Virtual Beach 3.0.4: User's Guide
Mike Cyterski1, Wesley Brooks2, Mike Galvin1, Kurt Wolfe1, Rebecca Carvin2, Tonia
Roddick2, Mike Fienen2, Steve Corsi2
'National Exposure Research Laboratory
USEPA
960 College Station Road
Athens, GA 30605
2U. S. Geological Survey
Wisconsin Water Science Center
8505 Research Way
Middleton, WI 53562
Virtual Beach 3
Software for Developing Empirical Models of
Pathogen Indicators in Recreational Waters
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TURNING
DATA
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DECISIONS
USEPA
Office of Research and Development
Ecosystems Research Division
Athens, GA
USGS
Wisconsin Water Science Center
Middleton, WI
Wisconsin DNR
Madison, WI
Release 3.0.4
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Table of Contents
1. Introduction 4
1.1 On Predictive Modeling 4
1.2 Recommended User Background 5
1.3 General Overview 5
1.3 History of VB 6
2. Composition and Installation 9
3. Operational Overview 10
4. Project Management 12
5. Location Interface 13
5.1 Finding a Beach 13
5.2 Defining the Beach Boundaries for Orientation Calculation 14
5.3 Saving Beach Information 15
6. Global Datasheet 16
6.1 Data Requirements and Considerations 16
6.2 Importing a Dataset 17
6.3 Validating the Imported Data 18
6.4 Working with a Dataset after Validation 22
Scatter Plot Interpretation 23
6.5 Computing Wind, Wave and Current Components 25
Notes on Component Calculations 26
6.6 Creation of New Independent Variables 29
6.7 Transforming the Independent Variables 31
Plotting Transformed IVs 33
6.8 Singular Matrices and Nominal Variables 34
6.9 Saving Processed Data 35
6.10 Proceeding to Modeling 35
7. Multiple Linear Regression Modeling 36
7.1 Selecting Variables for Model Building 36
7.2 Modeling Control Options 37
7.3 Linear Regression Modeling Methods 38
7.4 Using the Genetic Algorithm 41
7.5 Evaluating Model Output 42
7.6 Viewing X-Y Scatter plots 46
7.7 ROC Curves 47
7.8 Residual Analysis 47
Viewing the Data Table 51
7.9 Cross-Validation 53
7.10 Report Generation 53
8. Partial Least Squares 56
8.1 Data Manipulation 56
8.2 Selecting Variables for Model Building 57
8.3 The Regulatory Standard 58
8.4 Modeling Control Options 58
Dropping Unimportant Variables 59
Setting the Decision Threshold 59
8.5 Diagnostics 60
9. Generalized Boosted Regression Modeling 62
9.1 Data Manipulation 63
9.2 Selecting Variables for Model Building 63
9.3 The Regulatory Standard 64
9.4 Modeling Control Options 65
Dropping Unimportant Variables 65
Setting the Decision Threshold 66
9.5 Diagnostics 67
10. Prediction 69
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10.1 Model Statement 69
10.2 Model Evaluation Thresholds 69
10.3 Prediction Form 70
10.4 Column Mapping of Imported Data 7 0
10.5 Viewing Plots 74
10.6 Prediction Form Manipulation 75
10.7 Importation of EnDDaT Data 75
11. User Feedback 77
12. References 78
13. Acknowledgments 7 9
Appendices 80
A. 1 Transformations 80
A. 2 Singular Matrices and Nominal Variables 82
A. 3 MLR Model Evaluation Criteria 84
A. 4 Changes from version 3 to 3.04 85
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1. INTRODUCTION
Virtual Beach version 3 (VB3) is a decision support tool that constructs site-
specific statistical models to predict fecal indicator bacteria (FIB) concentrations at
recreational beaches. VB3 is primarily designed for beach managers responsible for
making decisions regarding beach closures or the issuance of swimming advisories due to
pathogen contamination. However, researchers, scientists, engineers, and students
interested in studying relationships between water quality indicators and ambient
environmental conditions will find VB3 useful. VB3 reads input data from a text file or
Excel document, assists the user in preparing the data for analysis, enables automated
model selection using a wide array of possible model evaluation criteria, and provides
predictions using a chosen model parameterized with new data. With an integrated
mapping component to determine the geographic orientation of the beach, the software
can automatically decompose wind/current/wave speed and magnitude information into
along-shore and onshore/offshore components for use in subsequent analyses. Data can
be examined using simple scatter plots to evaluate relationships between the response and
independent variables (IVs). VB3 can produce interaction terms between the primary IVs,
and it can also test an array of transformations to maximize the linearity of the
relationship between the response variable and IVs. The software includes search routines
for finding the "best" models from an array of possible choices. Automated censoring of
statistical models with highly correlated IVs occurs during the selection process. Models
can be constructed either using previously collected data or forecasted environmental
information. VB3 has residual diagnostics for regression models, including automated
outlier identification and removal using DFFITs or Cook's Distances.
1.1 On Predictive Modeling
Empirical/statistical modeling outperforms persistence models (using the most
recent FIB concentration as the sole predictor of the next FIB concentrations) at beaches
where conditions such as weather, water characteristics, and human/animal density levels
change significantly day to day (Frick et al. 2008, Brooks et al. 2013). Virtual Beach
constructs models that can predict a dependent or response variable (i.e., FIB) by using
variables to describe current environmental conditions that can be measured or estimated
in a timely manner. These are referred to as independent variables (IVs) and often
include beach water parameters such as turbidity, water temperature, specific
conductance, or wave height; parameters monitored and made available via the web such
as rainfall, stream flow, and stream water quality; and parameters estimated by
environmental models such as water currents, wave height and direction, and radar
rainfall.
In any predictive modeling endeavor, variability and uncertainty associated with
model output arise for a variety of reasons that are impossible to eradicate completely.
VB3 attempts to examine this variability and uncertainty in a transparent manner using a
probability of exceedance for any regulatory standard the user wishes to investigate.
Even so, there is no guarantee than every model prediction will be correct, and a situation
may arise in which the model predicts acceptable water quality for public recreation that
could be erroneous. Decisions to allow or disallow swimming at beaches must be made,
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however, and in the best case scenarios, regression models developed with VB3 will
outperform traditional persistence models based on just the previous day's FIB
concentrations.
1.2 Recommended User Background
For those using VB3, some experience with spreadsheet data manipulation
programs like Microsoft Excel is recommended, but not necessary. A familiarity with
multiple linear regression analysis is also helpful, but again not mandatory. Without this
background, VB3 will take longer to master, but it should not prohibit users from
producing and using models.
1.3 General Overview
VB3 has four major components:
Beach location map interface where users can define the orientation of the beach.
Interface that facilitates initial import and manipulation of data.
Multiple "method" tabs where the statistical modeling is done. Each tab has some
features identical to those seen in other method tabs and some that are unique. For
example, the multiple linear regression (MLR) tab allows examination of regression
residuals, elimination of highly influential data records, and viewing of receiver
operating characteristic (ROC) curves.
Prediction interface allowing entry of new data and subsequent estimation of
pathogen indicator concentrations with a selected model from any of the statistical
methods.
Each component is accessible from the application's main window via tabs at the
top and bottom of the main screen (Figure 1). The Location and Global Datasheet tabs
are always visible, while the statistical method tabs only become visible once data pre-
processing has been completed (i.e., clicking the "Go to Model" button on the Global
Datasheet ribbon). The Prediction tab appears when model-building on any method tab is
complete and a model is selected
Lastly, we note that statistical models are only as effective as the data used to
develop them. No statistician, however skilled, can turn a dataset of low-quality
independent variables (IVs) into a useful predictive device.
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Figure 1. The major components of VB3: "Location," "Global Datasheet," three "Method" tabs
(GBM, MLR, and PLS), and the "Prediction" interface. The Global Datasheet is currently active.
1.3 History of VB
VB3 is a direct descendant of Virtual Beach version 2, whose most recent release
is VB2.4. The original Virtual Beach Model Builder application (Virtual Beach version 1)
was developed by Walter Frick and Zhongfu Ge at the USEPA in Athens, Ga (Frick et al.
2008). VBi can be characterized as a linear regression model-building tool that supports
primarily manual analyses of datasets via visual inspection of data plots and manipulation
of variables (e.g., transformations, creating interaction terms), followed by an iterative
process of testing, comparing and evaluating models. The fitness of developed models is
computed and tracked, allowing comparison and eventual selection of a "best" model for
the dataset under consideration. This model then produces estimates of pathogen
indicator concentrations using current or forecasted environmental data from the site.
VB2 (Cyterski et al. 2012) enhanced the functionality of its predecessor by
performing similar functions (visual inspection of univariate data plots, manual
transformations of individual variables, MLR model building, prediction, etc.), but also
automated and extended functionality in several ways:
The Map component provided information on the location and availability of nearby
data sources through the map interface. These sources include the USGS National
Water Information System (NWIS) and the National Climatic Data Center (NCDC)
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which provide recently collected and/or forecasted data to generate predictions by a
chosen model.
The Map component provided a convenient method for defining beach orientation by
overlaying the beach on current shoreline layers (satellite images, Google Maps, MS
Virtual Earth, etc). Given the orientation, VB2 could calculate wind, wave, or current
components (the A-component is parallel to shore and the O-component is
perpendicular to shore) which can be important predictor variables.
Although manual processing and analysis of imported data (visual inspection of
univariate data plots and the transformations/interactions of variables) was retained,
the data-processing component of VB2 automated generation of all possible second-
order interaction terms among a set of IVs, formed more complex functions of
multiple columns, and automated testing of a suite of variable transformations that
improved model linearity. This functionality increased the number of models to
evaluate during later selection routines and removed the burden of manual assessment
that users of VBi encountered.
Within the linear regression analysis component, multi-collinearity among predictor
variables was handled automatically. Any model containing an IV with a high degree
of correlation with others (as measured by a large Variance Inflation Factor [VIF])
was removed from consideration during model selection.
During MLR model selection, models were ranked by a user-selected evaluation
criterion: R2, Adjusted R2, Akaike Information Criterion (AIC), Corrected AIC,
Predicted Error Sum of Squares (PRESS), Bayesian Information Criterion (BIC),
Accuracy, Sensitivity, Specificity, or the model's Root Mean Square Error (RMSE).
See Section A.3 for definitions of these criteria. Regardless of which criterion is
chosen, the software records the ten best models in terms of it. In comparison, VBi
had a single criterion choice, Mallow's Cp.
As the number of IVs in a dataset increases, possible MLR models increase
exponentially (considering transforms/interactions), resulting in trillions of possible
models from a modest number (12-13) of IVs. VB2 implemented a genetic algorithm
(GA) that efficiently searched for the best possible MLR model. Alternatively, VB2
users could perform exhaustive calculations in which all possible combinations of IVs
were tested if the number of possible models was reasonably small (< 500,000). Both
the GA and exhaustive approaches greatly expanded the model-building capabilities
of VB2, compared to VBi.
Users no longer had to enter data values in transformed, interacted, or component-
decomposed form to make a prediction with the selected MLR model. On the VB2
MLR Prediction tab, a user-selected model is coded into an input grid with data entry
columns matching main effects of the model. Any mathematical manipulation of
these IVs is then performed automatically prior to making predictions.
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VB3 primarily builds on VB2 by adding additional statistical methods that give
users more flexibility in modeling their datasets. In addition to MLR, users can now use
Partial Least Squares (PLS) regression and Generalized Boosted Regression Modeling
(GBM) to fit their data and make predictions. The redesigned software architecture
(using DotSpatial libraries) easily accommodates future expansions of the suite of
modeling tools. Possible future additions could be Binary Logistic Regression, Least-
Absolute Shrinkage (LASSO) and Neural Networks. The Prediction tab of VB3 also has
a button to allow direct interaction with the USGS's data acquisition system, EnDDaT
(http://cida.usgs.gov/enddat/). for automated dataset construction and ease of FIB
prediction from web-accessible data.
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2. COMPOSITION AND INSTALLATION
VB3 was developed with MS Visual Studio and written in C#, and uses multiple
public domain system components:
FLEE equation parser (http://flee.codeplex.com/)
Accord.Net math libraries (http://accord-framework.net/)
R statistical libraries (http://cran.r-project.org/web/packages/)
DotSpatial mapping libraries (http://dotspatial.codeplex.com/)
Weifen Luo Docking UI (http://sourceforge.net/projects/dockpanelsuite/)
ZedGraph (http://sourceforge.net/projects/zedgraph/)
GMap.Net (http://greatmaps.codeplex.com/)
No license or software purchase is required to install and run VB3, but an internet
connection is needed to display Geographical Information System (GIS) information.
Users must have Windows XP or 7 with DotNet Framework 4.0 to assure proper
installation and operation. Other versions of Windows (e.g., Vista) have caused various
errors to occur, thus are not recommended for use with VB3. Certain VB3 data
manipulation and model-building operations are computationally intensive, so faster
CPUs are better, but laptop or desktop systems with at least 2 GB RAM will be adequate.
Disk space requirements are about 140 MB for VB3 and 170 MB for the DotNet
Framework 4. The VB3 application installer will attempt to download and install the
DotNet Framework 4.0 if it is not already installed on the target system; this also requires
a network connection. If necessary, a user can obtain the DotNet Framework 4 installer
at no cost at:
http://www.microsoft.com/download/en/details.aspx?id=17851
The EPA's Center for Exposure Assessment Modeling (CEAM) web site
distributes VB at:
http://www2.epa.gov/exposure-assessment-models/virtual-beach-vb
Obtain and run the VB3 application installer and follow the on-screen instructions.
After installation, a shortcut will appear on the desktop.
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3. OPERATIONAL OVERVIEW
To make VB3 straightforward to operate, it has four functions, each with its own
interface:
Location - an optional mapping/GIS screen for calculating a beach orientation used for
later computation of orthogonal (alongshore and offshore/onshore) wind, current, and/or
wave components for the beach under consideration. Such components can be powerful
predictors of pathogen indicator concentrations at the beach, so defining the beach
orientation is recommended if the dataset under consideration contains wind, wave or
current data.
Global Datasheet - a way to support data manipulation on an imported dataset. In
addition to wind/current/wave component generation, users can generate new
independent variables that represent the products, means, sums, differences, minimums,
and maximums of other IVs, as well as investigate data transformations for the IVs.
Methods - there are three Method tabs - Multiple Linear Regression (MLR), Partial
Least Squares regression (PLS), and Generalized Boosted Regression Modeling (GBM).
Each has its own unique interface, but shares common elements. One common element
is a "variable selection" tab where the user chooses from a list of eligible IVs for
consideration in model-building and model-generation. Another common element is a
"Data Manipulation" tab which is initially populated with data from the Global
Datasheet. After initialization, however, the user can then modify "local" data for the
chosen statistical technique.
Prediction this tab is comprised of three spreadsheets/grids where users can enter or
import the IVs needed for the chosen model (left grid), enter or import the values of the
response/dependent variable that will be compared to model predictions (middle grid),
and examine model predictions and exceedance probabilities (right grid). Time series
and scatter plots of the measured dependent variable values versus predictions help users
gauge model effectiveness.
The following list attempts to provide an overall context for how a general, basic
modeling session using VB3 would be conducted (optional actions in green, required
actions in red):
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Open the Software
Location Tab is Visible
Use the GfS map to find beach of interest
Delineate beach shoreline
VB3 calculates the beach orientation angle
Click on the Global Datasheet Tab
Import data from a file
Validate the imported data
Click the "Go To Model" button
Click the MLR, PLS, or GBM Tabs
Set the method-specific modeling options
Run the model
Look at fitted results and choose a model to use
PLS/GBM - only a single model produced
MLR - returns the "best" ten models; user must choose
Take the Chosen Model to the Prediction Tab
Import data file, or manually enter new data
Make predictions using new data and the chosen model
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4. PROJECT MANAGEMENT
The user will often perform a number of pre-processing steps on an imported
dataset to prepare it for analysis, and then develop models from the resulting data. To
avoid repeating all of this work, a file can be saved (termed a "project" file) and re-
opened via the File -> Save and File -> Open menu selection. Project files have a
",vb3p" extension. Opening a saved project file will load the saved data into the Global
Datasheet and re-populate the methods tabs with the local data, as well as any modeling
results generated prior to the save. The beach orientation defined by the user on the
Location tab is also saved inside a project file. We suggest giving Project files a
descriptive name of the beach/site being modeled for later easy identification.
In addition to project files, "model" files can be saved by using "Save As
(prediction only)" under the "File" menu at the top of the VB3 interface. These files have
a ",vb3m" file extension. A model file contains information on the IVs, model
parameters, and other metadata for the currently selected models on each method tab.
When users open a saved model file within VB3, they are taken directly to the Prediction
tab (the only accessible tab) where they can use the model to generate predictions. Model
files allow the user to construct models and choose a "best" one for a site, save a model
file, and deliver this file to a beach manager. With this approach, a manager will not
need VB3 for full-scale model development, but only to input new data, generate
predictions, and make decisions about issuing swimming advisories.
If the user clicks the red "X" in the upper-right corner of the main VB3 window
(Figure 1), a prompt will ask if they wish to save their project before closing.
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5. LOCATION INTERFACE
On VB;; application startup, the "Location" tab is shown first (Figure 2). Because
use of this tab is optional, users can go directly to the "Global Datasheet" interface by
clicking that tab at the top or bottom of the screen.
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Figure 2. Location interface; the default map type is OpenStreet, but users have several other
options.
5.1 Finding a Beach
The location interface provides map controls (Figure 3) that let users navigate to a
beach site by panning and zooming (right-click and drag mouse to pan; use mouse wheel,
slider at the left of the map, or the two buttons in the top ribbon for zoom). Alternately, a
latitude/longitude can be entered at the top left, followed by a click on "GoToLat/Lng"
button.
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Zoom
Map Controls
44.0766102719193
-87,6560497283936
GoTo Lat/Lng
Beach Orientation
Lat
Lng
Map Settings
Type
Open Street Map ป
Add 1st Beach Marker
Add 2nd Beach Marker
Add Water Marker
Beach Orientation
Map Controls
Zoom Slider - drag slider up and
down to zoom in and out.
Map Controls - Add Lat/Long and
click "GoTo Lat/Lng" button.
Map Settings - Select map type from
drop down menu to change the
display in the map window.
Beach Orientation - use these
buttons to add or remove beach
boundary markers (shown as green
balloons) on the map. Once the
beach shoreline is delineated by
placing the 1st and 2nd beach
markers, click on the water and then
click "Add Water Marker," which will
lead to the correct orientation angle
being placed into the "Beach
Orientation" box.
Current Location
41.9676592036782
-87.6708984375
Loading
Lat
Lng
Current Location - click anywhere on
the map to display Lat and Long
values.
Loading - progress bar that shows
network download activity for map
images.
Figure 3. Location controls and their function.
5.2 Defining the Beach Boundaries for Orientation Calculation
The map control allows delineation of a beach's boundaries so that VB3 can
calculate its orientation (Figure 4), which is useful if wind, wave, and/or current flow
components are used in model-building. Maps provide less shoreline detail, so it is
recommended that a hybrid or satellite image be selected prior to adding point locations
that define beach boundaries. Once the beach of interest is found and the swimming area
is located, left-click on the map (a red marker will appear) and click the "Add 1st Beach
Marker" button; this represents one endpoint of the beach shoreline/swimming area.
Now left-click the other end of the beach on the map and click the "Add 2nd Beach
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Marker" button. Finally, left-click on the map to indicate where the water is, relative to
the shoreline, and click the "Add Water Marker" button. Marker points will turn from
red to green as they are identified. Once the water marker is added, a shaded box appears
and the beach orientation angle is displayed to the left of the map at the bottom of the
"Beach Orientation" box (Figure 4).
Map Controls
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44.0766102719193 Lat
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Map Settings
Type
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Beach Orientation
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Remove 1st Beach Marker |
mmkm
Remove 2nd Beach
| Remove Water Marker |
Beach Orientation 249.94
Current Location
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Loading
mm
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-
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Status:
Figure 4. Adding shoreline and water markers to define beach orientation.
These boundary points can be added or removed until the user is satisfied with the
beach representation. VB3 will pass the calculated beach orientation angle to the global
datasheet for wind/current/wave component calculations.
5.3 Saving Beach Information
As covered in Section 4, the File^Save menu selection will open a window that
allows the user to save the project information (such as placement of the beach/water
boundary markers and the calculated beach orientation) inside a VB3 project file.
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6. GLOBAL DATASHEET
6.1 Data Requirements and Considerations
VB3 can import .xls, .xlsx, and .csv files, but input data must conform to certain
standards:
The first row of any column must be a header specifying the column's name.
For error-free operation of the software, column names should be composed only of
letters, numbers, and/or underscores ("_").
Do not begin a column name with a number.
VB3 will issue an error statement if a dataset with spaces in a column name is
imported.
The left (first) column of the dataset must be an identifier for the observations
typically a date, time, or serial number that indicates when or where that row of data
was collected.
Each row MUST have a unique ID value (left-most column). If VB3 finds duplicate
IDs, it will issue an error statement.
If the ID column specifies a collection date or time, time series plots in VB3 will be
most interpretable if the rows are in chronological order, from the earliest to the most
recent data. VB3 will not re-arrange the data in chronological order on its own.
The second column of the dataset will initially be set as the response variable;
however, this can be changed after data are imported. Other columns will be
considered as IVs (besides the first ID column).
Variable measurement units are not considered by VB3, but certainly affect
predictions. Ensure that any data used for predictions are in the same units as those
used to build the models; for example, do not build a model with water temperature in
degrees Fahrenheit, then import water temperature in degrees Celsius for predictions.
It is prudent to include unit information in the column names (e.g., "WaterTemp C")
to remind the user of the proper unit when entering data to make predictions.
Missing data (blank cells) are permitted upon import, but must be dealt with (either
deleted or values filled in) prior to modeling.
If Excel data files are imported, cells with non-numeric values (i.e., symbols or text)
are converted to empty cells. Exceptions are the column names and the first column
of IDs. If such non-numeric characters are present in an imported .csv file, they will
be imported into VB3's datasheet. However, they will be flagged as anomalous
during the validation scan and they must be dealt with (deleted or populated) at that
time.
When the required validation scan is launched, VB3 will identify any column in the
dataset containing only a single value and ask the user to delete the column (because
such data columns are useless for predictive purposes).
There is no hard-coded limit on the number of IVs one can import; however, the VB3
datasheet is designed for a maximum of 300 columns. Beyond that number, the
application's performance will degrade significantly. Investigating 250+ IVs results
in over 2* 1020 possible IV combinations for MLR processing. The MLR genetic
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algorithm can handle this modeling task, but choosing "Run all combinations" would
likely take months or years to complete. Depending on how many additional IVs will
be created by the user, importing a dataset with less than 100 IVs should be
acceptable.
We note here that VB3 can be used as a powerful exploratory research tool,
allowing the user to investigate a great many IVs concurrently. However, this approach
can lead to models with spurious response/IV relationships (i.e., the association is only a
random statistical artifact, not a "real" phenomenon). To avoid this, the user could
restrict their analyses to only those IVs for which they have a prior, process-based,
theoretical expectation of influence on pathogen concentrations. A criticism of this
approach is that the researcher will never discover a relationship between the response
and a truly influential IV if they don't already expect it to exist. Discovery of
unexpectedly influential IVs can lead to process insight and advancements in
understanding of the physical system. If an exploratory approach is taken, there are
mechanisms within the statistical modules of VB3 (primarily cross-validation to ensure
that predictions on future data points are nearly as good as the model fits) to protect
against over-fitting a model using too many IVs and finding spurious correlations that
don't hold up when the model is used for prediction of future events.
6.2 Importing a Dataset
When users first click on the Global Datasheet tab, they can import a data file
using the "Import Data" button in the top ribbon (Figure 5). This opens a dialog screen
where a directory explorer can be used to find the data file. If the file is an Excel
workbook with multiple worksheets, the dialog box asks which worksheet to import.
17
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Once imported, the data are shown in a datasheet. The second column of this
datasheet will be highlighted in blue to indicate its status as the current response variable.
Information about the dataset, such as number of rows and columns, name of the ID
column and name of the response variable, appear at the left of the datasheet. At this
point, the datasheet cannot be edited or interacted with in any manner; to access
additional processing functionality, the data must be validated.
6.3 Validating the Imported Data
Validation options can be accessed by clicking the "Validate Data" button in the
top button ribbon. Validating the data launches a required scan to identify blank and non-
numeric cells in the imported spreadsheet (Figure 6). One can also find and replace other
specified values (e.g., a missing data tag like -999) in the dataset, using the "(Optional)
Find:" input box.
18
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9
37
tstamp
LogCFU
Disabled Row Count 0
Disabled Column Count 0
Hidden Column Count 0
Independent Variable Count 7
Global Datasheet
Project File Name:
Ready.
6/4/200511:02..
6/4/2005 3:07 PM
6/5/2005 7:55 AM
6/5/200511:02...
6/5/2005 3:07 PM
6/18/200511:02.
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Figure 6. Data validation required to begin data processing.
Clicking "Scan" begins the validation process. VB3 goes through the datasheet,
cell by cell, looking for blanks, non-numeric, or user-specified values entered in the
"(Optional) Find:" input box. If such a cell is found, the scan will stop and highlight it.
Users must then decide how to deal with that cell from choices in the "Action" section
(Figure 7): replace the cell with a specified value, using the "Replace With:" input box,
or delete the row or column containing the cell. The user must decide where to
implement the chosen action with the "Take Action Within" dropdown menu. Possible
choices are "Only this Cell," "Entire Row," "Entire Column," and "Entire Sheet." Items
in this menu are context-sensitive, i.e., they change with the Action selected. After
setting the "Take Action Within" menu, the user clicks the "Take Action" button, VR s
makes the specified changes to the datasheet, and the scan continues. Even if no cell
errors are found, VB3 may still report that a "Column has no distinct values" and prompt
the user to delete the column (see the second-to-last bulleted item in Section 6.1). When
the entire datasheet has passed inspection, VB3 reports "no anomalous data values found"
at the bottom of the Validation window.
19
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waisttemp WindS peed
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Figure 7. Context-sensitive choices for the "Take Action Within" drop-down menu.
After the data have been validated, but prior to clicking the "Return" button on
the Validation window, the user has the option to specify which columns in the dataset
are categorical variables. Why do this? VB3 will not attempt to transform categorical
data columns (transformations discussed later), because it generally does not make sense
to do so. Thus, identifying IV columns as categorical saves time later when
transformations are investigated. If the user clicks on the "Identify Categorical
Variables" button (Figure 7), a window pops up (Figure 8). A list of the datasheet's
independent variables is shown in the right-hand section of this window. VB3
automatically identifies columns with only two unique values as categorical variables
(i.e., they will already be in the left section of this window); if the user has other
categorical IVs with more than two categories, those should be moved from the right to
the left section using the I<" J button. The user can also move any currently-identified
categorical IV back to the right list using the tZO button.
20
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Categorical Variables
Identify Categorical Variables in Your DataSet.
(Categorical variables cannot be transformed.)
Variable Selection
Categorical Variables
WindS peed
I ndependent Variables
uv
airtemp
WaveH eight
~ I centershintemp
centerwaisttennp
1 WindDirection
Ok
Cancel
Figure 8. Pop-up window for identifying categorical variables.
21
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6.4 Working with a Dataset after Validation
After the dataset has passed the validation scan, the function buttons across the
top of the Global Datasheet tab ribbon are enabled (Figure 9).
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T stamp
LogCFU_Ecoli
T urbidity
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Wet_Bulb_F ~
Column Count 13
Row Count 709
D ate-T ime I ndex T stamp
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Figure 9. Post-validation enabling of the Global Datasheet functionality.
At this point, grid cells (other than the ID column) are editable - that is, users can
manually enter new numeric data with a left-double-click on a cell and typing in a new
value. VB3 does not allow a cell to be made blank or non-numeric. A right-click on an
IV column header presents additional options (Figure 10):
ty
Disable Column
Enable Column
Visibility
10
Set Response Variable
10
View Plots
Delete Column
Enable All Columns
2
10
4
10
1
10
1
10
Figure 10. Right-click options on columns that are not the response variable.
22
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"Disable Column" turns the text red and prevents the column from being passed
to the method tabs. Previously-disabled columns can be activated with "Enable
Column." "Set Response Variable" makes the chosen IV the new response variable (the
column becomes blue to indicate this change). "View Plots" shows a new screen with
column statistics at the far left and four plots for the chosen column (Figure 11): (1) a
scatter plot of the IV versus the response variable in the lower left panel; (2) a plot of the
IV values versus the ID column at the upper left (a time series plot if the ID is an
observation date); (3) a box-and-whiskers plot at the top right; and (4) a histogram for IV
values at the bottom right.
Variable uv
Data
Value
IVariable Name i
uv
Row Count
37
Maximum Value
1,834.00
Minimum Value
203.00
Average Value
1,058.97
Unique Values
35
Zero Count
0
Median Value
1,278.000
Data Range
1,831.000
KS Statistic 0.2128
KS Stat P-Value 0.0599
Mean Value 1,058.973
Standard Deviation 578.G36
Variance 334,819.138
Kurtosis 51.498
Skewness -0.354
Replot
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BoxWhisker Plot
29-May 8-Jun 18-Jun 28-Jun 8-Jul 18-Jul 28-Jul 7-Aig
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Frequency Plot
111
Figure 11. Four different plots available for evaluation of IVs.
Scatter Plot Interpretation
Curvature in the scatter plot (lower left) can indicate a non-linear relationship
between the IV and the response variable, problems with homogeneity of variance across
the range of the IV, or outliers. Ensuring that the IVs are linearly related to the response
variable raises the probability of producing a robust, meaningful MLR and PLS analysis
(GBM does not need linearity). If the relationship between the response and the IV is not
well-approximated by a straight line (a fundamental assumption of MLR and PLS), it
may be beneficial to transform the IV. Using VB3 to accomplish this will be explained
later (Section 6.7). The scatter plot also shows the best-fit linear regression line in red,
23
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along with the correlation coefficient (r) and the significance (p-value) of the correlation
coefficient at the top of the plot. In general, p-values below 0.05 are considered
statistically significant. While VB3 does not provide a plot of the residuals of the
regression line depicted in the scatter plot, this important diagnostic is given much
attention on the MLR tab (see Section 7.8).
Identifying odd values (potential outliers or bad data) of any IV can often be done
by visual inspection. If users move the mouse cursor over a data point in any plot (other
than the histogram), they will see the ID value of that observation (Figure 12). They can
then go back to the datasheet, find the outlying observation (data row), and disable that
row (described below) if justifiable.
BoxWhisker Plot
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Figure 12. Identifying an observation from within the XY scatter plot.
The "Delete Column" right-click column header option deletes a column from the
VB3 datasheet. Note that original columns of the imported data sheet (VB3 defines these
as "main effects") cannot be deleted. Rows can be disabled and enabled, but not deleted,
from the datasheet by right-clicking the row header (far left of each row) and making the
desired choice. Changes that the user makes can be undone and redone using the "Undo"
and "Redo" options under the VB3 "File" menu.
If the user right-clicks on the column header of the response variable, a different
set of choices is shown (Figure 13).
24
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Tstamp LogCFU_E
Transform ~
View Plots
UnTransfGrnn
Define Transform ~
T I.I
WaveH eight
36676.375
3.431
1
36677.375
2.006
1
36678.375
1.55
1
36682.375
2.74
55
4
36683.375
3.82
133
4
36684.375
2.686
88
1
36685.375
1.255
21
1
36688.375
2.833
20
3
Figure 13. Available choices when right-clicking the response variable.
Users can transform the response variable in three ways: logio, loge, or a power
transformation (raising the response to an exponent: y'*). They can also un-transform the
response, view the plots shown previously for the IVs, or define a transformation of the
response variable. This last option is used when a datasheet is imported with an already-
transformed response variable. For example, users could import a datasheet with logio-
transformed fecal indicator bacteria concentrations and should define the response as
logio-transformed. Doing this facilitates later comparisons with the fitted response
variable values, decision criteria, and regulatory standards. If this is not done, then later
plots and comparisons of model predictions to response variable values will be strange
and misleading. When users transform the response variable within VB3 using the
"Transform" option, VB3 automatically defines the response as having the chosen
transformation and, in doing so, synchronizes the units of measurement for later
comparisons.
6.5 Computing Wind, Wave and Current Components
Orthogonal wind, current, and wave components can be powerful predictors of
beach bacterial concentrations. Depending on the orientation of the beach, wind and
currents can influence the movement of bacteria from a nearby source to the beach, and
wave action can re-suspend bacteria buried in beach sediment. To make more sense of
this information, researchers typically decompose wind/current/wave magnitude and
direction data into A (alongshore) and O (offshore/onshore) components for analysis (see
equations at the end of this section).
If direction and magnitude (speed/height) data are available, A and O components
can be calculated with the "Compute A O" button in the ribbon (Figure 9). Clicking it
brings up a window with drop-down menus for users to specify which columns of the
datasheet contain the relevant magnitude and directional data (Figure 14). There is also
an input box at the bottom of the form for the beach orientation angle. If the user defined
the beach angle on the "Location" tab, that value will be seen. After clicking "OK," new
data columns are added to the far right of the grid, representing the A and O components
of the specified wind, current, or wave data. Unlike the originally-imported IVs, these
components can be deleted from the grid after creation. Names of these new columns
are: WindA_comp(X,Y,Z), CurrentO_comp(X,Y,Z), WaveA_comp(X,Y,Z), etc., where
25
-------
X is the name of the column of data used for direction, Y is the name of the column used
for magnitude, and Z is the beach orientation angle. Note that the IVs used to create the
A and O components are automatically disabled by VB3 once the components are created.
These columns can be re-enabled by right-clicking on their column header in the
datasheet and choosing "Enable Column." The "Compute A O" function is repeatable as
many times as the user wishes.
Wind/Current/Wave Components
Wind Data
Specify wind data columns:
Speed
Direction (deg)
Current Data
Specify current data columns:
Speed
Direction (deg)
Wave Data
Specify wave data columns:
Wave Height
Direction (deg)
Beach Angle (deg):
0.00
Ok
Cancel
Figure 14. Window for computation of alongshore and offshore/onshore components.
Notes on Component Calculations
Direction is an angular degree measure. Moving in a clockwise direction from
north (0 degrees), values are positive, and negative while moving counter-clockwise.
Wind and current speed (as well as wave height) can be measured in any unit. VB3
adheres to scientific convention: wind direction is specified as the direction from which
26
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the wind blows and current and wave directions are specified as the direction towards
which the current or waves move. Thus, wind blowing west to east has a direction of 270
degrees (or equivalently -90) degrees, while a current/wave also moving west to east has
a direction of 90 (or -270) degrees.
The A-component measures the force of the wind/current/wave moving parallel to
the shoreline (Figure 15). A positive A-component means winds/currents/waves are
moving from right to left as an observer looks out onto the water. A negative A-
component means winds/currents/waves are moving left to right as an observer looks out
onto the water. The O-component measures force perpendicular to the shoreline. A
negative O value indicates movement from the land surface directly offshore (unlikely to
be seen with wave action). A positive O indicates waves/wind/currents from the water to
the shore. These relationships apply no matter how the beach is oriented (Figure 16).
Negative O
Water
Land
T
Positive O ^
~
Positive A Negative A
Figure 15. A- and O-component definitions for wind, current, and wave data.
27
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Beach Orientation for Component Calculations
45 degrees 90 degrees
0 degrees
Water
315 degrees
Water
Water
270 degrees
Water
Water
225 degrees
Water
135 degrees
Water
180 degrees
Water
t
North
Figure 16. Principal beach orientations given in degrees.
The equations for calculation of Wind A/O components:
Wind A: -S * cosine ((D-B) * 71/180)
Wind O: S * sine ((D-B) * ji/180)
where S is wind speed, D is wind direction, B is the beach orientation (in degrees) and 31
~ 3.1416. Current A/O and Wave A/O are the same equations multiplied by -1 to account
for the difference in how these data are measured.
28
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6.6 Creation of New Independent Variables
Users may click the "Manipulate" button (Figure 9) to create new columns of data
(as functions of existing IVs) that might be useful IVs. On the pop-up screen (Figure 17),
there is a list (automatically populated by VB3 from the imported spreadsheet) of
available IVs on the far left under "Independent Variables." If users wish to create a new
term, they add the desired existing IVs to the "Variables in Expression" box by selecting
the IV and clicking the ">" button. Clicking and dragging, shift-clicking and control-
clicking in the "Independent Variables" list allow multiple IVs to be added at once.
Manipulate
Build Expression
Independent Variables
Variables in Expression
T urbidity
WaveHeight
Visibility
Dry_Bulb_F
Wet_Bulb_F
Dew_Point_F
RelJHumd
WindU
WincW
Station_Pressure
Precip_T otal
~
~
ฉ Sunn O Diff O Max O Min O Mean O Product
Add | | Remove |
2nd Order Interactions
OK ~| ( Cancel
Figure 17. Window for the formulation of "Manipulates" - arithmetic combinations of existing
columns within the datasheet.
For example, if users wish to create a new IV that is a row-by-row mean value of
the "Dry Bulb F" and "Wet Bulb F" variables, they add those two IVs to the
"Variables in Expression" box (Figure 18), choose the "Mean" function, "Add" that
expression to the lower box, then click "OK." A new column of data representing a row-
by-row average of those two IVs is then added to the end of the datasheet.
29
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Manipulate
Build Expression
Independent Variables
Variables in Expression
T urbidit^J
WaveHeight
Visibility
Dew_Point_F
Flel_Humd
WindU
WintW
Station_Pressure
Precip_T otal
~
~
D ry_B ulb_F
Wet Bulb F
O Sunn O Diff O Max O Win ฉ Mean O Product
MEAN[Drji_Bulb_F,Wet_Bulb_F]
Add
Remove
2nd Order Interactions
MEAN[Dry_Bulb_F,Wet_Bulb_F]
OK
Lancel
Figure 18. Creation of a new IV defined as the mean of two existent IVs.
Users can create a row-by-row sum, difference, maximum, minimum, mean, or
product from any number of IVs added to the "Variables in Expression" box. More than
one expression can be created before the "OK" button is clicked and IVs can be easily
moved in and out of the "Variables in Expression" box using "<" and ">" keys. Note
that creating a difference of more than two columns (e.g., XI, X2, X3, and X4) would
lead to this quantity:
Diff(Xl,X2,X3,X4) = XI - X2 - X3 - X4
Created expressions can be removed from the lower box with the "Remove"
button. No matter how many IVs are added to the "Variables in Expression" box,
clicking "2nd Order Interactions" will add the cross-products for all possible pairings of
those IVs (Figure 19). Thus, four IVs in the "Variables in Expression" box will produce
six 2nd second-order interactions; five IVs will produce ten interactions, and so on. Note
that the names of the columns used to create any new data columns are inside the
parentheses of those columns' names.
30
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Manipulate
~0ฎ
Build Expression
Variables in Expression
T urbidity
Visibility
Dry_Bulb_F
Station_Pressure
O Sum O Diff O Max O Min O Mean ฎ Product
PR 0 D [T urbidity .Visibility ,D ry_B ulb_F,S tation_Pressure]
Add ~] [ Remove | [ 2nd Order Interaction^
PR0D[T urbidity,Visibility]
PROD[Turbidity,Dry_Bulb_F]
PROD [T urbidity,S tation_Pressure]
PROD [Visibility ,Dry_Bulb_F]
F'R 0 D [Visibility ,S tation_Pressure]
PR 0 D [D ry_B ulb_F,S tation_Pressure]
Cancel
Figure 19. Formation of two-way cross-products of a set of four IVs.
VB3 does not allow previously created "manipulates" new columns of data
created through the "Manipulate" button to be further manipulated. Previously created
manipulates will not appear in the "Independent Variables" section at the left. They can,
however, be chosen as the response variable or deleted from the datasheet, using the
appropriate menu choices accessed by a right-click of the column header.
6.7 Transforming the Independent Variables
VB3 gives users the ability to transform non-categorical IVs to assist in linearizing
the relationship between the IVs and the response variable, a fundamental assumption of
an MLR/PLS analysis. VB3 transformations are described in section A.l. When users
click the "Transform" button (Figure 9) in the Global Datasheet ribbon, they are
presented with the window seen in Figure 20:
I ndependent Variables
WaveHeight
Wet_Bulb_F
Dew_Point_F
Rel_H umd
WindU
WincW
Precip_T otal
~
CD
OK
31
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E
Transforms to Perform
Available T ransforms
I I Log10
~ Ln
I I Inverse
I I Square
I I SquareRoot
I I QuadRoot
I I Polynomial
I I General Exponent 1.0
I I Select All
Dependent Variable:
LogCFU
Go
Cancel
Figure 20. The choices for IV transformations.
When users click "Go," the chosen transformations are applied to each and every
non-categorical IV (there is not an option to ignore transformation for particular IVs).
VB3 then opens a table (Figure 21) that compares the success of each transformation
using a Pearson correlation coefficient which is a measure of linear dependence between
the response variable and the IVs.
The table created byVB3 groups all transformed versions of each IV and specifies
type of transformation, the Pearson coefficient, and its statistical significance (p-value).
This includes the un-transformed version of the IV, denoted by "none." By default, the
transformation with the largest absolute value of the Pearson coefficient is highlighted in
black text. Users may override the default selection by left-clicking on the row header of
a transformed IV. They may also override the default by setting a percentage and
clicking "Go" under the "Threshold Select" box on the left side of the window. This will
select the un-transformed version of every IV unless the transformed IV with the highest
absolute value Pearson coefficient exceeds the un-transformed IV Pearson coefficient by
the specified percentage. In essence, the user is saying, "Unless the Pearson coefficient
of the transformed IV is some % greater than the Pearson coefficient of the un-
transformed IV, use the un-transformed IV." This can be useful because transforming
IVs makes interpreting model coefficients more difficult; unless a major improvement is
seen, transformation simply may not be worth the trouble. Users can also revert to the
default (selecting the transform with the largest absolute value Pearson coefficient) by
clicking "Go" under "Auto Select."
32
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Pearson Univariate Correlation Results - Maximum Pearson Coefficients (signed) in BOLD text
Help
Variables, possible variable
interactions, and their
transforms are shown. Select
variables for further
processing and modeling.
Auto-Select
The variable or one of its
transforms is selected by
maximum Pearson Coefficient.
(This is the default view shown.)
Threshold Select
Select a transformed variable only
if its Pearson Coefficient exceeds
the untransformed variable's
Pearson Coefficient by a
specified threshold.
Threshold[%) ST $]
I Gฐ I
Manual Select
Mouse-click on a row header to
select or deselect that variable.
At most one member from each
group can be selected.
[ Ok J | Cancel J | Print j
Dependent Variable: LogCFU_Ecoli
w . , | x , Pearson
Variable Transform Coefficient
Correlation
P-Value
A.
~
T urbidity
none
0.5261
0.0000
V
T urbidity
LOG10[T urbidity]
0.5187
0.0000
T urbidity
IN VE R S E [T urbidity ,0.65]
-0.3941
0.0000
T urbidity
S Q U AR E R 0 0 T [T urbidity]
0.5444
0.0000
T urbidity
P0LY[T urbidity;1.495695,0.022423027,-7.1765833e-05]
0.5289
0.0000
WaveH eight
none
0.4650
0.0000
WaveHeight
LOG10[WaveH eight]
0.4674
0.0000
WaveH eight
IN VE R S E [Wa veH eightjl 5]
-0.4587
0.0000
WaveHeight
SQUAREROOT[WaveHeight]
0.4681
0.0000
WaveH eight
PO LY[WaveH eight,1.070917,0.57236527,-0.049326044]
0.4565
0.0000
Visibility
none
-0.1339
0.1046
Visibility
L0G10[Visibility]
-0.1375
0.0957
Visibility
INVERSE [Visibility/1]
0.1299
0.1155
Visibility
S Q UAR E R 0 0 T [Visibility ]
-0.1368
0.0974
Visibility
POLY[Visibility,2.5246722,-0.15230621,0.0075484516]
0.1385
0.0933
Dry_Bulb_F
none
-0.0800
0.3335
Dry_Bulb_F
LOG10[Dry_Bulb_F]
-0.0792
0.3387
Dry_Bulb_F
IN VE R S E [D ry_B ulb_F,26]
0.0770
0.3520
Dry_Bulb_F
S Q U AR E R 0 0 T [D ry_B ulb_F]
-0.0798
0.3351
Dry_Bulb_F
P0LY[Dry_Bulb_F,2.6344881 ,-0.015402695,5.3124717e-05]
0.0779
0.3469
Figure 21. Pearson correlation coefficient scores for judging the efficacy of IV transformations.
Plotting Transformed IVs
Users may prefer to examine plots visually in determining which transformation
of IV to choose. Right-clicking on a row header in the correlation table provides an array
of scatter plots, time series plots, or frequency plots for each transformation of that IV
(Figure 22). Scatter plots show the best-fit regression line. In the table at the top of this
window, users are shown the correlation coefficient and its p-value, as well as the
Anderson-Darling test statistic for normality, and its p-value.
33
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งง Variable airtemp and its Transforms
airtemp LOG10 INVERSE SQUARE QUADROOT POLY
Pearson Coefficient -0.3772 -0.3706 0.3624
SQUAR Eplrtem p]
QUADROOTplrtomp]
pol V[air1ปm p.-3.ro4a932.o.3so2s&sa.-ซ.oor6rs2iss]
Figure 22. Scatter plots (Response vs. IV) for six different data transformations of a single IV.
After choosing a transformation for each IV, users click "OK." This populates
the datasheet with new columns representing transformed versions of the IVs. Notice
two things: if a transformation was chosen for an IV, the column representing the
untransformed version of that IV is disabled in the datasheet (it can be re-enabled by
using the right-click column header menu option) and the transformed versions of an IV
are put into the datasheet immediately after the original, un-transformed IV. Any
transformations put into the datasheet can be deleted with the "Delete Column" choice
(right-click on their column header). Transformed IVs will appear in the list of IVs on
the "Manipulate" screen, however, transformed IVs cannot be further transformed and
will not appear in the transform table if the user returns to the "Transform" window.
Also, transformed IVs cannot be the response variable. Finally, because transformations
are determined from the current response variable, all transformed IVs in the datasheet
are erased (a warning appears) when users change the response variable in the datasheet.
For the interested reader, further discussion of VB3 transformations can be found in
section A. 1.
6.8 Singular Matrices and Nominal Variables
Advice on avoiding singularities within the data matrix and handling nominal
categorical variables can be found in section A.2.
34
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6.9 Saving Processed Data
Changes made to the imported spreadsheet can be saved in a project file
(File-^Save). When it is re-opened, the datasheet will appear as it did when the project
was saved. Users also may highlight the entire datasheet or sections of the datasheet and
use Control-C and Control-V to copy and paste it into a word processing or spreadsheet
application.
6.10 Proceeding to Modeling
After data processing is complete, users must click the "Go to Model" button to
open the statistical method tabs. If they have already done some modeling and return to
the global datasheet to make changes, they will receive a message that the datasheet has
changed and any prior modeling results will be erased.
35
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7. MULTIPLE LINEAR REGRESSION MODELING
The MLR tab finds the best multiple linear regression model based on criteria
selected by the user. As the number of IVs increases, the number of possible models in
the solution space increases exponentially. Users may select all or a subset of the IVs for
consideration in the model to reduce the size of the solution space.
Notice that the MLR tab (as well as the PLS and GBM tabs) has its own datasheet
on the "Data Manipulation" sub-tab. When the user first moves over to the MLR tab
from the Global Datasheet, the data in the MLR Data Manipulation sub-tab is identical to
the data on the Global Datasheet. Once inside the MLR tab, the user can change the
"local" data to suit the MLR analysis. The local datasheet has all of the functionality of
the Global Datasheet discussed in Section 6. Changing the local data has no effect on the
Global Datasheet, however, going back to the Global Datasheet and making changes
causes local datasheets on the MLR, PLS, and GBM tabs to be overwritten.
7.1 Selecting Variables for Model Building
Under the "Model" sub-tab, two additional sub-tabs are found (Figure 23). On
the "Variable Selection" sub-tab, all eligible IVs are listed in the left column ("Available
Variables"). Any variable users wish to consider for model inclusion must be moved to
the right column list ("Indep. Variables") by highlighting the IV and clicking the ">"
button. IVs currently under consideration (in the right list) can be ignored by
highlighting them and clicking the "<" button. The user can hold down shift while left-
clicking or control while left-clicking to select multiple IVs at once.
B Virtual Beach v3.0
File
Location Global Datasheet GBM
MLR PLS
L 3C &
Compute Manipulate Transform
AO
Manipulate Data
Data Manipulation j
i-Model Settings
Variable Selection Control Options Number of Observations: 101
Dependent Variable: L0G10[Ecoli]
Available Variables (0) Indep. Variables (0]
ClearWater_y1_0
Turbid_y1_0
0paque_y1_CI
WaveHeight_ft
Gulls
RRAIN24
RRAIN48
Q24
Q48
Q72
CLDCV
DOY
WVHT
WTEMP
Figure 23. Selecting variables for MLR processing within the Modeling tab.
0
36
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7.2 Modeling Control Options
After choosing the set of IVs to investigate, the user should click the "Control
Options" sub-tab. The first decision to be made involves which evaluation criterion will
be used to judge model fitness (Figure 24). There are ten choices in the drop-down
menu:
Akaike Information Criterion (AIC)
Corrected Akaike Information Criterion (AICC)
R2
Adjusted R2
Predicted Error Sum of Squares (PRESS)
Bayesian Information Criterion (BIC)
RMSE
Sensitivity
Specificity
Accuracy
Evaluation Criteria
Akaike Information Criterion (AIC)
Maximum Number of Variables in a Model
Available: 7, Recommended: 4, Max: 7
Maximum VIF
Figure 24. Setting modeling options within the modeling interface.
Depending on the evaluation criteria, VB3 searches for a minimum or maximum
value. The minimum value for AIC, AICC, BIC, RMSE, and PRESS is used to choose a
model, while the maximum is used for R2, Adjusted R2, accuracy, specificity, and
sensitivity. A more detailed description of each criterion can be found in section A.3.
Sensitivity, specificity and accuracy are special cases requiring users to enter both
a Decision Criterion (DC) and Regulatory Standard (RS) so that true/false positives and
true/false negatives can be defined (Figure 25). The user chooses the DC value. Model
predictions above this threshold are considered exceedances/positives, and model
predictions below this value are considered non-exceedances/negatives. The RS is
typically a safety limit on fecal indicator bacteria (FIB) concentrations set by a state or
federal agency. The "Threshold Transform" radio buttons tell VB3 the units of DC and
RS to ensure a proper comparison to model predictions and observations. For example, if
"235" is entered into the DC box (representing the EPA standard for freshwater E.coli),
then "none" should be chosen. If 2.371 (= logio(235)) is entered as the DC, then "LoglO"
is used. The DC and RS should always use the same units. Improper setting of this
button choice will lead to problems later when comparing modeling predictions to
observations.
37
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Model EvaluationThresholds
Decision Criterion (Horizontal)
Regulatory Standard (Vertical)
235
235
Threshold Transfornn Current US Regulatory Standards
ฎ None E. coli. Freshwater: 235
O Log10 Enterococci, Freshwater: 61
O Ln
O Power Enterococci, Saltwater: 104
Figure 25. Setting evaluation thresholds and threshold transformation information within the
modeling interface.
The "Maximum Number of Variables in a Model" parameter tells VB3 the
maximum allowable size for any tested models. In general, one should have about 10
observations per estimated parameter in a model, otherwise model over-fitting and poor
estimation of regression parameters can occur. VB3 recommends this limit be set to (1 +
n/10) parameters, where n is the number of observations in the dataset. The maximum
allowable limit is n/5. The total number of available parameters is also shown.
The "Maximum VIF" (Variance Inflation Factor) is used to discard models
containing variables with a high degree of multi-collinearity, i.e., IVs that are highly
correlated with other IVs in the model. If any IV in a model has a VIF exceeding the VIF
threshold, that model will be ignored. The default VIF is 5, which means that 80% (1 -
1/VIF = 1 - 1/5 = 4/5) of the variability in an IV can be explained by the other IVs in the
model. A VIF of 10 means that 90% (1 - 1/10 = 9/10) of the IVs variability can be
explained, and so on. Raising the Maximum VIF means a higher degree of multi-
collinearity will be tolerated, but this can lead to poorly estimated regression coefficients
(i.e., large standard deviations of these coefficients).
7.3 Linear Regression Modeling Methods
Two buttons are at the bottom of the "Control Options" sub-tab to provide
different ways of exploring the regression solution space (Figure 26).
The Manual button is for a directed model search. If the 'Run all combinations' box
is not checked, only a single model that includes every IV that was added to the
"Indep. Variables" column will be evaluated. If the number of available IVs exceeds
the "Maximum Number of Variables in a Model" value, however, VB3 will show an
error. If 'Run all combinations' is checked, an exhaustive search is performed,
testing every model that can be constructed with the selected IVs, but does not
evaluate models with more parameters than the "Maximum Number of Variables in a
Model." For example, if there are 24 available IVs and the maximum number of IVs
is 8, the exhaustive routine will examine every 1-, 2-, 3-, 4-, 5-, 6-, 7- and 8-
parameter model. VB3 shows the total possible number of combinations below the
"Model Settings" box. As the number of IVs rises, the number of possible models
gets so large that the time needed to compute regression fits for each of them
becomes unreasonable. We advise switching to the genetic algorithm in this case.
38
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The genetic algorithm (GA) button explores solution spaces too large to handle
exhaustively. Genetic algorithms are loosely based on natural evolution in which
individuals in a population reproduce and mutate (Fogel 1998). Individuals with high
fitness (regression models that produce small residuals) are more likely to reproduce
and pass their genes (IVs) to the next generation. The goal is to find a good solution
without having to examine every possible option. The GA balances random and
directed searching.
Model Settings
Variable Selection Control Options
Number of Observations: 101
Evaluation Criteria
Akaike Information Criterion (AIC)
3
Maximum Number of Variables in a Model
Available: 17, Recommended: 11, Max: 17
Maximum VIF
Model EvaluationThresholds
[235 | Decision Criterion (Horizontal)
j 235 Regulatory Standard Vertical)
Threshold Transform
ฎ None
O Log10
O Ln
O Power |
Current US Regulatory Standards
E. coli, Freshwater: 235
Enterococci, Freshwater: 61
E nterococci, S alt water: 104
Manual Genetic Algorithm
~ Run all combinations
Run
There are 109294 possible variable combinations
Data Manipulation Model
Model Settings
J Variable Selection j Control Options
Evaluation Criteria
Number of Observations: 101
Akaike Information Criterion (AIC)
Maximum Number of Variables in a Model
Available: 14, Recommended: 11, Max: 14
Ma
nVIF
Model EvaluationThresholds
235 ] Decision Criterion (Horizontal)
1235 | Regulatory Standard (Vertical)
Threshold Transform Current US Regulatory Standards
ฎ Nฐne E. coli. Freshwater: 235
O Log10
O Ln
O Power |
Enterococci, Freshwater: 61
E nterococci, S altwater: 104
Manual 11 Genetic Algorithm
I I Set Seed Value:
Population Size:
Number of Generations:
Mutation Rate:
Crossover Rate:
100
100
0.05
0.50
Run
Figure 26. Model building interface using a manual search (left panel) or the genetic algorithm
(right panel).
Choosing between the exhaustive and the GA searches depends on the dataset, the
computer's available random access memory (RAM), and time constraints. On a dataset
of 101 observations and ten IVs, the exhaustive search was completed in approximately
6 seconds, using a Dell Precision T5400 (WinXP; dual Xeon 2.66 GHz processors; 4 GB
RAM). Every additional IV doubles the number of models to examine and, thus,
approximately doubles necessary computational time (Table 1).
39
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Table 1. Relationship between the number of IVs, number of possible models, and time required to execute
an exhaustive search using VB3
Exhaustive Search - Run AH Combinations
Number of IVs
Number of MLR models
Approximate Time Required to Generate
and Filter Models (seconds)
10
1023
6
11
2047
13
12
4095
27
In contrast, running the GA with 10 IVs, using a population of 100 for 100
generations, took 90 seconds to complete (90/6 =15 times slower than the exhaustive
routine for this number of IVs); the GA with 12 IVs takes about the same amount of time
- 90 seconds. So, as computational time of the exhaustive routine doubles every time an
IV is added, the time required to run the GA stays approximately the same. As the
number of IVs rises (here, to 14 or 15), the GA would be expected to save time and
provide a solution very close to optimal.
An alternative modeling strategy with a large number of IVs would be to run the GA
on the entire list of IVs initially, then switch to the exhaustive search on a subset of
initial IVs - any IV that appears in one of the best ten models found by the GA. This
two-step process is facilitated with the "IV Filter" list control (Figure 27).
Model Information
Best Fits:
-143.3235
A
-143.0920
-142.9118
-142.8249
-142.6259
' 1
-142.4560
-141.4349
V
IV Filter
Add to List
Clear List
~4
View
Report
Cross
Validation
Figure 27. Using the IV filter to select a subset of variables from the best-fit models.
When the GA finishes and the 10 best models are shown in the Model
Information box "Best Fits" window, clicking the "Clear List" button removes all IVs
from the selection list. Select a model from the "Best Fits" list and click "Add to List"
which adds any IVs in the selected model to the "Indep. Variable" list in the Model
Settings box. After doing this for each of the ten best models, users will have a more
manageable IV list and can run an exhaustive search to find the best combination of IVs.
Regardless of the method chosen to build models, the "Best Fits" window shows the top
ten models found, based on user-specified evaluation criterion.
40
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7.4 Using the Genetic Algorithm
Several parameters are used to adjust the performance of the GA (Figure 28):
Seed value: VB3 uses an internal random number generator to produce random
values. Setting the seed to a previously-used value will produce results identical to
that earlier run, allowing the analysis to be reproduced by other parties. Changing the
seed creates a new series of random values, possibly returning a different set of
identified regression models.
Population size: number of individuals in the population of each generation. A larger
population broadens the search at each generation, but slows processing time.
Number of generations: because individuals can reproduce and mutate once each
generation, the question is how long to run the search. Fitness of every individual in
the population is evaluated at the end of each generation.
Mutation rate: chance each individual has of undergoing random mutation in each
generation. The higher the mutation rate, the more random (less directed) the search
of parameter space is.
Crossover rate: the percent of each parent's genome that children receive. For
example, if crossover = 0.5, child 1 and child 2 each receive 50% of the genome of
parent 1 and parent 2. If crossover = 0.3, child 1 receives 30% of the parent 1
genome and 70% of the parent 2 genome, while child 2 receives 70% of the parent 1
genome and 30% of the parent 2 genome.
The best GA parameter values depend on the dataset being investigated, but
typical values of the mutation rate are between 0.001-0.1 and typical values of the
crossover rate are 0.25-0.5. For small datasets, a population size and generation number
of 100 are sufficient. Larger datasets may require increased numbers for optimal
solutions. The user must invoke an experimental approach for changing these parameters
and examining the results.
Manual Genetic Algorithm
I I Set Seed Value:
Population Size:
Number of Generations:
Mutation Rate:
Crossover Rate:
100
100
0.05
0.50
Run
Figure 28. Genetic algorithm options within the modeling interface.
41
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7.5 Evaluating Model Output
After selecting a method to build models (GA or Exhaustive) and an evaluation
criterion, click the "Run" button at the bottom of the "Control Options" sub-tab (Figure
25). Progress is displayed on the "Progress" sub-tab at the lower left of the MLR screen.
Note that the "Run" button changes to "Cancel" if the user desires to terminate the
process. Once model-building is completed, the ten best models are displayed in the
"Best Fits" window (Figure 29). Selecting a model from the list results in:
A list of selected IVs for the model, with associated regression coefficients and
statistics displayed on the "Variable Statistics" sub-tab (Figure 30).
A list of evaluation metrics for the selected model shown on the "Model Statistics"
sub-tab (Figure 31).
The "Results" sub-tab shows two data series - model fits and observations versus
observations (Figure 32). Observations that are chronologically ordered are similar
to a time series plot of the two data series, but ignore the possibility that time steps
between data points are not equally spaced.
The "Fitted vs Observed" sub-tab shows plots and tables based on fitted model
values versus the observations (Figure 33).
The "ROC Curves" sub-tab shows a plot of the Receiver Operating Characteristic
curve of each "Best Fits" model (Figure 34), as well as a table showing the
computed AUC (area-under-the-curve) for each ROC curve (see Section 7.7).
The "View Report" generates a text report of model and variable statistics for the
selected model.
The "Residuals" sub-tab allows access to residual analysis functions in VB3 (see
Section 7.8).
The "Prediction" tab appears at the top and bottom of the VB3 screen, allowing users
to proceed to the prediction component (Figure 29).
Note that selecting a different model from the "Best Fits" list will update the
Variable and Model Statistics tables, as well as the information displayed on the
"Results," "Fitted vs Observed," "ROC Curves," and "Residuals" sub-tabs.
42
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1 ง0 Virtual Beach v3.0
BBS
| Location Global Datasheet GBM MLR PLS Prediction
d
L*i
&
X
Compute Manipulate Transform
Manipulate Data
Data Manipulation Model
Model Settings
Variable Selection | Control Options [ Number of Observations: 101
Evaluation Criteria
Akaike Information Criterion (AIC)
GET
Maximum Number of Variables in a Model
Available: 12, Recommended: 11, Max: 12
|5 j MaximumVIF
Model EvaluationThresholds
[SI Decision Criterion (Horizontal)
[235 | Regulatory Standard (Vertical)
Threshold Transform Current US Regulatory Standards
ฎ None
O Log10
O Ln
O Power
E. coli. Freshwater: 235
Enterococci, Freshwater: 61
E nterococci, S altwater: 104
Manual || Genetic Algorithm
I I Set Seed Value: j
Population Size: 1100
Number of Generations: [ 100
Mutation Rate: 10.05
Model information
Best Fits:
ii Hi iii
-38.6157
-38.3333
-38.2914
-38.0716
-37,9430
-37.8625
Lv,|
View
Report
Cross
Validation
Variable Statistics - SelectedModel Model Statistics - SelectedModel
Parameter
Coefficient
Standardized Coefficient
Std. Error
(Intercept)
-1.3435
0.4832
072
0.0005
0.2421
0.0002
Gulls
-0.0011
-0.1686
0.0005
R RAIN 24
0.0322
0.3435
0.0072
WaveHeight ft
0.3843
0.3630
0.09G4
DOY
0.0128
0.5528
0.0020
ClearWater_y1_0
-0.1733
-0.1343
0.1107
Progress i Results j Fitted vs Observed ROC Curves Residuals
YPred Threshold!
YObs
Location Global Datasheet PLS MLR [ GBM Prediction
Figure 29. Modeling results after completion of a run using the genetic algorithm.
43
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Model Information
Best Fits:
7.2471
8.2076
warn
9.2219
9.2291
9.275S
10.1760
IV Filter
Add to List
Clear List
View
Report
Cross
Validation
Variable Statistics ฆ SelectedModel Model Statistics SelectedModel
Parameter Coefficient Standardized Coefficient
Std. Error t-Statistic
(Intercept] 2.3074
uv -0.0006 -0.4740
1.4679 1.5720
0.0002 -2.8611
WindDirection -0.0029 -0.4027
0.0010 -2.7783
airtemp -0.0194 -0.0582
0.0545 -0.3377
WaveH eight 1.7177 0.2287
1.0498 1.6362
_>|
Figure 30. Modeling Interface showing variable statistics for the selected model.
Model Information
Best Fits:
IV Filter
| Add to List
I Clear List
View
Report
Cross
Validation
7.2471
a
8.2076
ail
8.1112 i
8.2219
8.2231
8.2758
10.1760
V
Variable Statistics ฆ SelectedModel ! Model Statistics - SelectedModel
Metric
Value
A
R Squared
0.4216
Adjusted R Squared
0.3283
Akaike Information Crite...
9.1112
Corrected AIC
11.9112
Bayesian Info Criterion
-21.8342
PRESS
17.5160
RMSE
0.6373
Transformed DC
2.3711
Transformed RS
2.3711
False Positives
0.00
^ nprifirih i
i nnnn
Figure 31. Modeling interface showing model evaluation metrics for the selected model.
44
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Progress [ Results | Fitted vs Observed || ROC Curves || Residuals
Results
YObs
YPred
Threshold
6
5
4
3
2
1
0
0 5 10 15 20 25 30 35 40 45 50 55 60
Figure 32. Modeling interface showing a time series plot for the selected model.
Progress || Results [ Fitted vs Observed | ROC Curves || Residuals |
Select View
Plot: Pred vs Obs
Plot Thresholds
Decision Criterion (Horizontal)
235
235 J Regulatory Standard (Vertical)
Threshold Transform
O None
Update
ฉ LoglO
O Ln
O Power |NaN
Model Evaluation
False Positives (Type I):
7 '
Specificity:
0.9882
False Negatives (Type II):
81 J
Sensitivity:
0.2956
Accuracy:
0.8758
Fitted vs Observed
-2
Decision Threshold Regulatory Threshold |
i i ฆ i | i ฆ i i | i i ฆ i
ฆ
i ฆ i i | ฆ i i ฆ
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Observed
Figure 33. A scatter plot of fitted values versus observ ations of the selected model.
45
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Figure 34. The ROC curves and AUC table for the model chosen from the "Best Fits" window.
Progress Results Fitted vs Observed F! IJ C Curves Residuals
Model Fit AUC
351.9481 .905019
350.703G .90686
348.9373 .907469
348.6314 .908758
348.4565 .905903
348.3552 .910609
347.0891 .909103
347.0436 .907453
346.5043 .905427
346.4604 .910199
View Table
&
e>
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c
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(/)
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
Receiver Operating Characteristic Curves
for Best-Fit Models
i ' I 1 ' i . | . i i i i i i . | . i i i i . i . . | . i i i i . i . i | i i i i i i i i i'
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.
1 - Specificity
7.6 Viewing X-Y Scatter plots
On the MLR "Fitted vs Observed" and the MLR "Residuals" sub-tabs in the
Model Information box, users are shown a graph to compare observations to fitted values
from the model (Figure 33). Users can view different results from the pull-down tab
from the "Select View" box:
A plot of fitted values versus observations: "Pred vs. Obs"
A table summarizing model errors (false negatives/false positives) as the decision
criterion (DC) varies across the range of the response variable: "Error Table: DC
as CFU"
A plot of the percent of probability of exceedance (based on the current DC)
versus observations: "% Exc vs. Obs"
A table summarizing model errors as the percent of probability of exceedance is
varied: "Error Table: DC as % Exc"
On the two plots, a right-click in the plot area shows a menu of functions for
saving, copying, printing or manipulating the plot view. The plot area can be zoomed
and un-zoomed: the left-click on the mouse drags an area for zooming in; the right-click
selects "Un-Zoom" or "Set Scale to Default" to see the entire data set. To pan to a plot
area not in view, hold the Shift key down and use the left mouse button to drag the view.
Hovering the cursor over a data point shows the ID of the selected data point; if the
information does not appear, right-click on the graph and select "Show Point Values."
46
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Regarding interpretation of these plots, the green (Regulatory Standard or RS) and
blue (Decision Criterion or DC) lines allow model evaluation and provide information for
choosing a DC for later predictive purposes. On the plots, false positives represent data
points in the upper left quadrant of the graph, where the model fits/predictions exceed the
DC, but observations are below the RS. In such cases, a beach advisory would be
incorrectly issued based on the model's prediction, potentially leading to, for example,
economic losses. False negatives (points in the lower right quadrant) represent a more
serious scenario: model fits/predictions below the DC and observations that exceed the
RS. In other words, swimming at the beach may have been allowed when it should have
been prohibited due to elevated FIB concentrations.
A model that produces no false positives or false negatives would be an ideal
decision tool, but this is often unattainable with real data. Examining the two tables from
the "Fitted vs Observed" select view tab should allow users to set a robust DC, by using
units of the actual response variable or a percentage probability of exceedance that
minimizes both errors. In most cases, the RS is set by federal or state law and should not
be adjusted by the user; however, users are free to adjust the DC to minimize false
negatives and false positives.
7.7 ROC Curves
In addition to time series and scatter plots which show results for an individual
model, users may also compare all the "Best Fits" models using the ROC Curves tab
(Figure 34). A Receiver Operating Characteristic curve shows the true positive rate
(sensitivity) plotted against its false positive rate (1 - specificity) for a model, as the
Decision Criterion (DC) varies between its minimum and maximum predicted values.
Models can then be compared using the area under their ROC curves (AUC). Models
having the largest AUC values perform best over the entire decision space.
The model with the largest AUC appears in red text in the ROC tab's model list.
A single ROC may be plotted by selecting a model in the list and clicking the "Plot"
button. Multiple models can be selected in the usual Windows fashion with Shift-Click
(select all items between the first and second selection) or Control-Click (select only the
clicked items). The background cell color of models not selected for plot display will be
gray after "Plot" button is clicked.
Clicking the "View Table" button will replace the ROC plot with a table showing
false positives, false negatives, sensitivity, and specificity at every evaluated value of the
Decision Criterion for a single model. Users need only click on a model in the list at the
left of this table to see its results. The ROC plot returns to view after clicking the "View
Plot" button.
AUC calculations are performed and curves are plotted when the "ROC Curve"
sub-tab is selected. If this tab is active and new models are subsequently built, leaving
this tab and returning will generate the new plots and AUC values.
7.8 Residual Analysis
Users may click the "Residuals" sub-tab to view information about the residuals
of the selected model (Figure 35). There are three additional tabs on Residuals:
"Residuals vs Fitted," "Fitted vs Observed," and "DFFITS/Cooks" (DF/C).
47
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Progress Results Fitted vs Observed ROC Curves Residuals
r Rebuilds
:ielectedM odel
R esiduals vs Fitted Fitted vs 0 bserved D FFIT S /Cooks
A.D. Normality Statistic = 0.6167 "
A.D. Statistic P-value = 0.1082
Studentized Residuals vs Fitted
ฐ ฐ <ฃ
Fitted Values
Figure 35. Information available on the Residuals sub-tab, including a plot of externally-studentized
residuals versus model fits that shows results of the Anderson-Darling normality test.
The Residuals vs Fitted tab shows a plot of externally-studentized residuals (Cook
and Weisberg 1982) versus their fitted model values (Figure 35). In the upper-left corner
of the plot, the Anderson-Darling normality statistic (Anderson and Darling 1952) is
shown with its statistical significance (p-value). Linear regression assumes normally-
distributed residuals, so that if this A-D normality test fails (i.e., the p-value is less than
0.05), the user can transform the response variable, transform some of the IVs, or delete
high leverage observations, using the DF/C tab.
On the DF/C tab, observations are sorted by the largest (absolute value) measure
in a table (Figure 36). At the lower left, radio buttons can be used to toggle between
DFFITS and Cook's values, as well as change the view from a table of sorted values to a
plot of the DF/C values versus the Record ID (Figure 37). Data points with very large
DF/C values (i.e., lying outside the horizontal red boundaries on the plot) distort the
estimates and standard deviations of the regression coefficients. They are essentially
"outliers" and some thought to their removal from the dataset should be given.
48
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Progress Results |j Fitted vs Observed || ROC Curves [ Residuals
Rebuilds
Clear
| View Data |
View
ฎ Table
O Plot
Residuals
0 DFFITS
O Cook's
Residuals vs Fitted Fitted vs Observed I DFFITS/Cooks
Residual Table
Iterative Rebuild
2xSQR(p/n) = 0.3676
Auto Rebuild
Stop when all DFFITS values less than 0.3676
0 iterative threshold using 2"SQR(p/n) = 0.3676
O constant threshold 10 3308
Record
Date/Time
DFFITS
A
~
114
37447.375
-1.178758
36
36745.375
-0.744289
0
36676.375
0.722615
124
37483.375
0.632616
97
37410.375
0.494881
11
36696.375
0.443437
50
37041.375
-0.438701
95
37405.375
-0.431037
94
37404.375
-0.418439
V
Figure 36. A table of the DFFITS scores of the residuals.
Progress Results Fitted vs Observed ROC Curves Residuals
Rebuilds
SelectedModel
Clear
View
O Table
ฉ Plot
Residuals
ฎ DFFITS
O Cook's
Residuals vs Fitted Fitted vs Observed DFFITS/Cook:;:
Residual Plot
DFFITS Residuals for SelectedModel
0.6
ฃ 0.0
cutoff = 0.1987 -cutoff = -0.1987
n n
i I i i i i I i
100 200
300 400
Record
500
600
700
Figure 37. A plot of the DFFITS scores of the residuals.
When the grid of DF/C values is visible, clicking the "Go" button in the Iterative
Rebuild section removes the observation with the largest absolute value DF/C, re-fits the
regression, and calculates new DF/C values for the remaining observations (Figure 38).
This model is named Rebuild! and added to the "Rebuilds" window at the top left of the
sub-screen. Clicking the Iterative Rebuild "Go" button again produces a model called
Rebuild! which is calculated after removing the observation with the largest absolute
value DF/C remaining in the dataset. The user can continue to click "Go" and remove
49
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observations with the largest remaining DF/C, creating Rebuild3, Rebuild4, Rebuild5,
etc. VB3 will not allow users to delete any observations if 10 or fewer remain in the
dataset.
Whenever a rebuild model is created by pressing the "Go" button, the information
displayed in the Variable and Model Statistics tables, as well as the plots and information
on the "Residuals" sub-tab, is automatically updated to reflect it, even if another model is
highlighted in the "Best Fits" window. The user can select any model in the "Best Fits"
window list, however, to view its associated data and plots.
The user has freedom to remove outliers while toggling between DF/C measures.
For example, the first removal can be based on a DFFITS value, the next removal on a
Cook's Distance, the next two removals on DFFITS, etc. Users may clear models from
the "Rebuilds" window by clicking the "Clear" button.
Rather than using Iterative Rebuild, there are two other choices under the "Auto
Rebuild" box, both of which remove all observations above some threshold. The
"iterative threshold" radio button bases removals on a threshold that is updated whenever
an observation is deleted. For DFFITS, this threshold is 2*(p/n)0 5, where p is the
number of IVs in the model and n is the current number of observations in the dataset.
For Cook's Distance, the threshold is 4/n.
Residual Table
Iterative Rebuild Auto Rebuild
Stop when all DFFITS values less than 0.3676
2KSQR(p/n] = 0.3676 <*) iterative threshold using 2KSQR(p/'n) = 0.3676
O constant threshold
Go
Figure 38. DFFITS/Cook's Distance controls for removing highly influential data points.
When the "iterative threshold" radio button is invoked inside the "Auto Rebuild"
box, VB3 first checks if any DF/C values are above the threshold; if so, VB3 removes the
observation with the largest absolute DF/C and recalculates the regression model, the
DF/C values, and the threshold because n has been reduced by 1. VB3 then checks if any
of these new DF/C values are above the recalculated threshold. If so, the process repeats.
VB3 continues until no remaining DF/C values exceed the current threshold or until half
of the dataset has been removed, whichever comes first. For example, if a dataset has
100 observations, VB3 will allow 50 to be removed before it breaks the Auto Rebuild
removal loop. The user can then click the Auto Rebuild "Go" button again to remove
another 25 observations of the remaining 50. In practice, one should not remove more
than about 5% of the original dataset as outliers; removing more observations than this
indicates a poor regression fit and warrants a different analytical technique. Indeed,
under the assumption of normally distributed data, we expect 5% of the observations to
fit relatively poorly.
The "constant threshold" radio button option differs from the "iterative threshold"
only in that the threshold entered by the user to the input box remains the same regardless
of how many observations are deleted. Updated DF/C values are still calculated after
every removal. VB3 will also stop this process if half the number of starting observations
0.3308
Go
50
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has been deleted. There is an upper limit to the number that can be entered into the
"constant threshold" input box (DFFITS = 3; Cook's Distance = 16/n).
Upon completion of the Auto Rebuild process, multiple models may have been
added to the "Rebuilds" window (Figure 39). For example, if 10 observations were
removed, Rebuild 1 through RebuildlO will appear in that window.
When the user wants to move from the MLR tab to the Prediction tab, the model
carried forward is the one highlighted blue in the "Best Fits" window or "Rebuilds"
window. It is easy to confirm that the model selected will be carried forward by checking
the numbers shown within the "Variable Statistics" and "Model Statistics" sub-tabs
(Figures 30 and 31). Note that observations removed from the dataset using the
"Residuals" sub-tab are not removed from the local dataset shown on the MLR "Data
Manipulation" tab.
US Virtual Beach v3.0
BIBB
Location Global Datasheet GBM MLR PLS Prediction
tS>
L * G*
Compute Manipulate- Transform 1
AO
Manipulate Data
1 Data Manipulation |l Model j
Variable Selection j Control Options j Number of Observations: 148
Evaluation Criteria
Akaike Information Criterion (AIC)
11 Maximum Number of Variables in a Model
' Available: 11, Recommended: 11, Max: 11
|5 MaximumVIF
Model EvaluationThresholds
mn Decision Criterion (Horizontal)
1235 | Regulatory Standard (Vertical)
Threshold Transform Current US Regulatory Standards
ฎ None E. coli. Freshwater: 235
5 '"ฐ9^ Enterococci, Freshwater: 61
O Ln
O Power Enterococci, Saltwater: 104
Manual J | Genetic Algorithm
P1 Set Seed Value:
- 1
Population Size:
100
Number of Generations:
100
Mutation Rate:
0.05
Crossover Rate:
0.50
Run
There are 2047 possible variable combinations
Model Information
Best Fits:
34.4884
34.2375
34.0417
33.5026
33.1921
33.0653
I View |
| Report |
Variable Statistics - Rebuild4 Model Statistics - Rebuild4
Parameter
(Intercept)
WaveH eight
Precip_T otal
T urbidity
Visibility
Wet_Bulb_F
Coefficient Standardized Coefficient
-0.4789
0.2544 0.3506
14.8304 0.1146
0.0120 0.3894
-0.0431 -0.1380
0.0302 0.2905
Std. Error t-Statistic
0.6500 -0.7368
0.0603 4.2175
8.8095 1.6834
0.0027 4.4654
0.0217 -1.9885
0.0080 3.7759
Progress Results Fitted vs Observed ROC Curves I Residuals
Rebuilds
| View Data |
View
ฉ Table
O Plot
Residuals
ฉ DFFITS
O Cook's
Residuals vs Fitted Fitted vs Observed DFFITS/Cooks
Residual Table
Iterative Rebuild
2*SQR(pAi) = 0.3727
I Gฐ I
Auto Rebuild
Stop when all DFFITS values less than 0.3727
ฉ iterative threshold using 2*SQR(p/n) = 0.3727
O constant threshold j 0.3308 ~|
~ED
137776.375
37041.375
36738.375
I -0.467011
J-0.451878
0.389194
Global Datasheet
Figure 39. Residuals interface showing a list of rebuilt models resulting from observation deletions,
and their associated statistics and residual plots.
Viewing the Data Table
From the DFFITS/Cooks sub-tab, users can click the "View Data" button to
display a history of observation removal for the selected model. From this window, users
may export the dataset for external use or re-importation into VB3 (Figure 40).
51
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Records Eliminated from Model Data Set
Model
Value113' Residual Type Date
logEcoli
clouds
SQR(turbidity) SQR[Previous24h
~
Rebuildl
[-1.339716
DFFITS
8/16/2007
3.58546073
5
16.06237840420...
1.118033988749..
Rebuild2
-1.013314
DFFITS
6/1/2009
0.301029996
4
2.664582518894...
0
*
Rebuild3
0.685558
DFFITS
7/25/2008
2.939519253
3
5.540758070878...
0
Model Data Set - Inactive Records in Red
Date
logEcoli
clouds
SQR[turbidity]
SQR[Previous24hrr
POLY[airtemp] A
~
6/1/2007
1.230448921
4
1.717556403731...
0
1.507064992941.
6/2/2007
2.939519253
4
1.612451549659...
0
1.603774691988.
6/3/2007
1.897627091
2
6.606814663663...
0.223606797749...
1.783618147049.
6/4/2007
1.204119983
3
3.154362059117...
0
1.783618147049.
n QmnQQQQ?
A
1 QT^CMneiC?
n
1 77QAQ0C711OQ V
<
>
Figure 40. "View Data Table" window for examining the dataset after removal of influential data
points.
The "Fitted vs Observed" plot on the "Residuals" sub-tab is the same as that
introduced in Section 7.6 (Figure 41). There are two plots and two tables to examine,
along with controls to modify the Decision Criterion (blue horizontal line) and
Regulatory Standard (green vertical line).
Progress Results Fitted vs Observed ROC Curves Residuals
Rebuilds
Rebuild4
Clear
o elect edM ode!
FtebuHdl
Rebuilds
Residuals vs Fitted [ Fitted vs Observed j DFFITS/Cooks |
Select View
Plot: Pred vs Obs
| Update ]
Model Evaluation
False Positives (Type I): 9
Specificity: 0.9848
False Negatives (Type II): 77
Sensitivity: 10.3125
Accuracy: 0.8781
Plot Thresholds
|235 | Decision Criterion (Horizontal)
Regulatory Standard (Vertical)
Threshold Transform
O None
0 Log10
O Ln
O Power
Fitted vs Observed
| DecblQi Tirettott Reg i Story T> ret told |
*2 2
\ ii
fir--*
0
>
2
Observed
Figure 41. Fitted vs Observed plot on the Residual sub-tab with model evaluation threshold control
and model evaluation statistics.
52
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7.9 Cross-Validation
Clicking the "Cross-Validation" button in the "Model Information" box brings up
another window where the user can set two parameters: sample size for the testing data
(Ne) and number of random samples (Nr) taken (Figure 42). When the "Run" button is
clicked, a random sample of size Ne is taken from the modeling dataset and set aside.
Each "Best Fits" model is then re-fit to the remaining training data. The IVs in each
model stay the same, but the regression coefficients are adjusted to reflect the least-
squares fit to the training data. The Mean Squared Error of Prediction (MSEP) is then
calculated based on the Ne testing data points for each candidate model. This process is
done Nr times. A table then appears to show the average MSEP values for each of the 10
"Best-Fit" models.
Cross-validation is useful for examining the predictive power of models, i.e.,
ability to make predictions for data they have not seen before. For users wishing to
emphasize predictive ability of a potential model, cross-validation allows evaluation of
which candidate model consistently makes the best predictions, i.e., has the lowest
MSEP. Note that the PRESS statistic VB3 provides as a model evaluation criterion is a
cross-validation statistic with Ne, set to 1. The PRESS algorithm removes one
observation at a time from the dataset, re-fits the model regression coefficients, and
calculates the squared residual for the removed observation. It does this once for every
observation in the dataset to compute the model's PRESS value a somewhat cursory
look at a model's predictive potential.
We recommend that approximately 25% of the total number of observations be
used for testing, and that at least 1000 trials be performed.
US Cross Validation |^~|f5][5T|
Total Number of Observations: 225
Number of Observations Used for Testing: I ^
Number ol Trials: 100 i ^un il
Fitness
MSEP
Ind Var 1
Ind Var 2
Ind Var 3
Ind Var 4
Ind Var 5
Ind Var 6
Ind Var 7
~
ฆ143.323483044...
0.178258878933...
clouds
S Q R [turbidity]
SQR[Previous24...
P0LY[ai
rtemp]
POLY[dewpoint]
PO LY[atmpressure]
LOG[cuyahogariv..
143.092024887...
0.183755G17G10...
clouds
SQR[turbidity]
SQR[Previous24...
P0LY[ai
rtemp]
POLY[dewpoint]
POLY[atmpressure]
LOG[cuyahogariv..
-142.911814497...
0.189189307571...
clouds
SQR [turbidity]
SQR[Previous24...
P0LY[ai
rtemp]
POLY[dewpoint]
LOG[cuyahogariv...
P0LY[ucomp]
-142.824883297...
0.172544273813...
clouds
S Q Ft [turbidity]
SQR[Previous24...
P0LY[ai
rtemp]
POLYfdewpoint]
PO LY[atmpressure]
L0G[cuyahogariv..
-142.625947884...
0.184948801378...
clouds
SQR [turbidity]
SQR[Previous24...
P0LY[ai
rtemp]
POLY[dewpoint]
LOG[cuyahogariv...
POLY[rockyriverfl..
-142.45G0294B0...
0.178419303326...
clouds
SQR[turbidity]
SQR[Previous24...
P0LY[ai
rtemp]
POLY[dewpoint]
POLY[atmpressure]
LOG[cuyahogariv..
-141.434871829...
0.175263600776...
windspeed
clouds
SQR [turbidity]
SQR[Previous24...
POLY[airtemp]
POLY[dewpoint]
POLY[atmpressure
-141.33G885984...
0.178221812478...
windspeed
clouds
SQR [turbidity]
SQR[Previous24...
POLY[airtemp]
POLY[dewpoint]
PO LY[atmpressure
-141.288453099...
0.180921289930...
windspeed
clouds
SQR [turbidity]
SQR[Previous24...
POLY[airtemp]
POLY[dewpoint]
POLY[atmpressure v
<
>
Figure 42. Cross-validation results for each of the 10 best-fit models.
7.10 Report Generation
A text report of modeling results can be generated, copied to the system
clipboard, or saved to a text file using the "View Report" button in the middle of the
MLR-Model screen. From here (Figure 43), users can view the report by selecting the
desired models and clicking the "Generate Report for Selected Models" button. The
report contains descriptive statistics for each model variable and model evaluation
statistics. Any number of best-fit models can be selected for reporting.
53
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A recommended approach to saving the information in an external application is
to copy the report to the clipboard with the "CopytoClipboard" button and paste it into an
application such as Microsoft Word or WordPad. NotePad or other simple text editors
will also work, but column formats will likely be lost, making the report difficult to
interpret.
] MLR Model Building Report - Best Fits
106.8737
105.1724
103.9300
103.6889
103.6593
102.6959
102.4918
-101.6905
-101.0790
-100.7915
Generate Report
for Selected
Models
SaveToFile
CopyT oClipboard
View Evaluation
Graphs
Done
Select models for report:
MLR Model Building Report
VB2 Project Name:
VB2 Project File:
Imported Data Input File:
Independent Variable: logEcoli
Number of observations: 225
Models are listed in order of best-fit based upon selected evaluation criterion.
Model Evaluation Criterion: Akaike
Model: logEcoli = 1 3.1649e-01 - 25.41 04e-03*airtemp + 10.227e-03*turbidity + 87.1 553e-03
"clouds - 26.4922e-05*rockyriverflow + 1 8.4437e-03*windspeed + 18.7124e-05*cuyahogariverflow
+ 22.4786e-02*Previous24hrrainfall + 26.035e-03*dewpoint
Model Evaluation Score: -1.0687e02
All Evaluation Metrics:
R Squared: 4.789e-01
Adjusted R Squared: 4.596e-01
Akaike Info Criterion: -1.0687e02
Corrected AIC: -1.0585e02
Figure 43. A text report generated on the modeling results.
Comparative bar graphs can be displayed (Figure 44) to view evaluation criteria
for all top models by left-clicking and dragging the mouse to highlight selection and
clicking the "View Evaluation Graphs" button (Figure 43). Hover the mouse over any
plot to display the model evaluation criteria at the very top of the screen. Moving the
mouse over a bar on a plot will show that model's coefficients under the title at the top,
and a label will appear with that same information. Note that evaluation criteria graphs
are initially scaled to emphasize differences between model scores although those
differences may, in fact, be quite small on an absolute scale (Figure 45). With the cursor
over any graph, right-click the mouse and select "Set Scale to Default" to view the un-
sealed graph.
54
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S Model Evaluation Criteria
Adjusted R2
logEcoli = 13.0836e-01 - 23.3539e-03xairternp + 10.8332e-03xturbidity + 98.1067e-03xclouds - 28.6138e-05xrockyriverflow + 18.535e-05xcuyahogariverflow h
23.473e-02xPrevious24hrrainfall + 25.5045e-03xdewpoint
.l.lllllll
mm
Figure 44. Plots of various model evaluation metrics for the 10 best-fit models.
U Model Evaluation Criteria
! Model Evaluation Criteria
R2
R2
logEcoli = -14.2B08e00 + 50.1301 e-01 "POLY[[airtemp][dewpoint]] - 47.2897e-02"PC logE coli = -13.9053e00 * 48.3165e-0rP0LY[[airlemp][dewpoint]] - 51,8026e-02"PC
11.2129e-04"SQR[[airtemp][cuvahogaiiverflow]] + 14.3251 e-02"SQR[[Previous24hri 14.3141 e-02"SQR[[Previous24hrrainfaU][windspeed]] + 12.4374e-01 "POLY[[air(emp.
i ฐJ
+
Figure 45. Scaled versus un-scaled views of selected model evaluation criteria.
55
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8. PARTIAL LEAST SQUARES
Partial Least Squares (PLS) regression minimizes a problem that can arise in
MLR modeling: over-fitting in the presence of correlated predictors. To over-fit is to
match past data more closely than the real-world process being modeled. MLR is prone
to over-fitting because it makes the closest possible linear match to past data, even at the
cost of accuracy in predicting future observations.
As opposed to requiring the MLR user to be vigilant and proactive, PLS
regression (Brooks et al. 2013) inherently accounts for collinearity to suppress over-
fitting, and ranks the IVs by their influence in variable selection. Using PLS regression,
the user can include all available IVs in the model and let the algorithm sort out which
IVs are most useful, simplifying the sometimes laborious processes of variable selection
and comparing interactions.
A key feature of PLS (and GBM) modeling is the use of cross-validation to assess
real-world prediction accuracy. Model selection and threshold setting (section 8.4) are
done with reference to the true positive, true negative, false positive and false negative
counts, which are calculated by 5-fold cross validation. This means that the data are split
randomly and evenly into five subsets and five models are built to predict exceedances on
each of the five subsets. For each of these models, the subset predicted is left out of
model building, so the counts reflect prediction of novel observations, not accuracy in
fitting past observations. Greater detail about the PLS modeling method is available in
Brooks et al., 2013 and Hastie et al. 2009.
8.1 Data Manipulation
The MLR, PLS, and GBM modules all have "Data Manipulation" sub-tabs
(Figure 46). When the user first clicks on the PLS tab from the Global Datasheet, data in
the PLS Data Manipulation sub-tab is identical to data on the Global Datasheet. From the
PLS data tab, the user can change the "local" data to suit the PLS analysis. The local
datasheet has all of the functionality of the Global Datasheet discussed in Section 6.
Changing local data has no effect on the Global Datasheet; however, going back to the
Global Datasheet and making changes will overwrite local datasheets on each of the
modeling tabs.
56
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Global Datasheet
X ฎ #x ~
Compute Manipulate Transform
AO
Manipulate Data
Run Cancel Drop Variable(s)
Model Variable Selection
Data Manipulation Variable Selection j Model j Diagnostics
Make further manipulations if desired for modeling.
File
Column Count
Row Count
Date-Time Index
Response Variable
Testing_Dataset jds
50
100
DateTime
Ecolijog
Disabled Row Count 0
Disabled Column Count 0
Hidden Column Count 0
Independent Variable Count 48
DateTime
6/14/201011:(H
6/15/2010 9:55
6/16/20109:45
6/17/20109:45
6/21/2010 11:11
6/22/201011:3!
G/?R/7nm Q-RA
Figure 46. Data Manipulation: the first sub-tab on each of the method tabs.
8.2 Selecting Variables for Model Building
The "Variable Selection" tab is where IVs for model development are chosen
(Figure 47). Users may select all or a subset of the IVs for consideration in the model. All
eligible IVs are listed in the "Available Variables" window (left column). Any IVs that
users wish to include in the model must then be moved to the "Independent Variables"
window by highlighting the IV and clicking the ">" key. Any number of IVs can be
added or removed from this list. Once the desired IVs have been selected, click the
"Model" sub-tab.
I Location
Global Datasheet GBM
MLR PLS
I-
X
#X
~
Manipulate Data
Model
Variable Selection
Data Manipulation j Variable Selection j Model Diagnostics
Dependent Variable: logEcdi
Available Variables (11) Number of Observations: 225 (0)
Independent Variables
turbidity
Previous24hnairrfall
windspeed
airtemp
dewpoint
clouds
atmpressure
cuyahogariverflow
rockyriverflow
ucomp
vcomp
0
0
Figure 47. Selecting variables for PLS processing within the modeling module.
57
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8.3 The Regulatory Standard
To build a PLS ribbon, observations must be defined as exceedances or non-
exceedances; PLS and GBM models will not run if the dataset has no exceedances. This
is done by setting the Regulatory Standard (RS) at the top of the "Model" tab and then
specifying, using the radio buttons, units to enter into the RS. The default RS is the
USEPA's federal standard for E. coli in freshwater, 235 CFU per 100 mL. Because these
are raw units of measurement, the radio button transformation choice should be set to
"Value." However, users may be thinking of bacteria concentrations in logarithmic units;
if so, the RS is 2.371 (= logio(235)). To communicate this to VB3, enter 2.371 in the
"Regulatory Standard" box and click the "LoglO (value)" radio button (Figure 48).
HI Virtual Beach v3.0
File
Location Global Datasheet GBM MLR
PL5
m * ฎ #X E3
Compute Manipulate Transform
A 0
Manipulate Data
Run Cancel
Model
Drop Variable(s)
Variable Selection
Data Manipulation Variable Selection
Model
Diagnostics
Model Evaluation Threshold
Regulatory Standard
235
Threshold entry is transformed:
ฎ Value
O Logl 0 (value)
O Loge (value)
O Power (value) exp:
2013 US Regulatory Standards
E. coli. Freshwater: 235
Enterococci, Freshwater: 104
Enterococci, Saltwater: 61
Figure 48. Setting the Regulatory Standard and running models for PLS.
8.4 Modeling Control Options
Clicking "Run" on the PLS Model tab (Figure 48) will start model development.
There is some randomness built into the PLS/GBM solver (due to the aforementioned
randomly-created data folds), so running the PLS/GBM multiple times on the same
dataset will likely produce slightly different solutions. If the user wishes to later replicate
a given PLS/GBM modeling result, they should check the "Set Seed Value" box and put
some positive integer into the input box (Figure 49). If that seed is input again, the
PLS/GBM solver will return a solution identical to the previous solution using that seed
value. After a solution is reached, the "Drop Variable(s)" option on the PLS ribbon
becomes enabled and a Decision Criterion (DC) for the model can be chosen.
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Dropping Unimportant Variables
The "Model Summary" window (left side of Figure 49) lists IVs in descending
order of influence. For a PLS model, a variable's influence is its model coefficient
multiplied by its standard deviation. The influence measurements are then adjusted to
sum to one. The larger the influence of a variable (global sensitivity), the more its
variation drives the response. Low-influence variables (highlighted in red text) can be
dropped from the model by clicking on the variable's name in the list, then clicking the
"Drop Variable(s)" button on the ribbon. If any variables are dropped at this stage, the
model must be rebuilt by clicking the "Run" button.
Virtual Beach 3
Location
MLR
Global Datasheet GBM
M
W
Compute Manipulate Transform i Run Cancel Drop Variable(s)
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Model Evaluation Threshold
235 Regulatory Standard
Threshold entry is transformed:
# Value
Log 10 (value)
ฉ Loge (value)
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79.82 Decision Criterion
Model | Diagnostics
Decision Criterion: 79.82
2013 US Regulatory Standards
E. coli. Freshwater: 235
Enterococci, Freshwater: 104
Errterococci, Saltwater: 61
0 Set Seed Value:
Variable
Intercept
Turbidity
Re!_Humd
Fake_Wind_Dir
WindO
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1.3734
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-0.0006
0.0003
0.0006
-0.0003
0.0005
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0.8306
0.0606
0.0543
0.0244
0.0092
0.0092
0.0053
0.0031
0.0016
0.0010
0.0007
Model Validation
True Positives True Negatives False Positives False Negatives Sen
30 True positive
True negativ
Figure 49. Results after completion of a PLS model run.
Setting the Decision Threshold
Once the user has selected a model, the Decision Criterion (DC) is chosen (see
Section 7.2 for a description of the DC). The graph on the right side of the "Model" tab is
used for this purpose (Figure 49). To understand the plot, consider that a lower DC will
correctly identify more exceedances of the RS threshold, but also produce more "false
positives" by flagging predicted values when the actual water quality is below the RS.
Raising the DC has the opposite tradeoff: reducing false positives at the expense of
identifying fewer true exceedances.
59
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The blue line on the graph indicates true positives and the yellow line indicates
true negatives. Current model performance is indicated by the following:
The vertical dotted line indicates the current location of the DC. The arrow buttons at
the bottom are used to lower/raise the threshold by a small (< , >) or large (ซ , ป)
amount.
The "Model Validation" window (Figure 50) indicates the number of true positives,
true negatives, false positives, false negatives, sensitivity, specificity, and total
accuracy of the model using the current DC. These results are based on cross-
validation of the training data. These numbers will change after a short
computational delay as the DC is moved. Care should be taken when comparing
PLS/GBM model performance (in terms of false/true positives/negatives) with MLR
models. MLR model performance is based on fitted, not cross-validated results.
Cross-validation results are commonly thought to be more realistic in how well the
model will do in future predictions, while fitted values better indicate how well the
model fits previously-collected data. Cross-validation results are generated by
developing models with partial data sets and making predictions for data left out of
model development. For example, 5-fold cross validation would result in five
different sets of IV coefficients for a single model by using 4/5 of the data to develop
each set of IV coefficients, then predicting the remaining 1/5 of the data using those
coefficients. MLR, on the other hand, uses all available data points to fit the model
coefficients and then predicts the same data points. Look at cross-validated
performance of MLR models using the "Cross-Validation" button described in
Section 7.9.
The current numeric value of the DC is shown above the "Model Summary" window
(Figure 49).
The user can change the DC, drop variables, and re-run the model to fine-tune it.
After the model and DC have been chosen, the user can advance to the Prediction tab
(Section 10) to make predictions with the most recently computed model.
Model Validation
I TruePosi... ; TrueNe... :ฆ False P... ; False N... : Sensiti... Sp|
[111"""'" -ฆ'Yp'---" - ig- -jj' i!55 llf*
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Figure 50. Summary of PLS model performance metrics.
8.5 Diagnostics
There are four plots are offered on the "Diagnostics" sub-tab (Figure 51):
60
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The Time Series plot (upper left) displays predicted and observed values of the
response variable. This is a time-series plot if the ID values for the observations are
chronologically-ordered dates/times. If they are not, then this plot will look rather
messy and strange, and be of little interest to the user.
The Residuals vs. Fitted plot (upper right) shows the externally-studentized residuals
versus model-fitted values. The externally-studentized residuals are a way to flag
influential outliers. A common benchmark for a data point with undue influence on
the regression model is an externally-studentized residual (absolute value) greater
than 3.0.
The Residuals vs. Observed plot (lower left) graphs the externally-studentized
residuals against the observations.
The Fitted vs. Observed plot (lower right) shows observations versus model fits and
depicts the RS (green horizontal line) and current DC (blue vertical line).
Note that the fitted values plotted here are not cross-validated fits; rather they are
the model fits based on all the data. For this reason, model performance in this plot
(numbers of true negatives/positives) will likely be better than the model performance
metrics given in the "Model Validation" window on the "Model" tab. A perfect model
will fall along the 1:1 line. The more scatter in this plot, the worse the model fit.
Data Manipulation | Variable Selection | Model | Diagnostics |
Time Series Plot
Residuals vs. Observed
Residuals vs. Fitted
I Q Res duals I
Soft
2o a?
1.5 2.0 2.5 3.0 3.5 4.0
Fitted
Fitted vs. Observed
5 o
o o
Figure 51. PLS Diagnostic plots to help evaluate model fit and influential outliers.
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9. GENERALIZED BOOSTED REGRESSION MODELING
The Generalized Boosted Regression Model (GBM, also known as a gradient
boosting machine) is a machine learning method that uses decision/regression trees
instead of linear equations (Friedman, 2001). A decision/regression tree is a set of binary
decision rules. For example, "if turbidity is less than 15 NTU, go down the right branch,
otherwise go left." A "node" is the end of any branch and designates a continuous or
categorical predictive value for the response variable. The innovative aspect of GBM is
that it doesn't build a single, complex tree: it builds a hierarchical set of many simple
trees, with each subsequent tree fit to the remaining residual error in the data after
previous trees have all been fit. The default maximum number of trees in VB3 is 10,000.
Each tree is determined using a random set of the residual values from the dataset. This,
along with the fact that it sensibly weights the data to learn more about the most difficult-
to-predict cases, means GBM can make accurate predictions for new observations
without over-fitting the training data.
While each tree is a simple structure, the long, linear combination of regression
trees is more complicated. A negative aspect of a GBM model is that the model cannot
easily be inspected graphically or expressed mathematically - it's something of a "black
box." But what it lacks in interpretability and transparency can often be made up in
terms of prediction accuracy. Another noted aspect of GBM, unlike MLR and PLS, is
that it handles non-linear relationships between the response and IVs without having to
transform the IVs. However, GBM is best used on larger datasets (> 100 observations),
and odd results can occur if using GBM on small datasets.
In a GBM, variable selection (identifying and dropping unimportant IV's from the
model) is less important, compared to MLR. Even so, the "Drop Variables" button
(Figure 52) performs as described in Section 8.4. For a GBM model, an IV's influence is
the percentage of branches across all of the decision trees involving that variable, i.e., the
most important variables are those that are most often used to create the branches.
For the GBM analysis, VB3 implements the "gbm" package in R. Details of the
algorithm are provided in Hastie (2009). Despite very different underlying mathematics,
the GBM modeling interface in VB3 is almost identical to the PLS interface (Section 8).
A key feature of GBM and PLS modeling is the use of cross-validation to assess
real-world prediction accuracy. Model selection and threshold setting (Section 8.4) are
done with reference to true positive, true negative, false positive and false negative
counts which are calculated by 5-fold cross-validation. This means the data are split
randomly and evenly into five sections and five models are built to predict exceedances
on each of the five sections. For each, the section being predicted is left out of model
building, so the counts reflect prediction of novel observations, not accuracy in fitting to
past observations.
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9.1 Data Manipulation
The MLR, PLS, and GBM modules all have "Data Manipulation" sub-tabs
(Figure 52). When the user first clicks the GBM tab, the data in the GBM Data
Manipulation sub-tab are identical to data on the Global Datasheet. From here, the user
can change the "local" data to suit the GBM analysis. The local datasheet has all of the
functionality of the Global Datasheet discussed in Section 6. Changing the local data has
no effect on the Global Datasheet; however, going back to the Global Datasheet and
making changes will overwrite the local datasheets on each of the modeling tabs.
งnl Virtual Beach v3.0
File
Location Global Datasheet
GBM
MLR PL5
1*1 *ฃ <3
Compute Manipulate Transform
AO
Manipulate Data
wf
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Model
Drop Variable(s)
Variable Selection
Data Manipulation Variable Selection Model Diagnostics
Make further manipulations if desired for modeling.
File
Testing.xls
Column Count
13
Row Count
148
Date-Time Index
Tstamp
Response Variable
LogCFU_Ecoli
Disabled Row Count
0
Disabled Column Count
0
Hidden Column Count
0
Independent Variable Count
11
T stamp
LogCFU_Ecoli
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35
Figure 52. Data Manipulation: the first sub-tab on each of the Method tabs.
9.2 Selecting Variables for Model Building
The "Variable Selection" sub-tab is where IVs for model development are chosen
(Figure 53). Users may select all or a subset of IVs for the model. All eligible IVs are
listed in the "Available Variables" window (left column). Any IVs that users wish to
include in the model must be moved to the "Independent Variables" window by
highlighting the IV and clicking the ">" key. Any number of IVs can be added or
removed from this list. Once the desired IVs have been selected, click the "Model" sub-
tab.
63
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3 Virtual Beach v3.0
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Global Datasheet
GBM
MLR
PLS
L *
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Data Manipulation Variable Selection Model Diagnostics
D ependent Variable: LogCFU_E coli
Available Variables (11) Number of Observations: 148
Independent Variables
(0)
T urbidity
WaveH eight
Visibility
Dry_Bulb_F
Wet_Bulb_F
Dew_Point_F
Flel_Hunnd
WindU
WindV
Station_Pressure
Precip_T otal
CD
Figure 53. Selecting variables for GBM processing within the modeling module.
9.3 The Regulatory Standard
To build a GBM model, observations must be defined as exceedances or non-
exceedances; GBM and PLS models will not run if the dataset has no exceedances. This
is done by setting the Regulatory Standard (RS) at the top of the "Model" sub-tab (Figure
54) and then specifying with the radio buttons units to enter for the RS. The default RS is
the USEPA's federal standard for E. coli in freshwater, 235 CFU per 100 mL. Because
these are the raw units of measurement, the radio button transformation choice should be
set to "Value." When thinking of bacteria concentrations in logarithmic units, think of
the RS as 2.371 [= logio(235)]. To communicate this to VB3, enter 2.371 in the
"Regulatory Standard" box and click the "LoglO (value)" radio button (Figure 54).
64
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Virtual Beach 3
Location
L
1*1 32
Compute Manipulate Transforrr
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Global Datasheet
W
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GBM
MLR
PLS
~
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o Value
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104.73 Decision Criterion
2013 US Regulatory Standards
E. coli, Freshwater: 235
Enterococci, Freshwater: 1B4
Enterococci, Saltwater: G1
D Set Seed Value:
Figure 54. Setting the Regulatory Standard and running models for GBM.
9.4 Modeling Control Options
Clicking the "Run" button on the GBM ribbon (Figure 54) will start model
development. When modeling is finished, results are displayed (Figure 55). The "Drop
Variable(s)" option is now available on the ribbon and a Decision Criterion (DC) for the
model can be chosen. As described in Section 8.4, a GBM model solution can be
replicated using the "Set Seed Value" check and input box. After a solution is reached,
the "Drop Variable(s)" option on the PLS ribbon becomes enabled and a Decision
Criterion (DC) for the model can be chosen.
Dropping Unimportant Variables
The "Model Summary" window (left side of Figure 55) lists the IVs in
descending order of influence. For a GBM model, a variable's influence is the percentage
of the model's total branches based on the given variable. The larger the influence of a
variable (global sensitivity), the more its variation drives the response. Low-influence
variables (highlighted in red text) can be dropped from the model by clicking on the
variable's name in the list, then clicking the "Drop Variable(s)" button. If any variables
are dropped at this stage, the model must be rebuilt by clicking the "Run" button.
65
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Figure 55. Results after completion of a GBM model run.
Setting the Decision Threshold
Once the user selects a model, the Decision Criterion (DC) can be chosen (see
Section 7.2 for a description of the DC). The graph on the right side of the Model tab is
used for this purpose (Figure 55). To understand the plot, consider that a lower DC
correctly identifies more exceedances of the RS threshold, but also produces more "false
positives" by flagging predicted values when the actual water quality is below the RS.
Raising the DC has the opposite tradeoff: reducing false positives at the expense of
identifying fewer true exceedances.
The blue line on the graph indicates true positives and the yellow line indicates
true negatives. Current model performance is indicated by the following:
The vertical dotted line indicates the current location of the DC. The arrow buttons at
the bottom are used to lower/raise the threshold by a small (< , >) or large (ซ , ป)
amount.
The "Model Validation" window (Figure 56) indicates the number of true positives,
true negatives, false positives, false negatives, sensitivity, specificity, and total
accuracy of the model using the current DC. These results are based on cross-
validation of the training data. These numbers will change after a short
computational delay as the DC is moved. Comparing GBM (and PLS) model
66
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performance, in terms of false/true positives/negatives, with MLR models must be
done carefully. MLR model performance is based on fitted results, not cross-validated
results. Cross-validation results are commonly thought to indicate more realistically
how well the model will do in future predictions and fitted values better indicate how
well the model fits previously collected data. Cross-validation results are generated
by developing models with partial data sets and subsequently making predictions for
data that were left out. For example, 5-fold cross validation would result in 5
different sets of IV coefficients for a single model by using 4/5 of the data to develop
each set of IV coefficients, and then predicting the remaining 1/5 of the data using
those coefficients. MLR, on the other hand, uses all available data points to fit the
model coefficients and then predicts the same data points. Look at cross-validated
performance of MLR models using the "Cross-Validation" button described in
Section 7.9.
The current numeric value of the DC is shown above the "Model Summary" window
(Figure 55).
The user can change the DC, drop variables and re-run the model to fine-tune it.
After the model and DC have been chosen, the user can advance to the Prediction tab
(Section 10) to make predictions with the most recently computed model.
Model Validation
Tru... Tru... Fal... Fals... Sensiti... Specifi... Accur...
18 100 19 11 0.62 0.84 0.80
Figure 56. Summary of GBM model performance metrics.
9.5 Diagnostics
There are four plots offered on the "Diagnostics" sub-tab (Figure 57):
The Time Series plot (upper left) displays predicted and observed values of the
response variable over time if the ID values for the observations are dates/times.
The Residuals vs. Fitted plot (upper right) shows the externally-studentized residuals
versus model-fitted values. The externally-studentized residuals are a way to flag
influential outliers. A common benchmark for a data point with undue influence on
the regression model is an externally-studentized residual (absolute value) greater
than 3.0. Certain patterns seen in this residual plot can indicate the need for a
transformation of the response variable or model IVs. Refer to Meyers (1990) for
details.
The Residuals vs. Observed plot (lower left) graphs the externally-studentized
residuals against the observations.
67
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The Fitted vs. Observed plot (lower right) shows observations versus model fits and
depicts the RS (green horizontal line) and current DC (blue vertical line).
Note that the fitted values plotted here are not cross-validated fits; rather they are
the model fits based on all the data. For this reason, model performance in this plot
(numbers of true negatives/positives) will likely be better than the model performance
metrics given in the Model Validation table on the Model tab. A perfect model will fall
along the 1:1 line. The more scatter in this plot, the worse the model fit.
Data Manipulation [ Variable Selection | Model I Diagnostics
Residuals vs. Fitted
I A Residuals
Figure 57. GBM diagnostic plots to help evaluate model fit and influential outliers.
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10. PREDICTION
VB3's Prediction interface allows users to select a model from the PLS, GBM, or
MLR tabs and make predictions with it, but the prediction tab is hidden until a model is
chosen.
10.1 Model Statement
At the top left of the Prediction tab is the "Available Models" window.
Depending on how many statistical methods were performed on the data, the user could
see "MLR," "PLS," and/or "GBM" in this area. Once a model is chosen, an expression
with the IVs and coefficients in that model is shown in the "Model" window to the right
(Figure 58).
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Available Models:
Ecoli = Turbidity + WaveHeight + Visibility + Dry_Bulb_F + Wet_Bulb_F + Dew_Point_F + Rel_Humd + WindU + WindV +
Model: Stati o n_Pre s s u re + Pre ci p_T otal
Model Evaluation Thresholds
Decision Criterion (Horizontal)
Exceedance Probability
Regulatory Standard (Vertical)
Threshold Transform
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Location Global Datasheet PLS MLR GBM Prediction
Ready.
Figure 58. The VB.i Prediction interface.
10.2 Model Evaluation Thresholds
In the "Model Evaluation Thresholds" box, there are input boxes for the Decision
Criterion (DC), Exceedance Probability, and Regulatory Standard (RS). Setting these
allows model predictions to be evaluated and model specificity, sensitivity, and accuracy
to be calculated. The radio buttons inside the "Threshold Transform" box tells VB3 how
69
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to transform the DC and RS to compare to model predictions and observations (see
Section 7.2 for further guidance). Note that we define the "observations" as the measured
values of the model's response variable (e.g., E.coli CFU measurements). If the
threshold transform definition is set improperly, there can be problems when comparing
modeling predictions to observations, so exercise caution.
10.3 Prediction Form
The bottom half of the Prediction interface is occupied by three data panels (the
empty gray sections separated by blue vertical bars at the bottom of Figure 58): the left
holds IV data; the middle is for observations; and the right shows model predictions and
evaluation metrics. Each panel also contains a column for a unique ID for each row of
data, e.g., the date that data were collected. The panels have separate horizontal and
vertical scroll bars that become visible if the number of rows or columns exceeds the
viewable area. The three panels independently scroll horizontally, but as a group
vertically. Panels can be re-sized by clicking and dragging the blue vertical partitions.
The order of the columns in the left (IV) and right (Model Predictions) panels can be
changed by clicking and dragging the column headers left or right. If it is important to
save a re-arranged column order for the selected model, click on the "Save Column
Order" button just above the IV panel.
Users can import data from files using the "Import IV Data," "Import
Observations" and "Import Combined" (both IVs and observations) buttons on the top
ribbon, or type data directly into the left and middle grids. It is the user's responsibility
to ensure that IV data are in the same units as those used to construct the model.
Depending on the model selected for prediction, the left panel will contain one column
for every unique model IV plus a column for an ID. The middle panel has two columns:
one for the ID and one for the observations (note that the name of the observation column
is identical to the name of the model's response variable).
10.4 Column Mapping of Imported Data
When data are imported via one of the three import buttons (Figure 58), a
"Column Mapper" window opens (Figure 59). This allows users to tell VB3 which
columns in the imported datasheet should be used to fill in the row IDs, IVs and the
observations. By default, the first column of the imported file is mapped to the ID field,
but this can be overridden. If a column in the imported spreadsheet has a name identical
to a model IV or the response variable, VB3 will select it as the appropriate column for
that IV or the observations. If no identically-named column is found, the user must
specify which column of the imported file should be used for the IV and observations.
Once a user has gone through the mapping process for a model, that configuration
is saved. If another data file with the same column names is imported, the column
mapper will not appear. If a model has a saved mapping configuration, it can be viewed
and cleared by clicking "View Column Mapping" on the ribbon (Figure 58).
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11 Column Mapper
Model Variables
Imported Variables
waveheight
waveheight
WindDirection
WindDirection
Ok
Cancel
Figure 59. Importation of IV data using the "Column Mapper" window.
Column Mapper
T stamp
LogCFLLEcoli
LogCFU_Ecoli
Cancel
Figure 60. Importation of observational data using the "Column Mapper" window.
After observations have been imported or manually entered, users specify the
correct data transformation to ensure proper comparison to model predictions. This is
done by right-clicking on the observation column header (the right column of the middle
panel) and choosing an option from the "Define Transform" drop-down menu: none,
logio, loge, or a power transformation; "none" is the default choice. For example, if Logio
observations are imported, the user must change the "Define Transform" menu option to
"LoglO." If untransformed (raw) values of the observations are entered/imported, then
the appropriate "Define Transform" menu choice would be "none."
The IV data are automatically scanned for errors (e.g., blank or non-numeric
cells) when "Make Predictions" is clicked on the ribbon (however, this button is not
enabled until data are entered into the IV data panel). If bad data cells are found, VB3
will tell the user to run an IV data scan by clicking the "Scan IV Data" button on the
ribbon (Figure 61). The IV scan pop-up window is very similar to the one seen on the
Global Datasheet; however, "Delete Column" is not a choice. "Replace With" and
"Delete Row" are the only options for dealing with problems in the IV data grid.
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Virtual Beach v3.0
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Global Datasheet
Import Set EnDDaT Import From Import EnDDaT View Column Scan IV Data Make Predicitons Plot Clear Export
IV Data Observations Combined Data Source EnDDaT
JV 'Jo'-':
(Optional)
ฆ
Import Data
Available Models:
PLS
Model:
Ecoli = T urbidity + WaveHeight + Visibility + Dry_Bulb_F + Wet_Bulb_F + Dew_Poirit_F + Rel_Humd + WindU + WindV +
Station_Pressure + Precip_Total
Model Evaluation Thresholds
ฉ
139.4
Dl
o
50
4
235
rJ
Predictive Record
I id
T urbidity
5/30/2000 9:00:.
5/31/2000 9:00:.
6/1/2000 9:00:0.
6/5/2000 9:00:0.
6/6/2000 9:00:0.
6/7/2000 9:00:0.
6/8/2000 9:00:0.
6/12/2000 9:00
6/13/2000 9:00
6/14/2000 9:00
6/15/2000 9:00
6/19/2000 9:00
6/20/2000 9:00
6/21/2000 9:00
6/22/2000 9:00:... 18
(Optional) Find:
Replace With: ฃ
5/31/2000 9:00:.
j 6/1/2000 9:00:0.
j 6/5/2000 9:00:0.
6/6/2000 9:00:0.
i 6/7/2000 9:00:0.
| 6/8/2000 9:00:0.
6/12/2000 9:00:.
6/13/2000 9:00:.
| 6/14/2000 9:00:.
j 6/15/2000 9:00:.
6/19/2000 9:00:.
j 6/20/2000 9:00:.
j 6/21/2000 9:00:.
I 6/22/2000 9:00:.
Location Global Datasheet PLS MLR GBM Prediction j
Ready.
Figure 61. The scan IV window on the MLR Prediction tab.
Observational data need not be present to make predictions, but they are needed
for model evaluation (sensitivity, specificity, false negatives, false positives, accuracy).
After clicking "Make Predictions" on the ribbon, VB;; uses the model, IV data, and
observational data to fill the right panel with these data columns: ID, Model Prediction,
Decision Criterion, Exceedance Probability, Regulatory Standard, and Error Type (Figure
62).
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B Virtual Beach v3.0
BBB
9
Import Import Import 5et EnDDaT Import From Import EnDDaT View Column Scan IV Data
IV Data Observations Combined Data Source EnDDaT by Date Mapping (Optional)
iat
Plot Clear Export
As CSV
Available Models:
Model:
Ecoli = Turbidity + WaveHeight + Visibility + Dry_Bulb_F + Wet_Bulb_F + Dew_Point_F + Rel_Humd + WindU + WindV +
Station_Pressure + Precip_Total
Model Evaluation Thresholds
ฎ 1139.4
o
Decision Criterion (Horizontal)
Exceedance Probability
Regulatory Standard (Vertical)
Threshold Transform
ฉ None
O Log10
O Ln
O Power |o
Predictive Record
Save Column Order
Clear Column Order
ID
T urbidity
WaveHeight
1
5/30/2000 9:00:...
92
1
2
5/31/2000 9:00:...
12
1
3
6/1/2000 9:00:0...
7.7
1
4
6/5/2000 9:00:0...
55
4
5
6/6/2000 9:00:0...
133
4
6
6/7/2000 9:00:0...
99
1
7
6/8/2000 9:00:0...
21
1
8
6/12/2000 9:00:...
20
3
9
6/13/2000 9:00:...
35
2
10
6/14/2000 9:00:...
14
1
11
6/15/2000 9:00:...
11
1
12
6/19/2000 9:00:...
51
2
13
6/20/2000 9:00:...
20
1
14
6/21/2000 9:00:...
20
2
15
6/22/2000 9:00:...
18
2
16
c/?cnnnno.nn.
E A
>
5/30/2000 9:00:.
5/31/2000 9:00:.
6/1/2000 9:00:0.
6/5/2000 9:00:0.
6/6/2000 9:00:0.
6/7/2000 9:00:0.
6/8/2000 9:00:0.
6/12/2000 9:00:.
6/13/2000 9:00:.
6/14/2000 9:00:.
i 6/15/2000 9:00:.
6/19/2000 9:00:.
6/20/2000 9:00:.
6/21/2000 9:00:.
6/22/2000 9:00:.
6/26/2000 9:00:.
Ecoli
| 2700.000000000.
j 101.5000000000.
135.5 ~
1550.0000000000
6600.000000000
I 485.0000000000
j 18.00000000000
1680.0000000000
| 700.0000000000
159.9999999999
1143.5000000000
| 2100
1180
140.0000000000.
1190
16.00000000000.
Model_Prediction Decision_Criterion A
5/30/2000 9:00:...
2.672
2.144
5/31/2000 9:00:...
1.722
2.144
6/1/2000 9:00:0...
1.685
2.144
6/5/2000 9:00:0...
2.407
2.144
6/6/2000 9:00:0...
2.843
2.144
6/7/2000 9:00:0...
2.549
2.144
6/8/2000 9:00:0...
1.618
2.144
6/12/2000 9:00:...
2.389
2.144
6/13/2000 9:00:...
2.457
2.144
6/14/2000 9:00:...
2.194
2.144
6/15/2000 9:00:...
1.819
2.144
6/19/2000 9:00:...
2.652
2.144
6/20/2000 9:00:...
1.943
2.144
6/21/2000 9:00:...
2.063
2.144
6/22/2000 9:00:...
2.058
2.144
c nc nnnn o-nn.
1 g
11 **
>
Global Datasheet PLS
GBM Prediction
Status:
Figure 62. A prediction grid after I Vs and observational data have been imported, and model
predictions made.
The ID column of the model output panel is taken directly from the ID column of
the IV panel, not IDs in the middle panel. VB3 will make one model prediction per row
in the IV data panel, regardless of how many observations are entered in the middle
panel.
The Model Prediction column contains predicted values of the response variable,
initially displayed in the same units as the model's response variable. Right-clicking on
this column header changes how predictions are displayed in the table (raw, log, or power
units). The Decision Criterion and Regulatory Standard are set by the user. They are
displayed in the same units as the Model Predictions, and their column headers can be
right-clicked to change the displayed units. The Exceedance Probability (displayed as a
percentage, or 100 times the probability) is defined as the probability that the model's
prediction will be larger than the Decision Criterion, based on uncertainty bounds
(confidence intervals) of the model's predictions.
To compare model predictions to observations, VB3 looks at the prediction ID and
attempts to find an observation in the middle panel with the same ID. It does not require
unique IDs for each row in the observation panel, but a model prediction is compared to
the first observation found with the same ID. When comparing model predictions to
observations, an error ("False Negative" or "False Positive") will be reported in the
"Error Type" column.
We again emphasize that assessing model output correctly depends on the
synchronization of units of the Decision Criterion (DC), Regulatory Standard (RS),
73
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model predictions, and observations. VB3 will ensure this happens if the user correctly
specifies the units for the observations (using the right-click column header menu of the
right column of the middle panel) and for the DC and RS (using the radio buttons in the
"Threshold Transform" box of the prediction window).
10.5 Viewing Plots
After predictions are made, a scatterplot of observations versus predictions, or
observations versus the probability of exceedance, can be viewed by clicking "Plot" on
the ribbon (Figures 62 and 63). If no observational data were entered, a message asking
for observational data appears. The features of this plot are similar to those described in
Section 7.6. Plotted points are based on comparing model predictions (right pane of the
Prediction Form) with observations (middle pane) that share the same unique row ID.
Note that the plotted exceedance probabilities are not automatically re-computed because
the Decision Criterion is changed in this plotting window. To see updated exceedance
probabilities for a new Decision Criterion, users must close this plotting window, change
the DC in the "Model Evaluation Thresholds" box, re-click the "Make Predictions"
button on the ribbon, and then click the "Plot" button again.
Plot Thresholds
O 150 | Probability Threshold
ฎ m Decision Criterion (Horizontal)
|235 | Regulatory Standard (Vertical]
Threshold entry is transformed:
ฎ None I Fieplot |
O Log10
O Ln
O Power |1 1
Model Evaluation
False Positives (Type I):
Specificity:
False Negatives (Type II):
Sensitivity:
Accuracy:
0.S57S
17
0.4137
0.8513
Results
4 -ฆ
H 2 -
-1 -ฆ
Y
Regulatory Threshold
Decision Threshold
^
*
0
*
2-101 2
Observe
d
4 5 i
Close
Figure 63. Prediction interface plotting of the observations versus predictions, with model evaluation
threshold controls.
74
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10.6 Prediction Form Manipulation
Two other buttons found in the "Evaluate" section of the ribbon are "Clear" and
"Export as CSV." To view the table in a spreadsheet or word processing program,
"Export as CSV" saves the contents of the entire table (all three panels) in .csv format.
"Clear" deletes all information in every panel of the table. As with most tabular
information in VB3, data in individual panels can be selected with a left-click and drag.
Control-c and Control-v are then used to copy and paste the data into another application
such as Excel.
10.7 Importation of EnDDaT Data
The Environmental Data Discovery and Transformation (EnDDaT;
http://cida.usgs.gov/enddat/) service accesses data from a variety of sources, compiles
and processes it, and performs common transformations. The result is environmental data
from multiple sources sorted into a single table. EnDDaT is a tool for compiling datasets
prior to model development. Once models are developed, EnDDaT can create datasets for
the VB3 Prediction tab. The "Set EnDDaT Data Source," "Import from EnDDaT" and
"Import EnDDaT by Date" buttons on the ribbon (Figure 64) allow users to import data
directly from the EnDDaT web service to the prediction tab of VB3, avoiding manual
entry. See the EnDDaT user guide (available from the EnDDaT website link above) for
step-by-step instructions on obtaining data, specifying transforms, processing data and
developing a URL.
To import EnDDaT data to the IV panel of the prediction grid, click the "Set
EnDDaT Data Source" button and insert an EnDDaT-generated URL that calls for the
IVs needed to make predictions (Figure 65). Choose and activate the radio button for
whether to collect data from a specific time (e.g., the time the beach was visited) or from
the most recently available time. Users must also choose the desired time zone from the
dropdown list. After clicking "OK," the "Import from EnDDaT" and "Import EnDDaT
by Date" buttons are enabled on the ribbon. To import data for the current day, use the
former button. Clicking the latter button opens a calendar for retrieval of data from a
previous day (Figure 66). Whichever button is used, afterwards a pop-up window will
indicate EnDDaT is being accessed (Figure 67). Once data have been retrieved, the
"column mapper" window will open, allowing the user to specify which columns in the
imported EnDDaT data should be matched to each IV in the selected model (see Section
10.4 for more details on column mapping).
datasheet GBM MLR PL5 F
* S
Set EnDDaT Import From Import EnDDaT
Data Source EnDDaT by Date
Import Data
Figure 64. The three EnDDaT-related buttons on the prediction tab.
75
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Set EnDDaT URL -
[ = i S |- I
hrttps ://cida .usgs .gov/enddat/service/execute?&Beach Name=MemoriakDrive+Wayside+Beach+North&Beach Lat =44.138&Beach Lon =-87.58334 Lake =michiga
Timestamp for retrieving EnDDaT data
(<ง) Daily at 8:00:00
Most recently available
|-5;0B(CDT/ES^ 'I | Canoe, ] ( OK | |
Figure 65. Setting URL options for retrieval of (lata from EnDDaT.
Select a date to import
P
September, 2013 |
Sun Mon Tue Wed Thu Fri Sat I
25
26 27 28
28 30 31
1
2 3 4
5 6 7
8
8 10 11
12 EE! 14
15
16 17 18
18 20 21
22
23 24 25
26 27 28
29
30 1 2
3 4 5
~ Today: 9/13/2013
I
Cancel
OK l|
Figure 66. Choosing a previous day for EnDDaT data retrieval.
f
Working
The EnDDaT web service is
connection and the amount of
accessing remc
Jata you've reqi
Cancel
Jte data. Depending on your Internet
ested, this may take up to two minutes.
Figure 67. Pop-up window indicating that data have been requested from EnDDaT.
76
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11. USER FEEDBACK
The USEPA and USGS provide no warranty, expressed or implied, as to the
correctness of the furnished software or the suitability for any purpose. The software has
been tested, but as with any complex software, there could be undetected errors.
Suggestions and experiences from the user community are welcomed by the Virtual
Beach design/development team, and users are encouraged to report problems, issues and
likes/dislikes to:
Mike Cyterski, USEPA: 706.355.8142 (cvterski.mike@epa.gov)
Steve Corsi, USGS: 608.821.3835 (srcorsi@usgs.gov)
The USEPA has limited resources to assist users; however, we make an attempt to
fix reported problems and help whenever possible.
77
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12. REFERENCES
Anderson, T.W., Darling, D.A., 1952. Asymptotic theory of certain "goodness-of-fit"
criteria based on stochastic processes. Annals of Mathematical Statistics 23: 193-212.
Brooks, W.R., Fienen, M.N., Corsi, S.R., 2013. Partial least squares for efficient models
of fecal indicator bacteria on Great Lakes beaches. J Environ Manage 114:470-5. doi:
10.1016/j.jenvman.2012.09.033.
Cook, R., Weisberg, S., 1982. Residuals and Influence in Regression. Chapman and
Hall, New York.
Cyterski, M., Galvin, M., Parmar, R., Wolfe, K., 2012. Virtual Beach User's Manual -
version 2.2. USEPA/600/R-12/024/.
Efron, B., Tibshirani, R., 1986. Bootstrap methods for standard errors, confidence
intervals, and other measures of statistical accuracy. Stat. Sci. 1 (1), 54e77.
Fogel, D. (editor), 1998. Evolutionary Computation: The Fossil Record. New York: IEEE
Press.
Frick, W.E., Ge, Z., Zepp, R.G., 2008. Nowcasting and forecasting concentrations of
biological contaminants at beaches: a feasibility and case study. Environmental Science
and Technology 42, 4818-4824.
Friedman, J.H., 2001. Greedy function approximation: A gradient boosting machine. The
Annals of Statistics, 29:5, 1189-1232.
Hastie, T., Friedman, J., Tibshirani, R., 2009. The Elements of Statistical Learning.
Springer-Verlag, New York.
Myers, R., 1990. Classical and Modern Regression with Applications, 2nd Edition.
Duxbury Press, Belmont, California.
78
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13. ACKNOWLEDGMENTS
We would like to thank the following people, who generously donated their time
and expertise for software testing and review of this document, as well as general support
for the continued development of VB:
Adam Mednick
Wisconsin Department of Natural Resources
Madison, WI
David Rockwell
Cooperative Institute for Limnology and Ecosystems Research, University of Michigan
Center of Excellence for Great Lakes and Human Health
NOAA Great Lakes Environmental Research Laboratory
Ann Arbor, MI
Gerry Laniak
USEPA National Exposure Laboratory
Ecosystems Research Division
Athens, GA
Brett Hayhurst
USGS New York Water Science Center
Ithaca, NY
Diane Mas
Fuss and O'Neill, Inc.
Manchester, CT
Amie Brady
USGS Ohio Water Science Center
Columbus, OH
Donna Francy
USGS Ohio Water Science Center
Columbus, OH
Richard Zepp
USEPA National Exposure Laboratory
Ecosystems Research Division
Athens, GA
Any use of trade, product, or firm names is for descriptive purposes only and does not
imply endorsement by the U.S. Government.
79
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APPENDICES
A.l Transformations
VB3 provides the following transformations, where Xt is the transformed IV and
X is the original IV:
Logio: Xt = logio(X)
Loge: Xt = loge(X)
Inverse: Xt = 1/X
Square: Xt = X2
Square Root: Xt = X0 5
Quad Root: Xt = X0 25
Polynomial: Xt = a + bX + cX2
General Exponent: Xt = Xz where the user specifies the value of z
For the polynomial transformation, the Pearson coefficient is calculated as the
square root of the adjusted R2 value derived from the regression of the response on Xt.
Because this adjusted R2 value can be negative, an empirically-derived formula is applied
when adjusted R2 values fall below 0.1:
Polynomial Pearson Coefficient = (-6.67*REi2 + 13.9*REi- 6.24)*(R2)0 5
where
REi = 1.015 - 1.856*R2+ 1.862*adjR2 - 0.000153*N.
R2 and adjR2 are defined by the regression of the response on Xt, and N = number of
observations.
VB3 transformations (primarily converting x into xb) have specific processing for
certain data values and are not pure mathematical transformations they were designed
to maintain data order while helping to linearize the response-IV relationship. For the
SQUARE (b = 2), SQUAREROOT (b = 0.5), QUADROOT (b = 0.25), INVERSE (b = -
1) and GENERAL EXPONENT (user-defined b) transformations, VB3 uses the signed
equivalent of the mathematical function:
xb == sign(x)*(abs(x))b
For example: (-2)2 =-4 (-9)0 5 =-3 (-4)"0 5 =-0.5 (-2)"2 =-0.25
To avoid potentially undefined values (e.g., 1/x when x = 0), the INVERSE and
GENERAL EXPONENT (if the user sets b < 0) transformations have special processing:
If x = 0, VB3 will find the minimum value of abs(z) where z is the set of all non-
zero values for the IV in question. To compute the transformation after z is defined, VB3
80
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substitutes z/2 for x. From this definition, note that z can be a positive or negative
number.
LOGio and LOGe transforms are also the signed equivalent of their mathematical
functions:
loge(x) == loge(x)
loge(-x) == -l0ge(x)
logio(x) == logio(x)
logio(-x) == -logio(x)
In addition, if (-1 < x < 1), then loge(x) = 0 and logio(x) = 0
VB3 will not compute the INVERSE, GENERAL EXPONENT (with a negative
b), LOG10 and LOGe transformations for data columns if more than 10% of the IV values
are zero. Programmatically, zero is defined as any number whose absolute value is less
than 1.0e-21.
POLYNOMIAL transformations are the result of a linear regression of the
response variable on the IV and the square of the IV:
Poly(x) = a + b*x + c*x2
where a, b, and c are determined by a multiple linear regression of x and x2 on the
response variable.
In general, the name of the transformed column of data that VB3 creates is simply
the type of transformation, with the original data column name in parentheses. For
example, the logio of WaterTemp becomes LOG(WaterTemp); however, there are some
exceptions:
INVERSE(x,y) : x is the original data column name and y is the z/2 value
discussed in the last paragraph on page 80.
POWER(x,y) : when y is positive, x is the original data column name and y is the
exponent specified by the user.
POWER(x,y,z) : when y is negative, x is the original data column name, y is the
exponent specified by the user, and z is the z/2 value discussed earlier in this section.
POLY(x, a,b,c) : x is the original data column name and a, b, and c are the values
of the polynomial regression coefficients.
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A.2 Singular Matrices and Nominal Variables
The solution to least squares regression (MLR modeling is discussed in Section 7)
involves computing the inverse of the X'X matrix (the X matrix contains the IV values
for the model). When one IV is a linear combination of other IVs, the X'X matrix is
singular, and trying to invert it produces a mathematical quandary (i.e., division by zero).
Examples of variables that are linear combinations of other IVs:
Xi = 3.5 + 4.2*X2
Xi = 1 -X2-X3 -X4
X3 = Xi + x2
In these examples, it doesn't matter if the IVs are continuous (real numbers) or
categorical (0/1 values). In fact, VB3 allows the user to produce, using the "manipulate"
button described in section 6.6, IVs that are linear combinations of others (like example c
above). When VB3 evaluates MLR models, it checks each model for highly correlated
IVs because perfectly correlated IVs lead to a matrix singularity and throws out any
model with this condition (as measured by the Variance Inflation Factor, explained in
Section 7.2). Using example equation c: attempting to compute a regression model
involving Xi, X2, and X3, VB3 will issue an error message.
Singularities are often produced if an IV with several categories is being defined
using multiple indicator variables. Let's say there is an IV for cloud cover. One could
make this categorical measure a continuous variable by using a single column with values
ranging from 1 (no clouds) to 5 (completely overcast). This is acceptable because this IV
is "ordinal" there is a natural order to its values. As values increase from 1 to 5, it
implies more clouds.
There may be other categorical IVs that are "nominal," meaning there is no real
order to their values. An example is the species of bird most abundant at the beach on a
given day. If there are four possible species (A, B, C, D), it would be incorrect to code
this IV in a single column with values 1, 2, 3, and 4. A value of 2 doesn't imply any
larger mathematical quantity than a value of 1 or a smaller quantity than a value of 4. So
the bird species should be coded as a series of indicator variables, using 0's and l's
(Table A.l):
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Table A. 1. Example of using 0/1 indicator variables for a multi-category IV
ID
Species_A
Species_B
Specie s_C
Species_D
Day 1
1
0
0
0
Day 2
1
0
0
0
Day 3
0
1
0
0
Day 4
0
0
0
1
Day 5
1
0
0
0
Day ง
0
0
1
0
Day 7
0
0
1
0
Day 8
0
1
0
0
Day 3
0
1
0
0
Day 10
0
0
0
1
Day ii
0
1
0
0
Day 12
1
0
0
0
Day 13
Day 14
0
0
0
0
1
0
0
1
Day 15
Day 16
0
0
1
1
0
0
0
0
A "1" denotes when a species is dominant and "0" when it isn't. Looking closely,
we see that the four columns form a linear combination:
Species D = 1 - Species A - Species B - Species C
Given this relationship, VB3 cannot evaluate a MLR model that includes all four
columns (mathematically impossible due to a matrix singularity), but a model that
contains three or fewer of the columns is acceptable, as is including all four columns in
the dataset (but they will never occur together in a model). An advantage of PLS
(Section 8) and GBM (Section 9) modeling is that they are not constrained by the
collinearity of IVs and can compute solutions for models that include all four columns.
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A.3 MLR Model Evaluation Criteria
If p is defined as the number of parameters in a model, n as the number of
observations in the dataset, RSS as the residual sum of squares for a model, and TSS as
the total sum of squares for a model, then the evaluation criteria for any model can be
defined as:
Akaike Information Criterion (AIC): 2p + n*ln(RSS)
Corrected Akaike Information Criterion (AICC): ln(RSS/n) + (n+p)/(n-p-2)
R2: 1 - RSS/TSS
Adjusted R2: 1 - (l-R2)(n-l)/(n-p-l)
Bayesian (Schwarz) Information Criterion (BIC): = n*ln(RSS/n) + p*ln(n)
Root Mean Squared Error (RMSE): (RSS/n)1/2
Predicted Error Sum of Squares (PRESS): 1 - S(y;- y.;)2 / E(y; - ym)2
where yi is the ith observation, y_; is the model estimate of the ith observation when the model
coefficients are fitted with the ith observation removed from the dataset, and ym is the mean
value of y in the dataset
Accuracy: (true positives + true negatives) / number of total observations
Specificity: true negatives / (true negatives + false positives)
Sensitivity: true positives / (true positives + false negatives)
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A.4 Changes from version 3 to 3.04
Fixed bugs in the MLR Cross Validation routines that could lead to errors.
The splash screen, with version information, will now launch from the "File"
dropdown menu item "About."
Removed the help system and automated access to the user's manual.
New statistical libraries and methodologies implemented to optimize MLR and GBM
performance.
Non-functional base maps (Yahoo, Google, ESRI) removed from the map UI on the
"Location" tab.
Non-functional "Go to Place" button removed from the map UI on the "Location" tab.
Non-functional "Show Station Locations" button removed from the map UI on the
"Location" tab.
Fixed an issue where non-alphanumeric characters were removed from variable
names in the "influence" list on the GBM tab, which prevented use of the "Drop
Variable(s)" button.
Fixed an issue where variables were not properly ranked by declining influence in the
GBM results window.
Fixed a problem where re-opening a project saved with a PLS or GBM model
removed data from the MLR plugin.
Corrected a problem when importing EnDDaT rainfall data from EnDDaT by date.
Corrected spelling error on Prediction tab button.
Added Anderson-Darling p-value to "view plots" window of transformation table
from datasheet.
Corrected an error in the calculation of the Anderson-Darling test statistic assessing
the normality of independent variables.
Corrected an error induced by running transformations when the datasheet had a
column identical to the current response variable.
Added random seed control to GBM and PLS tabs.
Corrected plot anchoring on the GBM and PLS Diagnostics tab.
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