EPA-430/1-73-016
AN INTRODUCTION TO GUMBEL, OR
EXTREME-VALUE PROBABILITY PAPER
TRAINING MANUAL
US ENVIRONMENTAL PROTECTION AGENCY
WATER PROGRAM OPERATIONS
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EPA-430/1-73-016
September 1973
AN INTRODUCTION TO GUMBEL, OR EXTREME-VALUE
PROBABILITY PAPER
by
Joseph F. Santner
Mathematical Statistician
Environmental Protection Agency
Library Systems Branch, Room 2903
401 M Street, S.W.
Washington, :D.C. 20460
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OD
U. S. ENVIRONMENTAL PROTECTION AGENCY
Water Program Operations
NatJoaal Training Center
Cincinnati, OB 45268
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AN INTRODUCTION TO GUMBEL, OR EXTREME-VALUE,
PROBABILITY PAPER
The use,of probability paper as a rough, ready, and rapid graphical
hand tool in the analysis of frequency type data is an accepted technique,
generally. Although this is the computer age, in some situations manual
data analysis is justified.
Working with a small sample is one example, before the data cards
could be punched, the manual analysis would be completed.
Graphical manual analysis would be practicable in the study of a new
.product, process, etc., where no prior knowledge is available. The aim
is to uncover the underlying unknown distribution, which could entail much
trial and error.
A third use of probability paper analysis is to obtain rough estimates
of population parameters. As will be shown later, obtaining these is
simple once the data is plotted and the regression line of best fit is drawn.
There are many probability papers commercially available that can
be used for the purposes outlined above. Most of them are designed for
continuous type data. They are normal, log-normal, Weibull, Gumbel
or extreme value, log Gumbel or log extreme value, logistic, and
reciprocal.
Two others, of a different type, are the binomial and Poisson. The
latter is used for analyzing discrete data that are Poisson distributed.
The binomial paper is useful in testing different types of hypotheses, and
can be used also to handle other type problems too numerous to mention
here. Of the papers listed, the binomial has the greatest utility.
This paper is another in a series designed to introduce graphical
analysis, utilizing probability papers, to those who have not been exposed
to this technique.
The first usual attempt is to plot the data on normal probability
paper. If the plotted data fall in a straight line, the data are normal.
they are not normal, the analysis continues.
If
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An Introduction to Gumbel, or
The graph on normal paper generally points to the next move. If the
data plot as concave up (horizontal axis is percentage or probability), the
data are positively skewed or have a long-right tail. The next attempt at
linearization is to plot the data on log-normal probability paper. If the
data'plot concave up on log-normal, the data are more skewed than log-
normal, arid Weibull probability paper or log-extreme probability paper
should be used in the next attempt. If the data when plotted on log-normal
are concave down (or convex), the data are less skewed than log-normal,
and Gumbel or extreme-value probability paper should be tried next.
If the plotts on normal paper are concave down, the curve is skewed
left or has a long-left tail. Interchanging the order of the data will
'change them from skewed left to skewed right, and the interchanged data
can be plotted on log-normal or extreme-value paper.
''' In this paper we give an introduction to Gumbel, extreme-value dis-
tribution type I, or double-exponential distribution. It has been used
very successfully"in many disciplines. Some of the applications found in
reference 2 are (1) floods, (2) aeronautics, (3) breaking strength of ma-
terials, (4) life tables, (5) extention time for bacteria, (6) radioactivity,
and (7) stock market. Some other applications found in the literature are
(1) rainfall, (2) temperature, (3) crop yields, (4) crop-hail losses, and
(5) design of waterway bridges.
Gumbel (1954) cites a specific example of the use of extreme-value
paper in an analysis of the floods of the Colorado River at Black Canyon
from 1878 - 1929. From the analysis, he concludes that the return period
of a flood of 400, 000 cubic feet per second (design flood for the dam at that
point) is about 3, 500 years. The probability is 2/3 that the design flood
would occur between 1, 100 and 11, 000 years from 1929. This appears to
be a case of overdesign. •
The study of the statistical theory of extreme value has resulted in
three distributions, which have been labeled types I, II, and III by Gumbel
(1958).
The mathematical expression for the Gumbel type I cumulative dis-
tribution is
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Extreme-Value, Probability Paper
-e"y
F (y) = e e ,
(1)
where y is any real number and is a reduced variate. It plays the same
role as the reduced variate z does in normal theory. The transformation
from the original variate x to the unit normal is given by
z = (x - H.)
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An Introduction to Gumbel, or
By way of comparison, (3 ± = 0 for normal and p 2 = 3- One
that there is a marked difference in the value of these two parameters.
are:
Some values for key characteristic measures of the original variate
•\.
mode (x) = u
mean (x) = u+ 0. 57722/«
mediants:) = u + 0.3665/a
standard deviation =
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Extreme-Value, Probability Pager
relationship between y and x is linear in x and y; therefore, plotting y
versus x will result in a straight line (see Figure 2).
In sampling, « and u are unknown and, hence, y is unknown. In
place of y, one uses its probability scale. Its probability is a function
of its ordered value and sample size only. The value is determined by
the formula
plotting position (as a %) -
Here i = 1 for the smallest ordered value, and i = n for the largest
ordered value.
In Figures 3 and 4, the top horizontal scale is labeled "return period."
Formally, it is defined as follows:
return period = T .. .•..•:. .
^ 1 - probability
As stated by Gumbel, "it (return period) is the number of observations
such that, on the average, there is one observation equaling or exceed-
ing x. "
Since the probability for the median or less is 1/2, its return period
is 2.
return period = -r TJI
Given two observations, on the average, there is one observation
equaling or exceeding the median.
Another use which seems to justify the name is when observations
are made over constant time intervals. In these cases, return period
is related to time. In Figure 3, some fictional air pollution data were
plotted, and the regression line of best fit was drawn. Each point rep-
resents the maximum value of SC>2 measured in a large American city
during a 1-year period. Extrapolation of this or any data is hazardous,
but is often a necessary and reasonable request. In this case, the prob-
lem can be stated in two ways. The first is as follows. How long will it
be before SC>2 reaches 40 ppm?
In Figure 3, we draw a horizontal line from 40 ppm on the vertical
scale until it reaches the line of best fit. This point is projected vertical
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An Introduction to Gumbel, or
upward' to the return-period scale. The value there is 25. This means
on the average, within 25 years, we expect to reach a value of 40 ppm.
Precisely which year it will be, is lost. We are not treating these data
as a time series, but as a frequency distribution.
The phrase "on the average" should be carefully noted and interpreted
as repeated experiments of the time to reach 40 ppm would average 25
years. The elapsed time, therefore, could be as short as 1 year or occur
on successive years. For others the gap could be much longer than25
years.
A second use of the return period for extrapolation could be deter-
mining how high SO2 levels will go during the next 50 years. In Figure 3,
we start at return period 50, project vertical to the line of best fit, and
read the corresponding ppm from the vertical scale. Some time within
the next 50 years, on the average, one expects an SCX level of about 43
ppm.
We next consider plotting data on Gumbel paper.
considered.
Two cases are
First, the number of data points is small enough to hand plot values
obtained in Table 1. Table 1 gives the plotting position for each value in
the sample from size 2 to 50. These numbers represent the values re-
commended by Gumbel (1958).
Let the n original variates or measurements be X^, %2, ... Xn>
Order them from smallest to largest. This is written notationally as
X(D' X(2)'
X
(n)
where X(^ is the smallest value and X(n) is the largest. The plotting
positions for the ranked data are arranged in columns in Table 1. An
example of their use is given in Table 2 and plotted in Figure 5. A
straight line represents the data; hence, the distribution is Gumbel type I.
For rough and ready preliminary investigation and sample sizes
greater than 50, the following technique works well and can be done easily
by hand. If there are from 51 to 100 data points, plot every other one.
If there are from 101 to 150, plot every third one, etc. Plotting more
than 50 points for a preliminary investigation appears to be too much time
wasted. To some, 50 is too high and not more than x (the reader should
supply his own value for x) should be plotted.
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Extreme-Value, Probability Paper
Personally, if I had 134 points, then the plotting position for the third
ordered value, or first point plotted, is:
plotting position (i = 3; n • 134) ~
= 2.222.
Since the plotted points are all equally spaced, adding 2. 222 to the
first plotting position gives the plotting position for the 6th ordered value.
This repeated addition can be done easily by desk calculator or by hand.
After the 44th point (or the 132nd ordered value) is plotted, both ends
of the distribution can be plotted by calculating the individual plotting
positions. In this example, plot 1st, 2nd, 133rd, and 134th ordered
values by hand. This example is illustrated in Table .3.
Generally before plotting end values, one can decide subjectively
whether data are approximately linear or not, hence, those points can be
omitted.
Another labor saver that can be used is the rounding of the plotting
positions. Consider the problem of plotting 160 points. Every 4th value
would be plotted, and the first plotting position is
plotting position (i B 4, n = 160)
10Q
161
This is very close to 2. 50, so close that one cannot plot each separately.
The plotted points are put 2. 50 apart up to 50.0. Next, we work from
the other end to the middle by fixing the end point and subtracting until
the middle is reached. The largest errors are near 50.0 where small
errors (50.0 versus correct 49.69) cause essentially no effect. In this
case, nothing is added or lost by omitting the 84th ordered value. See
Table 4, column 2, for these values.
• The reader might object to starting at both ends and working toward
the middle since more work is involved by doing it this way. Some
answers to this objection are as follows.
In this particular example, the difference between the true value and
the approximation was very small. Hence, the difference in all plotting
positions between the two values (correct versus approximate) was
essentially the same whether one started at both ends and worked toward
the middle, or started at one end and worked toward the other. In some
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An Introduction to Gumbel, or
problems where the approximation requires much rounding,, the, difference
between the two techniques is very marked.
Comparison of columns 2 and 3 of Table 4 exhibits another objection
to starting at one end only. The point at the 160th ordered value cannot
be plotted since the 100 percent point is absent on probability paper. If
the rounding resulted in values greater than 100 percent, the result would
be more incorrect. . . .; . ,
Another point to be noted is that when one works from end points
toward the middle, there are no.errors at the tails and the largest error
is in the middle. This means that an error on tails represents a larger
horizontal distance or incorrect displacement. The same size error near
the 50 percent point would be a small horizontal displacement. This is
the nature and construction of the probability scale.
-,*.-. \ - ' . '•
The regression line is a function of the horizontal errors, and these
errors should be minimized since the parameters are graphically esti-
mated from the line. The line would.Jbe most affected by larger errors
at one end only. , . . .
There are no hard and fast rule's except speed, efficiency, and using
the minimum of data that you need to represent the true situation.
Sometimes one has to plot a frequency distribution where the indivi-
.dual values are lost. . One solution is to plot cumulative frequency versus
end of the class interval.. This is shown in Table 5, where the last inter-
val cannot be plotted. One plots A at 5 percent, 9 at 40 percent, etc. The
last interval, of course, is lost. Plotting of individual values is preferred.
After the data have ....been plotted and are subjectively determined as
linear, the regression line of best fit is drawn.
The well-known analytic technique of minimizing squared deviations
is normally used. This method cannot be applied when one subjectively
draws the line. Instead, the line must be drawn so that the sums of the
absolute values of the deviations above and below the line are equal.
In practice, this is more difficult to do than to describe. Ferrell
(1958) suggests, the following excellent iterative method. Use a trans-
parent,straight line (like the edge of a draftsman triangle) to obtain the
initial line of what looks like a.best fit. .This is the first iteration, and
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Extreme-Value, Probability Paper
the line should not be'( drawn. Fix a pivot point on this straight edge in
the iower left-hand corner of the graph near the smallest ordinal value.
Rotate the straight edge about the initial pivot. The pivoted transparent
straight edge is rotated to obtain a best fit for the right half of the data,
i.e., the data on the right side of the graph paper only; This is the
second iteration, and this line also should not be drawn. While keeping
the straight edge fixed after the second line of best fit has been deter-
mined, shift the pivot point to the upper right-hand corner nearest the
largest value. Rotate the straight edge around the second pivot point to
obtain a line of best fit for the left half of the data only. This is the
third iteration, and this line is drawn. The fourth iteration consists of
shifting the pivot from right to left. Again the straight edge is rotated
about the third pivot point, minimizing the distance on the right half of
the data again. This line is generally coincident with the previous one.
If not, it is drawn, and the previous one is discarded. Other iterations
are made until two coincident lines in a row are obtained. Generally,
three or four iterations, at most, are needed, since no further adjust-
ment is made on the drawn straight line. The big advantage of this method
is that- one cuts in :half the data points that are being minimized, by eye,
with respect to deviations.
- At this point, the goodness of fit is accepted subjectively, or a test
for-linearity is made. One nonparametric test appeals to run theory.
The method is described by Crow, et al. (1960). He used it in a different
situation, but it applies here also.
The graphical estimate of the parameters, after the regression line
of best fit has been drawn by eye or subjectively, is quite simple.
Solving for x from equation (2), we get
(3)
If y = 0, this equation reduces to
x = u.
In Figure 3 at y = 0 (0. 360 on the probability scale), one reads that
: ' ' " 26.4 = u. '
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10
, An Introduction to Gumbel, or
If one substitutes ,y = 1 in equation 3, the. result is
, 1'' ' ' : ' ' '• ' ' '
x = u • +,—
a
Since u is known and x for y = 1 can be read from the.regression line,
the only unknown is> l/a. From Figure 3 for y =, 1 (or probability = 0. 692),
we obtain . : •
.- .30.6 - 2.6.4 = -5- . '
; • .•• • -4.-2.f" :
• These, of course, are graphical estimates. Analytical estimates can
be, obtained by using the work sheets on pages 50 and 51 with Table 3. 3 on
page ;31 of Gumbel (1954). . . .
= 'i - Following the work sheets, one not only gets the regression line of
best fit but confidence bands for the .regression line, as well as expected
values for selected return periods.
There are cases when the original variate is not a Gumbel distribution,
but some transformation will make it so. -A log transformation is accom-
plished by plotting the original data on the graph paper (shown in Figure 4),
and then working with the transformed data.
If the data do not plot as a straight line, then the shape of the distri-
bution is still to be determined. Figures 6.to 12 give the results of plotting
on double exponential when the underlying distributions is not double ex-
ponential. A conclusion that can be drawn from these graphs is that a
finite distribution terminates at both ends in a horizontal line.
At times, data are recorded incorrectly, either being too large or
small. The results of graphing data with the presence of outliers is
shown in Figure 13. Curves A and C have outliers at one end, and in B
they are present at both ends.
Another type of error is a truncated distribution in which data above
or below a threshold value are lost or not recorded. This result is shown
in Figure 14 where A and C again depict a loss at one end and curve B shows
a lose at both ends. Misclassified data behave in the same way and would give
the same sort of picture. In curve A, themisclassificationis at the one end,
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Extreme-Value, Probability Paper
11
that is, data are recorded at a higher value than they should be. Analogous
interpretations should be given for curves B and C in Figure 14 for mis-
classified data.
There are times when the data plotted are actually the sum of two
Gumbel distributions. Such a result is shown in Figure 15. Curve A is
the result of adding two distributions whose ratio of sample size is roughly
2 to i as shown by the length of the two straight line segments. Each of
these could be used to estimate the respective parameters. In curve B,
the contribution of each distribution to the sum distribution is about equal.
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Table 2. EXAMPLE OF PLOTTING ON GUMBEL PAPER
oampj
Ordered original
variate
7.6
8.1
10.2
11.6
13.4
13.7
17.0
17.4
22.8
ie uaia.
Plotting
from Ta
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20
30
40
50
60
70
80
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position
ble 1, %
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Table 3. APPROXIMATE PLOTTING OF A SAMPLE OF SIZE 134
Ordered value
1
2
3
6 By
addition
9 I
: I
129 T
132
133
134
Plotting position
as a percent
100 (1)
135
100 (2)
135
2.222 =
4. 444
6.666
95.548*.;,
97.768
100 (133)
135
100 (134)
135
0.74a
1.48a
100 (3)
135
^ 98. 52a
= 99.26a
Calculated separately to plot end values
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Table 4. APPROXIMATING PLOTTING OF A SAMPLE OF SIZE 160
Plotting position as a Plotting position as a
Ordered value percent working from percent working from
both ends to the middle one end only
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80
84
88
*
156
160
100 (4) ^
P-tr " ° AfiAAT* % ° R Pv ° 'S
jay ifii ^.*tp*^i^^'
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Table 5. PLOTTING A FREQUENCY DISTRIBUTION
Class
interval
1
5
10
15
20
- 4
- 9
- 14
- 19
- 24
Frequency
10
70
84
32
4
Percent Cumulative
frequency frequency
5
35
42
16
2
5
40
82
98
100
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REFERENCES
1. Crow, E.L., et al. Statistics Manual. Dover Publications, Inc.
New York, New York. 1960.
2. Ferrell, E. B. Plotting Experimental Data on Normal or Log-
Normal Probability Paper. Industrial Quality Control. July 1958.
3. Gumbel, E. J. Probability Tables for the Analysis of Extreme-Value
Data. Natl. Bureau of Stand. Appl. Math. Ser. No. 22. 1953.
4. Gumbel, E. J. Statistical Theory of Extreme Values and Some
Practical Applications. Natl. Bureau of Stand. Appl. Math. Ser.
No. 33. 1954.
5. Gumbel, E. J. Statistics of Extremes. Columbia University Press.
New York, N.Y. 1959.
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