,LIFE
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Mouse Manual Page 96
Find the errors, if any, in the following Matrix Data
lines:
(a) The dimension statement is DIMENSION VELOC(2,3)
7. "
VELOC COEFFICIENTS IN SPEED EQUATIONS
10, ,20. ,30, ' „
5. ,4.,1.
(b) The dimension statement is DIMENSION VELOC(4,2)
8-
VELOC COEFFICIENTS IN SPEED EQUATIONS
10.,20.,5.,4.
5.,4.,40.,100.
REFERENCE
Law, A.M., "Statistical Analysis of Simulation Output Data",
Proceedings of 198Q Winter Simulation Conference. December 3-5,
Orlando, Florida, p. 361-370 (1980).
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CHAPTER 8
mnwTnBiT. unnQv PimrTTrsiic
AI/UXxxvIlALi JRK#%rS3JEji Jr%JJNV»JLJL>lvJR2s
8.1 Introduction
This chapter discusses two MOUSE functions that expand the
utility of the system. The first affords an easy way to obtain
certain graphical plots of pairs of variables, and the second
concerns sensitivity analysis.
8.2 The Plot Output Line
You night wish to know something about the functional rela-
tionship between two stochastic variables in a MOUSE program. If
the two variables are output variables, there is no way for you
to glance at the risk equations and easily deduce the nature of
this relationship. Even if one of the variables was an input
variable, there might be a considerable number of equations
between the introduction of the input variable and the output
variable of interest. Once again, it would be difficult to
deduce the nature of the relationship simply by looking at the
intervening equations.
MOUSE has an option whereby any variable can be plotted
against another. In addition, a total of up to eight regression
equations are automatically fitted to the plotted data. The Plot
Output Line has the following form:
CALL PLOT(xvar,'(Ixname)',yvar,'(lyname)',ti)
where the arguments are analogous to those defined for Table
Output Lines. Note that "xvar" is the "independent" or
exogenous variable and that "yvar" is the dependent or endogenous
variable. Also note that both the xvar and yvar variable names
must be decimal names. (A decimal name is one that starts with
any letter except I, J, K, L, M, or N. See Chapter 3, Section
3.2). In Figure 8.1 (Modification #1 of the Regulatory
Problem), a Plot Output Line has been used to plot FACTOR
against DEPTHL, the latter being considered as the independent
variable (which it certainly is in the Regulatory problem). This
line appears in Figure 8.1 as follows:
f DT/w /nTPt>rmrT ? \ t irar^rvvD ' f ps\ t pi \
LiJLi Jr Jjw 1 ^ UEfJrx JuLJLi f ^ Uu* 1.TlLjy p X Av JL vK f \ X AW* i vl\ ) f tr JL j
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c
C REGULATOR* PROBLEM MODIFICATION #1
C
DIMENSION TAB(46)
C
C PLOT DIMENSION LINE
C '
nTlTDlTCTAM n 1 / 1 AC1 \
DXHEN5XON PX(X0uf j)
c
COMMON/Q1/C80/Q2/JUMP/Q3/LOOP/Q4/SENS/Q5/IDATA/Q6/IOUT/Q7/IPASS/
&/IRAN/Q9/IREAD/Q10/ISCR1/Q11/ISCR2/Q12/ISNAP/Q13/ITER/Q14/ITOG/Q
&/IWRIT/Q16/IXXX/Q17/OK1/Q18/OK2/Q19/OK3/Q20/OK4/Q21/OK5/Q22/OK6
CALL NAME(•(REGULATORY PROBLEM - MODIFICATION #1)*)
ITER=5000
5 DO 10 IXXX=1,ITER
PRECIP-TRA(762.0,1016.0,1270.0,1524.0)
RLEACH=CUNI(.5,1.0)
C
C SENSITIVITY CONTROL LINES
C ..
CALL SEN(RLEACH,'(RLEACH)',50.,0.)
CALL SEN(RLEACH,1(RLEACH)»,0.,-50.)
C .
DEPTHL=TRA(3.0,6.1,9.1,15.2)
WD1NS=TRI(297.0,415.0,534.0}
FACTOR=70000*(PRECIP*RLEACH)/(DEPTHL*WDENS)
C
C PLOT OUTPUT LINE
C
CALL PLOT(DEPTHL,1(DEPTHL)»,FACTOR,1(FACTOR)1,PI)
CALL TABLE(FACTOR,•(FACTOR)',DUMO,DUM1,40,TAB)
10 CONTINUE
GOTO(5,15), JUMP
15 CONTINUE
CALL LIST(1(REG1.FOR)•)
END
FIGURE 8.1s
REGULATORY PROBLEM MODIFICATION #1
Along with the Plot Output line the program must contain a
Plot Dimension Line of the following form:
DIMENSION T1(100,3),T2(100,3),T3(100,3), etc.
(See Chapter 11, Section 11.6, for a discussion of TRAP, a MOUSE
utility that greatly facilitates the preparation of the Dimension
Block. Also, CHECKER, Chapter 11, Section 11.2, will examine a
-------
Mouse Manual Page 99
MOUSE source file to insure that arguments of Plot Output lines
are correct, and that the dimension of the array used for storing
the plot table agrees with that specified in the Dimension
Block.) Also note that the variable names Tl, T2, etc., must be
decimal names. (A decimal name is one that starts with any letter
except I, J, K, L, M, or N. See Chapter 3, Section 3.2).The Plot
Dimension line in Figure 8.1 is, for example,
DIMENSION PI(100,3)
The plot output for the program of Figure 8.1 is shown in
Figure 8.2. The points represent the average value of FACTOR for
the intervals shown on the DEPTHL axis. (They are plotted at the
midpoints of the intervals of the DEPTHL axis, however, rather
than the lower bound of the intervals as is done in MOUSE
tables.) Accordingly, the points represent different numbers of
observations. In order not to mislead the eye, points that repre-
sent less than 20% of the maximum number of observations repre-
sented by any one point are plotted as a "+" character? all other
are plotted as an . The dotted line represents the best fit of
eight models: linear, quadratic, cubic, reciprocal, square root,
logarithmic, power, and exponential. Underneath the graph in
Figure 8.2 is a summary of these fitted equations (fitted via a
weighted regression technique). The values of these parameters
are given, as well as the goodness-of-fit as measured by R-
square ( the coefficient of determination}. Note, however, that
some equations cannot be fitted to data sets having zero or
negative numbers. One cannot, for example, take the square root
of a negative number. In such cases, the inappropriate equations
will automatically be deleted from the regression summary.
In Figure 8.2, we find that the best-fitting equation is the
cubic (R-square = 0.56), the equation being
Y = 3743 - 0.0042X3
where Y is FACTOR and X is DEPTHL (the E+04 and E-02 in the line
for the cubic equation in Figure 8.2 stand for multiplication by
104 and 10~2, respectively). There were no zero values for X nor
were there negative values for X or for Y; accordingly, all eight
equations were fitted.
As with Table Output Lines, you can scale the variables in a
Plot Output Line (see chapter 8, Section 8.2). The scales of the
plot axes and the regression parameters, of course, will reflect
this scaling. Also, Plots require a minimum of 100 iterations.
Fewer than this will result in termination of the run. As has
been mentioned before (Chapter 6, Section 6.2), other than a
Program Listing Line the last executable statement in a MOUSE
program must be a Table output Line; it cannot be a Plot Output
Line.
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Mouse Manual Page 100
* PLOT OF QUANTITY FACTOR VERSUS OEPTML *
#**«********#************#•*********#*****•««******
FACTOR
.0000
16000.0000
--o
32000.0000
o
48000.0000
5.5000
OEPTHL
11.0000 o
16.0000 o
Point code: * = with in 20% of rasxima weight
+ = less than 20X of maxinun weight
Dotted line Is best fit, I.e., RECIPROCAL MG0EL
SUWttRY OF FITTED EQUATIONS, WEIGHTED LEAST SQUARES
A
a
R-SQUARE
LINEAR : Y=A*i*X
.44101E*>05
-.26920E*04
.86
QUADRATIC : Y=A<-B*X#X
.32996C+05
•.14056E+03
.74
CUBIC ; YaA*B*X*X*X
.292411*05
-.88362E-HJ1
.63
RECIPROCAL ; Y»A+B/X
-.33078E+03
.14972E+06
.99
SQUARE ROOT; Y*A+8*sqrtCX>
.66175E*05
-.15767E*05
.92
LOGARITHMIC: Y=A*ln(X>
.66376E*05
-.220696+05
.96
POWER : Y=A*X**8
•14997E*Q6
-.10098£*01
.99
EXPONENTIAL: Y>A*exp(8X)
»10938£+02
-.127646+00
.96
PLOT OUTPUT
FIGURE
FOR REGULATORY
8.2:
PROBLEM,
MODIFICATION #1
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Mouse Manual Page 101
Incidentally, IF statements can be used with plots to narrow
the information being plotted or for some other specific purpose.
As example, consider the following line:
I FtFACTOR .GT. 10000) CALL PLOT(D€PTHL,•COEPTHL)1.FACTOR,'(FACTOR)\P1>
In this case, only those points will be plotted where the factor
variable exceeds 70,000.
8.3 MOUSE Automated Sensitivity: Introduction
As was mentioned in Chapter 1, traditional sensitivity
analysis usually starts with a base value estimate, followed by a
perturbation or change in one of the input variables (holding
all other input variables at their previous values). The process
s repeated for as many variables and for as great a change as
is desired. When the sensitivity of parameters (i.e., temporal
constants) on the final output is involved, different values of
the parameters can be placed either in matrices or vectors and,
using the MOUSE Pass, sensitivity mans can be made. For example,
in Figure 7.1 the sensitivity of the present value, PVALUE, of
the project to the value of the slope in the YIELDl equation was
studied by placing three different values of the slope in a
matrix (a vector also could have been used).
The IF Statement discussed in Chapter 4, Sections 4.6 and
4.7 could also have been used in conjunction with the MOUSE Pass
to study the sensitivity of the present value of the project to
different assumptions of probability distributions for the input
variables. For example, it was assumed that the CONTAM variable
followed the discrete uniform distribution with parameters lower
* 1.0 and upper = 4.0. On three MOUSE passes we could have exam-
ined the effects of three distributions by replacing the CONTAM
equation with the following three equations (all three of the
following distribution have the same mean, i.e., 3, but their
shapes are quite different:
IF(IPASS .EQ. 1) CONTAM=DUNI(1.0,4.0)
IF(IPASS .EQ. 2) CONTAM=TRI(l.0,3.0,5.0)
IF(IPASS .EQ. 3) CONTAM=GAU(3.0,1.0)
The following Section describes yet another method for making
sensitivity runs in MOUSE.
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Mouse Manual Page 102
®.€ TJi« Sensitivity Control Line
With a single control line (the Sensitivity Control Line),
it is possible to alter either the mean or the standard deviation
of any stochastic input variable by a stated percentage (up or
down) to see what effect such a change has on the output varia-
bles. When this option is used, the following takes place:
1. First, a normal iteration sequence occurs, i.e., there is no
change in the original parameters or the form of the input
c4*n/^Vt ae^" i t»a v*t aK1 as e1 1 )IKa e*a mum 11 \ •
owwwitaoviv vaji layico ^ wiic &y wciijlcu yaoc iuh / r
2. Then, for every Sensitivity Control Line in the program, another
iteration sequence automatically occurs, incorporating the
change in parameter specified in that line and only in that
line (the so-called "sensitivity run").
It also should be noted that tables, vector tables, and plots
(the last two to be discussed shortly) are clearly labeled as to
whether they are "base" runs or "sensitivity" runs. Furthermore,
on sensitivity runs each table contains the percent change from,
the base table mean and standard deviation.
The Sensitivity Control Line has the following form:
CALL SEN(variable,1(lname)',m%,std%)
where "variable" is the stochastic input variable to be altered,
"lname" is a literal name (note the single quotes and
parentheses) to be assigned to the variable during printout
(usually this is the same as the variable name), "m%" is the
desired change in the mean, "std%" is the desired change in the
standard deviation. Also note that the variable name must be a
decimal name. (A decimal name is one that starts with any letter
except I, J, K, L, M, or N. See Chapter 3, Section 3.2). A maxi-
mum of 50 Sensitivity Control Lines may be used in any MOUSE
program. One can change both the mean and the standard deviation
of a variable simultaneously, but this is not recommended since
it would not be possible to sort out separately the two effects.
It should be mentioned that although changing the parameters of a
probability distribution may alter the shape of the distribution,
the underlying distribution remains the same, e.g., a normal
distribution still remains a normal, a uniform distribution still
remains a uniform, etc.
The following example is taken from Figure 8.1:
CALL SEN(RLEACH,'(RLEACH)',50.,0.)
CALL SEN(RLEACH, 1(RLEACH) ' , 0 • ,-50.)
Figure 8.1 shows the placement of these two Sensitivity Control
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Mouse Manual Page 103
* DISTRIBUTION FOR QUANTITY FACTOR *
NUMBER OF ITERATIONS » 5000
MEAN
MINIMUM
MAXIMUM
20361,21000
4219.64600
82200.32000
STANDARD DEVIATION = 12154.1600
COEFFICIENT OF VARIATION, X ¦ 59.6927
*************** ***********
* DISTRIBUTION FOR QUANTITY FACTOR *
THIS TABLE IS A RESULT OF A SENSITIVITY TEST FOR BLEACH :
MEAN OF RLEACH IS INCREASED BY 50.0 PERCENT.
STANDARD DEVIATION OF RLEACH REMAINS UNCHANGED.
NUMBER OF ITERATIONS = 5000 MEAN = 20737.69000 STANDARD DEVIATION = 12264,31C0
MINIMUM = 4716.75000 COEFFICIENT OF VARIATION, X = 59.1402
MAXIMUM « B923fi.35000
CHANCE FROM BASE FACTOR VALUES DUE TO THIS CHANGE IN RLEACH :
MEAN: 1.8490S
STANDARD DEVIATION: .9063%
• DISTRIBUTION FOR QUANTITY FACTOR *
THIS TABLE IS A RESULT OF A SENSITIVITY TEST FOR RLEACH ;
MEAN OF RLEACH REMAINS UNCHANGED.
STANDARD DEVIATION OF RLEACH IS DECREASED BY 50.0 PERCENT.
NUMBER OF ITERATIONS ¦ 5000
MEAN ¦
MINIMUM ¦
MAXIMUM *
20292.2Z000
5335.72300
83532.23000
STANDARD DEVIATION ¦ 11396.3800
COEFFICIENT OF VARIATION, X * 56.1613
CHANCE FROM BASE FACTOR VALUES DUE TO THIS CHANGE IN RLEACH
MEAN:
STANDARD DEVIATION:
-.3388*
*6.2347%
W JLCSCJRB 8 • 3 •
SENSITIVITY OUTPUT FOR TIE REGULATORY PROBLEM, MODIFICATION #1
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Mouse Manual page 104
Lines. (They can be placed, preferably, immediately after the
Stochastic Variable Sampling Line that generates the variable or,
in most cases, immediately after the last Stochastic Variable
Sampling Line•) The first line increases the mean of the varia—
ble RLEACH by 50% and the second decreases its standard devia-
tion by 50%. (Note: Any attempt to decrease the standard devia-
tion by more than 100% will result in an error message and the
run will be terminated.) A sample output for the first sensitiv-
ity run is shown in Figure 8.3.
Note that MOUSE does not require you to know either the
mean or the standard deviation of a stochastic input variable in
order to use Sensitivity Control Lines. All you have to specify
is a relative increase or decrease in these parameters. If Sensi-
tivity Control lines are used, the last executable statement in
any MOUSE program must be a Standard Table Output line.
8.S The sensitivity Level Line
MOUSE output is rather prodigious. The standard MOUSE table
discussed in Chapter 6 ordinarily fills one page? the Plot (see
next Section) and Vector Table (see Chapter 10) outputs generally
run to two pages each. If many sensitivity runs are scheduled or
if you are interested only in the major effects reflected in such
runs, such a level of output may not be desired. An option is
provided in MOUSE whereby you may select up to four levels of
output when sensitivity runs are to be made. The level desired
is selected by a Sensitivity Level Line which is placed in the
Specification Block of the program. This line has the following
form:
¦ SENS-c
where c, the sensitivity level, is one of three integers as
follows:
Level Description
No curtailment of plots or tables
All plots deleted (except for base iteration) ,*
All vector tables and plots deleted (except for
base iteration)#
All vector tables and plots deleted? standard tables
automatically converted to short form (except for
base iteration).
The 3 level of sensitivity is achieved by default, i.e. by the
absence of the Sensitivity Level line in the Drooran. Note that
the use of a Sensitivity Level line in no way affects the base
0
1
2
3
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Mouse Manual Page 105
output, i.e., the output of the first iteration sequence. Only
the sensitivity runs themselves will be affected.
See Chapter 11, Section 11.6, for a discussion of TRAP, a
MOUSE utility that greatly facilitates the preparation of the
Sensitivity Control and Level lines. Also, CHECKER, Chapter 11,
Section 11.2, will examine a MOUSE source file to insure that
arguments of these lines are correct.
EXERCISES
Which of the following are valid MOUSE calls?
1. CALL PLOT(XI,'(PRICE)«,X2,'(QUANTITY),T1)
2. CALL SEN(AREA,'(AREA)«,-120.,0.)
3. CALL PLOT(XI,»(PRICE)',X2/1000,'(QUANTITY)«,T1)
4. CALL SEN(AREA,'(AREA)«,0.,-120.)
5. CALL PLOT(XI,1(XI)',PVALUE,«(PVALUE)',T1)
Which of the following program line sequences, if any, are
valid?
6.
CONTAM=EDIS(Dl,5)
CALL SEN(CONTAM,•(CONTAM)',0.,50.)
PRICE=EDIS(D2,5)
COST«EDIS(D3,5)
CALL SEN(COST,•(COST)',0.,50.)
•
CONTAM =EDIS(D1,3)
PRICE=EDIS(D2,3)
CALL PLOT(CONTAM,'(CONTAM)1,PRICE,•(PRICE)',T1)
10 CONTINUE
GOTO(5,15), JUMP
15 CONTINUE
END
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Mouse Manual Page 106
CHAPTER 9
CORRELATED STOCHASTIC UfFCJT VARIABILES
9.1 'Introduction
It will be noted that in the Waste Oil Reclamation Problem of
Figure 2.1 (and also in the modification shown in Figure 7.1),
all of the stochastic input variables are independent of each
other. In many cases this is not a realistic situation since
correlated variables (i.e., variables which are likely to vary
together in a systematic way) are frequently encountered in
practice. In times of high inflation, such have occurred in
recent years, the independence of variable costs.and prices is
certainly questionable. Product or project life is related to
market share and market growth, and price and market size are
often related. Many examples from the sciences can also be
cited. As Pouliquen (1970) remarks: "Correlated variables are
difficult to detect, and even more difficult to measure, but
overlooking then may lead to a completely wrong interpretation in
the analysis".
9.2 Linear Correlation
The simplest kind of correlation between two variables is
that expressed in the form of the following linear equation:
Y = slope*X + intercept + error [9*11
Since the slope is a function of the correlation between the
variables X and ¥, equation 9.1 can be written as:
Y = (sx/sy)*rho*X + intercept + error [9.2]
where sx and sy are .the standard deviations of variables X and" Y,
respectively, and rho is the correlation coefficient between X
and Y. Usually, the error is assumed to be distributed normally,
with mean ¦ zero and standard deviation = std so that equation
9.2 can be written as:
Y = (sx/sy)*rho*X + intercept + GAU(0.0,std) [9 - 3]
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House Manual Page 107
As an example, suppose in the we replace the equation for
the throughput in production of motor oil grade 1, THRU1=10.0-
CONTAM, with the equation,
THRU1=- (1. 3/2 .0)*0. 8*CONTAM+3 . 0+GAU (0 . 0,0.5)
where 1.3 is the standard deviation of CONTAM, 2.0 is the stand-
ard deviation of THRU1, and the error term is Normal with mean =
0.0 and standard deviation = 0.5. The correlation coefficient
between X and Y is 0.8.
IMP (interactive Modeler for £robabilities) has already been
introduced in chapter 4. It can be used to analyze data pairs,
including fitting a variety of X-versus-Y-type equations. The
user has a choice between placing the (X,Y) observation pairs
into a data file (named IMP.DAT) accessible to the program or
entering the data directly from the keyboard. The first line of
the data file contains descriptive information (up to 60 charac-
ters) used to identify the file. (This line is mandatory. \ The
remainder of the file contains the sample data, one (X,Y) obser-
vation pair per line starting in column 1, with the use of the
decimal point mandatory. The X and Y values are separated by
either a comma or a space. Figure 9.1 shows part of such a file
for a 100-observation data pair set consisting of numbers gener-
ated by the following equation:
Y ¦ 5 + 10.*X*X + GAU(0.,20.)
where GAU(0.,20.) is a normally distributed error term and the
X-values were random over the interval (1,10).
As mentioned in Chapter 4, IMP prompts the user for instruc-
tions and offers a series of menus and options as follows:
1. After calling IMP (simply by typing "IMP"), you will be
asked to supply a title or comment line (not to exceed
60 characters) that will be placed in the IMP output
file. If the file IMP.OUT (i.e., the standard IMP output
file) exists, you will be asked whether you wish to
overwrite this file or not. If not, you will asked for
the name of an alternate output file.
2. The following menu is presented;
**********************************************
* FIT A DATA SET TO A CLASSICAL DISTRIBUTION *
* INTEGRATE A CONTINUOUS DISTRIBUTION *
* ANALYZE A BIVARIATS DISTRIBUTION *
* ANALYZE A DATA SET FOR AUTOCORRELATION *
**********************************************
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Mouse Manual Page 108
BIVARIATE ANALYSIS: QUADRATIC Y=5+10.*X*X+GAU(0.,20.)
5. 50
307.50
2.69
99.97
9.61
932.88
8.42
720.17
2.22
60.04
8.49
692.86
3.30
98.94
5.68
333.54
2.42
80.78
6.58
443.61
2.73
86.43
2.58
114.06
9.28
884.38
FIGURE .9-1:
PORTION OF EXAMPLE IMP DATA FILE
FOB ANALYSIS OF A BrVARXATE DISTRIBUTION
The choices on the menu are highlighted. For the purpose of
analyzing a bivariate data set, select the third option, "ANALYZE
A BIVARIATE DISTRIBUTION". The IMP program was run using 100 data
pairs generated by the quadratic model shown previously, and the
results are shown in Figure 9.2. (Omitted are the bivariate
normal and bivariate lognormal analyses that are part of this
option. See Section 9.3.) The best equation is the quadratic
model (R-square = .99), with estimated parameters A = 7.76
(actual value » 5) , B - 9.97 (actual value = 10), and error
standard deviation = 19.84 (actual value = 20). Since some X-
observations were negative, neither the power or the exponential
models could be fitted to this data set. Note that the
Accumulation Line (Chapter 4, Section 4.10) can be used in a
MOUSE program to accumulate data to be subjected to a
bivariate analysis.
9.3 Bivariate Normal Distribution
Suppose in the Waste Oil Reclamation problem of Figure 7.1
Waste Oil Reclamation problem of Figure ?.l that although it is
known that PR0D2 and PRICE1 are correlated, we do not know what
the cause-and-effect relationship is. We assume, however, that
PROD2 and PRICE1 are bivariate normally distributed with mean and
standard deviations of (50.0,10.0) and (3.0,0.6), respectively.
(These very closely approximate the discrete distributions de-
fined in Table 2.1 if we assume that the bounds of the discrete
distributions are roughly the plus or minus 2 standard deviation
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SUMMARY OF FITTED REGRESSION EQUATIONS
NUMBER OF OBSERVATIONS - 100
LINEAR MODEL: Y=A+B*X
A CONSTANT
R-SQUARE
-232.79310
.95027
B CONSTANT
ERROR
110.06930
63.90828
QUADRATIC MODEL: Y=A+B*X*X
A CONSTANT -
R-SQUARE ' ™
7.76751
.99521
B CONSTANT
ERROR
9.97774
19.84336
CUBIC MODEL: Y=A+B*X*X*X
A CONSTANT
R-SQUARE
98.55668
.97550
B CONSTANT
ERROR
1.01323
44.85876
RECIPROCAL MODEL: Y=A+B/X
A CONSTANT
R—SQUARE
660.85100
.51076
B CONSTANT
ERROR
•1149.52900
200.45450
SQUARE ROOT MODEL: Y=A+S*sqrt(X)
A CONSTANT
R-SQUARE
-687.53300
.88524
B CONSTANT
ERROR
466.27110
97.08292
LOGARITHMIC MODEL: Y-A+B*ln(X)
A CONSTANT
R-SQUARE
-330.14710
.78446
B CONSTANT
ERROR
447.97770
133.05100
POWER MODEL, Y=A*X**B, OMITTED SINCE ONE
OR MORE X- OR Y- VALUES LESS THAN ZERO.
EXPONENTIAL MODEL, Y=A*exp(B*X), OMITTED
SINCE ONE OR MORE Y-VALUES LESS THAN ZERO.
FIGURE 9.2: HIP OUTTOT, BTVARIATE DISTRIBUTION, DATA OF FIGURE 9.1
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Mouse Manual Page 110
points of the continuous distributions.) Since another parameter
of a bivariate normal distribution is rho, the correlation
coefficient, some practical way must be found to determine this
value.
We start by noting that the slope, bi, of a linear
regression of y on x can also be expressed (see Draper and Smith,
1966) as:
bi = rho(sy/sx) [9»4]
where sy and sx are the standard deviations of y and x,
respectively. Since the intercept term, bQ, is my - bimx, where
my and mx are the averages of y and x, respectively, the regres-
sion equation can be written as:
y = my + rho(sy/sx)(x- mx) [9.5]
Solving for rho we obtain:
rho - (y - my)(sx)/(x - mx)(sy) [9.6]
After my, mx, sy, and sx have been determined, the value of one
variable is estimated at two different values of the other,
preferably a value on each side of the mean. To illustrate,
suppose the analyst thinks that, if the price is 1.5 then the
production will be 67.5, and that if the price is 4.5 then the
production will be 27.5. These.values are then used in Equation
9.3 to determine rho as follows:
rho - (67.5 - 50)(0.6)/(1.5 - 3)(10) = -0.7
rho = (27.5 - 50)(0.6)/(4.5 - 3)(10) = -0.9
Averaging these two values we obtain a value of rho of -0.8.
(The procedure is due to Hillier, 1964.) Note: If data is
available, the correlation coefficient can also be estimated by
IMP (see Section 9.5).
The bivariate nox ma 1 distribution . function is called by •
BGAUl(mx,sx,Z)
BGAU2 (my, sy, rho, Z)
where the m's and s's denote the means and standard deviations of
the two variables, respectively, rho is the correlation coeffi-
cient, and Z is a dummy variable used to link the two distribu-
tions. It is not necessary to change the dummy variable designa-
tion of "Z" for different: bivariate normal distributions, but for
each bivariate normal distribution function called, no other
bivariate normal distribution line can fall between the BGAU2
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Mouse Manual Page ill
line and the BGAU1 line. For example, the Stochastic Variable
Sampling Lines Cor PROD2 and PRICE1 in Figure 3.1 are now re-
placed with:
PR0D2=BGAU1(50.0,10.0,Z)
PRICE1=BGAU2(3.0,0.6,-0.8,Z)
as shown in Figure 8.2. It does not matter which variable is
chosen to be placed in the BGAU1 function.
The bivariate normal distribution can be used to chain
variables in an interesting manner. Consider the following set
of three lines in a MOOSE program:
X1»BGAU1(200.,40.,Z)
X2=BGAU2(700.,140.,0.8, Z)
X3=BGAU2(400.,80,0.6,Z)
Note that only the first line utilizes the BGAU1 function. The
result is the production of three variables, each correlated with
one another. The correlation between XI and X2 is, as expected,
OR; fho rorrel At- i or* hpf"U0on YO ariH V"? i c f hat in the t"hi
• Or f %•* l>*SS WrVw# JL 4L J5S a CI V» JL W X & Jhr *3 V* WCSCSl 4 Aw Q I. 4nt Jt JL 9 Im> Uk ^5 Va JL J 1 A *5 V>JLA«I» JL \JL
or X3 line, i.e., 0.6. The correlation between XI and X3 is the
product of all the correlations, i.e., 0.8(0.6) = 0.48. You can
chain as many variables as is desired.
For changing means or standard deviations, Sensitivity
Control Lines can also be used with correlated variables
(dependency unknown) in the usual manner. Such Sensitivity
Control Lines should be placed after the last BGAU2 line in the
bivariate chain to which they refer. Sensitivity Control Lines
can also be used with correlation coefficients but the following
special procedure must be used:
a. Assign a name to the correlation coefficient in
question;
b. Assign a value to this name;
c. Place a Sensitivity Control line after the
assignment in (b) ,*
d. Place these two lines immediately before the
SGAU2 line in which the correlation coefficient
is to appear;
e. In the BGAU2 line, use the name assigned for the
correlation coefficient, rather than the original
number.
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Mouse Manual Page 112
For example, Consider the three Variables Chained Xn the last
example. Suppose we wish to, oil successive sensitivity runs,
increase the first correlation coefficient by 10% and the de-
crease the second by 30%. The lines would appear as follows:
Xl-BGAUl(200.,40., Z)
RH01=0.8
CALL SEN(RHOl,' RH01',10.,0.)
X2=BGAU2(7G0.,140.,RH01,Z)
RH02=0.6
CALL SEN(RH02RH02-300.)
X3=BGAU2(400.,80.#RH02,Z)
Warning: Correlation coefficients are numbers between -1 and i.
Be careful not to alter the value of a correlation coefficient
beyond this range or else an error message will be printed by
MOUSE and your run will be terminated automatically.
9.4 Bivariate Lognormal Distribution
MOUSE provides another bivariate distribution, the lognormal.
The general form of the bivariate lognormal distribution function
is:
BGAUL1(mx,sxr X,Y,Z)
BGAUL2(my,sy,rho,X, Y, Z)
Except for the additional dummy variables (X and Y), the
discussion of this. Section is identical to that of Section 9.3.
There are three slight differences:
a. The permissible range of the correlation coefficient
of a bivariate lognormal distribution is:
. + (sx)(sy)/(mx)(»y)] to +1
b. If the bivariate lognormal correlation coefficients
are of the same sign, the geometric mean is used to
average them, i.e.,
both positive then rho = [(rhol)(rho2)]0,5
both negative then rho = -[ (rhol) (rho2n0*5
different signs then rho = (rhol + rho2)/2
c. The procedure for estimating the correlation coeffi-
cient must be modified as follows:
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Mouse Manual Page 113
let mx » ln[mx'(1 + rx)-0-5] [9,7]
my = ln[my'(l + ry)~0-5] [9.8]
sx = [ln(l + rx)]"*5 [9.91
sy - [ln(l + ry)]0*5 [9.10]
where rx * (sx')2/(mx')2 [9.11]
ry ® (sy')2/(my')2 [9.12]
the primes denoting the logriormal means and standard deviations.
The x and y values in Equation 9.3 are modified also:
x = In x' [9.13]
y = In y' [9.14]
Equation 9.3 is used to calculate rho as before, and the
lognormal correlation coefficient is obtained by:
rho' = [(1 + rxy)rho - i]/rxy [9.15]
where
rxy = (sx') (sy') / (nix*) (my') [9.16]
Remember that the geometric mean is used to average bivariate-
lognormal correlation coefficients if they are of the same sign.
As an example, we use the data of Section 9.3 to calculate
the correlation coefficient for a bivariate lognormal distribu-
tion. These data are as follows:
mx' = 3
my' =50
sx' = 0.6
sy' =10
1st (x',y') pair = (1.5,67.5)
2nd (x',y') pair = (4.5,34.5)
(The last value of the 2nd pair of data points has been changed
to 34.5 in order to remain within the allowable limits of a
bivariate lognormal distribution.)
Using equations 9.8, 9.9, and 9.13:
rx = 0.62/32 = .04
ry <= io2/502 = .04
rxy - (0.6)(10)/(3)(50) - .04
Using equations 9.4, 9.5, 9.6, and 9.7:
mx = ln[3(1 + .04)"0*5] = 1.0790
my = ln[50(l + .04)-°-5] = 3.8336
sx = [ln(l + .04)]«•5 . ,1980
sy ¦ [ln(l + .04)j0*5 a .1980
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Mouse Manual Page 114
Using equations 9.10 and 9.11:
1st pair: x = In 1.5 = .4055 y = In 67.5 = 4.2121
2nd pair: x = In 4.5 = 1.5041 y » In 34.5 = 3.5410
Using the first pair of x/y values in equation 9.3 we obtain;
rho = (4.2121 - 3.8336).1980/(.4055 - 1.0790).1980 = -.5620
Using equation 9.12:
rho' = [(1 + , 04) ""°* 5620 • 1] /. 04 = -0.5450
Using the second pair of x/y values in equation 9.3 we obtain:
rho ® (3.5410 - 3.8336).1980/(1.5041 - 1.0790).1980 = -0.6883
Using equation 9.12:
rho' = [( 1 + ,04)-°*6883 - 1]/.04 = -0.6659
Averaging the values of rho' of -0.5450 and -0.6659 we obtain the
final value of rho':
rho' = .-[ (.5450) (.6659) ]0*5 = -0.6024.
Note: If data is available, the uncorrected correlation
coefficient, rho, can also be estimated by IMP (see Section 9.5).
9.5 Using IMP to Analyze Bivariate Normal
And Bivariate Lognoral Distributions
IMP can be used to analyze data pairs for bivariate normal
and bivariate lognormal distributions. The user has a choice
between placing the (X,Y) observation pairs into the data file
IMP.DAT or entering the data directly from the keyboard. The
first line of the data file contains descriptive information (up
to 60 characters) used to identify the file. (This line is manda-
tory. ) The remainder of the file contains the sample data, one
(X, Y) observation pair per line starting in column 1, with the
use of the decimal point mandatory. The X and Y values are
separated by either a comma or a space. Figure 9.3 shows part of
such a file for a 100-observation data pair set consisting of
numbers generated by the following pair of MOUSE calls:
X1=BGAU1(100.,25.,Z)
X2=BGAU2(50.,10.,0.8,Z)
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Mouse Manual Page 115
BIVARIATE NORMAL: X1=BGAU1(100.,25. ,Z), X2=BGAU2(50.,10.,0.8,Z)
100.00
1.00
89.48
45.93
106.23
49.66
107.13
58. 35
83 .86
40.39
112.08
54.78
121.52
57.94
84.66
47.18
131.95
60.41
123.32
59.19
87.83
47.37
81.05
44.93
FIGURE 9.3:
PORTION OF EXAMPLE IMP DATA FILE FOR ANALYSIS
OF A BIVARIATE NORMAL DISTRIBUTION
IMP prompts the user for instructions and offers the series .
of menus and options shown previously.
1. After calling IMP (simply by typing "IMP"), you will be
asked to supply a title or continent line (not to exceed
60 characters) that will be placed in the IMP output
file. If the file IMP.OUT (i.e., the standard IMP output
file) exists, you will be asked whether you wish to
overwrite this file or not. If not, you will asked for
the name of an alternate output file.
2. The following menu is presented:
**********************************************
* PIT A DATA SET TO A CLASSICAL DISTRIBUTION *
* INTEGRATE A CONTINUOUS DISTRIBUTION *
* ANALYZE A BIVARIATE DISTRIBUTION *
* ANALYZE A DATA SET FOR AUTOCORRELATION *
**********************************************
For the purpose analyzing a data set for bivariate normal or
bivariate lognormal distributions, select the third option,
"ANALYZE A BIVARIATE DISTRIBUTION". The IMP program was run using
100 data pairs generated by the bivariate normal model shown
above, and the results are shown in Figure 9.4. (Omitted are the
regression fits that are part of this option. See Section 9.2.)
Note that the mean and standard deviation parameters for varia-
bles XI and X2 have been estimated as (100.0,24.2) and
(49.2,10.4), respectively (the actual values were (100,25) and
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Mouse Manual Page 116
BIVARIATE DISTRIBUTION ANALYSIS
(Numbers in parentheses are significance levels of coefficients)
CORRELATION COEFFICIENT = .6990
STATISTICS FOR 1st VARIABLE:
MEAN = 100,0303
STANDARD DEVIATION = 24.2179
COEFFICIENT OF SKEWNESS (UNTRANSFORMED) = .03 (.458)
COEFFICIENT OF KURTOSIS (UNTRANSFORMED) = .43 (.183)
COEFFICIENT OF SKEWNESS (LOG TRANSFORM) = -1.20 (.999)
COEFFICIENT OF KURTOSIS (LOG TRANSFORM) = 3.77 (.001)
STATISTICS FOR 2nd VARIABLE:
MEAN - 49.2156
STANDARD DEVIATION « 10.4783
COEFFICIENT OF SKEWNESS (UNTRANSFORMED) = -.94 (.999)
COEFFICIENT OF KURTOSIS (UNTRANSFORMED) = 3.32 (.001)
COEFFICIENT OF SKEWNESS (LOG TRANSFORM) = -7.14 (.999)
COEFFICIENT OF KURTOSIS (LOG TRANSFORM) = 62.88 (.001)
Note:
* = .01 < p and p < or = .05
** = p < or = .01
FIGURE 9.4 :
IMP OUTPUT, BIVARIATE NORMAL DISTRIBUTION
ANALYSIS, DATA OF FIGURE 9.3
(50,10), and the correlation coefficient was estimated as 0.7
(the actual value was 0.8).
9.6 Time Series and Autocorrelation
If a stochastic process produces random variables such that
there is a random variable for each value of t (where t indi-
cates time), we observe a random function of time called a Time
Series. A common property of many time series related to economic
-------
House Manual Page 117
and technological phenomena is the relationship of x^+j and x^,
where j is the lag, i.e., the number of time intervals between
the respective values of the time series. If xt+j and xt are
independent, then their covariance, cov(x^,xt+j), will be zero.
If this covariance is not zero, then the tine series is said to
be Autocorrelated¦
Suppose we define an Autocorrelation Function as follows:
rho(j)—cov(xt,xt+j)/cov(xt,xt+0} [9.17]
where, since j»Q implies no lag, the cov(xt,xt+o) term = varfx^}.
We also assume that the expected or mean value of x^ is zero.
These conditions are valid for stationary time series.
It is not generally possible to' generate a time series with
an arbitrary autocorrelation function. There are two special
functions, however, which can be used with satisfactory results
and flexibility if the distribution of the xt variables is nor-
mal.
(a) Linear Autocorrelated Function
Let
rho(j) = 1 - j/m for j < m [9.18]
- 0 for j >« m
This is a linearly decreasing function of j and a model for
autocorrelated time series where zero autocorrelation can be
assumed for lags equal to or larger than m. A linearly
autocorrelated time series of normally distributed variables, x,
can be generated by the following M0US1 line:
CALL AUTOL(mean,std,k,value)
where "mean" and "std" are decimal variables representing the
mean and standard deviation of the variable, respectively, "Jc" is
an integer variable equal to 12/m, and "value" is the name of the
variable. AUTOL uses a fast approximation to the normal distri-
bution involving the generation of 12 uniform variables, thus the
k - 12/m identity and subsequent limitation of this subroutine to
12 lags. Because of the limits on the ranges of m in equation
9.18, AUTOL is limited to a maximum of 12 lags. Consequently, the
practical values of k are as follows:
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Mouse Manual Page 118
& i of lags
1 12
2 6
3 4
4 3
6 2
A typical call, therefore, would look like:
CALL AUTOL(100.,25.,2,PRICE)
PRICE, therefore, would be generated as a normally distributed,
linearly autocorrelated tine series (up to 6 lags) with mean
equal to 100 and standard deviation equal to 25.
(b) Exponential Autocorrelated Function
In an exponential autocorrelated function, the coefficient
of correlation with lag j is expressed as
rho(j) = lambda^ 0 < lambda <1 [9.19]-
It can be shown that this type of autocorrelation function exists
in time series obtained by exponential smoothing based on the
recursion relation,
xQ = (1 - lambda)r0
xt = lambdaxt-i + (1 - lambda)rt [9.20]
where the r^ numbers are mutually independent variables with zero
mean and variance sigma.
An exponential autocorrelated time series of normally dis-
tributed variables, x, can be generated by the following MOUSE
line:
CALL AUTOE(mean,std,lambda,value)
where "mean" and "stdw are decimal variables representing the
mean and standard deviation of the variable, respectively. A
typical call, therefore, would look like:
CALL AUTOS(100.,25.,0.8,PRICE)
PRICE would be generated as a normally distributed, exponential
autocorrelated time series with lambda equal to 0.8, mean equal
to 100, and standard deviation equal to 25.
IMP can be used to analyze a data set for autocorrelation
(Option four in the Main Menu). The user has a choice between
placing the observations into the usual data file IMP.DAT or
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Mouse Manual Page 119
entering the data directly from the keyboard. The first line of
the data file contains descriptive information (up to 60 charac-
ters) used to identify the file. (This line is mandatory. \ The
remainder of the file contains the sample data, one observation
per line starting in column 1, with the use of the decimal point
mandatory. Figure 9.5 shows part of such a file for a 100-obser-
vation data pair set consisting of numbers generated by the
following MOUSE call;
CALL AUTOE(100.,25,,0«8,X)
Partial results are shown in Figure 9.6. Note the exponentially
decreasing autocorrelation coefficient as the lag increases. Note
that the Accumulation Line (Chapter 4, Section 4.10) can be used
in a MOUSE program to accumulate data to be subjected to an
autocorrelation analysis.
EXERCISES
Which of the following are valid statements or sequences?
1. Y6=»BGAUL1(200. ,20.,X,Y,Z)
Y7=BGAUL2(100.,10.,0.7,X,Y,Z)
2. X1=BGAU1(50.,10.,Z1)
X2=BGAU2(100.,55.,0.8,Z2)
3. X1=BGAU1(200.,40,,Z)
X2=BGAU2(700.,140.,0.8,Z)
X3*=BGAU2(400.,80.,-0.6fZ)
X4-BGAU2(100.,20.,0.8, Z)
5. Given the following data, find the bivariate lognormal
correlation coefficient:
mx » 100, my - 200, sx - 10, sy - 40
when x = 80, y = 150 and when x «= 120, y = 250
p h ramwpBS
Draper, N.R., and Smith, H., Applied Regression Analysis. John
Wiley & sons, Inc., New York, N.Y., p. 35 (1966).
Hillier, F.S., The Evaluation of Riskv Interrelated Investments,
Technical Report No. 73, Contract Nonr-225(53) (NR-642-002) with
the Office of Naval Research (1964).
Pouliquen, L. Y., Risk Analysis in Project Appraisal. World Bank
Staff Occasional Papers No. Eleven, The Johns Hopkins University
Press, Baltimore, p. 1-95 (1970).
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Mouse Manual Page 120
AUTOCORRELATION: AUTOE(100.,25.,0.8,X)
100.00
83.75
104.19
97.69
121.86
125.43
141.76 FIGURE 9.5; PORTION OF EXAMPLE HIP DATA
150.28 FILE FOR ANALYSIS OF AH
161.361 ADTOOOKREIATION (EXPONENTIAL)
135.90 TUSE SERIES
109.79
112.41
AUTOCORRELATION ANALYSIS
SUMMARY OF AUTOCORRELATION COEFFICIENTS:
AUTOCORRELATION
LAG
COEFFICIENT
0
1.0000
1
.7694
(.001)**
2
.6234
(.001)**
3
.5129
(.001)**
4
.4013
(.002)**
5
3305
(.009)**
6
.2504
(.040)*
7
.2254
(.060)
8
.1382
(.173)
9
.0228
(.439)
10
-.0419
(.610)
11
-.0961
(.738)
12
-.0745
(.689)
13
-.0471
(.621)
14
.0161
(.458)
15
-.0001
(.500)
16
-.0514
(.630)
17
-.0389
(.598)
18
-.0679
(.668)
19
-.0878
(.711)
20
-.0779
(.689)
Note; * = .01 < p and p < or = .05
** = p < or = .01
FIGURE 9.6: IMP OUTPUT, ADTOCORRRLATED (EXPONENTIAL
TTHE SERIES, DATA OF FIGURE 9.5
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CHAFFER 10
HOK-OHIFOIH CASH FUNIS, VICTORS, MID
THE INTKRKAL-BATE-OF-RETORN FUNCTION
10.1 Introduction
With the exception of the single payment interest functions,
SCA, SPW, and CSPW, all of the interest functions described in
Chapter 5 assume either a constant cash flow for each time period
or else a constant change in cash flow (functions of the latter
type include UFWP, GCA, and GPW). The nature of these interest
functions makes it very easy to deal with a stochastic project
life in MOUSE as project life is merely one of the calling
arguments; it makes no difference to MOUSE whether or not these
arguments are stochastic variables or constants. For example, in
Figure 3.2 the annual net benefit, ANETB, is the same for all
years, i.e., .
ANETB=GREVl+GREV2+GREV3-70.0
so that the present value can be calculated by use of one of the
uniform series interest formulas, UPW, i.e.,
PVALUE=ANETB*UPW(0. 08,XLIFE,0.0) -XININV
However, suppose the annual net benefits are increased by 12%
each year (this is one of the assumptions made in modification #2
to the Waste Oil Reclamation Project problem shown in Figure
10.1), i.e.,
ANETB(LOOP)«SCA(.12,YEAR,0.)*(GREV1+GR1V2+GRSV3-70.)
using the SCA function for the increase and noting that YEAR and
LOOP are identical (going from 1 to XLIFE) except that LOOP is an
integer representation in MOUSE while YEAR represents a decimal
number. Clearly ANETB is now a vector (the index for a vector in
MOUSE is always LOOP) and UPW cannot be used. Each element of
ANETB could be brought back to its present value by the SPW
function; following this, all of the elements in ANETB would then
have to be summed.
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e
C WASTE OIL RECLAMATION PROBLEM MODIFICATION #2
C
DIMENSION D1(3,2),02(4,2),D3<4,2) ,
C VECTOR DIMENSION LINE
C
DIMENSION AXBEH(15)
£
C VECTOR TABLE BIMiMSIOM LINE
C
DIMENSION VTABS7,15)
c
ilHEHSIOS TAB(«)
CC«MON/ai/C80/O2/JUW>/a3/LO0P/ft4/SENS/Q5/It>A7A/a6/rOUT/07/IPASS/Q8
CALL R£AD(D1,3>
CALL READ<02,4>
CALL IEAJ)
ITE«®5D0O
5 00 10 IXXX»1,JTER
CQ«TAM=OUM id. 0.4,0}
HAN01=CUNI(-10.6,10.0)
PRICEl*E0IS(0t.3)
SL0tt»ECTSCD2,4)
PRCOZ*$TIP{D3,4>
RAN02»GAU(0.06,0.10>
PR]GE3®EIETA<6,I,S.0,9,5} .
PR003-EXPON<18.0>
XLIFE-OTRK5.0,10.0.15.0)
XININV=T8A<1600.0,2000.0,2200.0,2400.0)
THRU1=10.0-CONTAH
YJElDl"'10.0-Z.0*CONTAM '
PRaO!»T(tRU1*TI£lD1*RAND1
' GREV1«PRICE1*PR001
PR I CE2=10.0- SL0PE*PR002»RAW>2
Gt£V2=PRICE2*PR002
CREV3=PRICEJ»PK»3
L!«»XLIFE
C
C VECTOR LOOP CONTROL LINE #1
C
BO 100 L00P»1,LIFE
y£AR=LOOP
AN8EN{LOOP)=SCA{.12,TEAR,0.)*(GREV1+(*EV2*GSEV3-70.>
C VECTOR TABLE DEFINITION LINE
C
CALL STAAL£(ANBEN,'(ANBEM)',LIFE,VTAB)
C VECTOR LOOP CONTROL LINES #2 AND *3
C
100 CONTINUE
LOOP*!
C
C PRESENT WORTH SUK FUNCTION
C
PVALUE
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10.2 The Present Worth sum Function
A function that does all of this in a single line is the
MOUSE Present Worth Sun Function. PWSUM. This function is called
by:
PWSUM(vector,i,upper,inv)
where "vector" is an m-dimensiona1 vector, i is the
interest rate, "upper" (an integer or integer variable) is the
upper index of the vector loop, and "inv" is the initial
investment. Also note that the vector name must be a decimal
name. (A decimal name is one that starts with any letter except
I, J, K, L, M, or N. See Chapter 3, Section 3.2). The PWSUM
function cannot be used within vector loops (see Section 10.3
for an explanation of the vector loop). As with all MOUSE inter-
est related functions, "i" and/or "upper" can be stochastic
variables. Thus, as is shown in Figure 10.1, the present value
of the annual benefits when these cash flows are non-uniform over
time is calculated in MOUSE simply as:
PVALUE=PWSUM(ANBEN,.08,LIFE)-XININV
(In order to distinguish between a vector and a scalar quantity,
ANETB has been renamed to ANBEN in Figure 10.1.) The vector
ANBEN must be dimensioned using a Vector Dimension Line which has
the following general form:
DIMENSION namel(ml),name2(m2),name3(m3),etc.
where "naaei" is the name of the ith vector and "mi" is its
dimension. {See Chapter 11, Section 11.6, for a discussion of
TRAP, a MOUSE utility that greatly facilitates the preparation of
the Dimension Block.) Also note that the variable name must be a
decimal name. (A decimal name is one that starts with any letter
except I, J, K, L, M, or N. See Chapter 3, Section 3.2). In
Figure 10.l this line appears as:
DIMENSION ANBEN(15)
since, in Modification #2 to the Waste Oil Reclamation Project
problem, the maximum life of the facility has been stated to be
15 years. Note, however, that when MOUSE encounters the PWSUM
line in the model it sums ANBEN only from the first year to LIFE,
LIFE (being stochastic in this problem) varying from iteration to
iteration.
Another feature of this function is that the user can
specify the use of continuous compounding simply by entering the
interest rate as a negative number, e.g.,
PVALUE-PWSUM(ANBEN,-.0 8,LIFE)-XININV
-------
Mouse Manual Page 124
When MOUSE sees the negative interest rate, it automatically
assumes that the user wants the interest calculated on a
continuous basis (and, of course, uses the absolute value of the
interest rate in its calculations). The true interest rate,
however, must be used.
Finally, when "iM in the PWSUM function is set to zero, the
PWSUM function serves as a convenient method for the simple
summation of the terms in a vector. CHECKER, Chapter 11, Section
11.2/ will examine a MOUSE source file to insure that arguments
of the PWSUM function are correct, and that the dimension of the
vectors in the function agree with those specified in the Dimen-
sion Block.
10.3 Calculating the Elements of a Vector
There remains the question of how the individual terms in a
vector are calculated. If the equation for the calculation of
these terms does not change from time period to time period (save
for the numbers themselves), this calculation is simplified
greatly by means of a vector loop. This loop is similar to that
used in the MOUSE pass (see Chapter 7) , but one more control line
is involved. The loop need only surround those equations in the
model which are affected by time. (This will be more fully
explained later.) In Modification #2 to the Waste Oil Reclama-
tion project problem there is only one such equation in the
model, x.e•,
ANBEN(LOOP)=SCA(.12,YEAR,0.)*(GREV1+GREV2+GREV3-70.)
The vector loop consists of three vector loop control lines. one
preceding the equation(s) affected by time and two immediately
afterwards. These control lines are of the following form:
DO 100 L00P=1,upper
100 CONTINUE
L00P=1
where "upper" is an integer or integer variable representing the
number of elements or coordinates to be filled in the vector.
Note that this can be a deterministic number (such as 3) or a
stochastic variable (such as LIFE). For Modification #2 to the
Waste Oil Reclamation Project problem, the full vector loop is as
follows:
-------
Mouse Manual Page 125
LIFE=XLIFE
DO 100 LOOP=l,LIFE
YEAR-LOOP
ANBEN(LOOP)=SCA(.12,YEAR,0.)*(GREV1+GREV2+GREV3-70.)
100 CONTINUE
LOOP=l
Since the upper limit of the loop is the life of the facility,
which is a stochastic variable in this problem, and since the
upper limit must be either an integer or an integer variable, it
is necessary to define a new integer variable by setting
LIFE=XLIFE (see Chapter 3, Section 3.2) prior to the vector loop.
Furthermore, since in this problem we also wish to use the index
LOOP as an argument in the SCA function and since the SCA
function takes decimal variables only, it is necessary to define
a new decimal variable by setting YEAR=LOOP. (Note that neither
XLIFE or LOOP are altered by these statements? all that is being
done is that two new variables are being created.)
Note that since XLIFE is used in the Irrigation Project
problem only to compute the annual net benefits, the LIFE-XLIFE
line in the program of Figure 10.1 could have been omitted by.
defining the variable on the left-hand side of the Stochastic
Variable Sampling line directly as an integer, i.e.,
LIFE=DTRI(5.0,10.0,15.0)
10.4 The Use of the IF statement in Vector Loops
In Chapter 4, Sections 4.6 and 4.7, the IF statement was
introduced in conjunction with truncated and composite
probability distributions. The IF statement also finds
application within a vector loop when special conditions are
imposed upon some coordinates and not upon others. For example,
assume for Modification #2 to the Waste Oil Reclamation Project
problem that there is a fixed salvage value of $5000 at the end
of the project life. This condition is very easily met by the
following addition placed immediately after the existing
ANBEN(LOOP) loop:
IF(LOOP .EQ. LIFE) ANBEN(LOOP)=ANBEN(LOOP)+5000.
The use of this IF statement insures that the $5000 will «b© QClcifiCl
only to the last coordinate of the ANBEN vector. There are many
models in which conditional calculations, such as this one, are
involved. The IF statement can be very helpful in these
instances.
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Mouse Manual Page 126
10.5 Vector Loops and Order of Statements
Consider the following three examples,:
Example 1: X-CUNI(20. ,30.)
DO 100 L00P=1,3
YEAR=LOOP
VEC(LOOP)=QUAN*SCA(X,YEAR,0.)
100 CONTINUE
LOOP=l
EXAMPLE 2: DO 100 LOOP=l,3
X«eUNI(20.,30.)
YEAR=LOOP
VEC(LOOP)=QUAN*SCA(X,YEAR,0.)
100 CONTINUE
LOOP=1
EXAMPLE 3: DO LOOP=l,LIFE
YEAR=LOOP
VEC(LOOP)=QUAN*SCA(CUNI(20.,30.)YEAR,0.)
100 CONTINUE
LOOP=l
Examples 1 and 2 produce quite different results. In Example 1,
for every time period the same value of X is used as the
percentage increase argument in the SCA function in the vector
VEC. In Example 2, the value of X changes from time period to
time period. Which is correct? The answer to this depends upon
your model. The model of Example 1 assumes that the percentage
increase in the quantity QUAN does not vary from year to year; in
Example 2 the model assumes that it does.
Finally, Example,3 produces the same results as Example 2.
Example 3 is an illustration of the direct method of
introducing stochastic variables in MOUSE. (See Chapter 3,
Section 3.4.) Some users may find the direct method useful as it
eliminates a line of code in a program; however, it does
complicate the line in which it is used.
10.6 The Vector Table
Although Sensitivity Control and Table and Plot Output lines
can be used within a vector loop, only under special circum-
stances can the last two be used with vectors. However, a spe-
cial kind of table, called a "vector Table", has been provided
for this purpose. The Vector Table Output Line has the form:
CALL VTABLE(variable,• (Iname)upper,ti)
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c
C VECTOR DIMENSION LINE
C
DIMENSION ANBEN(15)
C
C VICTOR TABLE DIMENSION LINE
C
PRECISION 12(7,15)
C
C LOOP FOR ANNUAL NET BENEFIT VECTOR
C
L1F|=XLJFE
DO 100 IOOP»1,LIFE
YEAS=LOOP
C
ANBEN(LOOP)=SCA(.12,YEAR,0.)*(6REV1*GREV2*GRE¥3-7Q„3
CALL VTA8LE(AMBEN,¦(ANBEN)1.L1FE.T2)
C
C END OF LOOP FOR ANNUAL NET BENEFIT VECTOR
C
100 CONTINUE
LOOP=1
C
PVALUE=PWSUM{ANBEN,.08,LIFE,XIN1NV3
Key;
¦ Vector Name
¦ Index for Vector Loop, Integer or Integer Variable
(used in Vector Loop Control Line 1 and PWSUM function)
FIGURE 10.2: EXCERPT FROM WASTE OIL RECLAMATION PROBLEM,
MODIFICATION #2, SHOWING LINES RELEVANT TO VECTORS,
VECTOR TABLES, AND PRESENT VALUE SUM FUNCTION.
**•**•****•#•******•*•••*«****•#•
* SUMMARY FOR VECTOR ANBEN *
STANDARD
COEF. OF
COEF. OF
COEF. OF
PERIOO
COUNT
MEAN
DEVIATION
VARIATION
MINIMUM
MAXIMUM
SKEWNESS
KURT0S1S
1
5000
651.05860
311.58850
47.86
-128.76200
2140.54400
.38
.16
2
5000
729.18620
348.97740
47.86
-144.21350
2397.40900
,38
.16
3
5000
816.68740
390.85830
47,86
-161.51910
2685.09800
.38
.16
4
5000
914.69040
437.76010
47.86
-180.90140
3007.31000
.38
.16
5
5000
1024.45300
490.29250
47.86
•202.60960
3368.18700
.38
.16
6
5000
1147.38800
549.12430
47.86
-226.92270
3772.37000
.38
.16
7
4811
1288.70400
615.96390
47.80
-254.15340
4225.05400
.37
.15
8
4419
1442.26800
690.56230
47.88
-284,65190
4732.06100
.37
.17
9
3765
1614.51100
766.17130
47.46
-318.81010
5299.90800
.34
.17
10
2962
1804.77200
852.72930
47.25
-354.06310
5935.89600
.37
.22
11
1982
2037.24500
950.96720
46.68
-396.55070
5900.91400
.35
.08
12
1141
2261.82700
1058,32700
46.79
-436.68860
5935.98300
.39
.11
13
571
2509.24500
1172.23800
46.72
-376.88600
6615.00100
.37
.06
14
185
2699.94800
1269.92400
47.04
168.61490
7408.80100
.41
-.03
FIGURE 10.3: VECTOR OUTPUT
RECLAMATION
FOR HASTE OIL
PROBLEM, MODIFICATION #2
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AVERAGE OF AN8EN
650.0000
e-----
1 o*
1300.0000
2000.0000
-o---
2700.0000
-- o
'2 *
3
4 ii
5 V
6 it
7 ,i
6 11
9 u
10 ii
11 ii
12 ii
13 i.
14 ii
FIGURE 10.3 Continued:
VECTOR OUTPUT FOR WASTE OIL
RECLAMATION PROBLEM, MODIFICATION #2
-------
Mouse Manual Page 129
where "variable" is the vector to be tabulated, '"Inane"' is an
alphanumeric name for the table (note that this is a literal name
and the single quotes and the parentheses must accompany the name
exactly in the order shown), "upper" is the same as the "upper"
specification in the vector loop (i.e., either an integer or an
integer variable), and ti is the usual dummy argument used for
storing the table between iterations of the program. Note that
the variable name must be a decimal name. (A decimal name is one
that starts with any letter except I, J, K, L, M, or N. See
Chapter 3, Section 3.2). Also note that the variable can be
scaled as in regular MOUSE tables (see Chapter 6, Section 6.6).
In Figure 10.1 the Vector Table Output line appears as
CALL VTABLE(ANBEN,«...ANBEN',LIFE,T2)
since 15 is the dimension of the ANB1N vector and LIFE is the
upper limit of the vector loop. Note that scaling (see Chapter
6, Section 6.6) is permitted in Vector Table Output lines.
As with all tables in MOUSE, the Vector Table must be dimen-
sioned. The Vector Table Dimension Line is of the form:
PRECISION Tl(7,nl),T2(7,n2),T3(7,n3),etc.
(See Chapter 11, Section 11.6, for a discussion of TRAP, a MOUSE
utility that greatly facilitates the preparation of the Dimension
Block,) Also note that the variable names Tl, T2, etc., must be
decimal names. (A decimal name is one that starts with any letter
except I, J, K, L, M, or N. See Chapter 3, Section 3.2), For
example, in Figure 10.1 this line is
PRECISION T2(7,15)
Decimal points are, as in all dimension lines, not used. Figure
10.2 shows how the various vector-oriented lines interrelate.
The Vector Table produced by the program of Figure 10.l is
shown in Figure 10.3. The usual table statistics of mean, stand-
ard deviation, coefficient of variation, minimum, and maximum are
produced for each time period as well as two new statistics, the
coefficients of skewness and kurtosis. (If, within a coordinate,
the values do not vary or if the number of observations in any
coordinate is less than 4, only the mean will be reported, howev-
er.) The "COUNT" in the Vector Table is equal to ITER only for
those vector coordinates filled at every iteration. (On only
1613 iterations in Figure 10.3, for example, were coordinates 11
through 15 filled since there was only a one-third chance that
the life of the Waste Oil Reclamation Project would reach these
years.)
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House Manual Page 130
Q
."hi i 11 n .-i-rrliT
Lk
POSITIVELY SKEWED WEOATIVELY SKEWED
NEGATIVE KtlRTOSIS
"hn-
POSITIVE KUttTOSIS
FIGURE 10.4: EXAMPLES OF SXETOE8S AND mTOTOSIS
-------
Mouse Manual Page 131
For a normal distribution, the coefficient of skewness is
equal to zero. A negative number indicates that the distribution
is skewed to the left, i.e., that it has a long tail to the left;
a positive number indicates that it is skewed to the right, i.e.,
a long tail to the right (see Figure 10.4). The coefficient of
kurtosis for a normal distribution is also zero. A negative
value indicates that the distribution is flat-topped, and a
positive value indicates that the distribution has a pronounced
peak (see Figure 10.4).
Finally, as an added utility, following the Vector Table is
a plot of the average value of the vector versus the time period.
In Figure 10.3, the sharply rising curve reflects the 8% yearly
increase in net benefits programmed into Modification #2 to the
Waste Oil Reclamation problem. CHECKER, Chapter 11, Section
11.2, will examine a MOUSE source file to insure that arguments
in the Vector Table Definition line are correct, and that the
dimensions of the dummy argument used for storing the table be-
tween iterations agree with those specified in the Dimension
Block.
10.7 vectors, The mouse Fass, And Plots
In Figure 7.1, the Waste Oil Reclamation Problem, Modifica-
tion #1, used a matrix to store three different values of the
coefficient (COEF) in the YIELD1 equation. This program produced
three separate tables, one for each value of COEF. An entirely
different approach has been taken in Figure 10.5 (Waste Oil
Reclamation Problem, Modification #3) where the standard table
has been replaced with a vector table. COEF is now a vector with
index LOOP instead of IPASS; the Pass Control cards are no longer
needed. Also, rather than bother reading the values of this
vector, they are specified directly in the program. Finally,
PVALUE has been dimensioned as a three-element vector. Since
tables are not used in this program, a Dummy Table (with argument
» i, i.e., output requested) is the last executable statement in
the program.
The vector table produced is shown in Figure 10.6. This is
a concise summary of the effects on present value (PVALUE) of
varying this coefficient. Indeed, the standard vector table plot
below the table is tantamount to a plot of coefficient versus
present value since the coefficients were selected at equal
intervals. Although in general the standard table provides more
information than does the vector table, the fact that present
value at any coefficient is approximately normally distributed
(as indicated by the coefficients of skewness and kurtosis, and
their significant levels, shown in the vector table) negates most
of the advantage of the standard table in this problem. Further-
-------
c
C UASTE Oil RECLAMATION PROBLEM MODIFICATION #3
C
DIMENSION D1(3,2),D2(4,2),03(4,2)
C
C VECTOR TABLE DIMENSION LINE
C
DIMENSION PVEC(7,3)
C
C VECTOR DIMENSION LINE
C
DIMENSION C0EF(3),PVALUE(3)
C
C PLOT DIMENSION LINE
C
DIMENSION P1(100,3)
C
COMMON/O1/C80/02/JUNP/Q3/L00P/04/SENS/Q5/IDATA/06/1OUT/07/1PASS/08
1/IRAN/Q9/IREAD/Q10/ISCR1/Q11/ISCR2/Q12/ISNAP/Q13/1TER/Q14/IT0G/Q15
i/IURIT/a16/IXXX/017/OK1/Q18/OK2/Q19/OK3/020/OK4/021/OK5/Q22/OK6
C
C NAME LINE
C
CALL NAMECOIASTE OIL RECLAMATION PROBLEM - MODIFICATION #3)')
C
CALL READ(D1,3)
CALL READ(D2,4)
CALL READ(D3,4)
C
C SPECIFICATION OF VECTOR VALUES
C
C0EF(1)-1.
C06F(2)*2.
COEF(3)-3.
C
C ITERATION NUMBER LINE
C
ITER-5000
C
5 DO 10 IXXX-1.1TER
C
CONT AM ¦DUNK 1.0.4.0)
RAND1*CUNI(-10.0.10.0)
PRICE1-EDIS(D1,3>
SLOPE«ECTS(D2,1)
PROD2-STEP(D3.4)
RAND2*GAU(0.00,0.10)
PRICE3>EBETA(6.5,8.0,9.5)
PROO3»EXPON(1B.0>
XL IFE-OTRK5.0,10.0.15.0)
XININV-TRA(1606.0,2600.0,2200.0,2400.0)
C
C VECTOR LOOP CONTROL LINE 1
C
DO 100 L00P«1,3
C
THRUUIO.O-CONTAM
YIELD1>10.0-C0EF(L00P)*C0NTAM
PR001=THRU1*YIELDWRAND1
GREV1=PRICE1*PR001
PRICE2-10.0-SLOPE*PROD2+RAND2
GREV2=PRICE2*PR0D2
GREV3=PRICE3*PR003
ANETB=GREV1+GREV2+GREV3-70.0
PVALUE(LOOP)=ANETB*UPW(0.08,XLIFE,0.0)-XININV
CALL VTABIE(PVALUE,•(PVALUE)•,3,PVEC)
C
C VECTOR LOOP CONTROL LINES 2 AND 3
C
100 CONTINUE
LOOP=1
C
C PLOT DEFINITION LINE
C
CALL PLOT(PVALUE(1),'(PVALUE)1, PR ICE2,' ( PR 1CE 2)'
C DUMMY TABLE
C
CALL DTABLE(I)
C
10 CONTINUE
G0T0(5,15), JUMP
15 CONTINUE
CALL LIST('(M03.FOR)')
END
FIGURE 10.5: WASTE OIL RECLAMATION PROBLEM, MODIFICATION «3
-------
Mouse Manual Page 133
- *•***•**•******•**•**•••*•***•**•
* SUMMARY FOR VECTOR PVALUE •
*»•#•»*•»#*»*»»•«••»**»•»•»#»*«»*
PERIOD COUNT
MEAN
STANDARD COEF. OF
DEVIATION VARIATION
MINIMUM
MAXIMUM
COEF. OF
SKEUNESS
COEF. OF
KURTOSIS
1 5000 2815.71900
2 5000 1841.81700
3 5000 867.91000
1B18.11800 64.57
1942.75500 105.48
2127.63600 245.14
•1834.04600 11838.67000
-3190.54000 11204.57000
-4988.61500 10570.46000
.55
.50
.42
( 12.24}
.57
.33
.04
.62)
. AVERAGE OF PVAIUE
870.0000 1500.0000 2100.0000 2S00.0000
1 o *
I
I
I
PERIOO 2 e *
FIGURE 10.6:
VECTOR OUTPUT OF TEE WASTE OIL RECLAMATION PROBLEM,
MODIFICATION #3
-------
House Manual Page 13 4
*•#***#*¦* #**#**#*•*##**********#***#*****#**#*#*~##
* PLOT OF QUANTITY PRICE 2 VERSUS PVAUIE *
PRICE 2
•1.8000
o-----
1.3000
4.4000
"Q
7.4000
--o
270.0000 o
PVALUE
4700.0000 o
9000.0000 o
Point code: * = within 20% of maximun weight
* = Less than 203 of maxinun weight
dotted line is best fit, i.e., LINEAR MOGEl
SUMMARY OF FITTED EQUATIONS, UEISHTEO LEAST SQUARES
LINEAR :
QUADRATIC :
CUBIC :
RECIPROCAL ;
SQUARE ROOT:
LOGARITHMIC:
POWER ;
EXPONENTIAL:
r=A+B*X
r=A*B*X*X
r=A*8«x*x*x
Y«A*B/X
Y«A*i*sqrtCX)
l*Mn m
Y»A*X**B
Y*A*expCBX)
A
,191801*01
¦26672E+01
,32092E*01
.355631*01
.OOOOOE+OO
.O000OE*0O
,OOOOOE+OO
.OOOOOE+OO
B R-SQUARE
.60267E-03 .76
.67051E-07 .44
.77559E-11 .29
.149971*03 .04
.000001+00 .00
.OOOOOE+OO .00
.oooooe+oo ¦ .00
.OOOOOE+OO .00
FIGURE 10.7:
PLOT OUTPUT or TIB WASTE OIL RECLAMATION PROBLEM,
MODIFICATION #3
-------
Mouse Manual Page 135
more, the vector version of the program runs ouch faster. Howev-
er, if the present value distributions had been non-normal, then
it would have been better to have used the approach of Figure
7.1.
Another change has been incorporated into the Waste Oil
Reclamation Problem, Modification #3, i.e., the Plot Definition
line which involves a vector, PVALUE(2). This leads to consid-
eration of the conditions under which standard tables and plots
can be used in connection with vectors. Standard tables and plots
cannot be used within vector loops. Used outside of the vector
loop, the vector can be entered in the standard table or plot
call as a specific coordinate, e.g., PVALUE(2). In the Waste Oil
Reclamation Problem, Modification #3, the Plot Definition line has
been placed after Vector Loop Control Line #3. The plot produced
from this program is shown in Figure 10.7.
1XERCI81S
Which of the following are valid constructions in MOUSE?
1. X1=PWSUM(PROFIT,.10,XLIFE,XININV)
2. DIMENSION ANBEN(30)
3. DO 100 L00P=1,TIME
•
100 CONTINUE
LOOP=l
4. CALL TABLE(PROFIT,'(PROFIT)1,LIFE,T10)
5. PRECISION Tl(7,20),T2(7,10)
6. CALL VTABLE(SPEED,'(SPEED)',LAPS,T2)
7. Assume, in the Modification #2 to the Waste Oil Reclama-
tion problem, that there is a salvage value at the end
of the project life which varies according to a continu-
ous uniform distribution over the interval $4000 to
$6000. Using an IF statement, what lines should be
altered and/or added to the program?
The following two problems are optional. They show how
MOUSE may be used as a more general purpose simulation language.
-------
Mouse Manual Page 13 6
8. (This exercise involves the use of the IF and GOTO
statements described in Chapter 4, Sections 4.6 and
4.7.) A rat is placed into a cage containing three
doors. The first door leads to freedom, the second
leads to a long tunnel, and the third leads to a short
tunnel. Both tunnels, however, return the rat to the
cage. The doors all look alike to the rat. The time
spent, traversing the first tunnel is exponentially
distributed with a mean of 3 time units and that the
time spent traversing the second tunnel is lognormally
distributed with a mean of 6 and standard deviation of
2. Suppose that the rat's remaining time to live is
normally distributed with a mean of 10 and standard
deviation of 2. Write a MOOSE program that will, on the
basis of the experience of 1000 rats, determine: (a) The
probability that the rat reaches freedom before it dies,
and (b) The probability distribution of the time the
rat spends in trying to escape, regardless of whether it
lives or dies.
9. (This exercise involves the use of the Vector loop and
the Vector Table.) There are three square peg boards,
each containing 100 holes arranged in a symmetric 10 x
10 pattern. Board #1 has a peg in each hole; the other
boards are empty. Every hour, the following sequence
occurs ?
(1} 15 locations are chosen at random on the first board.
If a peg is in one of the holes chosen, it is moved to
its analogous position on Board #2.
(2) After the selections are made for Board #1, 5 locations
are chosen at random on. the second Board. If a peg is
in one of the holes chosen, it is moved to its analogous
position on Board #3.
Write a MOUSE program that plots, for 20 hours of opera-
tion, the number of pegs remaining on each board at the
end of every hour.Use 100 20-hour simulations in your
T cHED • *
-------
Mouse Manual Page 137
CHAtTER 11
TROUBLESHOOTING, ERROR MESSAGES, AMD HELP
11.1 Introduction
Regardless of the care taken in writing an uncertainty
analysis program, errors are unavoidable. The five most common
types are; (a) FORTRAN syntactical errors (i.e., errors caused by
improper coding of the HOUSE program that are caught by the
FORTRAN compiler), (b) MOUSE syntactical errors (i.e., errors
caused by improper coding of the MOUSE program that are not
caught by the FORTRAN compiler), (c) errors in confusing decimal
numbers with integers, (d) errors in the input data, and (e)
logical errors in the model. An example of a FORTRAN syntactical
error is:
Y=(A+B)/(C+D
The error is in the fact that the parentheses are unbalanced,
i.e., there are two left-hand parentheses in the equation but
only one right-hand parenthesis, something that FORTRAN does
not permit. The FORTRAN compiler (see Appendix A for a discussion
of compilers and what they do) will catch this kind of error and,
indeed, some FORTRAN systems (e.g., Microsoft see Appendix A,
Section A.3) have an option whereby a source program can be
checked for errors before it is compiled, thus saving considera-
ble time since compiling is a time-consuming process.
All computers are limited as to the size of the numbers
they can manipulate, the limitation varying with the particular
computer. The multiplication of large numbers or the division of
a number by another close to zero can easily exceed the capacity
of a computer. Either output will be deleted (usually by
printing asterisks where numbers are expected) or error messages
(such as "FLOATING POINT ARITHMETIC OVERFLOW") will be sent to
the user. This is most likely to occur when Vector Tables are
employed since the arithmetic involves taking numbers to their
fourth power; other MOUSE utilities that are susceptible to such
overflow conditions include the Plot routine and the standard
MOUSE Table. Fortunately, scaling (see Chapter 6, Section 6.6
and Chapter 8, Section 8.2) by the user will correct these
errors.
11.2 CHECKER, A Utility for Catching MOOSE Syntactical Errors
An example of a MOUSE syntactical error is:
CALL TABLE(VALUE,'(VALUE) •,DUMO,DUM1f TAB)
CALL TABLE(COST,'(COST)',DUM2,DUM3,TAB)
-------
Mouse Manual Page 138
The error here is in the fact that the storage variable, TAB, has
been used for two different tables. A FORTRAN compiler would not
catch this error; the program might run successfully but bizarre
results would be obtained (see Section 11.3).
A utility called "CHECKER" can be used to parse MOUSE pro-
gram files and check for a number of MOUSE syntactical errors. To
use CHECKER# simply type CHECKER and the following screen will
appear;
********************************* ***********************
CHECKER
(MOUSE Source File Checking Program)
VERSION 1.00+
US ENVIRONMENTAL PROTECTION AGENCY
written by ALBERT J. KLEE
*******************************************************
********************
* INTERACTIVE MODE * <—.
* MUTE MODE *
********************
*************************************
* ENTER NAME OF FILE TO BE CHECKED; *
* *
* » W02.FOR *
*************************************
If the program is found, CHECKER will parse the file for certain
errors (see below) and, if any are found, will report them in a
file called CHECKER.OUT. Selection is made in the
"Interactive/Mute" menu by using the up or down arrow keys and
then pressing the Enter key. In Interactive Mode, each time an
error is found the line in which the error occurs is printed on
the screen, after which the program will continue if the Enter
key is pressed. In Mute Mode, only the number of lines scanned
and the number of errors found to date are reported on the screen
(this information also appears in the Interactive Mode). Interac-
tive mode is useful for small source files since one frequently
can identify the error when the erroneous line appears on the
screen. However, for large source files the Mute Mode is recom-
mended as otherwise the constant interruption would make the
checking process lengthy and the user must be present for the
program to continue.
CHECKER first examines the Dimension Block and, if the
Dimension Block does not pass CHECKER'S tests, either of the
following messages will be reported;
ERROR IN DIMENSION LINE X
Parentheses are unbalanced, i.e.,
Number of Left Parentheses - y
Number of Right Parentheses = z
PROGRAM ABORTED - FIX DIMENSION BLOCK!
-------
Mouse Manual Page 139
ERROR IN DIMENSION LINE X
Wrong number of commas, i.e.,
Number of Paired Parentheses = y
Number of Separating Commas = z
The number of Separating commas
should be one less than the number of
Paired Parentheses.
PROGRAM ABORTED - FIX DIMENSION BLOCK!
If the Dimension Block successfully passes CHECKER'S tests, the
program goes on to check the following MOUSE line types:
Decimal Trace Line (i.e., CALL DTRACE)
Dummy Table Output Output Line (i.e., CALL DTABLE)
Empirical Variable Read Line (i.e., CALL READ)
Integer Trace Line (i.e., CALL ITRACE)
Matrix Read Line (i.e., CALL MREAD)
Name Line (i.e., CALL NAME)
Plot Output Output Line (i.e., CALL PLOT)
Program Listing Line (i.e., CALL LIST)
Sensitivity Control Line (i.e., CALL SEN)
Table Output (MOUSE) Output Line (i.e., CALL TABLE)
Table Output (USER) Output Line (i.e., CALL TABLE)
Vector Table Output Output Line (i.e., CALL VTABLE)
Also checked are all MOUSE probability and interest function
calls, the Pass Control line, and the Specification Block User
Control lines. A check is also made that all checked lines are in
the proper order (e.g., that all Specification Block Control
lines are placed before MOUSE Control Line #4). Finally, CHECKER
will ascertain whether the eight standard MOUSE Control Lines are
correct and in the proper order and, if used, whether the three
standard Vector Loop Control Lines are correct and in the proper
order.
CHECKER'S primary function is to see that dimensioned varia-
bles are properly defined and used; its secondary function is to
insure that arguments to probability and interest calls are
correct, although a number of other checks are made as well. A
list of such checks are as follows;
AGAUL Mode greater than Median.
BGAU1 dummy variable not a decimal name.
BGAU2 dummy variable not a decimal name.
BGAULl dummy variable not a decimal name.
BGAUL2 dummy variable not a decimal name.
Decimal point omitted in number x.
Dimension Block dimension does not agree with first
Matrix Read line dimension x.
Dimension Block dimension does not agree with second
Matrix Read line dimension x.
-------
Mouse Manual Page 140
Dimension Block dimension does not agree with
Empirical Variable Read line dimension x.
Dimension Block dimension of x is not 6 more than
Table Output line dimension of y.
Dimension of 999 exceeded in dimensioned name y.
Duplication of dimensioned variable y.
Duplication of dummy, variable y.
Erroneous Column Output Line.
Erroneous Loop Control Line.
Erroneous Output Destination Line.
Erroneous Pass Control Line.
First Dimension Block dimension should equal 7 instead
of x for Vector Table storage variable y.
First Dimension Block dimension should equal 100
instead of x for Plot storage variable y.
GAU standard deviation not greater than zero.
Illegal characters, illegal decimal name, or integer
name exceeds 6 characters in PWSUM argument.
Illegal characters or integer name, or decimal name
exceeds 6 characters in probability argument.
Illegal characters in PWSUM name.
Illegal characters or integer name, or decimal name
exceeds 6 characters in,interest argument. ¦
Illegal decimal point in number x.
Illegal decimal point in PWSUM period argument.
Illegal integer name.
Illegal integer PWSUM name.
Improper range for correlation coefficient argument.
Improper range for probability argument.
Iteration number out of range.
Interest argument not greater than zero.
Interest period argument less than zero.
Length of 6 exceeded in dimensioned name y.
Length of 6 exceeded in name.
Length of 6 exceeded in PWSUM name.
Length of argument x illegally or unreasonably long.
Minus sign omitted in dimension x.
Missing or incorrect MOUSE Control Line x.
Missing or incorrect Vector Loop Control Line x.
Missing parenthesis in literal name y.
Missing quote in literal name y.
No decimal point in interest argument.
Ho decimal point in probability argument.
No dimension has been declared for storage variable x.
Non-integer for IPASS upper limit.
Non-integer or non-integer variable for LOOP range.
Number of trials greater than population.
Permissible length exceeded in literal name y.
Probability argument not a positive integer.
Probability argument not greater than zero.
Probability argument should be rounded off.
Probability arguments not in increasing order.
-------
Mouse Manual Page 141
PWSUM interest argument not greater than zero.
PWSUM investment argument less than zero.
Second Dimension Block dimension should equal x
instead of y for Plot storage variable z*
Seed Reset number not an odd integer.
Sensitivity Level number out of range.
Snap Interval number out of range.
Specification number not an integer.
The following two lines are in the wrong order:
This variable has been dimensioned more than once.
User and MOUSE dimensions confused.
Vector Loop Control Lines in wrong order,
x interest parameters found where y were expected,
x parameters found where y were expected,
x probability parameters found where y was expected.
JkT -mm
CHECKER will identify suspect line orders, arguments, dimensions,
numbers, names, and variables and, in most cases, the line in
which the error occurred will also be printed.
11.3 A Case History
Figure 11.1 shows a short MOUSE program that was written by
a colleague (an economist) of the author. The program compiled,
1inked, and ran (See Appendix A, Section A.1), without any diffi-
culty. Output was obtained and is shown in Figure 11.3. A second
table (that for variable SALES TX) was also produced but it was
identical to that shown for the variable TAXREV in Figure 11.3.
The tables were a complete disaster. The frequency distributions
reported clearly were nonsense; other table entries made no sense
either. For example, in Figure 11.2 the "Cumulative % Entries"
column reached 190.10%. The "Number of Entries" column totaled
1901 instead of 1000. Also, a minor point but the name of the
first table was reported as "TAXREV(T" instead of "TAXREV".
Judging by the number of entries that appeared in the table,
it was clear that two tables were being combined. A quick glance
at the program revealed the answer. My colleague had dimensioned
only one table (Tl) and had used the same dummy variables, ATI,
BT1 in both tables. When CHECKER was run on the file, the follow-
ing was reported:
NAME OF FILE CHECKED: TAXI.FOR
1. Missing parenthesis in literal name * (TAXREV*
in line CALL TABLE(TOTXRV,•(TAXREV,ATI,BT1,40,Tl)
2. Duplication of dimensioned variable T1
in line CALL TABLE(SALETX,'(SALES TX)1,ATI,BT1,40,Tl)
and CALL TABLE(TOTXRV,•(TAXREV1,ATI,BT1,40,Tl)
-------
House Manual Page 142
PRECISION T1(46)
COMMON/Q1/C80/Q2/JUMP/Q3/LOOP/Q4/SENS/Q5/IDATA/Q6/IOUT/Q7/IPASS/
&/IRAN/Q9/IREAD/Q10/ISCR1/Q11/ISCR2/Q12/ISNAP/Q13/ITER/Q14/ITOG/Q
&/IWRIT/Q16/IXXX/Q17/OK1/Q1B/OK2/Q19/OK3/Q20/OK4/Q21/OK5/Q22/OK6
CALL NAKE(1(TAX PROBLEM)*)
ITER=1000
5 DO 10 IXXX*=1, ITER
CONSUM=CUNI(95.,105.)
PI=CUNI(120.,130.)
PROFIT=CUNI(12.,14 .)
SALETX=.05*CONSUM
XYTX=.10*PI
PROTX=.10*PROFIT
TOTXRV=SALETX+XYTX+PROTX
CALL TABLE(TOTXRV,»(TAXREV»,ATI,BT1,4 0,T1)
CALL TABLE(SALETX,•(SALES TX)1,ATI,BT1,40,T1)
10 CONTINUE
GOTO(5,15), JUMP .
15 CONTINUE
END
FIGURE 11.15 ORIGINAL CASE HISTORY PROGRAM
PRECISION T1(46),T2(46)
COMMON/Q1/C8 0/Q2/JUMP/Q3/LOOP/Q4/SENS/Q5/IDATA/Q6/IOUT/Q7/IPASS/
&/IRAN/Q9/IREAD/Q10/ISCR1/Q11/ISCR2/Q12/ISNAP/Q13/ITER/Q14/ITOG/Q
&/IWRIT/Q16/IXXX/Q17/OK1/Q18/OK2/Q19/OK3/Q20/OK4/Q21/OK5/Q22/OK6
CALL NAME('(TAX PROBLEM)*)
ITER=1000
5 DO 10 IXXXs1,ITER
CONSUM=CUNI(95.,105.)
PROFIT=CUNI(12.,14.)
PI-CUNI(120.,130.)
SALETX-.05*C0NSUM
XYTX=.10*PI
PROTX=.10*PROFIT
TOTXRV=SALETX+XYTX+PROTX
CALL TABLE(TOTXRV,'(TAXREV)',ATI,BT1,40,Tl)
CALL TABLE(SALETX,'(SALES TX)',AT2,BT2,40,T2)
10 CONTINUE
GOTO(5,15), LEAVE
15 CONTINUE
END •
FIGURE 11.2: CORRECTED CASE HISTORY PROGRAM
-------
Mouse Manual Page 143
3. Duplication of dummy variable ATI
in line CALL TABLE(SALETX,«(SALES TX)',ATI,BT1,40,Tl)
and CALL TABLE(TOTXRV,'(TAXREV~,ATI,BT1,40,T1)
4. Duplication of dummy variable BT1
in line CALL TABLE(SALETX,»(SALES TX)«,ATI,BT1,40,Tl)
and CALL TABLE(TOTXRV,1(TAXREV1,ATI,BT1,40,T1)
In short, CHECKER caught all of the MOUSE syntactical error in
thisi program.
Figure 11.2 shows the corrected version of the program, and
Figure 11.4 shows the results for the first table. The "improve-
ment" is noticeable! This example is especially instructive
since it shows that a program can be accepted and run by a com-
puter without reservation and still produce incorrect answers.
11.4 MOUSE Run-time Error Checks
Because MOUSE is a modular system, i.e., a system consisting
of a collection of functions and subroutines, it is necessary -to
transfer information between a main program (which the user
writes) and these functions and subroutines. In most, if not
all, high-level languages (such as FORTRAN, in which MOUSE is
written) integer data are entered, stored, and written
differently than are decimal data. If a function or subroutine
is structured to receive or transmit decimal data it cannot
substitute integer data or vice versa. Thus it is vital to
adhere to the rules regarding decimal points in MOUSE (see Table
3.1 in Chapter 3). Failure to do so is a common cause of errors
in MOUSE programs. Although a MOUSE program will successfully
compile and run with such errors, incorrect results will be
obtained.
Because of the vastly different manner in which integer and
decimal data are stored in a computer, positive decimal variables
without decimals are generally interpreted as extremely small
positive numbers or else as negative numbers when passed in
functions or subroutines. Integer variables, on the other hand,
when mistakenly used with decimal points, are generally
interpreted as numbers larger than 1000. MOUSE uses this fact to
test decimal and integer numbers passed in certain functions or
subroutines. For decimal numbers, MOUSE checks to see that the
number is not less than a small number, OK6 (the distribution
version of MOUSE sets OK6 equal to 0.0000000001 but this can be
altered easily? see Appendix A, Section B.l). The call for the
Poisson random variable is, for example, POI(mean). If the user
mistakenly omitted the decimal point in the call, e.g., POI(5),
MOUSE would detect this error easily since the mean of a Poisson
distribution must be greater than zero. In this case, "zero1' is
considered by MOUSE to be the value of OK6. For the triangular
-------
Mouse Manual Page 144
DISTRIBUTION FOR QUANTITY TAXREVIT *
NUMBER OF ITERATIONS • 1000
KEAN
MINI WW
MAXIMUM
21.95454
4.73014
19.52732
STANDARD DEVIATION ¦ 11.75831
COEFFICIENT OF VARIATION, X - 53.55799
lOUER
NUMBER OF
PERCENT
CUMULATIVE
CUUl AT IVE
DISTRIBUTIONS
UMIT
ENTRIES
ENTRIES
X ENTRIES
COMPLEMENT
« s FREQUENCY DISTRIBUTION 0 a CUMULATIVE DISTRIBUTION
4.3500
211.
21.10
21.10
7B.90
******•*»•*««g*#*
4.8600
788.
78.80
99.90
.10
5.S700
0.
.00
99.90
.10
¦
5.8800
0.
.00
99.90
.10
6.3900
0.
.00
99.90
.10
*
6.9000
0.
.00
99.90
.10
7.4100
0.
.00
99.90
.10
7.9200
0.
.00
99.90
.10
8.4300
0.
.00
99.90
.10
8.9400
0.
.00
99.90
.10
9.4500
0.
.00
99.90
.10
9.9600
0.
.00
99.90
.10
10.4700
0.
.00
99.90
.10
10.9800
0.
.00
99.90
.10
11.4900
0.
o
o
99.90
.10 .
*
12.0000
0.
.00
99.90
.10
12.5100
0. ,
.00
99.90
.10
13.0200
0.
,00
99.90
.10
*
13.5300
0.
o
o
99.90
.10
14.0400
0.
.00
99.90
.10
*
14.5500
0.
.00
99.90
.10
15.0600
0.
.00
99.90
.10
15.5700
0.
.00
99.90
.10
16.0800
0.
.00
99.90
.10
*
16,5900
0.
.00
99.90
.10
m
17.1000
0.
.00
99.90
.10
17.6100
7.
.70
100.60
-.60
**
18.1200
287.
28.70
129.30
•29.30
18.6300
465.
46.50
175.80
*75.80
19.1400
143,
14.30
190.10
-90.10
************
19.6500
0.
.00
190.10
-90.10
*
20.1600
0.
.00
190.10
-90.10
20.6700
0.
.00
190.10
-90.10
*
OVERFLOW
0.
.00
190.10
-90.10
m
FIGURE 11.3! TABLE OOTSOT OF ORIGINAL CASE HISTORY PROGRAM
-------
Mouse Manual Page 145
• distribution for quantity taxrev *
»»«»****•**«***•*¦*******•
NUUBE' QF ITERATIONS • 1000
MEAN «
MINIHUK «
MAXIMUM »
18.80238
17.99231
19.47256
STANDARD DEVIATION
COEFFICIENT OF VARIATION, X
.32029
1.70347
LOWER
NUMBER OF
PERCENT
. CUMULATIVE
CUHULATI
LIMIT
ENTRIES
ENTRIES
% ENTRIES
cohpleme
UNDERFLOW
1.
.10
.10 ¦
99.90
18.0000
1.
.10
.20
99.80
ia.os2o
6.
.60
.80
99.20
18.1040
5.
.50
1.30
98.70
18.1560
15.
1.50
2.BO
97.20
IB.2080
15.
1.50
4.30
95.70
18.2600
24.
2.40
6.70
93.SO
18.3120
28.
2.80
9.50
90.50
18.3640
39.
5,90
13.40
86.60
18.4160
37.
3.70
17.10
82.90
18.4680
45.
4.50
21.60
78.40
18.5200
51.
5.1Q
26.70
73.30
18.5720
50.
5.00
31.70
68.30
IB.6240
47.
4.70
36.40
63.60
18.6760
51.
5.10
41.50
58.50
10.7280
65.
6.50
48.00
52.00
18.7800
51.
5.10
53.10
46.90
18.8320
55.
5.50
58.60
41.40
18.8840
46.
4.60
63.20
36.80
18.9360
54.
5.40
68.60
31.40
18.9880
56.
5.60
74,20
25.80
19.0400
47.
4.70
78.90
21.10
19.0920
42.
4.20
83.10
16,90
19.1440
43.
4.30
87.40
12.60
19.1960
40.
4.00
91.40
8.60
19.2480
26.
2.60
94.00
6.00
19.3000
23.
2.30
96.30
3.70
19.3520
19.
1.90
98.20
1.80
19.4040
12.
1.20
99.40
.60
19.4560
6.
.60
100.00
.00
0 * CUMULATIVE DISTRIBUTION
DISTRIBUTIONS
* » FREQUENCY DISTRIBUTION
0*
0*
***••**~*#
•*«**•«***«***••*«*•*»«•*
*«*+*•*•***«*••**••**••**•*#**•••**•*•** Q
FIGURE 11.4S TABLE OUTPUT OF CORRECTED CASE HISTORY PROGRAM
-------
Mouse Manual Page 146
distribution the call is TRI(a,b,c) where a and b are the lower
and upper bounds, respectively, of the distribution. If the user
entered TRI(3,5,9) instead of the correct TRI(3.,5.,9.), MOUSE
would catch this since it would test to see that the quantity
(c - a) was greater than 0K6. The integer argument check to see
if the integer passed exceeds 1000 is performed only when it is
illogical that the number be this great. It is acceptable, for
example, that the number of iterations exceeds 1000 but it is not
acceptable that a table be dimensioned this large.
MOUSE provides 20 run-time error checks. If any of these
errors are found by MOUSE, an error message is reported and the
run is terminated. A list of these error messages can be found
in Appendix C. MOUSE makes almost all of the integer versus
decimal point checks necessary and, in addition, also tests for
other common errors such as placing certain function parameters
in the wrong order (e.g., in the continuous uniform distribution,
the first parameter is the lower bound and the second is the
upper bound and not vice versa), exceeding the permissible range
of the correlation coefficient in the bivariate normal or bivari-
ate lognormal distribution, decreasing the variance by more than
100% in the Sensitivity Control Line, etc. Other MOUSE require-
ments are also checked, such as the need to dimension tables 100
or more when MOUSE-dimensioned tables are used or having the
number of iterations exceed the snap interval number.
The above error checks are static checks, i.e., they are
made at the very beginning of the run (i.e., at IXXX=1) and, if
any errors are found, the run is terminated. If the MOUSE call
at IXXX-l is correct, however, static errors will not be
discovered. The interest rate in interest functions, however, is
frequently supplied as a random variable, e.g.,
RATE=GAU{0.10,0.4)
Xl=COST*SCA(RATE,XLIFE,0.)
It is very possible, under these conditions, for RATE to be
positive for values of IXXX=1 and negative at some other values.
Interest functions, therefore, have a special error trap to catch
such dynamic errors. Each call to the function is monitored,
regardless of the value of IXXX, and, if negative interest rates
or negative time periods are found, an error message will be
reported, along with the value of IXXX at which the error was
found, and the run will be terminated.
Although CHECKER will find and report the integer/decimal
errors described above (and it will find them before compilation,
thus saving much labor), if the arguments are in the form of
variables rather than numbers, e.g., POI(VAR), CHECKER skips the
-------
Mouse Manual Page 147
integer/decimal check (other than checking to see that an integer
or decimal variable name has been used) because it has no way of
knowing what value these variables will take during a MOUSE run.
The MOUSE run-tine error checks, therefore, act as a kind of
11.5 The HOUSE Trace
As has been mentioned, when errors are incurred that are a
result of an invalid instruction to the computer (i.e., errors in
syntax), messages are reported to the user by the system's FOR-
TRAN compiler that specify where in the main program such errors
occurred. This, of course, provides a help to the user in track-
ing down the source of the error. CHECKER and the MOUSE run-time
error messages also are of help in this regard. However, there
are errors that can go undetected by the FORTRAN system, by
CHECKER or by MOUSE, We might, for example, enter the value of a
constant as 1000 instead of 10; to the FORTRAN compiler or to
MOUSE this is not an error and it will obediently use the incor-
rect value in its calculations without sensing that anything is
wrong. These sorts of errors, which can involve logic as well as
data or parameters (e.g., a minus sign instead of a plus sign in
an uncertainty equation or perhaps an incorrect uncertainty equa-
tion) , can be quite difficult to find in complex problems. In
fact, the user might not be aware that such errors exist and
unfortunately accept the MOUSE output as delivered.
with regard to empirical variables or matrix data which are
read into MOUSE, the echo listing automatically produced in every
MOUSE run provides a convenient error check. One common method
used to uncover other sorts of errors is to do a single sample
calculation by hand to see whether the numbers produced are
reasonable. This kind of "debugging" often can become tedi-
ous. Help is available, however, in the form of Trace Lines. A
Trace Line has either of the following two forms:
CALL DTRAC1(dvariable,'(lname)')
or CALL ITRACE(ivariable,'(lname)')
where "'lname*'1 is as defined for Table Output Lines. The only
difference between the DTRACE and ITRACE Trace lines is that the
former are for decimal variables, i.e., "dvariable", and the
latter, i.e., "ivariable", are used for integer variables (see
Chapter 3, Section 3.2).
To illustrate their use, Figure 11.5 shows the placement of
Trace Lines into the Modification #2 of the Haste Oil Reclamation
problem. (For clarity, the Trace lines are shown indented.) Since
LIFE is an integer variable, an integer Trace line was used for
this variable. The results are shown in Figure 11.6. A trace is
-------
Mouse Manual Page 148
produced at the following number of iterations: 1, 20, 50, and
100. At iteration = 1, all variables are evaluated at the average
value of the input variables. Thus, the values of the output
variables at iteration = 1 are their non-stochastic values (i.e.,
the values under certainty using the average values of the input
variables). At all other iterations, the values produced are
those as a result of a true random sample at those iterations.
Considering the input data and the model itself, these are all
reasonable values for the variables traced.
As can be seen, a vector can also be used in a Trace line,
6 « CJ • ^
CALL DTRACE(ANBEN(LOOP),'(ANBEN)')
Notice that the index LOOP must be used when vectors are the
object of Trace lines.
The following rules apply to the placement of trace lines in
a MOUSE program;
1. A Trace line for a variable can appear only after the
first appearance of that variable in a program;
2. Trace lines for variables (including vectors) that
appear within vector loops must be placed within the
loop itself. The values printed out for these
variables, however, will always be for LOOP=l,
3. For the optimal operation of the trace, it is advisable
to have at least 100 iterations, otherwise output at
iterations 100, 50, and 20 may not be produced.
Trace lines are very useful for checking the performance of
various MOUSE functions which require decimal numbers as
arguments. If, by mistake, the decimal point is omitted, the
function frequently will not indicate that an error has occurred
but simply will return a value of zero. The use of a Trace line
will indicate whether or not a reasonable number is being
returned by the MOUSE function.
Any number of Trace lines can be used in a program but the
presence of a single Trace line will override the user's
Iteration line specification (which, as previously mentioned,
should be 100 or greater in order to obtain output at iterations
20, 50, and 100). Since no tables or plots are created during a
trace, the running time for a program containing one or more
Trace lines is extremely short. Removal of the Trace lines
restores the program to its full user-specified status.
Placing Trace lines into a MOUSE source program can be
tedious. However, a MOUSE utility called TRACER can be used to
automatically place Trace lines into a program or even remove
-------
Mouse Manual Page 149
C WASTE OIL RECLAMATION PROBLEM MODIFICATION #2
C
01 KENS ION 01(3,2>,D2(4,2)>03(4,2>
DIMENSION ANBENdS)
DIMENSION VTAB(7,15)
DIMENSION TAB(46)
COWON/01/CBO/02/JUMP/Q3/LOOP/Q4/SENS/05/1 DATA/06/1 OUT/07/1 PASS/08
&/IRAN/09/IREAD/010/ISCRl/011/1SCR2/Q12/ISMAP/013/ITER/flt4/ITOG/015
4/1WR IT/016/1XXX/017/OK! /Q18/0K2/Q WOK3/Q20/0K4/821 /0KS/Q22/0K6
CALL NAME('(WASTE OIL RECLAMATION PROBLEM - MODIFICATION #2)')
CALL READCD1,3)
CALL READ(D2,4)
CALL READ(03,4)
[TER=50Q0
5 DO 10 IX10(*1 ITER
CONTAM-OUNH1.0,4.0)
CALL OTRACE{CONTAM,'(CONTAM)')
RAND1=CUNI<-10,0,10.0)
CALL DTRACECRAND1,11RAND1)1)
PRICE1=EDIS(D1,3)
CALL DTRACECPRICE1,'(PRICED')
SLOPE=ECTS(D2,4)
CALL DTRACE(SLOPE,'(SLOPE)')
PR0D2*STEP(D3,4)
CALL OTRACE(PR002,1(PR002)')
RAND2-GAIH0.00,0.103
CALL DTRACE(RAND2,'(RAND2)')
PRICE3=EBETA(6.5,8.0,9.5)
CALL DTRACE(PRItE3,'(PR1CE3)')
PR003«EXPON(1S.0)
CALL DTRACECPR003,'(PR003)')
XLIFE*OTRI(5.0,10.0.1S.O)
t"V I rvi \ • V>| IW4W. i,jr •
CALL OTRACE(XLIFE,
XININV=TRA<1600.0,2000.1
CALL DTRACE(XININV
THRU1=10.0-CONTAM
CALL 0TRACE(THRU1,
Y1ELD1=10.0-2.0*CONTAM
CALL DTRACE
GREV3=PRICE3*PR003
CALL DTRACE(GREV3,'(CREV3)1)
LIFE-XL IFE
CALL ITRACE(LIFI,'(LIFE)')
DO 100 LOOP=1,LIFE
YEAR=LOOP
CALL DTRACE(YEAR,'(YEAR)')
AMBEN(LOOP)=SCA(.12,YEAR,0.)* < ANBEN)",LIFE,VTAB)
CONTINUE
100P=1
PVALUE=PWSUM(ANBEN,0.M,LIFE,XININV>
CALL DTRACE(PVALUE,'(PVALUE)')
CALL TABLE(PVALUE,'(PRES VAL)',DUN0,DUN1,40,TAB)
CONTINUE
COTO(5,15), JUMP
CONTINUE
CALL LIST(1(U02.FOR)')
END
FIGtTlE 11.5; PLACEMENT 01 TRACE LINES IK MODIFICATION #2
OF THE WASTE OIL RECLAMATION PROBLEM
-------
Mouse Manual Page 150
TRACE AT im « 1 (Average values): , TRACE AT IXXX ¦ 20:
CONTAM
s
2.50000
CONTAH
RAND1
M
.00000
RANDl
H
PRICE1
st
8.45000
PRICE1
«
SLOPE
s
.12500
SLOPE
B
PR002
a
52.50000
PRQ02
=
RAND2
v
.00000
RAN02
a
PRICE3
s
8.00000
PRICE3
a
PR0O3
a
18.00000
PR003
m
XIIFE
m
10.00000
XLIFE
m
XIN1NV
8
2100.00000
XININV
' ¦
THIHI1
S
7.50000
THRU1
3
YIELD!
=
5.00000
TIELD1
, =
PI! 001
a
37.50000
PR001
a
GREV1
a
316.87500
GREV1
m
PRICE2
9
3.43750
PRICES
a
GREV2
=
130.46880
GREV2
=
GREV3
=
144.00000
GREV3
=
LIFE
=
10 .
LIFE
=
TEAR
= '
1.00000
" TEAR
=
ANBEN
a
639.90500
ANBEN
a
PVALUE
a
4916.69700
PVALUE
3
RATRET
=
.38900
RATRET
=
TRACE AT IXXX = 50; TRACE AT 1XXX = 100;
CONTAX
a
2.00000
CON TAN
RANDl
=
1.26925
RANDl
a
PRICE1
=
7.50000
PRICE1
S
SLOPE
= '
.07756
SLOPE
3
PR002
s
64.68204
PR002
¦
RAND 2
=
-.23779
RAND2
s
PRICE3
=
8.23146
PR1CE3
s
FR0C3
¦
4.66683
PRC03
8
XLIFE
=
10.00000
XLIFE
. s
XININV
=
2020.78800
XININV
-
THRU1
=
8.00000
TH8U1
=
YIELD1
¦
6.00000
YIELD!
a
PRODI
s
49.26925
PR001
s
SREV1
s
369.51940
0REV1
3
PR1CE2
=
4.74562
PRICE2
s
GREV2
. s
306.95620
GREV2
¦
CREV3
a
38.41480
GREY3
8
LIFE
a
10
LIFE
C
YEAR
a
1.00000
TEAR
=
ANBEN
• =
722.27720
ANBEN
s
PVALUE
a
5899.13700
PVALUE
=
RATRET
a
.45000
RATRET
a
1.00000
-3.46499
8.00000
.17997
64.25715
.05501
7.34273
21.00083
10.00000
2022.62300
9.00000
8.00000
68.53501
548.28010
-1.50923
-96.97889
154.20340
10
1.00000
599.76510
4553.93100
.37900
1.00000
-2.13976
8.00000
.07618
50.17829
-.02676
7.48467
29.73042
9.00000
2183.47500
9.00000
8.00000
69.86024
558.88190
£.15046
308.61930
222.52240
9
1.00000
1142.42700
8676.06500
.62400
FIGURE 11.6: OUTPUT OF TEE TRACED PROGRAM OF FIGURE 11.5
-------
Mouse Manual Page 151
them. (The Trace lines in Figure 11.5 were actually placed there
by TRACER.) Simply type TRACER and you will be queried for the
name of the source file and the output file. If the output file
exists, you will be asked whether you wish to write over the file
or not. The next query asks whether you wish to add Trace lines
or to remove them. TRACER will satisfactorily place Trace lines
in a source file in most cases, but there are two instances where
you will have to make small corrections manually:
(1) If you have an equation that uses more than one line
(i.e., where a continuation character is used in column
six see Chapter 3, Section 3.2), TRACER will place the
Trace line immediately after the first line. The line
should be moved manually (usi.ng your text editor) to the
last line of the continuation series.
(2) If there is an evaluation of a variable after an IF
statement, e.g.,
IF(RAN(IRAN) .LT. 0.5) PRICE-CUNI(100.,2000.)
TRACER will not produce a Trace line for this variable,
and one will have to be inserted manually.
In either case, however, TRACER will tell you if any Continuation
lines or IF statements requiring Trace lines were found, and how
many of them there were.
11.6 TRAP, A Utility For Preparing MOOSE Source Programs
The overhead information (i.e., control, Dimension, Read,
Specification, and Output Lines) required by MOUSE is sometimes
complex, and users usually must resort to various chapters in
this book, the Summary Tables in Appendix B or the one-line
MSHELP program (described in the next Section) in order to re-
fresh their memories as to the form in which it must be supplied.
For example, Waste Oil Reclamation Problem, Modification #2,
requires the following overhead lines even before the stochastic
sampling takes place and the model is defined:
DIMENSION D1(3,2),D2(4,2),D3(4,2)
DIMENSION ANBEN(15)
DIMENSION VTAB(7,15)
DIMENSION PTAB(46),RTAB(26)
COMMON/Q1/C80/Q2/JUMP/Q3/LOOP/Q4/SENS/Q5/IDATA/Q6/IOUT/Q7/IPASS/Q8
&/IRAN/Q9/IREAD/Q10/ISCR1/Q11/ISCR2/Q12/ISNAP/Q13/ITER/Q14/ITOG/Q15
&/IWRIT/Q16/IXXX/Q17/OK1/Q18/OK2/Q19/OK3/Q20/OK4/Q21/OK5/Q22/OK6
CALL NAME('(WASTE OIL RECLAMATION PROBLEM - MODIFICATION #2)*)
CALL READ(D1,3)
CALL READ(D2,4)
CALL READ(D3,4)
ITER=5000
5 DO 10 IXXX«=1, ITER
-------
Mouse Manual Page 152
In addition, the user must ultimately also prepare the following
lines.
CONTAM=DUNI(1.0,4 .0)
RAND1=CUNI(-10.0,10.0)
PRICE1=£DIS(D1,3)
SLOPE-ECTS(D2,4)
PROD2=»STEP(D3,4)
RAND2=GAU(0. 00,0.10}
PRICE3=EBETA(6.5,8.0,9 . 5)
PR0D3=EXP0N(18.0)
XLIFE=DTRI(5.0,10.0,15.0)
XININV=TRA(1600.0,2000.0,2200.0,2400.0)
CALL VTABLE(ANBEN,«(ANBEN)»,LIFE,VTAB)
CALL TABLE(RATRET,'(INT ROR)1,DUM2,DUM3,20,RTAB)
CALL TABLE(PVALUE,'(PRES VAL)1,DUM0,DUM1,40,PTAB)
CALL LIST('(W03.FOR)')
DO 100 LOOP®81 f LIFE
100 CONTINUE
LOOP=l
10 CONTINUE
GOTO(5,15), JUMP
15 CONTINUE
END
Typing all of this information into a MOUSE source file clearly
can be tedious. TRAP is a miniature preprocessor (TRAnsformation
Preprocessor) that simplifies this task considerably. It is a
stand-alone interactive program that, like IMP, is entirely
separate and distinct from MOUSE. The following is an example,
using Waste Oil Reclamation Problem, Modification #2, of a TRAP
session.
A*****************************************
* TRAP *
* (TRAnsformation Preprocessor) *
* VERSION 1.00+ *
* US ENVIRONMENTAL PROTECTION AGENCY *
* Written by ALBERT J. KLEE *
******************************************
********************************
* PREPARE MOUSE DATA FILE *
* PREPARE MOUSE SOURCE PROGRAM * <—
********************************
ENTER OUTPUT FILE NAME (not to exceed 12 characters, including
extension) or press Return key for default of PROGRAM.FOR:
» W02.FOR
If the file W02.for already exists, you will be asked whether you
wish to overwrite this file or not. The other option, "PREPARE
-------
Mouse Manual Page 153
HOUSE DATA FILE", will be discussed in section 11.7. TRAP then
asks whether you wish to have comment lines placed into the
output file:
******************************
* FILE W02.FOR YES * <—
* EXISTS... OVERWRITE? NO *
******************************
*******************************
* DO YOU WISH TO HAVE I Y!S * <—
* THE OUTPUT COMMENTED? j NO *
*******************************
TRAP then starts its queries with the Specification Block. There
are three options« The first peznsits the user to modify any of
the MOUSE default values for this block, the second accepts
the default values (e.g., number of iterations - 1000, sensitivi-
ty level = 3, etc.), and the third produces a basic MOUSE shell
consisting of the eight standard MOUSE Control Lines, a Name
Line, and a Dummy Table Output Line. The menu is as follows:
*****************************************
* 1. MODIFY DEFAULT SPECIFICATION BLOCK * <—
* 2. ACCEPT DEFAULT SPECIFICATION BLOCK *
* 3. PRODUCE BASIC MOUSE SHELL *
*****************************************
The following example uses the "MODIFY" option.
*************************************************
* SPECIFICATION AND MISCELLANEOUS SECTION *
* — *
* Code is as follows: Y*YES or press Return key *
* to skip the query or to accept noted default. *
*************************************************
ENTER NUMBER OF ITERATIONS (or Return to default to 1000):
» 5000
80 COLUMN OUTPUT? (Y or Return to skip):
»
PMfFP QT7MQTrr,TX?TT,V T TTtnTT 11 Pofnrn <=V"i n\ •
Lli JL JmiJEv dEiIiDl JtlVJLi X iujVJ^iji % «•»/ <&/ J Oi KBUUrn vu aJvip / •
>>
ENTER SNAP INTERVAL (or Return to skip):
»
SEND MOUSE OUTPUT TO THE SCREEN? (or Return to skip):
»
-------
Mouse Manual Page 154
***************************************************
* SPECIFICATION & MISCELLANEOUS SECTION CONTINUED *
* _______________________________________________ *
* Y—YES or press Return key to skip the query. *
***************************************************
ENTER NEW RANDOM NUMBER SEED (or Return to skip):
»
ENTER NUMBER OF PASSES (or press Return to skip):
»
ENTER LISTING FILE NAME, not to exceed 12 characters,
including extension (or Return to skip):
» W02.F0R
Note that the program shows you how to enter the required data.
If you make an error, TRAP will notify you of the fact (by beep-
ing) and ask you to reenter the information. In the Waste Oil
Reclamation Problem, Modification #2, only the listing file and
the number of iterations was specified, so only this information
needed to be entered.
With regard to stochastic variables, tables, plots, vectors
and vector tables, matrices, and data accumulations, the follow-
ing menu provides these options.
************************************
* 1. PROCESS STOCHASTIC VARIABLES * <—
* 2. PROCESS TABLES *
* 3. PROCESS PLOTS *
* 4. PROCESS VECTORS/VECTOR TABLES *
* 5. PROCESS MATRICES *
* 6. PROCESS DATA ACCUMULATIONS *
* 7. FINISHED *
******************************4*****
Only the queries for one stochastic variable (PRICE1), one table
(PVALUE), and the vector (ANBEN) with its associated vector table
are shown in the following example.
TRAP presents a menu of 23 probability distributions for
stochastic variable sampling purposes. The following example
shows the procedure for the stochastic variable PRICE1, which is
generated from an empirical discrete probability distribution
evaluated at three points.
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*******************************************************
STOCHASTIC VARIABLE SAMPLING MENU
1. Beta, Standard Form 9. Empirical Beta 17.
*******************
2. Binomial
3. Bivariate Lognormal
4. Bivariate Normal
5. Continuous Uniform
6. Discrete Trapezoidal
7. Discrete Triangular
8. Discrete Uniform
10. Empirical Cts
11. Empirical Discrete 18.
12. Erlang 19.
1*. Gamma 20,
14. Exponential 21.
15. Hypergeometric 22.
16. Lognormal 23.
Lognormal,
Alternative Form
Normal
Pascal/Geometric
Poisson
Step Rectr
Trapezoidal
Triangular
NUMBER OF PROBABILITY FUNCTION DESIRED? (or Return to exit): » 11
*************************************************************************
Empirical Discrete Distribution - Example: NAME=EDIS(VAR1,5)
Enter VARIABLE NAME, and DISTRIBUTION DIMENSION
separated by commas or spaces, e.g., NAME,5
(or Return to select new probability distribution):
» PRICE1,3
The next query involves Sensitivity Control Lines:
*****************************************************
* 1. SKIP SENSITIVITY CONTROL LINE *
* 2. SELECT MIAN SENSITIVITY ANALYSIS *
* 3. SELECT STANDARD DEVIATION SENSITIVITY ANALYSIS *
* 4. SELECT BOTH * < —
*****************************************************
Although the Waste Oil Reclamation Problem, Modification #2, did
not involve Sensitivity Control Lines, for illustrative purposes
this example will ask for both mean and standard deviation Sensi-
tivity Analysis Lines (50% increase in the mean and 20% increase
in the standard deviation):
ENTER PERCENTAGE CHANGE IN MEAN: » 50
ENTER PERCENTAGE CHANGE IN STANDARD DEVIATION: » 20
Although MOUSE requires decimal points in all probability func-
tions (except for the three empirical types, continuous discrete,
and step), TRAP will place trailing decimal points into the
stochastic variable sampling lines if you choose to omit them in
the TRAP input.
********************************************************************
* TABLE SECTION
* ______—_____________ _________—___________ —_
* Sample Entry: COST,40
* where COST is variable to be tabled with 40 histogram intervals.
* _
* Note: Press Return key to exit to next section.
ft#*****#*#*****#****#*#****#**#****#************#**#***#*******#****
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ENTER VARIABLE NAME AND NUMBER OF INTERVALS, SEPARATED BY A COMMA:
» PVALUE,40
***************************************************
VICTOR & VECTOR TABLE SECTION
Sample Entry: NETBEN,50
where NETBEN is vector dimensioned at 50.
Note: Press Return key to exit to next section.
****************************************************
ENTER VECTOR NAME AND DIMENSION, SEPARATED BY A COMMA:
» ANBEN,15
# * * it # * it ir ifc it it it it it * it * it it
* TABLE THIS
* VECTOR?
YES,*
NO *
********************
The output, which is fully commented, from this TRAP example is
shown in Figure 11.7. Provided with the TRAP output, all that,
remains in order to obtain a completed MOUSE source file is for
you to enter the stochastic sampling lines and the model itself.
Note that TRAP provides standard names for empirical stochastic
variable, table, vector table, matrix, and plot dimensions
(these are, respectively, VARi, TABi, VTABi, MTABi, PTABi, where
the i's are successive integers). Also, for simplicity, TRAP
makes the literal names equal to the variable names in Sensitivi-
ty Control lines, and in Table, Plot, and Vector Table Output
lines.
11.7 using TRAP To Prepare mouse Data Files
Option l on the first TRAP menu is for preparing MOUSE
empirical variable data files:
+ it it it it it it it it it it it it it it it it it it it it it it it it it it it it it it it
* PREPARE MOUSE DATA FILE * <—
* PREPARE MOUSE SOURCE PROGRAM *
********************************
By default, TRAP writes data information to the file MOUSE.DAT,
but if the latter already exists you will asked for permission to
overwrite it or to select a new target data file name.
********************************************************
* 1. DISCRETE EMPIRICAL VARIABLE * <—
* 2. CONTINUOUS EMPIRICAL OR STEP RECTANGULAR VARIABLE *
* 3. QUIT PROGRAM *
********************************************************
-------
Empirical Variable Dimension Lines
DIMENSION VAR1(3),VAR2(4),VAR3(4)
Table Dimension Lines
DIMENSION TAB1(46)
Vector Table Dioension Lines
DIMENSION VTAB1(7,1S)
Vector Dimension Lines
DIMENSION ANBEN<15)
Mouse Control Lines 1,2, and 3
C0MM0N/Q1/C80/Q2/JUMP/Q3/L00P/Q4/SENS/05/IDATA/06/I0UT/Q7/IPASS/08
1/IRAN/09/IREAD/O10/ISCR1/O11/ISCR2/Q12/ISNAP/013/ITER/014/1T0G/Q15
1/1URIT/O16/1XXX/Q17/0K1/Q18/OK2/O19/OK3/O20/OK4/Q21/OK5/Q22/OK6
Nane Line
CALL NAMECUASTE OIL RECLAMATION PROBLEM - MODIFICATION #2)')
Empirical Variable Read Lines
CALL READ(VAR1,3)
CALL READ(VAR2,4)
CALL READ(VAR3,4)
Iteration N inter Line
ITER'5000
Mouse Control Line 4
DO 10 IXXX*1,ITER
C Stochastic Variable Sampling Lines
C (and Sensitivity Analysis Lines, if specified)
C
CONTAM=OUNI(1.,4.>
RAND1°CUNI(-10.,10.)
PRICE1=EDIS(VAR1,3)
CALL SEN(PR1CE1,¦(PRICE1)',50..0.)
CALL SEN(PRICE1,• (PRICED1,0.,20.)
SLOPE*ECTS(VAR2,4)
PROD2=STEP(VAR3,4)
RAND2=GAU(0.,. 1)
PRICE3=EBETA(6.5,8.,9.5)
PR0D3*EXPON(1B.)
XLIFEzOTRI(5.,10.,IS.)
XININV°TRA(1600.,2000.,2200.,2400.)
C
C Vector Loop Control Lines Uith Vector Table Output Lines
C
DO 100 L00Pa1,LlFE
CALL VTABLE(ANBEN,'(ANBEN),,LIFE,VTAB1)
100 CONTINUE
LOOP-1
C
C Table Output Lines
C
CALL TABLE(PVALUE,'(PVALUE)1,DIM1,DUM2,40,TAB1)
C
C House Control Lines 5, 6, 7, and 8, Uith Listing Line
C
10 CONTINUE
G0T0(5,15), JUMP
IS CONTINUE
CALL LIST('(W02.F0R)1)
END
FIGURE 11.7: OUTPUT FROM SAMPLE TRAP RUM FOR UASTE OIL RECLAMATION PROBLEM, NQDIFICATIOH «2
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Until you elect the option to quit, you will be returned to this
menu after every variable information set has been entered.
The processing of empirical variable data information
proceeds in three steps:
1. Entering a Name Line;
2. Entering the values of the variable;
3. Entering the probability of each value (for discrete
empirical variables) or the probability between each
value (for continuous empirical and step rectangular
variables).
If the total probability sums to less than 0.9999 or greater than
1.0001 (indicating that you nay have made an error or else your
probability rounding is not equal to this level of precision),
you will be asked whether TRAP should adjust the total probabili-
ty to one or abort processing the information about the variable.
(If the total probability sums to less than 0.9999 or greater
than 1.0001, you will not be queried, but the adjustment is
automatically made if the sum is not.one.) A summary graph of the
adjusted probabilities is also displayed, e.g.,
ft*******************#****#****#******#*********#**##**************#*!********
* PRICE1 (PRICE OF MOTOR OIL GRADE 1} •
• VALUE PROBABILITY
• 7.5000 .3500
• 8.0000 .4000
• 10.5000 .2500 MmmmmmtmMtimmm *
CUMULATE AND WRITE THIS DISTRIBUTION TO DATA FILE (T/N>?
Press Enter for yes, M for No: »
11.8 MSHELP - Another MOUSE Help Utility
MOUSE overhead information must be supplied exactly as
specified in this Manual or else errors will result when an
attempt is made to run a program. In addition, MOUSE has many
probability and interest calls available, and users usually must
either resort to various chapters in this book or, as a better
alternative, the Summary Tables in Appendix B in order to re-
fresh their memories as to the form in which all of this informa-
tion must be supplied. Neither of these alternatives may be
convenient, however. A utility that provides another alternative
is MSHELP. This is on-line, memory resident utility (about 30K of
memory is required) that you can call up at any time while edit-
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ing a MOUSE source program via a "hot key'
i.e.
Alt-H. This
provides access to six different screens of information as fol-
lows:
Mouse Control, Pass Control and Vector Loop Control Lines
Mouse Interest Functions
(3) Mouse Probability Functions
(4) Mouse Probability Calls, (Truncated and Composite)
(5) Mouse Line Types (Dimension, Read, &
Specification Blocks)
(6) Mouse Line Types (Program Block, Listing, Name,, & Review
Lines)
(1)
(2)
This is the same information contained in the summary tables of
Appendix B. The information appears in a window that temporarily
overwrites your file (if you are editing a file), and disappears
when the escape key is pressed. You can page up or down among the
screens, or transfer immediately to any selected screen.
MSHELP involves two files: MSHELP.COM and an auxiliary file,
MOOSE.HLP. Both files should be placed into the same subdirectory
on your hard dxsk,¦ and your ATJTOEXEC. B^^T f x 1 © must conta x n an
appropriate PATH statement if you do not plan to edit your MOUSE
source programs from this subdirectory. MSHELP is installed
merely by typing "MSHELP". This is a memory resident program, but
it can be removed simply by typing "MSHELP /U". The /U parameter
uninstalls the program from memory. Note that MSHELP should be
the last memory resident program installed in your computer.
Finally, if the default hotkey of Alt-H conflicts with any
other hotkey, it may be changed by calling MSHELP as follows:
MSHELP /H#
where If xs any alphanumeric key, the minus sign key or the equals
sign key, e.g., MSHELP /HT. In this example, the hot key becomes
Alt-T.
EXERCISES
Which of the following three sets of Trace lines, if any, are
valid?
1. CALL DTRACE(COST,'(COST)')
CALL ITRACE(NUMBER,1(NUMBER)')
3. CALL DTRACE(PRICE,'(PRICE)')
CALL DTRACE(QUANT,«(QUANT)1)
CALL DTRACE(X6,•(STOCK)•)
2. DO 100 LOOP=l,LIFE
YEAR=LIFE
X1=GAU(100.,50.)
X2(LOOP)=X1*SCA(.08,YEAR,0.)
100 ^ CONTINUE
LOOP=l
CALL DTRACE(X2(LOOP),•(X2)')
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CHAPTER 12
nroniG probability distributions to sample data
12.1 Introduction
An important part of uncertainty analysis is the determina-
tion of the appropriate probability distributions for the sto-
chastic input variables. There are two basic approaches: (a) fit
a probability distribution to sample data, and (b) elicit a
subjective probability distribution. These approaches are not
mutually exclusive; if the available data are few in number or
of poor quality, then a combination of both methods may be used.
This chapter is concierned with the first of these methods, i.e.,
fitting a probability distribution to sample data.
If the functional form of a probability distribution is
known, the distribution is completely defined by its parameters.
For example, given its mean and standard deviation, we need know
nothing more about a normal distribution in order for MOUSE to
generate random variables from the distribution. If, for exam-
ple, the mean is known to be 5 and the standard deviation is
known to be 2, then a call to GAU(5.,2.) in the MOUSE program is
all that is required. If we postulate that the distribution is
Poisson, then a call to POI(5.) is indicated. For some distribu-
tions, such as the beta, binomial, uniform, gamma, erlang, and
pascal/geometric, the parameters of the distribution may be
calculated directly from statistics such as the mean and standard
deviation of the distribution. For example, the mean and standard
deviation of a binomial distribution are given as:
Solving Equations 12.1 and 12.2 for the binomial parameters p and
n we obtain:
Substituting the values from the previous example of mean = 5 and
std = 2 in Equations 12.3 and 12.4 we obtain p = 0.2 and n = 25.
A call to BIN(0.2,25.) is all that is required then for MOUSE to
generate random values from the distribution.
mean = np
std = [np(l - p)]0*5
(12.1]
[12.2]
p » (mean - std2)/mean
n = mean2/(mean - std2)
[12.3]
[12.4]
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Some cases do pose more substantial difficulties. Consider
the two-parameter Weibull distribution, for example, a distribu-
tion of considerable importance in life testing and reliability
studies:
f(x) = (c/b)(x/b)c-1exp[-(x/b)c] [12.5]
where b is a scale parameter, and c is a shape parameter. The
mean and standard deviation of the Weibull distribution are:
mean « bG£(l/c) +1] [12.6]
std = {b2G[(2/c) + 1] - mean2}0-5 [12.7]
where G is the gamma function. As can be seen, the gamma function
is involved in the determination of the parameters of the Weibull
distribution from its mean and standard deviation. To evaluate
the gamma function in equations 12.6 and 12.7, however, we first
must know the value of c (which we are trying to find). Accord-
ingly, there is no direct algebraic solution for xts parameters.
Other methods (see Ruzicka, 1962) , primarily of a graphical or
trial-and-error nature,, are generally used for this purpose.
Some distributions, such as the hypergeometric, have more than
two parameters so that knowledge only of the mean and standard
deviation is not sufficient to estimate the parameters. For most
distributions, however, the parameters may be estimated by simple
statistical quantities readily calculated from sample data;
once the parameters of a distribution are determined, they may be
used with the standard MOUSE classical probability functions to
generate the required random variates for the uncertainty analy-
sis.
The problem now arises as to which form of probability
distribution should be selected. Knowing certain sample statis-
tics or facts about the sample data may help us to eliminate
certain possibilities. Negative data, for example, precludes the
exponential distribution (which only is defined for zero or
positive values of the variable)j for a lognormal distribution
the median must be greater than the mode, etc. Graphing the
sample data may provide insights as to the reasonableness of a
postulated distribution but the truth of the matter is that,
unless we have a great deal of sample data (200 or more
points), it is difficult to determine the shape of a distribution
with very much confidence simply by viewing a histogram of the
sample data. This is where statistical procedures known as
••goodness-of-fitM tests enter into the picture.
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12.2 ¦ Goodness-Of-Fit Tests
A goodness-of-fit test sets up a null hypothesis about the
identity of a probability distribution that has been fitted to a
set of sample data. Since the alternative hypothesis is broad,
rejection of the null hypothesis does not tell us what the dis-
tribution is, only what, it is not. Furthermore, goodness-of-fit
tests based upon estimates of parameters from sample data are
only approximate. Like all statistical tests, they require that
the user select a level of error called the "level of signifi-
cance", i.e., the probability that the null hypotheses will be
rejected when it is true. Prom a decision-theoretic point of
view, this level of error should be selected according to the
consequences of falsely rejecting the null hypothesis. Unfortu-
nately, these consequences are hardly ever known and are diffi-
cult to ascertain. The classical statistical approach to good-
ness-of-fit tests, in which strict adherence ' is made to signifi-
cance levels selected beforehand, therefore, makes little sense
for the purposes of uncertainty analysis. They can be used,
however, in conjunction with other information (objective or
subjective) we may have about the distribution in question for
we presumably have examined the data, not only by calculating
statistics such as mean and standard deviation, but also by
looking at histograms of the plotted data. In addition, we may
, also have subjective a priori opinions about the form of the
distribution (see Section 12.3).
Another reason why caution should be used with goodness-of-
fit tests is that some truncated distributions very closely
approximate data sets that come from very different distribu-
tions. For example, data sets from exponential distributions can
be fitted quite well to a normal distribution since the portion
of a normal distribution one-half standard deviation to the left
of its mean looks very much like an exponential distribution. By
looking at a plot of the fitted versus the original data, howev-
er, it would be clear that the data were not symmetric and that
therefore the full normal distribution would not be appropriate.
(The truncated normal distribution - see Chapter 4, Section 4.7
- might be appropriate but the truncation point would have
to be determined, perhaps simply by equating it to the minimum of
the data set.) A goodness-of-fit test in an uncertainty analysis
context, therefore; --can be used to lend support, mitigate or
alter the suspicions we already may have about the data set in
question, but the final decision is up to the decision-maker
himself.
The two best-known goodness-of-fit tests are the chi-square
and Kolmogorov-Smirnov (K-S) tests. The former is specifically
designed for use with discrete distributions, while the latter is
intended for use with continuous distributions. (See Gibbons,
1976.) The chi-square test is based upon vertical deviations
between the observed and hypothesized histograms of the data
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Mouse Manual Page 163
whereas the K-S test is based upon vertical deviations between
the observed and hypothesized cumulative probabilities. There is
another test, referred to here as the "Foutz test", which, al-
though similar to the K-S test, is not as well-known. (See
Foutz, 1980.) The Foutz test is also based upon vertical devia-
tions between the observed and hypothesized cumulative probabili-
ties but the Foutz test compares a much less restricted set of
probabilities than does the K-S test. In simulations by the
author, the Foutz test performed better than the K-S test for the
classical distributions tested (i.e., continuous uniform, normal,
lognormal, exponential, beta, gamma, and erlang). One disadvan-
tage is that it is much more complex to calculate but in IMP, the
computerized system for determining probability distributions
from sample data, this is not a concern for the user.
12.3 Subjective Determination Of Probability Distributions
In the previous sections of this chapter, two basic ap-
proaches to the determination of the probability distributions of
4 nniif" c? 4" a csi" i r* irat*! aKI ae uafa 4" *1 nctrl TK o toida i a 4* Hhl1* i cs
input SwvwuaSulv VC&Jl XaOlco W6ic UciifieU* 1116 ieluai,lluBi v/JL
chapter is concerned with the latter, i.e., the subjective deter-
mination of such distributions. The importance of the subjective
approach increases as the quantity or quality of the historical
or sample data available decreases t its value increases as the
expertise of the estimator increases.
The basic assumption underlying the subjective approach is
that the analyst knows more about the input variable in question
than simply some measure of central tendency or "best" estimate.
Any subject matter specialist has some store of knowledge, albeit
"fuzzy" at times, that can be translated into some sort of proba-
bility distribution. The "fuzzier" such knowledge, of course,
the greater the uncertainty reflected in the probability distri-
bution so derived. Furthermore, the approach is not without risks
since people are not always accurate assessors of probabilities;
sufficient experimental evidence exists that estimation biases do
exist.
The main emphasis now is to review the methods whereby we can
obtain subjective probability distributions that most closely
reflect the assessor's beliefs about the distribution in ques-
tion and, at the same time, to minimize estimation biases. (For
a review of these methods and the issues involved, see Huber,
1974.)
12*4 The Variable Interval Method
The variable interval method involves the identification of
the points on a continuum that correspond to the endpoints of
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Mouse Manual Page 164
intervals containing specified proportions of the probability
distribution. ("Probability distribution", as used here, means
the probability density function.) This is usually explained in
the form of a dialogue between a questioner (Q) and the analyst
(A):
Q: To help us get started, what, in your view is the lowest
possible unit price for steam?
A: Well, it hardly seems likely that the price could ever go
below $2.
Q: How about the highest price?
A: I can't see it ever going above $18.
Q: Let's talk about more likely prices. What is the unit
price of steam such that the actual price is just as
likely to fall above that number as below it? Or, put
another way, what is the steam price that you would bet
even money on for being either exceeded or not exceeded?
A: Well, I'm not really sure but I suspect that it would be
about $10.
Q: All right. Let's consider now how likely it is that
other unit steam prices can occur. For the moment,
forget about prices above $10 and let us concentrate on
prices below that figure. What is the steam price that
would divide this lower region (the region below $10)
into two equal parts such that it would be just as likely
that the actual price would fall below this price as it
would to fall between this price and $10?
A: Maybe $8.
In the course of the above dialogue, the questioner has begun
subdividing the continuum of steam prices into intervals of
variable width containing equal proportions of the probability
distribution, i.e.,
.25 .25 .50
2 -8 -10 18
The process would continue, subdividing the quartiles on the left
and proceeding afterwards to the right side of the $10 price. It
should be mentioned that the process also involves checks for
consistency:
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Q: You have, in effect, specified that there is a one in
four chance that the price will fall between $2 and $8,
and that there is a one in four chance that the price
will fall between $8 and $10. Do you really think that
these two outcomes are equally likely?.
After considering this question, the analyst will often make
revisions to his estimates. As the analyst gains experience with
this technique, the revisions will become smaller. This
example is a version of a psychological scaling technique,
known as the "method of bisection", in which the intervals of the
continuum are bisected with regard to probability.
Although a probability distribution is ultimately obtained
by the variable interval method, the responses of the analyst are
not probabilities but units of a quantity or measure with which
he is familiar. By altering the questions slightly, the tech-
nique can be used to elicit cumulative probability distributions
which, of course, are the forms required for direct use by MOUSE.
However, studies have shown that people prefer the probability
function form to the cumulative probability function form.
12.5 The Fixed Interval Method
In this method the continuum is divided into equal intervals
and the analyst is asked to estimate the probabilities (or odds)
associated with each interval. Using the example of Section
12.4, let us assume that the questioner and the analyst have
agreed to work with intervals as follows;
2 ^ ^10 —** 12 14 16 18
The dialogue might proceed as follows;
Q: Which of these outcomes is least likely to occur?
A: Hell, I'd say that a steam price of more than $16 is the
least. 1 cicely»
Q: OK. Now how likely is that outcome? What probability
would you assign to it?
A: Hmmmm. Say about one chance in twenty or 5%.
Q; Let's hold that thought for a moment. Which of the
remaining events is now least likely, and what do you
think its probability is?
A; That's hard to say. I kind of feel that a price of $2 to
$4 is just as likely as one of $14 to $16? maybe a 1 in
ten chance for either.
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What the questioner has determined so far is:
.10 .10 .05
2 4 -6 a 10 12-; 14 16 18
The questioning would proceed as before, with review and feedback
to iron out any inconsistencies that night arise (such as the
probabilities adding up to more than 1).
The fixed interval method deliberately highlights events of
low probability and so tends to mitigate somewhat the problem of
obtaining subjective probability estimates with too small a
variance, a characteristic of subjective methods.
12.6 Other Methods
For certain probability distributions, such as the binomial,
some investigators (see Winkler, 1967) have used an "equivalent
prior sample" method where the analyst assessing the parameters
of a probability function is asked for an r (successes) and an n
(sample size) that, as a pair, represents his estimate of the
population parameters and his uncertainty about his estimate;
Q: You have some knowledge of the proportion of defects
produced by this process. Can you give me two numbers, r
and n, such that your knowledge would be roughly
equivalent to having observed r defects in a random
sample of n items from the production line? Remember,
the ratio of r/n should be close to your assessment of
the proportion of defects and that, for a given ratio of
r/n, a larger n corresponds to a greater confidence in
your assessment.
hi Well, I can imagine a sample of 20 showing l defect so
make r = 1 and n =20.
Since the parameters of a binomial distribution are n (=20 here)
and p (= r/n = .05 here), the distribution required has been
determined. The method is suitable only for discrete distribu-
tions for which knowledge of a postulated sample provides a
unique determination of its parameters. Unfortunately, such
distributions are not common.
By modification of the equivalent prior sample technique,
the parameters of a standard beta distribution can be determined.
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Mouse Manual Page 167
The method depends upon the fact that, in Bayesian terns, the
conjugate distribution to the beta is the binomial distribution.
The standard beta distribution (i.e., defined over the interval 0
to 1) with parameters a = 1 and b = 1 happens to be the continu-
ous uniform distribution over the interval 0 to 1. Such a dis-
tribution implies complete lack of information since all values
of the distribution are equally likely. In the Bayesian approach
to be described (see Bierman, et al, 1961), this is considered
the "prior distribution". The mean of the standard beta distri-
bution is a/(a + b) = r/n where n = a + b. If we obtain a sample
from such a distribution, the application of Eayes Theorem tells
us that the revised sample parameters are;
r' ¦ 1 + r [12.1]
n' ¦ 2 + n [12.2]
where r' and n' refer to the revised parameters, and r and n
refer to the equivalent prior sample parameters. Thus, for our
above sample if it were a standard beta distribution that was
being estimated,
r' = 1 + 1 = 2
n' = 2 + 20 + 22
Accordingly, the parameters of the elicited standard beta dis-
tribution would be a = r' = 2, and b = n' - r' = 22 - 2 = 20.
Another method involves a series of bets or gambles. For
example, the expected value (E) of a gamble is:
E - pW + (1 - p)L [12.3]
where p is the probability of winning, W is the amount to win,
and L is the amount to lose. To enter this gamble, the analyst
pays a sum, B, determined by the analyst. The analyst, there-
fore, presumably considers the expected value of the bet to be
worth the amount B to him. If W and L are known, then p can be
determined. For example:
Q. I'll bet you that the steam price is between $4 and $5.
If I am right I will pay you $25; if wrong you pay me
$10. To take this gamble, however, you have to pay me
something. How much will you pay me to take this gamble?
A. I'll take that bet for $15.
From Equation 13.3 and letting E - $15, p (which is, in effect,
the probability that the analyst thinks that the steam price lies
between $4 and $6) can be calculated, i.e.,
15 ¦ p(25) + (1 - p)(10) or p - 1/3
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The probabilities for other intervals are determined in a similar
manner.
The difficulty with the equivalent prior sample method is
that it is useful only when the form of the probability distribu-
tion is known. This begs the question since the emphasis in this
chapter is on the subjective determination of the probability
distribution, which includes not only its parameters but its form
as well. The betting or wagering method depends upon a number of
assumptions, e.g., that the analyst has a linear utility for
money and that decision makers tend to maximize expected values.
Furthermore, the method requires responses to three stimuli (W,
L, and the outcome that decides the bet, in this instance the
steam price interval of $4 to $6 postulated). For these reasons,
these alternative approaches are not recommended by the author.
12.7 Improving Elicitation Methods
An important consideration involved, in probability elicita-
tion methods is how such methods can be made more accurate.
There are three basic ways in which greater accuracy may be
achieved: (1) train the estimators, (2) improve the elicitation
technique and (3) aggregate the judgments of several estimators.
Training includes the improvement of the statistical knowledge of
the analysts (especially with regard to the concepts of probabil-
ity distributions), review with the analysts of the experimental
evidence that suggests the presence of bias in the tails and
understating of the variance of subjectively elicited distribu-
tions, and actual practice in such elicitation, including feed-
back for consistency. With regard to biases, there is substantial
evidence that people tend to rely primarily on specific informa-
tion, even when it is unreliable, and that they give insufficient
weight to general information. The result of this bias is to
shift the entire elicited probability distribution to the left or
to the right. Consider, for example, the following problem. A
man has been selected at random from the adult population and
described as "meek, tidy, and with a passion for detail". The
question is then asked, "What odds would you give that this man
is a librarian rather than a farmer?" The immediate intuitive
answer is that the man in question is very likely to be a librar-
ian, However, the population from which the man has been random-
ly drawn is the whole adult population and, in this class, there
are many more farmers than librarians. Adjusting the estimate to
take this into consideration would yield a much lower probability
of the man being a librarian than just the intuitive guess. Some
other facts about biases and understatement of the variance of
the subjective probability distribution:
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a. Recent information, being more available than old infor-
mation, if often given much too much weight.
b. Initial estimates may serve as "anchors" for subsequent
responses. Analysts, for example, S6GM to produce a
central bias when they start the determination of a
subjective probability distribution by focusing on the
median or the mean of the distribution. In such cases,
it is better to start first by looking at events with
smaller probabilities (e.g., the tails of the
distribution).
c. Laymen and experts alike exhibit considerable
overconfidence. In general, the observed surprise rate
is about 50* instead of 20%, i.e., when people are "80%
sure", they are wrong about 50% of the time.
The elicitation method can be improved by the motivation of
the participants and the rigid structuring of the elicitation
(often referred to as the "encoding") procedure. An excellent
discussion of this second approach can be found in Stael von
Holstein (1972). Such rigid structuring, however, does have its
disadvantages. It takes time and it often makes the probability
elicitation procedure seem more complex than it really is. With
training and practice, however, there is no reason why the ana-
lyst should not be able to derive acceptable subjective probabil-
ity distributions by himself, using tools no more complicated
than pencil and graph paper.
The third approach is to aggregate the judgments of several
estimators, either mathematically or by group decision proce-
dures. Mathematical methods include combining the individual
probability distributions by means of some sort of weighted
average. There are at least three approaches to the problem of
weights:
1. Equal Weights: In this case the decision-maker has no
reason to think that there is much difference in ability
among the analysts, so he is willing to assign equal
weights.
2. Weights Based on Previous Experience: This method may be
subjective or objective, depending upon the quantity and
quality of hard data obtained from prior experience with
the analysts.
3. Weights Proportional £2 a Self-Rating; Have each analyst
rate himself on a scale from 0 to M (conveniently M is
either 10 or 100). Then assign a weight to each analyst
proportional to his self-rating, where the constant of
proportionality is such that the weights sum to one (the
constant is obviously the reciprocal of the sum of the
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Mouse Manual Page 17 0
self-ratings). The rationale behind this rule is that
although a person may be able to make an assessment of a
probability distribution, the degree of belief in that
assessment may vary from analyst to analyst. Also,
although an analyst may be an expert in a field, his
expertise may vary from topic to topic within it.
Any of these weighted approaches are easy to implement in
MOUSE. Consider the CONTAM variable in the Waste Oil Reclamation
Problem. Assume that three analysts have been polled for their
probability distributions for this variable and further that
their respective distributions, Al, A2, and A3 are of the three-
point empirical discrete form. Suppose that we have weighted the
experts in the ratio 0.2, 0.2, and 0.4, respectively. The CONTAM
call in MOUSE would appear as;
CONTAM-0.2*EDIS(Al,3)+0.2*EDIS(A2,3)+0.4*EDIS(A3,3)
Group Decision approaches can be of several types, in the
first, termed "Group Reassessment" (GR), the analysts meet as a
group and discuss the matter, ultimately arriving at a group
consensus. This is, of course, nothing more than the familiar
"Committee", "Panel" or "Workshop" method. The outcome of the GR
method is apt to be a compromise among divergent views, arrived
at under the undue influence of those with the greatest supposed
authority or even merely the loudest voice. The group decision
is affected on the one hand by the bandwagon effect of majority
opinion and on the other by the unwillingness to abandon publicly
expressed opinions. One way to reduce these effects is to allow
each analyst to reassess his judgment in the absence of the
others, a procedure known as Group Feedback and Individual Reas-
sessment (GFIR). An interesting version of GFIR is the Delphi
technique (Dalkey, 1975). In general, Delphi procedures have
three features:
1. Anonvraitv. During a Delphi sequence the analysts are not
made known to each other?
2. Controlled Feedback: The exercise is conducted in a
sequence Of rounds between which a summary of the results
of the previous round is communicated to the
participants?
3. Statistical Group Response: The procedure includes a
response prepared by the administrator that includes the
opinions of each member of the group, summarized by
measures of central tendency (means, medians, etc.) and
variability (quantiles, variances, ranges, etc.)
The most common form of Delphi is the paper and pencil
version in which an administrator designs a questionnaire that is
sent to each of the analysts. After the questionnaire is re-
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turned, the administrator summarizes the results, which becomes
the basis for a new questionnaire for the group. This consti-
tutes a "round". The analysts are given at least one opportunity
to reevaluate their original subjective probability distributions
based upon examination of the responses of the others. In gener-
al, those analysts whose responses are far from the mean of the
group are asked to state their reasons why their responses are
significantly different from the majority. In actual experience,
placing an onus of justifying relatively extreme positions has
the effect of causing those without strong convictions to move
their distributions closer to the majority opinion, while those
who feel that they have a good argument retain their original
estimate and defend it.
TABU3 12.1
SUMMARY OP CONSENSUS TECHNIQUES
Group Feedback
Mathematical Group And Individual
Combination Reassessment Reassessment
Can be implemented yes
via mail questionnaire
Pressure to arrive at no
consensus
Presence of other adverse no
social pressures
Consensus can be manipulated no
by decision maker
Degree of difficulty for low
decision maker
Degree of difficulty for low
analysts
Incorporates feedback no
Analysts can be yes
independently weighted
Introduction of new factors no
previously unthought of by
some analysts
Commitment to group low
consensus
no
yes
yes
no
low
low to high
yes
no
yes
high
yes
yes
no
yes
high
medium
yes
yes
partially
high
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Table 12.1 compares- the characteristics of the Mathemati-
cal Combination, Group Reassessment, and Group Feedback and indi-
vidual Reassessment procedures. -As-can be-seen,-no one technique
dominates the others, The primary weakness of the Mathematical
Combination technique is that it lacks' feedback - from the other
analysts and is also low on"commitment to group consensus, an
important factor with regard to "selling" or-'otherwise implement-
ing an uncertainty analysis. The primary advantage of Group
Reassessment is the possibility of introducing new factors or
considerations not previously thought of by all analysts. Its
primary disadvantage is, of course, lies in the adverse social
pressures present during group interaction. Group Reassessment
and Group Feedback and Individual Reassessment are 'similar with
respect to feedback and commitment. The latter is an improve-
ment over the former in that adverse social pressures are absent.
It is, however, a more complex procedure.
12-8 Same Useful Distributions for Subjective Methods
The easiest quantities for any analyst to estimate include:
(a) measures of central tendencies such as median, mode, and
mean, and (b) measures of dispersion such as minimum, maximum,
and range. The following probability distributions lend them-
selves well when only this sort of information is available:
1» Uniform ' Distribution ^(Discrete or Continuous!; The
uniform distribution requires a knowledge only of the
minimum and the maximum of the variable. Use of the
uniform distribution implies, however, that the analyst
has no knowledge of the relative likelihood of occurrence
of the values of the random variable between these two
extremes. It is, therefore, a distribution of extremely
low information content.
2. Distribution: The triangular distribution
requires, in addition to the minimum and the maximum, an
estimate of the mode or most likely value of the
variable. In spite of its simplicity, there is a great
deal of information contained in this kind of a
distribution. Under many conditions, it very
satisfactorily approximates more complex distributions
such as the normal, lognormal, and beta.
3. Trapezoidal Distribution: The trapezoidal distribution
is a useful substitute for the triangular when there is a
great deal of uncertainty regarding the most likely
value.
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Mouse Manual
• = 3 b=2
fix)
¦s1 b»3
1
Kxi
0
¦ = 3
1
fix]
0
0
1
fix]
0
0
• =l
1
ftxi
oL
FIGURE 12.1:
SHAPES OF SOME BETA DISTRIBUTIONS
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4. Beta Distribution; The beta distribution is one of the
most frequently employed classical distributions for
fitting subjective probability distributions (Winkler,
1972), perhaps because it fits an impressive number of
different shapes (see Figure 12.1) The standard beta
distribution can be fit easily using the Bayesian tech-
nique illustrated in Section 12.6. The beta distribution
need not be limited to values of the random variable
lying between 0 and 1, however. Any bounded interval can
be rescaled to this interval. This is why MOUSE has been
provided with the Empirical Beta Distribution for which
the user need supply only three estimates: the most
likely, and the lowest and highest value the variable
can reasonably assume. These are identical to the inputs
required for fitting the triangular distribution.
Another distribution of great value for subjective estimates
is the step rectangular distribution. As Pouliquen (1970)
states: "The distribution is an attractive one for several rea-
sons. In the first place it takes explicit advantage of the fact
that the quantification of subjective probability judgments, in
both theory and practice, is based upon preference ranking. If
also has the advantage that it can be drawn up by the appraiser
himself. He has the freedom to choose whatever intervals he
wants and to divide them into as many sub-intervals as he wants.
This complete freedom of initiative, which he lacked in the case
of the portrait approach (Author's note; a variation on the
method of drawing the probability distribution directly), seems
to help him considerably in the expression of his judgment."
RgJfJSKCIflpltS
Bierman, H. , et al. , Quantitative Analysis for Business
Decisions. Richard D. Irwin, Homewood, Illinois, Revised Edition,
p. 1-456 (1965).
Dalkey, N.C., "Toward a Theory of Group Estimation", in H.A.
Linstone and M. Turoff, Eds., The Delphi Method: Techniques and
Applications. Addison-Wesley Publishing Co., p. 236-262 (1975).
Foutz, R.V., MA Test For Goodness-Of-Fit Based Upon An Empirical
Probability Measure", TM Annual fif statistics, Vol. 8, No. 5,
p. 989-1001 (1980).
Gibbons, J.D., Nonparametric Methods for Quantitative Analysis.
Holt, Rinehart and Winston, New York, p. 1-463 (1976) .
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House Manual Page 175
Huber, G.P., "Methods for Quantifying Subjective Probabilities
and Multi-Attribute Utilities", Decision Sciences. Vol. 5, p.
430-458 (1974) .
Pouliquen, L.Y., Risk Analysis in Project Appraisal, World Bank
Staff Occasional Papers Number Eleven, The John Hopkins
University Press, Baltimore, p. 1-79 (1970).
Ru2icka, R.K., "Weibull Distribution Analysis - Short Method",
Journal of the Electronics Division, Axner. Soc. for Quality
Control. Vol. 1, No. 1, p. 38-54 (1962).
Stael von Holstein, C.-A. S., & Manual for Encoding Probability
Distributions. prepared for the Defense Advanced Research
Projects Agency by Stanford Research Institute International (SRI
Project 7078), p. 1-201 (1979).
Winkler, R.L., "The Assessment of Prior Distributions in Bayesian
Analysis", Jj. Amer. Statistical Assoc. 62(319), p. 776-800
(1967).
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APPENDIX A
RUNNING MOOSE, HIP, AND TRAP PROGRAMS
A.l Introduction
A number of terms will be used in this Appendix that may be
new to readers with little or no experience with high-level
computer languages. MOUSE was written in FORTRAN because it is
the most nearly universal of all scientific programming lan-
guages. There are, in the opinion of the author, much better
languages but if programs are to be widely used they must be
portable, i.e., it must be possible to move them from one comput-
ing environment to another with a minimum of alterations. In
this regard, FORTRAN is highly standardized? in comparison, BASIC
is just beginning to become standardized, and the problem with
PASCAL is that it has at least three major "standards". MOUSE
has been written in a subset of FORTRAN that is handled in the
same way by essentially all known computer systems.
Source files are the English-like representation of a
program, i.e., the text you type in on the keyboard of a terminal
through an editor or on the keyboard of a keypunch machine. If
this text conforms with certain syntactical rules (the rules of
MOUSE, in this case), the computer's FORTRAN compiler will turn
this text into code that the computer is able to understand.
Computers cannot understand source code; it must first be
compiled into a form (i.e., "machine language") that can be
understood by the machine being used. Machine code, which
generally is not human-readable, resides in a so-called object
file. In other words, the source file is what you wrlte, and the
object file is the destination of the translated (for the
computer) source code.
Although every program is different, there are still
commonalities. Most programs, for example, contain input and/or
output operations, simple arithmetic functions such as addition
or subtraction, and more complicated routines such as logarithms
and square roots. In order to avoid having to compile these
common elements over and over again, every FORTRAN system has an
internal library which contains compiled versions of these
operations. Before a compiled program can be run, the FORTRAN
system must examine the program, determine which of these
standardized operations are called for, and attach the required
modules from the system library to the compiled main program.
This process is called "linking", and the final product is a file
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w^xxch xs executable , x.e. , xt xs in Biachxne—readable code vtxth
all the required operations, functions, and subroutines appended.
Note that not all of the system's library is linked with the
user's main program; only those modules that are actually re-
quired are linked. Note that since MOUSE is a collection of
functions and subroutines, in its compiled form it can be thought
of an external library of standardized modules available during
the linking process.
Error messages can be sent by the computer during any of the
processes described above, i.e., during compiling, linking,
and/or running. The last-named are referred to as "run-
time" messages. Messages issued by the compiler describe the
OiTTAT* a Tl f-ko 1 Atl 4- V% yw Ofmi" 4 w vsy-ArtVani T f
UiilsS IbGVp WtSSvol fcS JL 4 Ua. ul ImL L>i J. Uvu w iUl 1 UJL l*,! ktS 6 JL 1> Ul JL11 L>1 iCSS ^#li> WnJ JL ulu • 11 ^
at the time of compilation, a listing file is requested, the
listing file will write out these error messages immediately
after the line in your program in which the error occurred. This
is, of course, most useful in debugging programs. Run-time
messages are sent to your terminal or will appear on your output
listings if batch mode is used. These are usually more difficult
to interpret; although the type of error is reported by the
computer, its exact location is generally not clear. However,
as described in Chapter 11, Section 11.3, MOUSE itself provides
run-time error messages that most likely will prove more helpful
than those provided by the computer's FORTRAN system. A summary
of MOUSE run-time error messages can be found in Appendix C.
A.2 Running CHECKER, IMP, TRACER, and TRAP
On PC—DOS or MS—DOS Computers
To run these programs, simply type "CHECKER", "IMP", "TRAC-
ER", or "TRAP". TRAP takes input only from the keyboard, CHECKER
and TRACER require an input file, the name of which you will be
required to supply, and IMP takes input either from the keyboard
or a data file called "IMP.DAT". The default output files for
CHECKER, IMP, TRACER, and TRAP are "CHECKER.OUT", "IMP.OUT",
"TRACETST.FOR", and "PROGRAM.OUT", respectively. If these files
already exist, you will be asked whether you wish them to be
overwritten or not. If not, you can supply any name you choose.
A. 3 Ruiraing MOUSE On PC-DOS or MS-DOS Computers
Since MOUSE consists of various functions and subroutines,
not all of which are needed for every program, these functions
and subroutines are placed into a MOUSE library file (MOUSE.LIB)
where they can be linked with a compiled source file. (See
Appendix 6, Section B.2.) The procedure is as follows:
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Step #1. Using a text editor, place your program into a
source file with extension FOR, e.g., PROGRAM.FOR.
Step #2. Using your FORTRAN compiler, compile this source
program, producing an object file with extension
OBJ, e.g., PROGRAM.OBJ
Step #3. Using your FORTRAN linker, link the object file
with the MOUSE library file, producing an executa-
ble file with the extension EXE, e.g., PROGRAM.EXE.
Step #4. Run the executable file by typing the name of the
executable file, e.g., "PROGRAM".
The actual operation will vary from compiler to compiler, but the
following is a typical example using the Microsoft FORTRAN sys-
tem, assuming that we have a MOUSE source file called
PROGRAM.FOR:
Compilation: FL /c /Fs /FPi PROGRAM.FOR
where FL is the Microsoft compilation command, /c is a switch
that instructs the compiler to compile without linking, /Fs will
produce the listing file PROGRAM.LST, and /FPi is a switch that
instructs the compiler to compile generating in-line instructions
while selecting the emulator math package (meaning that the
program will run whether or not the computer has a math coproces-
sor, i.e., if the computer does have a math coprocessor, it will
be used.)
Linking: LINK PROGRAM,,,MOUSE.LIB
Running: PROGRAM
Most MOUSE problems using the Microsoft FORTRAN system are com-
piled, linked, and run in his manner, although for certain very
large problems other switches may have to be used (such as the
/Gt[number] compiler switch). In the above example, the compiling
step will produce files PROGRAM.OBJ AND PROGRAM.LST, and the
linking step will produce will produce file PROGRAM.EXE.
The Microsoft FORTRAN system has an option whereby a source
program can be checked for errors before it is compiled, e.g.,
Testing: FL /c /Zs PROGRAM.FOR
where the /Zs option is the switch that selects program syntax
without compilation. I have a batch file (called TEST.BAT) that
permits me to test a source file quickly, regardless of its name.
It is a one-line file consisting of the following:
FL /c /Zs %1.for
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To use this batch program on the file PROGRAM.FOR, for
example, I enter:
TEST PROGRAM
A typical compiling step (the linking and running steps are the
same) for the IBM (Ryan-McFarland) FORTRAN system are:
compilation: PROFORT PROGRAM.FOR /B /L >PROGRAM.LST
The /B option will cause all adjustable arrays to be assumed to
be larger than 64K bytes, and the /L > PROGRAM.LST option will
generate a standard program listing in the file called
PROGRAM.LST. (Note: If you use the IBM (Ryan-McFarland) FORTRAN
system, the /B option must be used.)
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APFKNDUt B
IKSTALL&TIOK AMD CUSTOMIZATION NOTES fOR HOUSE
B.l Installation
MOUSE is designed to be used with a FORTRAN-77 compiler,
i.e., ANSI Standard FORTRAN(X3.9-1978), containing the following
extension: Hexadecimal constants must be allowed in a DATA state-
ment to initialize character values. Both the Microsoft and IBM
(Ryan-McFarland) FORTRAN allow such constants.
The distribution disk contains the source file for MOUSE
called "MOUSE.FOR". MOUSE.FOR contains a number of functions and
subroutines written in FORTRAN-77 and adheres closely to that
standard except for the manner of utilizing hexadecimal constants
in DATA statements as mentioned above. The distribution version
of MOUSE contains the form, #h, where h is a string of hexadeci-
mal characters 0 through 9 or A through F. For example, a car-
riage return is represented as #0D (this is the form used by the
Microsoft FORTRAN). Some compilers use a Z before a quote, e.g.,
Z'OD' (this is the form used by the IBM or Ryan-McFarland FOR-
TRAN) .
The following are the necessary Precompilation Steps.
However, if anything hss to be modified, it most likely will be
the hexadecimal constants noted in Step #1.
Step #1: Examine the following DATA statement which can be found
at the beginning of the subroutine FFREAD module in
MOUSE.FOR:
DATA CR,LF/#0D,#0A/
If necessary, change the form of the hexadecimal con-
stants to conform to the requirements of your FORTRAN
compiler as discussed previously. This distribution
form is that used by the Microsoft FORTRAN system.
Step #2: Examine the SUBROUTINE DATA routine in MOUSE.FOR and
find the following lines:
ITERM-5
JTERM=6
This assigns the terminal input and terminal output.
respectively, to logical unit number 5. If it is
different from your system (both the Microsoft and IBM
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or Ryan-McFarland FORTRAN use these assignments),
change the numbers accordingly. It is not necessary to
change the logical unit number assignments found in
other lines.
Step #3: The seed for the random number generator in MOUSE is
found in the module SUBROUTINE DATA in the line:
IRAN-23931
This can be changed to suit the user's preferences (any
large positi^^e, odd integer generally wi.ll do).
Step #3: The value of 0K6 (see Chapter 11, Section 11.2) is set
in the SUBROUTINE DATA subroutine in MOUSE.FOR by;
OK6=.0000000001
This can be changed to suit the computer being used.
However, the default number works well on 8088 and
80286 machines.
B.2 Compilation and Linking With Libraries
As explained in Appendix A, during linking an executable
file is formed with the extension EXE. After linking, the object
file is no longer needed and may be deleted. Since it does not
make sense to link all of MOUSE to your program when only a small
number of nodules may be called for by that program, it is ad-
visable to form what is known as a "library". This is a file
containing the MOUSE modules in object form, and has the exten-
sion LIB.
There are'two ways to create a MOUSE library. The easiest is
to compile the file MOUSE.FOR in one step. A typical Microsoft
FORTRAN system compilation step is as follows:
Compilation: FL /c /Z1 MOUSE.FOR
where the /Z1 option suppresses the use of library names in
object files (a useful option since MOUSE library routines are
not required to contain this information). The next step is to
place this module into a library called MOUSE.LIB. A typical
Microsoft FORTRAN system library inclusion step is as follows:
Library; LIB MOUSE +MOUSE;
This step will produce the library file, MOUSE.LIB. The object
file formed by the compilation step, MOUSE.OBJ, is of no further
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use now and nay be deleted. A typical compilation step using the
IBM (Ryan-McFarland) FORTRAN system (the library step is the
same) is as follows:
Compilation: PROFORT MOUSE.FOR /B
In the second way to produce a MOUSE library file, you must
first individually compile each of the MOUSE functions and sub-
routines that are present in the file MOUSE.FOR, Each of these
modules must be removed from MOUSE.FOR and placed in its own file
with extension FOR. The normal distribution module, GAU, for
example, must be placed into a file called GAU.FOR. Again, the
form of the compilation command will differ with the FORTRAN
system you are using. A typical Microsoft FORTRAN system compila-
tion step is as follows:
Compilation: FL /c /Z1 GAU.FOR
where the /Z1 option suppresses the use of library names in
object files (a useful option since MOUSE library routines are
not required to contain this information). The next step is to
place this module into a library called MOUSE.LIB. A typical
Microsoft FORTRAN system library inclusion step is as follows:
Library: LIB MOUSE +GAU;
The object file formed by the compilation step, GAU.OBJ, is of no
further use now and may be deleted. When you are finished
compiling all the MOUSE modules and including them into the MOUSE
library, you will have a complete MOUSE library file, MOUSE.LIB.
A typical compilation step using the IBM (Ryan-McFarland) FORTRAN
system (the library step is the same) is as follows;
Compilation: PROFORT GAU.FOR /B
The second method has the advantage that individual modules can
be removed, modified, and re-added to the library file. [Note:
Since constructing a library via the second method involves quite
a bit of work (although it can be automated to some extent using
batch files), the distribution version of MOUSE contains a
ready-to-use, individual-module library file, MOUSE.LIB, in two
versions: MOUSEMS.LIB and MOUSEIBM.LIB (for Microsoft and IBM or
Ryan-McFarland FORTRAN, respectively). Eliminate the unneeded
file and rename the remaining; one MOUSE.LIB. ]
In 'order to prevent accidental erasure of the MOUSE.FOR- and
MOUSE. LIB files, it is advisable to set the RO ("read-only1*)
attribute using the DOS ATTRIB program as follows:
ATTRIB +R MOUSE.FOR
. ATTRIB +R MOUSE.LIB
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JBt • 3 Adding mCHJSl ti*
Since MOUSE is a modular-oriented system, i.e., it is com-
posed of separate nodules or subprograms, it is easy to add
user-specific nodules such as probability or interest functions
or to modify modules provided that the library was formed using
the individual-module method. This section describes how such
modules can be added. For this section only, it is assumed that
the reader has a knowledge of FORTRAN.
Suppose we wish to add a two-parameter Weibull probability
distribution generator to MOUSE. Following Naylor, et al.
{1966), Weibull variates can be generated by the following
expression:
X *= b(-log r) Vc [A.l]
where b is a scale parameter and c is a shape parameter. The
standard call for a random number in MOUSE is made by RAN(IRAN)
and the argument is always found in COMMON location Q8. Thus,
the following function would be added at any convenient location
in MOUSE.FOR:
C
C GENERATION OF RANDOM VARIATES FROM A WEIBULL DISTRIBUTION
C
FUNCTION WEI(B,C)
C0MM0N/Q8/IRAN
R=RAN(IRAN)
WEI=B*(-ALOG(R))**1./C
RETURN
END
Thus a call to WEI(2.,6.) would return a random value from the
Weibull distribution with scale parameter equal to 2 and shape
parameter equal to 6.
Let us now consider a financial function. Doane, et al.
(1976) discuss the calculation of "annualized cost", in which one
of the factors used is the depreciation factor, DFF. Defined for
straight line depreciation, this factor is given as;
DPF - [1 - (1 + ipn]/ni [A.2]
where n is the depreciation lifetime and i is the interest rate.
In order to allow the use of nominal interest rates, MOUSE
provides for a conversion of nominal to true interest rate by the
following formula (which is the coded FORTRAN version of equation
5.10):
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Mouse Manual Page 184
XI=(1.+XI/C0MF)**C0MP-1. [A.3]
where XI is the interest rate and COMP is the number of
compounding periods. If the nominal interest rate is entered as
XI in the right-hand side of equation A.3, then the value of XI
returned in the left-hand side will be the true interest rate.
Any interest formula, therefore, that may be compounded should
first test for the value of COMP; if greater than zero, then
equation A.3 should be used. Thus, the function desired is
/ uViot'ci YM 4 «s fhfl Hdnrdo «| a+» 4 aa 1 4 fAt* 4 tdo \ •
1 Iff j| JitCS JL C2 tAJL*
-------
Mouse Manual Page 18 5
APPENDIX C
SUMMARY TRBT35S
MOUSE CONTROL LINES
Standard Mouse Control Lines
#1
COMMON/Q1/C80/Q2/JUMP/Q3/LOOP/Q4/SENS/Q5/IDATA/Q6/IOUT/Q7/IPASS/Q
#2
4/1RAN/Q9/IREAD/Q10/ISCR1/Q11/ISCR2/Q12/ISNAP/Q13/ITER/Q14/ITOG/Q1
#3
&/IWRIT/Q16/IXXX/Q17/OK1/Q18/OK2/Q19/OK3/Q20/OK4/Q21/OK5/Q2 2/OK6
#4
5
DO 10 IXXX=1,ITER
#5
10
CONTINUE
#6
GOTO(5,15), JUMP
#7
15
CONTINUE
#8
END .
Pass Control Line
DO 15 IPASS=I,K [K is an integer)
Vector Loop Control Lines
#1
#2
#3
DO 100 LOOP=l,L
100 CONTINUE
LOOP=l
[L is an integer or an integer variable]
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Mouse Manual Page 186
TUBUS C.2
ALPHABET! CAIj LIST OF MOOSE INTEREST FUNCTIONS
Function
compound interest:
CAPREC(i,n,period)
SCA(i,n,period)
SFW(i,n,period)
OPW(i,n,period)
UPWP(i, n, j,period)
UCA(i,n,period)
SINK(i,n,period)
Gradient Series:
GCA(i»n,period)
GPW(i,n,period)
Continuous Compounding:
CSPW(i,n,period)
CUPW{i,n,period)
Miscellaneous:
PWSUM(vector,i,upper,inv)
Type Page
capital recovery 73
single payment compound amount 71
single payment present worth 72
uniform series present worth 72
same but with constant
percentage change 72
uniform series compound amount 73
sinking fund 73
gradient series equivalent
uniform annual amount 74
gradient series present worth 74
single payment present worth,
continuous 76
uniform series present worth,
continuous 76
present worth sum 123
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Mouse Manual Page 187
TABLE C.3
ALPHABETICAL LIST OF MOUSE PROBABILITY FUNCTIONS
Function
AGAUL(mode,median)
BETA (a, b)
BGAU1(ml,s1,Z)
BGAU2(m2,s2,rho,Z)
BGAUL1(ml,si,X,Y,Z)
BGAUL2(m2,s2,rho,X,Y,Z)
BIN(n,p)
CUNI(lower,upper)
DTRA(a,b,c,d)
DTRI(a,b,c)
DUNI(lower,upper)
EBETA(l,m,u)
ECTS(name,n)
EDIS(name,n)
ERL(a,k)
EXPON(mean)
GAM(a,b)
GAU(mean,std)
GAUL(mean,std)
H¥F(N,n,p)
PAS(k,p)
POI(mean)
STEP(name,n)
TRA(a,b,c,d)
TRI(a,b,c)
Type
lognormal, M-form
beta, standard form
bivariate normal (1)
bivariate normal (2)
bivariate lognormal (1)
bivariate lognormal (2)
binomial
uniform, continuous
discrete trapezoidal
discrete triangular
uniform, discrete
beta, empirical form
empirical, continuous
empirical, discrete
erlang
exponential
gamma
normal
lognormal
hypergeometric
pascal/geometric
poisson
step rectangular
trapezoidal
triangular
IF statements for truncated distributions:
IF(variable .LT. lower) GOTO n
IF(variable .GT. upper) GOTO n
IF(variable .LT. lower .OR. variable .GT. upper) GOTO n
IF statement construction for composite distributions:
variable=functionl
IF(RAN(IRAN) .LE. v) variable=function2
CALL AUTOE(mean,std,lambda,value)
Exponential Autocorrelation Function
CALL AUTOL(mean,std,k,value)
Linear Autocorrelation Function
Page
46
46
110
110
112
112
49
45
51
50
50
46
40
40
49
45
47
45
46
50
50
49
43
4 8
47
52
53
11B
117
-------
Mouse Manual Page 188
TABLE C.4
MOUSE LINE TYPES
^2DS Iffit £m
1. Dimension Block
DIMENSION name(n,m) matrix, dimension 90
DIMENSION name1(n),etc. vector dimension 91
DIMENSION name1(n1,2),etc. empirical variable dinwn. 41
DIMENSION 11(100,3),etc. plot dimension 98
DIMENSION TlCn1),etc. table dimension SO
DIMENSION T1(7,n1),etc. - " vector table dimension 91
2. Read Block
CALL READ(namei,ni) empirical variable read 41
CALL HREAfi(n«nei,n,m) matrix read 90
3. Specification Block
CS0=B0 column output 89
S£NS=c sensitivity level 104
I00T»IUR1T output destination 88
ISNAP=s snap 86'
[TER=n iteration 85
IRAN=seed seed reset 86
4. Program Block
CALL DTABLE(switch) dunny table output 82
CALL DTRACE('(Iname)1.variable) decimal trace 147'
CALL 1 TRACEC'CInane)1,variable) integer trace 147
CALL PL0T
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Mouse Manual Page 189
APPENDIX D
MOUSE ERROR MESSAGES
10 FUNCTION PARAMETERS IN WRONG ORDER: Pertains to probability
functions. Example: In the continuous uniform distribution,
the call is CUNI(lower,upper). This error message would be
returned if the user coded CUNI(20.,10.) instead of
CUNI(10.,20.).
11 DECIMAL ARGUMENT LESS THAN OR EQUAL TO ZERO: This message
is most often returned when the user has omitted the decimal
point in a MOUSE function or subroutine call, e.g., if the
user had coded POI(5) instead of POI(5.)• (See note at end
of this Appendix.) Some arguments, of course, must be
positive, e.g., the standard deviation in calls such as
GAU(mean,std). A code of GAU(5Q.,-10.) would also result in.
this error message.
12 ITERATIONS LESS THAN 100: Plots require a minimum of 100
iterations.
13 CORRELATION COEFFICIENT RANGE EXCEEDED: The permissible
range for the bivariate normal or bivariate lognormal
probability distribution has been exceeded. See Chapter 9,
Sections 9,3 and 9.4.
14 DECIMAL ARGUMENT LESS THAN ONE: Some probability function
arguments are required to be at least 1, e.g., the N and n
in HYP(N,n,p).
15 INTEGER ARGUMENT GREATER THAN 1000; This message is most
often returned when the user has used a decimal point or a
floating point variable when an integer or integer variable
is required, e.g., PWSUM(ANBEN,.08,XLIFE,XININV) instead of
PWSUM(ANBEN,.08,LIFE,XININV). (See note at end of this
16 BOTH CHANGE ARGUMENTS EQUAL TO ZERO: Pertains to the
Sensitivity Control line. This message is most often
returned when the decimal point has been omitted from the
m% and std% arguments.
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Mouse Manual Page 190
17 ILLEGAL USE OF PLOT OUTPUT LINE WITHIN LOOP: A Plot Output
Line was used within a Vector Loop. This is riot permitted.
18 ILLEGAL USE OF TABLE OUTPUT LINE WITHIN LOOP: A Table Output
Line was used within a Vector Loop. This is not permitted.
19 SNAP INTERVAL EXCEEDS ITERATIONS: The snap interval must be
less than the number of iterations.
20 MISSING DATA OR NAME LINE OR DATA MISENTERED: Either a line
is missing in an Empirical Variable, Correlation or Matrix
data set, or else data has been entered in an incorrect
f 1 fill d "TK i g pa f"| Si 1 c A Off IIT 1 f A T*/** hi anV 1 4 mae
•w JL» VA • JL. JL inr JL JL \hf JL» 4 SSfV*r \mF\u0 Ml«Im> JL %*J. JL wL JL. X25 JmJf JL CX JL XJrt* «!» JL*# I.X2S St
at the beginning of the MOUSE.DAT file.
21 BOTH LIMITS ARE EQUAL TO EACH OTHER: Pertains to probability
functions and generally means that the decimal points have
been omitted.
22 MODE EQUAL TO OR LARGER THAN THE MEDIAN: In a lognormal
distribution, the median must be greater than the mode.
23 INTEREST RATE OR INTEREST PERIOD LESS THAN ZERO: Pertains
to most interest functions and is an obvious error since
neither of these quantities can be negative.
24 INTEREST PERIOD LESS THAN OR EQUAL ZERO: Pertains to
the Internal-Rate-of-Return function and generally means
that a decimal point or a floating point variable has been
used for the "upper" argument instead of an integer or an
integer variable.
25 HISTOGRAM INTERVAL EQUAL TO ZERO: Pertains to user-
dimensioned tables and generally means that the decimal
point has been omitted.
26 DECREASE EXCEEDS 100%: In the UPWP function,
cannot be decreased by more than 100%, nor can
deviation be decreased by more than 100% in the
Control line.
the amount
the standard
Sensitivity
27 ITER LESS THAN 2: A MOUSE run requires
iterations.
at least 2
-------
Mouse Manual Page 191
Note: Because of the vastly different manner in which
integer and decimal data are stored in a computer, decimal
variables without decimals are generally interpreted as
either extremely small positive numbers or as negative
numbers when passed in functions or subroutines. Integer
variables, on the other hand, when mistakenly used with
decimal points, are generally interpreted as numbers larger
than 1000. For decimal numbers, MOUSE tests to see whether
the number passed is less than OK6 {which, in the distribu-
tion version of MOUSE, is 0.0000000001). If it is less than
this number, an error is declared.
-------
Mouse Manual Page 192
AfPENDIX S
ANSWERS TO EXERCISES
Chapter 3:
1. Invalid: Variable names must start with a letter of the
alphabet.
2. Invalid: Two arithmetic operators cannot appear next to
each other.
3. Valid.
4. Invalid: Only a single variable name can appear on the
left-hand side of an equation.
5. Invalid: Q1 is a restricted variable name.
6. Valid.
7. Valid.
8. Invalid: The right-most parenthesis is missing.
9. Valid.
Chapter 4:
2.. Val id •
Invalid: If 100 is the most likely value, 200 cannot be
the lowest value.
Invalid: Decimal points are not used in the n argument.
Valid.
Invalid: The probability is the first argument in the
binomial probability function.
Valid.
Invalid: The lognormal probability distribution cannot
have a negative mean.
Valid.
Valid.
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Mouse Manual Page 193
10. Invalid: The statement should be CALL READ(D7,30).
11. There are no errors in this data line set.
12. The probability of the first value in a discrete
empirical probability distribution must be greater than
zero? otherwise, why put it in?
13. The first cumulative probability in a continuous
empirical probability distribution must be zero.
14. 200 X=TRI(10.,20.,70.)
IF(X .LT. 15. .OR. X .GT. 60.) GOTO 200
15. PROD=GAU(100.,10.)
R=RAN(IRAN)
IF(R .LE. .3333) PROD=GAU(75., 5.)
IF(R .GI. .6667) PR0D=GAU(55.,5.)
16. Data Set #1: The closest fits are the exponential,
lognormal, and gamma. The data were actually taken from
a lognormal distribution. This exercise shows the
dangers in relying solely upon goodness-of-fit tests
when small data sets are involved.
Data Set #2: The best fit is the Poisson distribution.
(The data were actually taken from a Poisson distribu-
tion.)
17. The 25-point integration provides a substantially
better fit than does the 13-point integration.
Chapter S s
1. Invalid: The period argument lacks a decimal point.
2. and 3. Both are valid.
4. Invalid: The interest rate and period arguments are
reversed.
5. Invalid: The second argument in the UPWP function is the
time period, not the fraction change.
6. Invalid: For continuous compounding, set the switch to
any positive number,
7. Valid.
8. Invalid: Interest rates cannot be negative. (An
exception is the i argument in the PWSUM function? see
Chapter 10, Section 10.2.)
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Mouse Manual Page 194
9. Invalid: The period must be an integer.
Chapter 6:
1. Valid.
2. Invalid: The number of intervals argument should be an
integer.
3. Invalid: The parentheses around the literal name are
omitted, and the variable name exceeds 6 characters.
4. Valid.
5. Valid.
6. Invalid: For MOUSE-dimensioned tables, the minus sign in
front of the number of intervals should not be used.
7. Valid: The scaling factor lacks a decimal point.
8. Invalid: The table interval must be greater than zero.
9. Invalid: The dimension specification should be T1(40).
10. Invalid: The number of intervals for short-form tables
should be set egual to zero.
11. Valid.
chapter 7;
1. Valid.
2. invalid: Seed lines must contain odd integers.
3. Valid.
4. Invalid; The first index of any matrix must be IPASS.
5. Invalid: Either the row or column argument is missing.
6. The index IPASS must be used instead of I.
7. There are no errors in this data set.
8. Since there are four rows in the matrix, the second row,
containing the data pair (5.,4.), must start on a new
line, etc.
-------
House Manual Page 195
Chapter 8 *
1. Valid.
2. Valid.
3 • Valid•
4. Invalid: The standard deviation cannot be decreased by
more than 100%.
5. Valid.
6. Valid.
7. Invalid: The last executable statement in a MOUSE
program must be a Table Output Line.
Chapter 9s
1. Valid.
2. Invalid: The dummy variable must be the same in both
C3L36S •
3. Valid.
4. rx = .01 mx = 4.6002 sx = .0998
ry = .04 my = 5.2787 sy = .1980
rxy - .02
rhol = .6199 rho2 = .6534
rhol' = .6176 rho21 = .6512
rho' = .6342
Chapter 10:
1. Invalid; The "upper" argument must be an integer number
or an integer variable.
2. Valid.
3. Invalid: The "upper" index in the first Vector Control
Line must be an integer or an integer variable.
4. Invalid: The call is VTASLE, not TABLE*
5. Valid.
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Mouse Manual Page 196
6.
7.
8.
DIMENSION T1(6),T2(26)
COMMON/Q1/C80/Q2/JUMP/Q3/LOOP/Q4/SENS/Q5/IDATA/Q6/IOUT/Q7/IPASS/Q8
&/IRAN/Q9/IREAD/Q10/ISCR1/Q11/ISCR2/Q12/ISNAP/Q13/ITER/Q14/ITOG/Q15
4/IWRIT/Q16/IXXX/Q17/OK1/Q18/OK2/Q19/OK3/Q2 0/OK4/Q21/OK5/Q22/OK6
IT1R=1000
5 DO 10 IXXX=ITER
C
C INITIALIZE REMAINING LIFE AND TIME IN TUNNEL
C
REMLIF=GAU(10.,2.)
TIME=0.
200 R=RAN(IRAN)
IF(R .LE. .3333) GOTO 220
IF(R .LE. .6667) TIME=TIME+EXP(3.)
IF(R .GT. .6667) TIME=TIME+GAUL(6.,2.)
C
C CHECK TO SEE IF RAT DIES BEFORE ESCAPING
C
IF(TIME .GT. REMLIF) GOTO 210
GOTO 200
C
C THIS RAT DIED IN A TUNNEL
C
210 TIME=REMLIF
SURV=0.
GOTO 230
C
C THIS RAT ESCAPED
C
220 SURV=1.
230 CALL TABLE(SURV,«(SURVIVOR)¦,,,0,T1)
CALL TABLE(TIMEj•(TIME)l,0.,l.,20,T2)
10 CONTINUE
GOTO(5,15), JUMP
15 CONTINUE
END
Note that this is a terminating simulation (see Chapter 7, Sec-
tion 7.3). The simulation (for one rat) ends when either the rat
dies or escapes the cage. The initial condition in this problem
is that only one rat is present in the cage at the start. Addi-
Valid.
The following line is placed immediately after the
ANBEN(LOOP) line in the vector loop:
IF(LOOP.EQ.LIFI) ANBEN(LOOP)=ANBEN(LOOP)+CUNI(40006000.)
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Mouse Manual Page 197
tional rats at the start would affect the results if, for exam-
ple, a rat is riot permitted to enter a tunnel if another rat is
already there.
9.
DIMENSION A1(20),A2(20),A3(20)fBl(100),B2(100),B3(100)
DIMENSION T1(7,20),T2(7,20),T3(7,20)
COMMON/Q1/C80/Q2/JUMP/Q3/LOOP/Q4/SENS/Q5/IDATA/Q6/IOUT/Q7/IPASS/Q8
&/IRAN/Q9/IREAD/Q10/ISCR1/Q11/ISCR2/Q12/ISNAP/Q13/ITER/Q14/ITOG/Q15
&/IWRIT/Q16/IXXX/Q17/OK1/Q18/OK2/Q19/OK3/Q20/OK4/Q21/OK5/Q22/OK6
ITER=100
5 DO 10 IXXX=ITER
C
C INITIALIZE NUMBER OF PEGS
C
DO 200 J=l,100
B1(J)=1.
B2(J)=0.
200 B3(J)=0.
DO 100 L00P=1,20
C
C SELECTION FOR BOARD #1 TO BOARD #2
C
DO 300 J=l,15
I=DUNI(1.,100.)
IF(B1(I) .NE. 0.) B2(I)=1.
300 IF(B1(I) .NE. 0.) B1(I)=0.
C
C SELECTION FOR BOARD #2 TO BOARD #3
C
DO 400 J«=l, 5
I»DUNI(1.,100.)
IF(B2(I) .NE. 0.) B3(I)=1.
400 IF(B2(I) .NE. 0.) B2(I)=0.
C
C SUM PEGS ON EACH BOARD
C
A1(LOOP)=PWSUM(B1,0,,100.)
A2(LOOP)-PWSUM(B2,0100.)
A3(LOOP)-PWSUM(B3,0.,100.)
C
C ACCUMULATE NUMBER OF PEGS/BOARD PER HOUR IN A VICTOR
C
CALL VTABLE(A1,•(BOARD1)',20,T1)
CALL VTABLE(A2,1(BOARD2)',20,T2)
CALL VTABLE(A3,1(BOARD3)',20,T3)
100 CONTINUE
LOOP=l
CALL DTABLE
10 CONTINUE
END
-------
Mouse Manual Page 198
This is another terminating simulation since the operation defi~*
nitely is over when all of the pegs are moved to Board #3. The
initial conditions are the number of pegs on each board at the
start of the simulation. Note that it is not necessary to use
MOUSE Control lines 6 and 7 in this problem.
Chapter 11
1. Valid
2. Invalid: Trace lines for variables that appear within
a vector loop must be placed within the loop itself.
3. Valid.
-------
Mouse Manual Page 199
APPENOIX ¥
HOUSE RANDOM NUMBER TESTER
F.l Introduction
The results of any simulation based upon "random" numbers
are reliable only to the extent that the numbers are truly ran-
dom. Unfortunately, many of the random number generators in use
today are not very satisfactory, either because of intrinsic
weaknesses or because of the selection of an inappropriate seed
number. RANDOM is a MOUSE utility that subjects any set of
random numbers (up to 5000 numbers) to a series of empirical and
theoretical tests for randomness.
Input to the program consists of a file, called RANDOM.DAT,
containing the random numbers (i.e. numbers continuously dis-
tributed uniformly, 0 < X < 1) in any arbitrary format. It is-
suggested that the set contain between 1000 and 5000 numbers.
The following is a MOUSE program that will generate 5000 random
numbers for testing:
C
C RANDOM NUMBER GENERATOR
€
COMMON/Q1/C80/Q2/JUMP/Q3/LOOP/Q4/SENS/Q5/IDATA/Q6/IOUT/Q7/IPASS/Q8
&/IRAN/Q9/IREAD/Q10/ISCR1/Q11/ISCR2/Q12/ISNAP/Q13/ITER/Q14/ITOG/Q15
&/IWKIT/Q16/IXXX/Q17/QK1/Q18/OK2/Q19/OK3/Q20/OK4/Q21/OK5/Q22/OK6
CALL DATA
ITER-5000
5 DO 10 IXXX=1,ITER
R=RAN(IRAN)
CALL ACCUM(R,0.,»(RANDOM.DAT)•,»(RANDOM NUMBER SET FOR TESTING)•)
CALL DTABLE(2)
10 CONTINUE
GOTO(5,15), JUMP
15 CONTINUE
END
-------
Mouse Manual Page 200
This program will place the title, ."RANDOM NUMBER SET FOR TEST-
ING", and 5000 random numbers in a file called RANDOM.DAT.
The program is called merely by typing "RANDOM",and starts
by requesting three pieces of information:
(1) A title or comment line (this is different from the one
that is in the data file, RANDOM.DAT):
(2) The name of an output file (the default is RANDOM.OUT).
If the file exists, you will asked whether the file is
to be overwritten or not.
RANDOM will report in the output file the first 10 numbers that
were in the RANDOM.DAT file (for a quick validation that the
numbers were correctly read in). An equidistribution test, which
tests whether the numbers are indeed uniformly distributed over
the interval (0,1), is then automatically carried out. This test
involves dividing the interval (0,1) into five equal divisions
and computing the chi-square statistics with 4 degrees of freedom
and (in parentheses after the chi-square statistic) the probabil-,
ity level of the statistic.
A menu then is presented, offering the following nine op-
tional random number tests;
(1) Gap and Two-dimensional serial test
(2) Kolmogorov-smirnov test
(3) Poker test
(4) Permutation test
(5) Runs test
(6) Maximum-of-T test
(7) D-square test
(8) Serial correlations and correlograms
(9) Lag test
F.2 Gap Test
A gap is defined as 0
-------
Mouse Manual Page 201
F.3 Tiro-dimensional Serial Test
This test concerns the degree of randomness between succes-
sive pairs of numbers. It calculates a chi-square statistic with
24 degrees of freedom (and, in parentheses after the chi-square
statistic, the probability level of the statistic) and requires
no additional input from the user.
F.4 Rolmogorov-Smirnov Test
This is a nonparametric test and is applied in three differ-
ent ways. First, a KS test is performed on the entire sequence
and KN+ and KN- statistics are computed. Then the entire se-
quence is subdivided into ten subsequences and KS test is applied
to each of the subsequences, resulting in ten KN+ and ten KS-
statistics. Finally the KS test is applied to the sequence of
these ten KN+ and KN- statistics. No additional input is re-
quired of the user for this test. Following each KN statistic
the probability level of the statistic is reported (in parenthe-
ses) . Note: This is a lengthy test.
F.5 Poker Test
The poker test the randomness of digits of random numbers.
This test can be performed either on a single digit or on more
than one digit at a time. The routine performs the poker test on
the first digit (the most significant digit in a uniform random
number), the second digit, and the first two digits combined and
the first three digits combined. In order to carry out this test
the entire sequence is subdivided into n-tuples. In the RANDOM
program, n can be any number between 2 and 11. The program
requests this number from the user and can be repeated for dif-
ferent values of n. Chi-square statistics (and their associated
probability levels) are reported.
F.6 Permutation Test
This test is also performed on n-tuples (n =<6) and is based
upon the premise that in each n-tuple the elements may have n!
possible orderings. The number of times each ordering is met
with is recorded and a chi-square test with n! degrees of freedom
is applied. For example, for n=3, a,b and may have six possible
orderings (e.g., a
-------
Mouse Manual Page 202
F.7 Runs Test
This is one of the most often applied tests. The version
supplied in RANDOM is based on the length of the monotone subse-
quences of the original sequence, i.e., segments that are in-
creasing or decreasing. It is of two types: Runs Up and Runs
Down. For example, in the sequence 1,2,4,7,2,1,3, Runs Up are
/l,2,4,7/2/1,3/ and Runs Down are /1/2/4/7,2,1/3. The length of
the segments are called the "length of the runs". Chi-square
statistics (and their associated probability levels) are report-
ed. The test requires no additional input from the user.
F.8 Maximua-of-T Test
This test is based upon the distribution of
Vj=max(rtj ,rtj+1,.,rti+t-l) where the r's are a segment of the
set of random numbers to be tested. The KS test is then applied
to the sequence, Vq, Vif# v2,' • • • vn-l• The maximum test of T
uniform random numbers is'also'called the test of t-tuples and is
considered to be a more stringent test than the basic equidistri-
bution test. Kolmogorov-Smirnov statistics (and their associated
probability levels) are reported. The test can be repeated with
different values of T. Note: Like the Kolmogorov-Smirnov test,
this is a lengthy test.
F.9 D-square fast
This test is based on the fact that consecutive random
numbers may be considered as coordinates of two points and the
Euclidian distance between the two points can be obtained. Chi-
square statistics can be computed based on the differences be-
tween the observed and the expected frequencies of these dis-
tances. chi-square statistics (and their associated probability
levels) are reported for the test. The test requires no addition-
al input from the user.
7.10 Serial Correlations and correlograms
Serial tests are used to check the degree of randomness
between successive numbers in a sequence. In RANDOM, the serial
correlation (a number between -1 and 1), its expected value and
95% confidence interval are computed. Also computed are the
autocorrelations for a specified lag, K (K=<50). These autocor-
relation values may be plotted by the user against the lag length
to prepare the correlogram. Different values of K can be tested
by the user.
-------
Mouse Manual Page 203
F.11 Lag Test
Somewhat related to the serial correlation (which is a
measure of the amount that rj+i depends upon r j)), the lag test
involves the sum of lagged products, r^ri+fc. If there is no
correlation between r^ and ri+fc, the test statistic is distribut-
ed normally. RANDOM reports the test statistic (and, in paren-
theses, its associated probability level), and its expectation
and variance.
If a set of random numbers does not pass all or most of the
tests available in RANDOM, the user might try changing the seed
value, IRAN (see Appendix B, Section B.l). RANDOM was applied to
5000 random numbers generated by the random number generator
distributed with MOUSE, using the program shown at the beginning
of Appendix E and the seed value of 23971. It passed all tests
except the Kolmogorov-Smirnov test for the KN+ subsequence
number 10 (significance « 0.07) and for KN+ on the Maximum-of-T
test (significance level = 0.02).
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Mouse Manual Page 204
INDEX
Accumulation Line, 63,108,116,119,199
Adding Modules (see Modules, Adding)
AGAUL, 46
Algebraic Equations, 1
Arithmetic Operations, 35-38
Autocorrelation
Exponential Autocorrelated Function, 118-119
General, 116
IMP Analysis, 118-119
Linear Autocorrelated Function, 117-118
Average Opportunity Loss, 79
Average Value Approach, 2
Bayesian Methods, 167
Best Value Approach (see Average Value Approach)
BETA, 46
Bets (see Gambles)
BGAU1, 110
BGAU2, 110
BGAUL1, 112
BGAUL2, 112
BIN, 49
Bivariate Distributions
Analysis, IMP, 107-109
Analysis, Bivariate Normal & Lognormal, 112-116
Lognormal, 112-114
Normal, 108-112
Blocks, 31-32,35
C80, 89
CALL ACCUM, 63,108,116,119,199
-------
CALL
AUTOE, 118
CALL
AUTOL, 117
CALL
DTABLE, 82
CALL
DTRACE, 147
CALL
ITRACE, 147
CALL
LIST, 95
CALL
MREAD, 90
CALL
NAME, 93
CALL
PLOT, 97
CALL
READ,41
CALL
SEN, 102
CALL
TABLE, 80-82
CALL
VTABLE, 126
Camp-
•Meidell Inequality, 81
CAPREC, 73
Case History, 141
Cash Flow, Non-Uniform, 121
CHECKER, 137-141
Chi-Square Test, 162
Classical Probability Distributions
(see Probability Functions)
Coefficient of Determination, 99
Coefficient of Variation, 23,79
Column Output Line, 88
Comment Lines, 31
Compiling, 176,181-182
Complete Enumeration (see Direct Enumeration)
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Conservative Approach, 2
Continuation Lines, 38
Control Lines
MOUSE, 31
Pass, 92
Summary, 185
Vector Loop, 124
Correlated Stochastic Variables
Bivariate Lognormal, 112-114
Bivariate Normal, 108-112
Chaining, 111
Linear, 106-108
Correlation (see Correlated Stochastic Variables)
Correlograms, 202
Correlation Coefficient
Bivariate Lognormal, 112
Bivariate Normal, 110
Linear, 106
Correlation Function, 106,110-114
CSPW, 76
CUNI, 45
CUPW, 76
Customization of MOUSE, 180-181
Data Files, IMP, 56-57
Data Files, MOUSE, 41-43
Data, F3ttt^ng Dist nbut,xons to
(see Probability Distributions, Fitting)
Debugging, 147
Decimal Points, Use of in MOUSE, 35-36
Delphi Method, 170
Device Number (see Logical Unit or Device Number)
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Mouse Manual Pa^e 207
DIMENSION
Block, 31
Empirical Variable, 41
Matrix, 90
Plot, 98
Table, 80
Vector, 91
Vector Table, 129
Direct Enumeration, 5
D-Square Test, 202
DTRA, 51
DTRI, 50-51
Dummy Tables, 82
DUNI, 50
EBETA, 46-47
ECTS, 40-42
ED1S, 40—42
Elicitation of Subjective Probability Distributions, 163-168
Empirical Probability Functions
(see Probability Functions)
Empirical Variable
Data Line, 41-42
Dimension Line, 41
Read Line, 41
Enumeration (see Direct Enumeration)
Equations, MOUSE, 37-38
Equidistribution Test, 200
Equivalent Prior Sample Method, 166
ERL, 49
Errors, 137
Error Messages, 137,189-191
Executable File, 176-177
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Exercises, Answers to, 192-198
Expected Value of Perfect Information, 79
EXPON, 45
Fitting Probability Distributions to Sample Data
(see Probability Distributions, Fitting)
FORTRAN, 13—14,176
Foutz Statistic and Critical Values, 60-61
Frequency Distributions (see Histograms, Examples of)
Functions, Algebraic, 1
GAM, 47
Gambles, 167
Gap Test, 200
GAU, 45
GAUL, 46
GCA,74
Goodness-of-fit Tests, 60,99,162-163
GOTO Statement, 52
GPW, 74
Group Decision Processes, 170
Group Feedback and Individual Reassessment, 170
Group Reassessment, 170
Histograms, Examples of, 26,30
HYP, 50
IF Statements, 52-53,101,125
IMP
Autocorrelation Analysis, 63
Bivariate Distribution Analysis, 114-116
Fitting Distributions to Sample Data, 55-63
Integrating Empirical Continuous Distributions, 63-66
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Mouse Manual Page 209
Installation of MOUSE, 180
Interest Functions
Capital Recovery Factor, 73
General, 71
Gradient Series Equivalent Uniform
Annual Amount Factor, 74
Gradient Series Present Worth Factor, 74
Present Worth Sum, 123-124
Single Payment Compound Amount Factor, 71
Single Payment Present Worth Factor, 72
Single Payment Present Worth Factor with
Continuous Compounding, 76
Sinking Fund Factor, 73
Uniform Series Compound Amount Factor, 73
Uniform Series Present Worth Factor, 72
Uniform Series Present Worth Factor with
Constant Percentage Change, 72
Uniform Series Present Worth Factor with
Continuous Compounding, 76
Interest Rates
Compound,74
Continuous, 75-76
Effective, 74-77
Force of Interest, 76
Nominal, 74-77
IOUT, 88
IPASS, 92,101
IRAN, 86
ISNAP, 86
ITER, 85
Iteration Number Line, 85
Kolmogorov-Smirnov Test, 61,162,201
Kurtosis, 129-131
Lag Test, 203
Languages, Computer
General Purpose, 14
Simulation, 9
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Level of Significance (see Significance, Level of)
Libraries, 176-177,181-182
Line Types, MOUSE, 188
Linking, 176,181-182
Listing a Program File, 95
Listing File, 177
Literals, 83
Logical Unit or Device Number, 88,180
Loop, Vector, 121,124
Mathematical Combination, 169
Matrix
Data Lines, 90-91
Dimension Line, 90
Read Line, 90
Reading Data from Program, 90
Maximum-of-T Test, 202
Model Sampling, 7
Models, Mathematical, 1
Modifications to Distribution Programs
Hexadecimals, 180
Logical Unit or Device Numbers, 180
MOUSE Random Number Seed, 181
OK6, 143,181
Modules, Adding
Probability, 183
Interest, 183
Monte Carlo Simulation, 7
MOUSE.HLP, 159
MSHELP, 158
Multiple Runs, 89-93
Name Line, 93
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Names
Literal, 83
Program, 93
Restricted, 38
Variable, 35-36
Non-linear Functions, 3,15
Non-Uniform Cash Flows, 121
Null Hypothesis, 61
object File, 176
0K6, 143,181
Output Destination Line, 88
Overflow, Arithmetic, 137
Parameter, 89,101
PAS, 50
Pass,
General, 92-93,101
Control Line, 92
Permutation Test, 201
Plots
Dimension Line, 98
IF Statements in, 101
Output Line, 97
Scaling, 99
Vectors in, 13 5
POI, 49
Point Estimates, 1
Poker Test, 201
Present Worth Sum Function, 123-124
Probability Calculus, 7
Probability Distributions, Fitting, 55-63,160
Probability Distributions, Integrating, 63-66
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Mouse Manual Page 212
Probability Functions
Beta, Empirical Form, 46-47
Beta, Standard Form, 46,173
Binomial, 49
Bivariate Lognormal, 112-114
Bivariate Normal, 108-112
Composite, 52-54
Empirical, Continuous, 40-42,172
Empirical, Discrete, 40-42,172
Erlang, 49
Exponential, 45
Gamma, 47
Hypergeometric, 50
Lognormal, 46
Lognormal, M-form, 46
Normal,45
Pascal/Geometric, 50
Poisson, 49
Pseudo-Continuous, 63
Step Rectangular, 43-44,174
Trapezoidal, 48,172
Trapezoidal, Discrete, 51,172
Triangular, 47,172
Triangular, Discrete, 50-51,172
Truncated, 51-52
Uniform, Continuous, 4 5
Uniform, Discrete, 50
Weibull, 183
Probability Generators, Methods, 55
Program Block, 31-32,35
Program Listing Line, 95
Programs, Examples (see Waste Oil Reclamation
Project Problem and Regulatory Problem)
Project Life, Stochastic, 124
Propagation of Error, 7
Pseudo-Continuous Probability Distribution, 63
PWSUM, 123-124
R-Square, 99
RANDOM, 199
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Random Number
Generator, 199
Seed, 86
Tester, 199
Random Variables
Continuous, 40-43
Discrete, 40-43
Read Block, 31-32,35
Read-Only Files, 182
Regression, 99
Regulatory Problem
Basic Problem, 22-30
Modification #1, 98
Restricted Names, 38
Rules of MOUSE, 35-38
Running Programs
CHECKER, 177
IMP, 177
MOUSE, 177-179
TRACER, 177
TRAP, 177
Run-Time Error Messages, 143
Runs Test, 202
Salvage Value, 125
Sample Data, Fitting Distributions to, 160
SCA, 71
Scaling
Plots, 99
Tables, Standard, 83
Tables, Vector, 129
Seed Reset Line, 86 .
Semi-Empirical Probability Function
(see Step Rectangular Function)
SENS, 104
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Mouse Manual Page 214
Sensitivity Analysis, 2,9,101-105
Sensitivity Control Line,
General, 102
Use with correlated variables, 111-112
Sensitivity Level Line, 104
Serial Correlations and Correlograms, 202
Significance Levels, 60,162
Simulation
Monte Carlo,7
Steady State, 85-86
Terminating, 85-86
Single Value Approach, 1
SINK, 73
Skeleton Program Lines, 157
Ske«mess, 129-131
Snap Interval Line, 88
Source File, 176
Specification Block, 31-32,89,95
SPW, 72
Steady State Simulation, 85-86
STEP, 43—44
Stochastic Variable
Data Line, 41-42
Dimension Line, 41
Direct Method, 39,125,126
General, 39-40
Indirect Method, 39
Read Line, '41
Subjective Probability Distributions
Biases, 163,168-169
Elicitation Methods, Improving, 168-172
Fixed Interval Methods,. 165
Miscellaneous Methods, 166
Portrait Approach, 174
Useful Distributions for, 172-174
Variable Interval Methods, 163
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Mouse Manual Page 215
Tables, Dummy, 82
Tables, Mixing, 84
Tables, Standard
Dimension Line, 80-81
General, 79
MOUSE-dimensioned, 79-81
Output Line, 80-81
Scaling, 81
Short Form, 82
User-dimensioned, 79,81-82
Vector, use in, 135
Tables, Vector
Dimension Line, 129
Output Line, 126
Scaling, 129
Tchebycheff Inequality, 81
Terminating Simulation, 85-86
Titles (see Name Line)
TRA, 48
Trace Lines
Decimal, 147
u^t^icj^^r, 147
TRACER, 151
mpip Tics i nn iki.i ca
xlvnlr | wsXil^j f X3x Xvv
TRI, 47
Troubleshooting, 137
Two-dimensional Serial Test, 201
UCA, 73
Uncertainty
Failure to Deal With, 1
Nature of, 9
Solutions to, 5
Traditional Approaches, 2-3
Uncertainty Analysis, 2-9
UPW, 72
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Mouse Manual Page 216
UPWP, 72
Variables
Decimal, 35-36
Integer, 35-36
Stochastic, 39
Variance, 3
Vector
Dimension Line, 91,123
Elements, Calculating of, 124-125
Plots, Using Vectors in, 128,131
Substitution for Standard Table, 131-135
Vector Loop
IF Statements in Vector Loops, 125
LOOP, 124
Order of Statements in Vector Loops, 124,126
Standard Tables and Plots, prohibited in, 135
Waste Oil Reclamation Project Problem
Basic Problem, 18-25,33-34
Modification #1, 87
Modification #2, 122
Modification #3, 134
Weighing of Several Estimators, 169-170
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