Model Building Training
Max Kuhn
Kjell Johnson
Global Nonclinical Statistics
1
Overview
• Typical data scenarios
– Examples we’ll be using
• General approaches to model building
• Data pre-processing
• Regression-type models
• Classification-type models
• Other considerations
2
Typical Data
R e sponse A
(continuous)
1
-24.0
2
4.4
3
34.9
4
-55.5
5
-54.0
6
-64.7
7
13.7
8
-25.3
9
51.9
-8.3
10
ID
R e sponse B
A1 . . .
(ca te gorica l)
Active
-29.8 . . .
Active
4.8 . . .
Active
75.3 . . .
Inactive
33.1 . . .
Inactive
118.1 . . .
Inactive
1.3 . . .
Active
97.9 . . .
Inactive
24.7 . . .
Active
-13.5 . . .
Inactive
-3.5 . . .
A92 B1 . . . B347 . . . K1 . . . K32
71.4
115.6
56.4
131.3
104.4
124.6
62.4
-27.3
35.6
-0.1
3
4
0
0
1
4
4
2
1
0
...
...
...
...
...
...
...
...
...
...
0
0
12
0
5
1
6
0
3
4
• Response may be continuous or categorical
• Predictors may be
– continuous, count, and/or binary
– dense or sparse
– observed and/or calculated
3
...
...
...
...
...
...
...
...
...
...
0
0
0
0
0
0
0
0
1
0
...
...
...
...
...
...
...
...
...
...
1
0
0
0
0
0
0
0
1
0
Predictive Models
• What is a “predictive model”?
A model whose primary purpose is for prediction
(as opposed to inference)
• We would like to know why the model works, as
well as the relationship between predictors and
the outcome, but these are secondary
• Examples: blood-glucose monitoring, spam
detection, computational chemistry, etc.
4
What Are They Not Good For?
• They are not a substitute for subject specific
knowledge
Science: Hard
(yikes)
Models: Easy
(let’s do these instead!)
• To make a good model that predicts well on
future samples, you need to know a lot about
– Your predictors and how they relate to each other
– The mechanism that generated the data (sampling,
technology etc)
5
What Are They Not Good For?
• An example:
An oncologist collects some data from a small clinical
trial and wants a model that would use gene expression
data to predict therapeutic response (beneficial or not)
in 4 types of cancer
There were about 54K predictors and data was
collected on ~20 subjects
• If there is a lot of knowledge of how the therapy
works (pathways etc), some effort must be put into
using that information to help build the model
6
The Big Picture
“In the end, [predictive modeling] is not a
substitute for intuition, but a
compliment”
Ian Ayres, in Supercrunchers
7
References
• “Statistical Modeling: The Two Cultures” by Leo
Breiman (Statistical Science, Vol 16, #3 (2001),
199-231)
• The Elements of Statistical Learning by Hastie,
Tibshirani and Friedman
• Regression Modeling Strategies by Harrell
• Supercrunchers by Ayres
8
Regression Methods
• Multiple linear regression
• Partial least squares
• Neural networks
• Multivariate adaptive regression splines
• Support vector machines
• Regression trees
• Ensembles of trees:
– Bagging, boosting, and random forests
9
Classification Methods
• Discriminant analysis framework
– Linear, quadratic, regularized, flexible, and partial least squares
discriminant analysis
• Modern classification methods
– Classification trees
– Ensembles of trees
• Boosting and random forests
– Neural networks
– Support vector machines
– k-nearest neighbors
– Naive Bayes
10
Interesting Models We Don’t Have Time For
• L1 Penalty methods
– The lasso, the elasticnet, nearest shrunken centroids
• Other Boosted Models
– linear models, generalized additive models, etc
• Other Models:
– Conditional inference trees, C4.5, C5, Cubist, other tree models
– Learned vector quantization
– Self-organizing maps
– Active learning techniques
11
Example Data Sets
12
Boston Housing Data
• This is a classic benchmark data set for regression. It
includes housing data for 506 census tracts of Boston
from the 1970 census.
•
crim: per capita crime rate
•
•
Indus: proportion of non-retail
business acres per town
dis: weighted distances to five
Boston employment centers
•
rad: index of accessibility to
radial highways
•
tax: full-value property-tax rate
•
ptratio: pupil-teacher ratio by
town
•
b: proportion of minorities
•
Medv: median value homes
(outcome)
•
•
nox: nitric oxides concentration
•
rm: average number of rooms
per dwelling
•
13
chas: Charles River dummy
variable (= 1 if tract bounds
river; 0 otherwise)
Age: proportion of owneroccupied units built prior to
1940
Toy Classification Example
• A simulated data set will be
used to demonstrate
classification models
– two predictors with a correlation
coefficient of 0.5 were simulated
– two classes were simulated
(“active” and “inactive”)
• A probability model was used to
assign a probability of being
active to each sample
– the 25%, 50% and 75%
probability lines are shown on
the right
14
Toy Classification Example
• The classes were randomly
assigned based on the
probability
• The training data had 250
compounds (plot on right)
– the test set also contained 250
compounds
• With two predictors, the class
boundaries can be shown for
each model
– this can be a significant aid in
understanding how the models
work
– …but we acknowledge how
unrealistic this situation is
15
Model Building Training
General Strategies
16
Objective
To construct a model of predictors that
can be used to predict a response
Data
Model
Prediction
17
Model Building Steps
• Common steps during model building are:
– estimating model parameters (i.e. training models)
– determining the values of tuning parameters that
cannot be directly calculated from the data
– calculating the performance of the final model that will
generalize to new data
• The modeler has a finite amount of data, which
they must "spend" to accomplish these steps
– How do we “spend” the data to find an optimal model?
18
“Spending” Data
• We typically “spend” data on training and test data sets
– Training Set: these data are used to estimate model parameters
and to pick the values of the complexity parameter(s) for the
model.
– Test Set (aka validation set): these data can be used to get an
independent assessment of model efficacy. They should not be
used during model training.
• The more data we spend, the better estimates we’ll get
(provided the data is accurate). Given a fixed amount of
data,
– too much spent in training won’t allow us to get a good
assessment of predictive performance. We may find a model that
fits the training data very well, but is not generalizable (overfitting)
– too much spent in testing won’t allow us to get a good assessment
of model parameters
19
Methods for Creating a Test Set
• How should we split the data into a training and
test set?
• Often, there will be a scientific rational for the split
and in other cases, the splits can be made
empirically.
• Several empirical splitting options:
– completely random
– stratified random
– maximum dissimilarity in predictor space
20
Creating a Test Set: Completely Random Splits
• A completely random (CR) split randomly partitions the
data into a training and test set
• For large data sets, a CR split has very low bias towards
any characteristic (predictor or response)
• For classification problems, a CR split is appropriate for
data that is balanced in the response
• However, a CR split is not appropriate for unbalanced
data
– A CR split may select too few observations (and perhaps none) of
the less frequent class into one of the splits.
21
Creating a Test Set: Stratified Random Splits
• A stratified random split makes a random split
within stratification groups
– in classification, the classes are used as strata
– in regression, groups based on the quantiles of the
response are used as strata
• Stratification attempts to preserve the distribution
of the outcome between the training and test
sets
– A SR split is more appropriate for unbalanced data
22
Over-Fitting
• Over-fitting occurs when a model has extremely good
prediction for the training data but predicts poorly when
– the data are slightly perturbed
– new data (i.e. test data) are used
• Complex regression and classification models assume
that there are patterns in the data.
– Without some control many models can find very intricate
relationships between the predictor and the response
– These patterns may not be valid for the entire population.
23
Over-Fitting Example
• The plots below show classification boundaries
for two models built on the same data
Predictor B
Predictor B
– one of them is over-fit
Predictor A
24
Predictor A
Over-Fitting in Regression
• Historically, we evaluate the quality of a
regression model by it’s mean squared error.
• Suppose that are prediction function is
parameterized by some vector 
25
Over-Fitting in Regression
• MSE can be decomposed into three terms:
– irreducible noise
– squared bias of the estimator from it’s expected value
– the variance of the estimator
• The bias and variance are inversely related
– as one increases, the other decreases
– different rates of change
26
Over-Fitting in Regression
• When the model under-fits,
the bias is generally high and
the variance is low
• Over-fitting is typically
characterized by high
variance, low bias estimators
• In many cases, small
increases in bias result in
large decreases in variance
27
Over-Fitting in Regression
• Generally, controlling the MSE yields a good
trade-off between over- and under-fitting
– a similar statement can be made about classification
models, although the metrics are different (i.e. not
MSE)
• How can we accurately estimate the MSE from
the training data?
– the naïve MSE from the training data can be a very
poor estimate
• Resampling can help estimate these metrics
28
How Do We Estimate Over-Fitting?
• Some models have specific “knobs” to control
over-fitting
– neighborhood size in nearest neighbor models is an
example
– the number if splits in a tree model
• Often, poor choices for these parameters can
result in over-fitting
• Resampling the training compounds allows us
to know when we are making poor choices for the
values of these parameters
29
How Do We Estimate Over-Fitting?
• Resampling only affects the training data
– the test set is not used in this procedure
• Resampling methods try to “embed variation” in
the data to approximate the model’s performance
on future compounds
• Common resampling methods:
– K-fold cross validation
– Leave group out cross validation
– Bootstrapping
30
K-fold Cross Validation
• Here, we randomly split the data into K blocks of
roughly equal size
• We leave out the first block of data and fit a
model.
• This model is used to predict the held-out block
• We continue this process until we’ve predicted all
K hold-out blocks
• The final performance is based on the hold-out
predictions
31
K-fold Cross Validation
• The schematic below shows the process for K = 3
groups.
– K is usually taken to be 5 or 10
– leave one out cross-validation has each sample as a
block
32
Leave Group Out Cross Validation
• A random proportion
of data (say 80%) are
used to train a model
• The remainder is
used to predict
performance
• This process is
repeated many times
and the average
performance is used
33
Bootstrapping
• Bootstrapping takes a random sample with
replacement
– the random sample is the same size as the original
data set
– compounds may be selected more than once
– each compound has a 63.2% change of showing up at
least once
• Some samples won’t be selected
– these samples will be used to predict performance
• The process is repeated multiple times (say 30)
34
The Bootstrap
• With bootstrapping,
the number of heldout samples is
random
• Some models, such
as random forest, use
bootstrapping within
the modeling process
to reduce over-fitting
35
Training Models with Tuning Parameters
• A single training/test split is
often not enough for models
with tuning parameters
• We must use resampling
techniques to get good
estimates of model performance
over multiple values of these
parameters
• We pick the complexity
parameter(s) with the best
performance and re-fit the
model using all of the data
36
Simulated Data Example
• Let’s fit a nearest neighbors model to the
simulated classification data.
• The optimal number of neighbors must be chosen
• If we use leave group out cross-validation and set
aside 20%, we will fit models to a random 200
samples and predict 50 samples
– 30 iterations were used
• We’ll train over 11 odd values for the number of
neighbors
– we also have a 250 point test set
37
Toy Data Example
• The plot on the right shows the
classification accuracy for
each value of the tuning
parameter
– The grey points are the 30
resampled estimates
– The black line shows the average
accuracy
– The blue line is the 250 sample
test set
• It looks like 7 or more
neighbors is optimal with an
estimated accuracy of 86%
38
Toy Data Example
• What if we didn’t resample
and used the whole data
set?
• The plot on the right
shows the accuracy
across the tuning
parameters
• This would pick a model
that over-fits and has
optimistic performance
39
Model Building Training
Data Pre-Processing
40
Why Pre-Process?
• In order to get effective and stable results, many
models require certain assumptions about the
data
– this is model dependent
• We will list each model’s pre-processing
requirements at the end
• In general, pre-processing rarely hurts model
performance, but could make model
interpretation more difficult
41
Common Pre-Processing Steps
• For most models, we apply three pre-processing
procedures:
– Removal of predictors with variance close to zero
– Elimination of highly correlated predictors
– Centering and scaling of each predictor
42
Zero Variance Predictors
• Most models require that each predictor have at
least two unique values
• Why?
– A predictor with only one unique value has a variance
of zero and contains no information about the
response.
• It is generally a good idea to remove them.
43
“Near Zero Variance” Predictors
• Additionally, if the distributions of the predictors
are very sparse,
– this can have a drastic effect on the stability of the
model solution
– zero variance descriptors could be induced during
resampling
• But what does a “near zero variance” predictor
look like?
44
“Near Zero Variance” Predictor
• There are two conditions for an “NZV” predictor
– a low number of possible values, and
– a high imbalance in the frequency of the values
• For example, a low number of possible values
could occur by using fingerprints as predictors
– only two possible values can occur (0 or 1)
• But what if there are 999 zero values in the data
and a single value of 1?
– this is a highly unbalanced case and could be trouble
45
NZV Example
• In computational chemistry we
created predictors based on
structural characteristics of
compounds.
# 11-Member Rings
• As an example, the descriptor
“nR11” is the number of 11member rings
• The table to the right is the
distribution of nR11 from a
training set
– the distinct value percentage is
5/535 = 0.0093
– the frequency ratio is 501/23 = 21.8
46
Value
Frequency
0
501
1
4
2
23
3
5
4
2
Detecting NZVs
• Two criteria for detecting NZVs are the
– Discrete value percentage
• Defined as the number of unique values divided by the number of
observations
• Rule-of-thumb: discrete value percentage < 20% could indicate a
problem
– Frequency ratio
• Defined as the frequency of the most common value divided by the
frequency of the second most common value
• Rule-of-thumb: > 19 could indicate a problem
• If both criteria are violated, then eliminate the predictor
47
Highly Correlated Predictors
•
Some models can be negatively affected by
highly correlated predictors
– certain calculations (e.g. matrix inversion) can
become severely unstable
•
How can we detect these predictors?
– Variance inflation factor (VIF) in linear regression
or, alternatively
1. Compute the correlation matrix of the predictors
2. Predictors with (absolute) pair-wise correlations
above a threshold can be flagged for removal
3. Rule-of-thumb threshold: 0.85
48
Highly Correlated Predictors and Resampling
• Recall that resampling slightly perturbs the
training data set to increase variation
• If a model is adversely affected by high
correlations between predictors, the resampling
performance estimates can be poor in
comparison to the test set
– In this case, resampling does a better job at predicting
how the model works on future samples
49
Centering and Scaling
• Standardizing the predictors can greatly improve
the stability of model calculations.
• More importantly, there are several models (e.g.
partial least squares) that implicitly assume that
all of the predictors are on the same scale
• Apart from the loss of the original units, there is
no real downside of centering and scaling
50
Model Building Training
Regression-type Models
51
Setting
Variables
Pred. 1
Obs 1 3.231
Obs 2 5.249
Obs 3 7.534
...
Obs n 6.878
Pred. 2
99.30
63.78
84.53
77.21
...
Pred. p
20104
30128
10021
Response
8.322
5.995
7.756
50249
3.490
Response is continuous
52
Objective
To construct a model of predictors that
can be used to predict a response
Data
Model
Prediction
53
Regression Methods
• Multiple linear regression
• Partial least squares
• Neural networks
• Multivariate adaptive regression splines
• Support vector machines
• Regression trees
• Ensembles of trees:
– Bagging, boosting, and random forests
• Each of these methods seek to find a relationship
between the predictors and response that minimizes
error between the observed and predicted response
54
Additive Models
In the beginning there were linear models:
EY   0  1 X1    p X p
And Nelder and Wedderburn (1972) said, “Let there be
Generalized Linear Models”:
gEY   0  1 X1    p X p
and link functions appeared.
And Hastie and Tibshirani (1990) said, “Let there be
Generalized Additive Models”:
EY   f 0  f1  X1    f p X p 
55
and scatterplot smoothers and backfitting
algorithms appeared.
Families of Additive Models
GLM
-
Flexibility
Recursive
Partitioning
(Trees)
PLS
Bagging
Boosting
+
Multivariate
Adaptive
Regression
Splines*
Random
Forests
GAM
* Additivity depends on model parameters
56
Neural
Nets
Support
Vector
Machines*
Assessing Model Performance
57
Assessing Model Performance
• How well does a regression model perform? Answering
this question depends on how we want to use the model.
Possible goals are:
– To understand the relationship between the predictor and the
response.
– To use the model to predict future observations’ response.
• In either case, we can use several of different measures
to evaluate model performance. We will focus on two:
– Coefficient of determination (R2)
– Root mean square error (RMSE)
• However, the set of data that we use to evaluate
performance will change depending on our purpose.
58
Which Set of Data to Use to Evaluate Performance?
• If we are only interested in understanding the underlying
relationship between the predictor and the response,
then we can compute R2 and RMSE on the data for
which the model was built (i.e the training data).
– However, these values will be overly optimistic of the model’s
ability to predict future observations.
• If we are interested in understanding the model’s ability to
predict future observations, then we need to compute R2
and RMSE on data for which the model was not built (i.e.
a test set or cross-validation set).
– For a held-out set of data, R2 is commonly referred to as Q2 and
RMSE is commonly referred to as root mean squared prediction
error (RMSPE)
59
Root Mean Squared Error (RMSE) and
Root Mean Squared Prediction Error (RMSPE)
• RMSE measures the average deviation of an observation
to the best-fit plane
SSE
RMSE 
n   p  1
• RMSPE measures the average deviation of an
observation to its predicted value for the test or crossvalidation set
n*
RMSPE 
2
ˆ


y

y
 i i
i 1
n
*
n* = the number of observations in the test or cross-validation set
60
Computing Q2
• Process:
– Partition the data into
• a training and testing set, or
• blocks to be used for training and testing
– Build the model on the training data and predict the
testing data
• Q2 = R2 of the relationship between the observed
and predicted values for the testing data.
61
Multiple Linear Regression:
A Quick Review
62
Multiple Linear Regression
Variables
Pred. 1
Obs 1 3.231
Obs 2 5.249
Obs 3 7.534
...
Obs n 6.878
Pred. 2
99.30
63.78
84.53
77.21
...
Pred. p
20104
30128
10021
Response
8.322
5.995
7.756
50249
3.490
Objective: Find the plane through the data that minimizes the
sum-of-squares error.
63
The Best Plane
• To find the best plane, we solve:
min Y  X
2

– where Ynx1, Xnx(p+1) and β(p+1)x1
• The best β is:
 ˆ0 
 
 ˆ1 
1 T
T
  X X X Y
  
 ˆ 
 p

64

Aside: A Bit More About (XTX)
• (XTX) is a critical matrix for many statistical
modeling techniques
• A few fun facts…
– (XTX) is proportional to the covariance matrix, S
– S contains the variances and covariances of all
predictors
– Techniques that depend on (XTX) also require that it is
invertible
65
Assumptions: Diagnostic Plots
66
When Does Regression Fail?
• When a plane does not capture the structure in the data
• When the variance/covariance matrix is overdetermined
– Recall, the plane that minimizes SSE is:


1 T
T
ˆ
 X X X Y
– To find the best plane, we must compute the inverse of the
variance/covariance matrix
– The variance/covariance matrix is not always invertible. Two
common conditions that cause it to be uninvertible are:
• Two or more of the predictors are correlated (multicollinearity)
• There are more predictors than observations
67
A (Trivial) Example of Multicollinearity
Suppose that we have one observation (3,5), and we wish to find the ‘best’ line for the
data. In this example, the number of observations (1) is less than the number of
parameters (2: slope and intercept). When the number of parameters is greater than
the number of observations, we can find an infinite number of ‘best’ solutions.
Solution 1
7
6
Solution 2
5
Y
4
Solution 3
3
2
In the presence of multicollinearity, the best
solution will be unstable.
1
0
0
1
2
3
4
5
X
68
6
7
8
9
10
Boston Housing Data
• Let’s use a linear regression model to predict the median
house price in Boston.
• Process:
– Split the data into a training set (n = 337) and testing set (n = 169)
– For the training set, use the bootstrap to determine the RMSPE
and Q2
– For the test data determine RMSPE and Q2
• If the underlying model is stable, the values of RMSPE
and Q2 should be similar between the bootstrap and
testing data
69
Results
Training Data
(bootstrap)
Linear Reg
Test Data
RMSE
Q2
RMSE
R2
5.23
0.691
4.53
0.742
• The results are fairly similar, at least within the variation of
resampling
• One reason you may see differences: multicollinearity
– Multicollinearity in the predictors can produce somewhat unstable
solutions for each resample
– When the data are slightly changed, the model can drastically
change
• The test set is a single, static set of data for verification
– The bootstrap estimate of performance may be better with
collinearity
70
Partial Least Squares Regression
71
Solutions for Overdetermined Covariance Matrices
• Variable reduction
– Try to accomplish this through the pre-processing
steps
• Partial least squares (PLS)
• Other methods
– Apply a generalized inverse
– Ridge regression: Adjusts the variance/covariance
matrix so that we can find a unique inverse.
– Principal component regression (PCR)
• not recommended—but it’s a good way to understand PLS
72
Understanding Partial Least Squares:
Principal Components Analysis
• PCA seeks to find linear combinations of the
original variables that summarize the maximum
amount of variability in the original data
– These linear combinations are often called principal
components or scores.
– A principal direction is a vector that points in the
direction of maximum variance.
73
Principal Components Analysis
• PCA is inherently an optimization problem, which
is subject to two constraints
1. The principal directions have unit length
2. Either
a.Successively derived scores are uncorrelated to previously
derived scores, OR
b.Successively derived directions are required to be orthogonal
to previously derived directions
• In the mathematical formulation, either constraint implies the
other constraint
74
Principal Components Analysis
5
Direction 1
4
3
Score
Predictor 2
2
1
0
-6
-5
-4
-3
-2
-1
0
-1
-2
-3
-4
Predictor 1
75
http://pfizerpedia/index.php/Image:PCAmovie.gif
1
2
3
4
5
Mathematically Speaking…
• The optimization problem defined by PCA can be solved
through the following formulation:
 aTX 
,
arg max Var 
T 
a
 a a
subject to constraints 2a. or b.
• Facts…
– the ith principal direction, ai, is the eigenvector corresponding to
the ith largest eigenvalue of XTX.
– the ith largest eigenvalue is the amount of variability summarized
by the ith principal component.
– a iT X are the ith scores
76
PCA Benefits and Drawbacks
• Benefits
– Dimension reduction
• We can often summarize a large percentage of original variability
with only a few directions
– Uncorrelated scores
• The new scores are not linearly related to each other
• Drawbacks
– PCA “chases” variability
• PCA directions will be drawn to predictors with the most variability
• Outliers may have significant influence on the directions and
resulting scores.
77
Principal Component Regression
Procedure:
1. Reduce dimension of predictors using PCA
2. Regress scores on response
Notice: The procedure is sequential
78
Principal Component Regression
Dimension reduction is
independent of the objective
Predictor
Variables
PCA
PC Scores
MLR
Response
Variable
79
First Principal Direction
Scatter of Predictors
5.00
3.00
Predictor 2
PD1
1.00
-1.00
-3.00
-5.00
-5.00
-4.00
-3.00
-2.00
-1.00
0.00
Predictor 1
80
1.00
2.00
3.00
4.00
5.00
Relationship of First Direction with Response
Scatter of First PCA Scores with Response
2.50
2.00
1.50
Response
1.00
0.50
0.00
-0.50
-1.00
-1.50
R2 = 0.001
-2.00
-6.00
81
-4.00
-2.00
0.00
2.00
First PCA Scores
4.00
6.00
8.00
PLS History
• H. Wold (1966, 1975)
• S. Wold and H. Martens (1983)
• Stone and Brooks (1990)
• Frank and Friedman (1991, 1993)
• Hinkle and Rayens (1994)
82
Latent Variable Model
Predictor1
1
Predictor2
Response1

Predictor3
Predictor4
Response2
Response3
2
Predictor5
Predictor6
Predictors
Latent Variables
Responses
Note: PLS can handle multiple response variables
83
Comparison with Regression
Predictor1
Predictor2
Predictor3
Predictor4
Predictor5
84
Response1
PLS Optimization
(many predictors, one response)
• PLS seeks to find linear combinations of the
independent variables that summarize the
maximum amount of co-variability with the
response.
– These linear combinations are often called PLS
components or PLS scores.
– A PLS direction is a vector that points in the direction
of maximum co-variance.
85
PLS Optimization
(many predictors, one response)
• PLS is inherently an optimization problem, which
is subject to two constraints
1. The PLS directions have unit length
2. Either
a.Successively derived scores are uncorrelated to previously
derived scores, OR
b.Successively derived directions are orthogonal to previously
derived directions
• Unlike PCA, either constraint does NOT imply the other
constraint
• Constraint 2.a. is most commonly implemented
86
Mathematically Speaking…
• The optimization problem defined by PLS can be solved
through the following formulation:
2

T

Cov a X, Y
arg max
,
T
a a
a
subject to constraints 2a. or b.
• Facts…
– the ith PLS direction, ai, is the eigenvector
corresponding to the ith largest eigenvalue of ZTZ,
where Z = XTy.
– the ith largest eigenvalue is the amount of co-variability
summarized by the ith PLS component.
– a iT X are the ith scores
87
PLS is Simultaneous Dimension Reduction and
Regression
2


T
Cov a X, Y
arg max
a Ta
a
var a T X varY corr2 a T X, Y
 arg max
T
a a
a
var a T X corr2 a T X, Y
 varY  arg max
a Ta
a
varscorescorr2 scores,response
 varresponse arg max
T
a a
a



88





PLS is Simultaneous Dimension Reduction
and Regression
max Var(scores) Corr2(response,scores)
Dimension Reduction
(PCA)
89
Regression
PLS Benefits and Drawbacks
• Benefit
– Simultaneous dimension reduction and regression
• Drawbacks
– Similar to PCA, PLS “chases” co-variability
• PLS directions will be drawn to independent variables with the most
variability (although this will be tempered by the need to also be
related to the response)
• Outliers may have significant influence on the directions, resulting
scores, and relationship with the response. Specifically, outliers can
– make it appear that there is no relationship between the
predictors and response when there truly is a relationship, or
– make it appear that there is a relationship between the
predictors and response when there truly is no relationship
90
Partial Least Squares
Simultaneous dimension
reduction and regression
Predictor
Variables
PLS
Response
Variable
91
First PLS Direction
Scatter of Predictors
5.00
First PLS
Direction
Predictor 2
3.00
1.00
-1.00
-3.00
-5.00
-5.00
-4.00
-3.00
-2.00
-1.00
0.00
Predictor 1
92
1.00
2.00
3.00
4.00
5.00
Relationship of First Direction with Response
Scatter of First PLS Scores with Response
2.50
2.00
1.50
Response
1.00
0.50
0.00
-0.50
-1.00
-1.50
R2 = 0.93
-2.00
-2.00
-1.50
-1.00
-0.50
0.00
0.50
First PLS Scores
93
1.00
1.50
2.00
2.50
PLS in Practice
• PLS seeks to find latent variables (LVs) that
summarize variability and are highly predictive of
the response.
• How do we determine the number of LVs to
compute?
– Evaluate RMSPE (or Q2)
• The optimal number of components is the
number of components that minimizes RMSPE
94
PLS for the Boston housing data:
Training the PLS Model
• Since PLS can handle
highly correlated
variables, we fit the model
using all 12 predictors
• The model was trained
with up to 6 components
• RMSE drops noticeably
from 1 to 2 components
and some for 2 to 3
components.
– Models with 3 or more
components might be
sufficient for these data
95
Training the PLS Model
• Roughly the same
profile is seen when
the models are judged
on R2
96
Boston Housing Results
• Using the two component model, we can predict
the test set
• PLS training statistics are similar to those from
linear regression
• Both methods perform about the same in the test
set
Training Data
(bootstrap)
97
Test Data
RMSE
Q2
RMSE
R2
Linear Reg
5.23
0.691
4.53
0.742
PLS
5.25
0.689
4.56
0.739
PLS Model Fit – Test Set Results
98
PLS Optimization (2)
(many predictors, many responses)
• PLS seeks to find linear combinations of the
independent variables and a linear combination
of the dependent variables that summarize the
maximum amount of co-variability between the
combinations.
– These linear combinations are often called PLS Xspace and Y-space components or PLS X-space and
Y-space scores.
– Likwise, X-space and Y-space PLS directions point in
the direction of maximum co-variance between the
spaces.
99
PLS Optimization (2)
(many predictors, many responses)
• PLS is inherently an optimization problem, which
is subject to two constraints
1. The X-space and Y-space PLS directions have unit
length
2. Either
a.Successively derived scores in each space are uncorrelated
to previously derived scores, OR
b.Successively derived directions in each space are orthogonal
to previously derived directions
• Constraint 2.a. is most commonly implemented
100
Mathematically Speaking…
• The optimization problem defined by PLS can be
solved through the following formulation:
2

T
T

Cov a X, b Y
arg max
,
T
T
a a b b
a,b
  
subject to constraints 2a. or b.

T
   
  
T
2
T
T
var a X var b Y corr a X, b Y
 arg max
T
T
a a b b
a,b
101

PLS is Simultaneous Dimension Reduction
and Regression
max Var(X-scores) Corr2(X-scores,Y-scores)Var(Y-scores)
X-space Dimension
Reduction (PCA)
102
Regression
Y-space Dimension
Reduction (PCA)
Neural Networks
103
Neural Networks
• Like PLS or PCR, these models create
intermediary latent variables that are used to
predict the outcome
• Neural networks differ from PLS or PCR in a few
ways
– the objective function used to derive the new variables
is different
– The latent variables are created using flexible, highly
nonlinear functions
– The latent variables usually do not have any meaning
104
Network Structures
• There are many types of neural network structures
– we will concentrate on the single layer, feed-forward network
One hidden layer of
latent variables
Predictor1
Hidden Unit 1
Predictor2
Hidden Unit 2
Predictor3
…
Predictor4
Predictor5
105
Hidden Unit k
Response1
From Predictors to Hidden Units
• The transition from this
sub-model to the hidden
units is nonlinear
– sigmoidal functions, such
as the logistic function, are
typically used
106
From Hidden Units to the Outcome
• The hidden units are then
used to predict the
outcome using simple
linear combinations
• Clearly, the parameters are not identifiable and
the hidden units have no real meaning (unlike
PCA)
107
Training Networks
• It is highly recommended that the predictors are
centered and scaled prior to training
• The number of hidden units is a tuning
parameter
• With many predictors and hidden units, the
number of estimated parameters can become
very large
– with a large number of hidden units, these models can
quickly start to overfit
• Random starting values are typically used to
initialize the parameter estimates
108
Weight Decay
• This is a training technique that attempts to
“shrink” the parameter estimates towards zero
– large parameter estimates are penalized in the model
training
• This leads to smoother, less extreme models
– the effect of weight decay is demonstrated for
classification models
109
Boston Housing Data
• The model seems to
do well with fewer
components (not
typical)
• For these data, larger
amounts of weight
decay is better for the
model fit
110
Boston Housing Results
• The final model used high value for weight decay
and 1 hidden unit
• This model seems to be an improvement
compared to the others
Training Data
(bootstrap)
111
Test Data
RMSE
Q2
RMSE
R2
Linear Reg
5.23
0.691
4.53
0.742
PLS
5.25
0.689
4.56
0.739
Neural Net
4.60
0.757
4.20
0.780
Support Vector Machines
112
Support Vector Machines (SVMs)
• SVMs are predictive statistical models developed
in 1963 by Vapnik that were significantly
expanded in the 90’s
• These models were initially developed for
classification models, but were later adapted for
regression models
113
Objective Functions
• Recall that linear
regression estimates
parameters by
calculating:
– the model residuals
– the total sum of the
squared residuals (SSR)
• The parameters with
the smallest SSR are
optimal
114
Objective Functions
• Support vector machine
regression models create a
“funnel” around the
regression line
– residuals within the funnel are
not counted in the parameter
estimation
– the sum of the residuals
outside the funnel are used as
the objective function (no
squared term)
• A funnel size is set to 1 SD
of the outcome is not a bad
place to start
115
The SVM Model Optimization
• Like Huber-type robust
regression, outliers have a
linear effect on the
objective function
• Overfitting can be
controlled by using a
penalized objective
function (more later)
• Quadratic programming
methods are needed to
solve these equations
116
Support Vectors and Data Reduction
• The points that are outside
the funnel (or on it’s
boundary) are the support
vectors
• It turns out that the prediction
function only uses the
support vectors
– the prediction equation is more
compact and efficient
– the model may be more robust
to outliers
117
Support Vectors and Data Reduction
• The model fitting routine produces values () that
are non-zero for all of the support vectors
• To predict a new sample, the original training data
for the non-zero  values are needed:
118
Nonlinear Boundaries
• Nonlinear boundaries can be computed using the
“kernel trick”
• The predictor space can be expanded by adding
nonlinear functions of the predictors
• Common kernel functions are:
119
Nonlinear Boundaries
• The “trick” is that the computations can operate
only on the inner-products of the extended
predictor set
• In this way, the predictor space dimension can be
greatly expanded without much computational
impact
120
Cost functions
• Support vector machines also include a regularization
parameter that controls how much the regression line can
adapt to the data
– smaller values result in more linear (i.e. flat) surfaces
• This parameter is generally referred to as “Cost”
• For example, this link show the effect of the cost function
for a highly nonlinear problem
– SvmRegMovieA.gif
• This one shows the robustness of SVM regression
models
– SvmRegMovieB.gif
121
Boston Housing Data
• As previously
mentioned, there is a
way to analytically
estimate the tuning
parameter for the RBF
– here, a fixed value of
0.0219 is used
• The remaining
parameter (cost) shows
a clear optimum
122
Summary
• Currently, the SVM model is best at prediction (but
worst at interpretation)
Training Data
(bootstrap)
123
Test Data
RMSE
Q2
RMSE
R2
Linear Reg
5.23
0.691
4.53
0.742
PLS
5.25
0.689
4.56
0.739
Neural Net
4.60
0.757
4.20
0.780
SVM (radial)
3.79
0.834
3.28
0.861
Multivariate Adaptive Regression Splines
124
Multivariate Adaptive Regression Splines
• MARS is a nonlinear statistical model
• The model does an exhaustive search across the
predictors (and each distinct value of the
predictor) to find the best way to sub-divide the
data
• Based on this “split” value, MARS creates new
features based on that variable
• These artificial features are used to model the
outcome
125
MARS Features
• MARS uses “hinge” functions
that are two connected lines
• For a data point x of a
predictor, MARS creates a
function that models the data
on each side of x:
• These features are created in
sets of two (switching which
side is “zeroed”)
126
x
h(x-6) h(6-x)
2
0
2
4
0
4
8
8
0
10
10
0
Prediction Equation and Model Selection
• The model iteratively adds the two new features and uses
ordinary regression methods to create a prediction
equation. The process then continues iteratively.
• MARS also includes a built-in
feature selection routine that
can remove model terms
– the maximum number of retained
features (and the feature degree)
are the tuning parameters
• The Generalized CrossValidation statistic (GCV) is
used to select the most
important terms
127
Sine Wave Example
• As an example, we can use
MARS to model one predictor
with a sinusoidal pattern
• The first MARS iteration
produces a split at 4.3
– two new features are created
– a regression model is fit with
these features
– the red line shows the fit
128
Sine Wave Example
• On the second iteration, a split
was found at 7.9
– two new features are created
• However, the model fit on the left
side was already pretty good
– one of the new surrogate predictors
was removed by the automatic
feature selection
• The model now has three
features
129
Sine Wave Example
• The third split occurred at 5.5
• Again, only the “right-hand”
feature was retained in the model
• This process would continue until
– no more important features are found
– the user-defined limit is achieved
130
Higher Order Features
• Higher degree features
can also be used
– two or more hinge functions
can be multiplied together
to for a new feature
– in two dimensions, this
means that three of four
quadrants of the feature can
be zero if some features are
discarded
131
Boston Housing Data
• We tried only additive
models
– the model could retain
from 4 to 36 model terms
• The “best” model used
18 terms
132
Boston Housing Data
• Since the model is additive, we can look at the
prediction profile of each factor while keeping the
others constant
133
Summary
• SVMs are still optimal, but the respectable
performance and interpretability of MARS might
make us reconsider
Training Data
(bootstrap)
134
Test Data
RMSE
Q2
RMSE
R2
Linear Reg
5.23
0.691
4.53
0.742
PLS
5.25
0.689
4.56
0.739
Neural Net
4.60
0.757
4.20
0.780
SVM (radial)
3.79
0.834
3.28
0.861
MARS
4.29
0.791
3.98
0.804
Regression Trees
135
Regression Trees
• A regression tree searches through each
predictor to find a value of single predictor that
best splits the data into two groups.
– the best split minimizes the mean squared error of the
model.
• For the two resulting groups, the process is
repeated until a hierarchical structure (a "tree") is
created.
– in effect, trees partition the predictor space into
rectangular sections that assign a single average to
compounds within the rectangle.
136
Computational Difficulties
• Suppose we have n observations and p
predictors.
– For each level of the tree, there are at most p(n-1)
possible splits
• As tree depth increases, the number of possible
split combinations multiplies
– The total number of possible split combinations is
bounded above by [p(n-1)]depth
– Suppose we have 100 observations and 100
dimensions.
– The number of possible trees is bounded above by
10400!
137
A Greedy Approach
• Instead of trying to find the best global set of
regions for which the responses are similar, we
recursively partition the data to find an optimal
set of decision rules.
• A regression tree searches through each
predictor to find a value of a single predictor that
best splits the data into two groups.
138
Objective at Each Split
• Let [Xnxp|Ynx1] represent the data matrix
• We seek a predictor, Xj, and split point, s, that solve:


2
2

yi  c1   min   yi  c2  
min

c2
xij R2
 c1 xij R1

where xij  R1  x  xij  sand xij  R2  x  xij  s,
for i  1,2,...,n and j  1,2,..., p.
• The best c1 and c2 are the average responses for the
observations in each region
• For the two resulting groups, the process is repeated
139
Splitting Example – Boston Housing
• We start with all of the
training data
• Searching through all
the data yields the first
split
– a lower status value of
9.6% provides the best
decrease in MSE
140
Splitting Example – Boston Housing
• Searching though the
first left split (), the
best split again uses
the lower status %
• In the initial right split
(), the split was
based on the mean
number of rooms
• Now, there are 4
possible predicted
values
141


Tree Fitting Process
• This process would continue until some criterion
for stopping is met
– such as the minimum number of compounds in a node
• The largest possible tree may over-fit
• “Pruning” is the process of iteratively removing
terminal nodes
– looking for drops in resampling performance
142
Tree Fitting Process
• There are many possible pruning paths
– how many possible trees are there with 6 terminal
nodes?
• We can index the possible trees by a complexity
parameter, Cp.
– Cp = 0 is the largest tree possible
– as Cp increases, the tree shrinks
– there are a discrete set of Cp values for a data set
• Algorithmically, we can control the complexity by
setting the maximum tree depth
143
Comparison
• For these data, we tried 6
possible tree sizes
• For each value, resample the
data and calculate
performance
• After a depth of 4, the model
cannot improve performance
Training Data
(bootstrap)
Single Tree
144
Test
RMSE
Q2
RMSE
R2
5.18
0.700
4.28
0.780
Boston Housing Example
• A depth of 4 was
optimal (see righthand branch)
• This model has a test
set performance of
0.78
– so far the best is 0.86
• However, we can
clearly get a sense of
what the model is
saying
145
Single Trees
• Advantages
– can be computed very quickly and have simple
interpretations.
– have built-in predictor selection: if a predictor was not
used in any split, the model is completely independent
of that data.
• Disadvantages
– instability due to high variance: small changes in the
data can drastically affect the structure of a tree
– data fragmentation
– high order interactions
146
Ensemble Methods
147
Ensemble Methods
• Ensembles of trees have been shown to provide
more predictive models than individual trees and
are less variable than individual trees
• Common ensemble methods are:
– Bagging
– Random forests, and
– Boosting
148
Bagging Trees
• Bootstrap Aggregation
– Breiman (1994, 1996)
Bootstrap
Sample
Bootstrap
Sample
Prediction
Prediction
...
Bootstrap
Sample
– Bagging is the process of
1. creating bootstrap samples
of the data,
2. fitting models to each
sample
3. aggregating the model
predictions
– The largest possible tree is
built for each bootstrap
sample
149
Final
Prediction
Prediction
Bagging Model
Prediction of an observation, x:
M
F ( x) 
150
 f x 
m 1
m
M
Comparison
• Bagging can significantly increase performance of trees
– from resampling:
Training Data
(bootstrap)
Test
RMSE
Q2
RMSE
R2
Single Tree
5.18
0.700
4.28
0.780
Bagging
4.32
0.786
3.69
0.825
• The cost is computing time and the loss of interpretation
• One reason that bagging works is that single trees are
unstable
– small changes in the data may drastically change the tree
151
Random Forests
• Random forests models are similar to bagging
– separate models are built for each bootstrap sample
– the largest tree possible is fit for each bootstrap sample
• However, when random forests starts to make a
new split, it only considers a random subset of
predictors
– The subset size is the (optional) tuning parameter
• Random forests defaults to a subset size that is
the square root of the number of predictors and is
typically robust to this parameter
152
Random Predictor Illustration
Randomly select a
subset of variables
from original data
Dataset 1
Dataset 2
Dataset M
|
|
|
Build trees
Predict
Predict
Final Prediction
153
Predict
Random Forests Model
Prediction of an observation, x:
M
F ( x) 
154
 f x 
m 1
m
M
Properties of Random Forests
• Variance reduction
– Averaging predictions across many models provides
more stable predictions and model accuracy
(Breiman, 1996)
• Robustness to noise
– All observations have an equal chance to influence
each model in the ensemble
– Hence, outliers have less of an effect on individual
models for the overall predicted values
155
Comparison
• Comparing the three methods using resampling:
Training Data
(bootstrap)
Test
RMSE
Q2
RMSE
R2
Single Tree
5.18
0.700
4.28
0.780
Bagging
4.32
0.786
3.69
0.825
Rand Forest
3.55
0.857
3.00
0.885
• Both bagging and random forests are “memoryless”
– each bootstrap sample doesn’t know anything about the other
samples
156
Boosting Trees
• A method to “boost” weak learning algorithms
(small trees) into strong learning algorithms
– Kearns and Valiant (1989), Schapire (1990), Freund
(1995), Freund and Schapire (1996a)
• Boosted trees try to improve the model fit over
different trees by considering past fits
157
Boosting Trees
• First, an initial tree model is fit (the size of the
tree is controlled by the modeler, but usually the
trees are small (depth < 8))
– if a sample was not predicted well, the model residual
will be different from zero
– samples that were predicted poorly in the last tree will
be given more weight in the next tree (and vice-versa)
• After many iterations, the final prediction is a
weighted average of the prediction form each
tree
158
Boosting Illustration
Stage
2
1
n=200
Build
weighted
tree
X1 > 5.2
n=90
X1 < 5.2
n=110
n
2
e
 i  32.9
Compute
stage weight
βstage 1 = f(32.9)
159
M
n=200
Compute
error
Reweigh
observations
(wi=1,2,..., n)
...
i 1
Determine weight of
ith observation:
The larger the error,
the higher the weight
X27 > 22.4
n=64
n=200
X27 < 22.4
X6 > 0
X6 < 0
n=136
n=161
n=39
n
2
e
 i  26.7
i 1
βstage 2 = f(26.7)
Determine weight of
ith observation
n
2
e
 i  29.5
i 1
βstage M = f(29.5)
Boosting Trees
• Boosting has three tuning parameters:
– number of iterations (i.e. trees)
– complexity of the tree (i.e. number of splits)
– learning rate: how quickly the algorithm adapts
• This implementation is the most computationally
taxing of the tree methods shown here
160
Final Boosting Model
Prediction of an observation, x:
M
F ( x)    m f m x 
m 1
where the βm are constrained to sum to 1.
161
Properties of Boosting
• Robust to overfitting
– As the number of iterations increases, the test set
error does not increase
– Schapire, et al. (1998), Friedman, et al. (2000),
Freund, et al. (2001)
• Can be misled by noise in the response
– Boosting will be unable to find a predictive model if the
response is too noisy.
– Kriegar, et al. (2002), Wyner (2002), Schapire (2002),
Optiz and Maclin (1999)
162
Boosting Trees
• One approach to training is
to set the learning rate to a
high value (0.1) and tune
the other two parameters
• In the plot to the right, a grid
of 9 combinations of the 2
tuning parameters were
used to optimize the model
• The optimal settings were:
– 500 trees with high complexity
163
Comparison Summary
• Comparing the four methods:
Training Data
(bootstrap)
164
Test
RMSE
Q2
RMSE
R2
Single Tree
5.18
0.700
4.28
0.780
Bagging
4.32
0.786
3.69
0.825
Rand Forest
3.55
0.857
3.00
0.885
Boosting
3.64
0.847
3.19
0.870
Model Building Training
Model Comparisons
165
Which Model is Best?
• The “No Free Lunch Theorem”:
– over the set of all possible problems, each algorithm
will do on average as well as any other
or, in other words,
– if one model is better than another, it is because of the
particular problem at hand; no one method is uniformly
best
• Despite this statement, the next slide has some
(subjective) ratings of models
166
Top Level Comparisons
Model
Speed Performance Interpretability Robustness
Boosted Tree
Random Forest
Linear Model
PLS
MARS
Neural Net
SVM
RDA
FDA
Naϊve Bayes
Excellent
167
Very Good
Average
Fair
Poor
Top Level Comparisons
Model
Boosted Tree
Random Forest
Linear Model
PLS
MARS
Neural Net
SVM
RDA
FDA
Naïve Bayes
Missing
#Param Pre-Process P > N ?
Data ?
2-3
0-1
0
1
2
2
2-3
2
2
0-1
None
None
ZV, NZV, HCP
CS
ZV, NZV, HCP
ZV, CS, HCP
CS
ZV
None
ZV
Yes
Yes
No
Yes
Yes
Yes
Yes
No
Yes
Yes
ZV = zero var predictor, NZV = near-zero var predictor,
CS = center+scale, HCP = highly correlated predictor
* Depends on implementation
168
Yes*
Yes*
No
No
Yes
No
No
No
Yes
Yes
Boston Housing Data
• The correlation between the results on the training set
(n=337) via cross-validation and the results from the test
set (n=169) were 0.971 (RMSE) and 0.965 (R2)
169
Some Advice
• There is an inverse relationship between
performance and interpretability
• We want the best of both worlds: great
performance and a simple, intuitive model
Interpretability
Tree
Regression
PLS
MARS
• Try this:
– Fit a high performance model to get an
idea of the best possible performance
– Move up the line and see if a less
complex model can keep performance
up with some interpretability
NNet
Boosted
Tree
SVM
RF/Bagging
Performance
170
Regression Datasets
171
Internet Move Data Base
• IMDB is an on-line resource that catalogs movies and TV
programs from many countries.
• Basic information about the program is maintained and
users can rate each program on a five point scale.
• We extracted information about movies and captured:
– the average vote
– the number of votes
– basic information: run time, rating (if any), year of release, etc
– genre: drama, comedy etc and
– keywords: based on novel, female lead, title spoken by character…
• Can we predict the movie rating based on these data?
172
Tecator Spectroscopy Data
• From Statlib:
“These data are recorded on a Tecator Infratec Food and Feed
Analyzer working in the wavelength range 850 - 1050 nm by the
Near Infrared Transmission (NIT) principle.
Each sample contains finely chopped pure meat with different
moisture, fat and protein contents.
For each meat sample the data consists of a 100 channel
spectrum of absorbances and the contents of moisture (water), fat
and protein.
The absorbance is -log10 of the transmittance measured by the
spectrometer.
The three contents, measured in percent, are determined by
analytic chemistry.”
173
Tecator Spectroscopy Data
• The variables are spectral
measurements at specific
wavelengths and are
highly autocorrelated.
• We wish to predict the
percent fat for each
sample.
174
Towson Home Sales
• Information about homes sold in the Towson, Maryland
area (north of Baltimore) were collected.
• The area encompasses the northern border of Baltimore
city (Idlewydle), suburban areas (Annelsie, Rodgers
Forge, Wiltondale) and more expensive areas (Stoneleigh,
Ruxton).
• Variables include:
– The lot size
– The sale date and
– Square footage
– The year built
– Number of baths
• Can we accurately predict the sale price of a home?
175
Regression Backup Slides
176
SVM Model Fit – Test Set Results
177
MARS Model Fit – Test Set Results
178
Regression Tree Model Fit – Test Set Results
179
Boosting Tree Model Fit – Test Set Results
180
Variable Importance for PLS
• To understand the
importance of each factor,
we can look at a weighted
sum of the absolute
regression coefficients
– the weights are based on
the decrease in error as
more components are
added
• We can also look at the
loadings to get a more
detailed assessment
181
Variable Importance for PLS
• Here, we can look at the
increase in R2 as model
terms are added
• If the variable is never
used in a term, it has an
importance of zero
182
Variable Importance for Regression Trees
• Here, we can look at the
decrease in MSE as
model terms are added
• If the variable is never
used in a split, it has an
importance of zero
183
Variable Importance for Random Forests
• A permutation approach is
used
• Each training data for
variable is scrambled in
turn and the % increase in
the out-of-bag MSE is
tracked
184
Boosting, Formally…
• Boosting fits a forward stagewise additive model
(Hastie, Tibshirani and Friedman, 2001) through
the following steps:
1. Let f 0 x   0
2. For m  1, 2,, M do stepsa and b
N
a.  βm , hm   arg min ,h   yi  f m1 xi   hxi 
i 1
where   R, and h is a tree.
b. f m x   f m1 x    m hm x 
185
2
Boosting’s Underlying Model
• λ acts as a shrinkage parameter and is called the
learning rate.
– a parameter that controls the rate of learning of observations
that overlap on a decision boundary (Friedman, 2001)
• Shrinkage boosting can be viewed as fitting this
additive model:
f M x  

hm H d
Hd
h x      m
m m
m 1
where hm(x)  Hd , and Hd represents a dictionary of
trees of depth d. (Hastie, 2001)
186
Linear Regression Pre-Processing
• Linear regression models will fail if there are zerovariance predictors included
– They will also fail during cross-validation if any nearzero variance predictors are in the data
• As just discussed, removing highly correlated
predictors is strongly suggested
• Centering and scaling are not required, but can
greatly increase the numerical stability of the
model
187
PLS Pre-Processing
• Because of its dimension reduction abilities, PLS
is resistant to zero- and near-zero variance
predictors
• Also, since PLS can handle (and perhaps exploit)
correlated predictors, it is not necessary to
remove them
• Centering and scaling are extremely important for
PLS models
– otherwise, the predictors with large variability can
dominate the selection of components
188
Neural Network Pre-Processing
• Neural network models will not fail with zero-variance
predictors
• However, these models use a large number of parameters
and near-zero variance predictors may lead to numerical
issues such as a failure to converge
• Highly correlated predictors should be removed;
multicollinearity can have a significant effect on model
performance
• Centering and scaling are required
189
MARS Pre-Processing
• MARS models are resistant to zero- and near-zero
variance predictors
• Highly correlated predictors are allowed, but this can lead
to significant amount of randomness during the predictor
selection process
– The split choice between two highly correlated predictors becomes
a toss-up
• Centering and scaling are not required but are suggested
190
Tree Pre-Processing
• A basic regression tree requires very little preprocessing
– missing predictor values are allowed
– centering and scaling are not required
• centering and scaling do not affect results
– highly correlated predictors are allowed
• Including highly correlated descriptors can cause instability
and make descriptor importance rankings somewhat random
– zero- and near-zero variance predictors are allowed
191
Model Building Training
Classification-type Models
192
Setting
Variables
Pred. 1
Obs 1 3.231
Obs 2 5.249
Obs 3 7.534
...
Obs n 6.878
Pred. 2
99.30
63.78
84.53
77.21
...
Pred. p
20104
30128
10021
Response
50249
Inactive
Inactive
Active
Active
Response is categorical
Response may have more than two categories
193
Objective
To construct a model of predictors that
can be used to predict a response
Data
Model
Prediction
194
Classification Methods
• Discriminant analysis framework
– Linear, quadratic, regularized, flexible, and partial least squares
discriminant analysis
• Modern classification methods
– Tree-based ensemble methods
• Boosting and random forests
– Neural networks
– Support vector machines
– k-nearest neighbors
– Naive Bayes
• Each of these methods seek to find a partitioning of the
data that minimizes classification error
195
Evaluating Classification Model Performance
• Like regression models, we desire to understand the
predictive ability of a classification model.
• We can evaluate a model’s performance by using crossvalidation or a test set of data.
• For regression models, the measure of performance was
RMSE (or RMSPE)—a function of the deviation of the
observed value from the predicted value.
– This is a valid measure of performance when the response is
continuous, but not when the response is categorical.
• Instead, we need a measure of predictive ability that is
appropriate for categorical data.
196
Objective
• Minimize classification error (or maximize accuracy)
– Determine how well the model prediction agrees with the
actual classification of observations.
Predicted
197
Active
Inactive
Total
Active
A
B
A+B
Inactive
C
D
C+D
Total
A+C
B+D
N=A+B+C+D
Intuition
• An intuitive measure of accuracy is
(A + D) / N
– When the actual classes are balanced, this is an
appropriate measure of model performance.
• But, this measure produces the same values for
different tables:
Active
Inactive
Active
50
50
Inactive
50
4850
vs
Active
Inactive
Active
95
5
Inactive
95
4805
Accuracy for both tables is 0.98
Does one table show more agreement than the other?
198
Another Measure: Kappa
• To provide a measure of agreement for unbalanced
tables, Cohen (1960) proposed comparing the observed
agreement to the expected agreement
• To compute Kappa, we need
– The observed agreement: O = (A + D) / N
– The expected agreement

A  C  A  B   B  D C  D 
E
N2
• Kappa is defined as: k = (O – E) / (1 – E)
199
Kappa Properties
• Generally: -1  k  1
– values close to 0 indicate poor agreement
– values close to 1 indicate near perfect agreement
• for complete disagreement, k = -1
– “Values of 0.4 or above are considered to indicate moderate
agreement, and values of 0.8 or higher indicate excellent
agreement.” (Stokes, Davis, and Koch, 2001)
• Can be generalized to > 2 classes
k = 0.49
Active
Inactive
k = 0.65
Active
Inactive
Active
50
50
Active
95
5
Inactive
50
4850
Inactive
95
4805
Note: When the observed classes are balanced, kappa = accuracy
200
Another Measure:
Receiver Operating Characteristic (ROC) Curves
• ROC curves can be used to assess a
classification model’s performance or to compare
several models’ performance
• Building an ROC curve requires that the model
produces a continuous prediction
• For each predicted value of the response, we
construct a 2x2 table using the predicted value
as the cutoff.
201
ROC Curves
Observed
Class
Positive
Negative
Predicted Class
Positive Negative
TP
FN
FP
TN
• Terminology:
– Sensitivity = True Positive Rate = TP / (TP + FN)
– Specificity = True Negative Rate = TN / (FP + TN)
• An ROC curve is a plot of 1 – specificity versus
sensitivity for each predicted value of the response
– false positive rate versus true positive rate
• A perfect classification model has both a sensitivity and
specificity of 1.
202
ROC Example
Predicted Observed
Prob
Class
0.05
0.35
0.37
0.60
+
0.61
0.63
0.83
0.88
+
0.89
+
0.99
+
Cutoff =
0.99
Observed
Class
Predicted Class
+
-
+
0
0
4
6
Sensitivity Specificity
0/4
6/6
Sensitivity
1
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
1 - Specificity
Cutoff =
0.89
Observed
Class
Predicted Class
+
-
+
1
0
3
6
Sensitivity Specificity
1/4
6/6
Cutoff =
0.61
Observed
Class
Predicted Class
+
-
+
3
3
Sensitivity Specificity
3/4
3/6
All observations with predicted probabilities ≤ the cutoff are classified as negative.
203
1
3
Classification Model Predictions
• Several classification models generate a predicted value
for each class in the original data
– PLSDA, FDA, and NN
• The class with the largest predicted outcome is the
predicted class
– Predictions from the model are generally between 0 and 1, but are
not guaranteed to be within this range.
• The softmax technique is used to transform the predicted
outcomes to “probability-like” values that can be
interpreted as class probabilities
– On the [0, 1] scale and add up to 1
204
Softmax Function
• Let gik be the classification score of the ith
observation into group k.
• The probability that the observation is in group k
g ik
is:
e
K
e
g ip
p 1
where K is the total number of groups
205
Discriminant Models
206
Classical Discriminant Models
• These models form a discriminant function that
can be used to classify samples
• The discriminant function is a linear function of the
predictors that attempts to:
• This is a latent variable method similar to PLS and
others that we have seen
– how the latent variable is created differs between
methods
207
Linear Discriminant Analysis
• Assumption: the within group variability is the same for
each group.
• For a two-class problem, the classification boundary is a
straight line
– The function uses the within-class means and the overall
covariance structure to create the latent variable
• Because it uses the covariance matrix, there must be
– at least as many compounds as predictors
– no zero-variance or linearly dependent predictors
• LDA is not optimal for groups separated by curvature
208
Example where LDA works
• The plot on the right
shows a three class
example where a linear
method like LDA is most
effective
209
Aside: LDA and Logistic Regression
• It turns out that LDA and logistic regression are fitting models that are
very similar
– LDA assumes that the predictors are measured with error and that the
classification of the observations is known
– LR assumes that the predictors are known and that the classification of
the observations are measured with error
• Assuming that the response error is Normal, the optimal separating
plane for logistic regression is:
• LDA estimates a large number of parameters and has fairly strict
constraints on the data
• Also, logistic models may be more forgiving of skewed predictor
distributions
210
Example Data
• For our example data
set, LDA doesn’t do a
very good job since
the boundary is
nonlinear
• The linear predictor is
determined to be
(1.18  Predictor A)
 (0.25  Predictor B)
211
Aside: LDA and Large Number of Predictors
• Some classification models are not drastically
affected by large numbers of predictors
– In many cases, a number of predictors will be noise
• LDA has the potential to overfit
– LDA class probability estimates become more extreme
as the number of predictors becomes large even when
there is no underlying difference
• A similar issue occurs in LR
– For LR, at some point a random predictor will perfectly
split the classes
212
Aside: LDA and Large Number of Predictors
• For example, we simulated a
data set that was complete noise
• For a small number of predictors,
the posterior probabilities were
grouped around 0.50
• As the number of predictors was
increased, the “certainty” of
these probabilities became more
extreme
213
PLS for Discrimination
• In regression PLS seeks to find linear
combinations of the original variables
(scores) that are highly correlated with
the response.
• For classification problems we can use
PLS to find linear combinations of the
original variables that optimally
separate the data.
– Unlike regression, the response for
classification is a binary matrix, with each
column indicating the class of the
observation
214
Response Matrix
1
1

0
Y
0
0

0
0 0
0 0
1 0

1 0
0 1

0 1
PLS Optimization
(many predictors, many responses)
• Like the regression setting, we must solve an
optimization problem that is subject to
constraints:
1. The X-space and Y-space PLS directions have unit
length
2. Either
a.Successively derived scores in each space are uncorrelated
to previously derived scores, OR
b.Successively derived directions in each space are orthogonal
to previously derived directions
215
Solution:
Same as PLS for Regression
• The optimization problem defined by PLS can be
solved through the following formulation:
2

T
T

Cov a X, b Y
arg max
,
T
T
a a b b
a,b
  
subject to constraints 2a. or b.

T
   
  
T
2
T
T
var a X var b Y corr a X, b Y
 arg max
T
T
a a b b
a,b
216

Facts
• Barker and Rayens (2003) showed:
– The PLS directions are the eigenvectors of a modified
between-class covariance matrix, B.
– Coding of the response matrix does not matter
• either g columns or g-1 columns provides the same answer
– The constraint in the Y-space does not make sense
• Why constrain a response that denotes class membership?
– If the Y-space constraint is removed, the PLS
directions are exactly the eigenvectors of the betweenclass covariance matrix, B.
– LDA is optimal if dimension reduction is not necessary
• The optimal directions for LDA are the eigenvectors of W-1B.
217
PLS Discriminant Analysis Example 1
The softmax function is used to determine classification boundaries.
218
PLS Discriminant Analysis Example 2
PLSDA
219
LDA
Quadratic Discriminant Analysis
• Assumption: the within group variability is different for
each group.
• The decision rule is
– where k represents group k.
– The class with the largest score is the predicted class
– A function of squared distance of each observation from each
group’s center
• The decision rule depends on the covariance matrix for
each group
220
Quadratic Discriminant Analysis
• QDA extends the LDA
model by using quadratic
(i.e nonlinear) classification
boundaries
• However, the data
requirements are more
stringent
– at least as many compounds
as predictors in each class
– no zero-variance or linearly
dependent predictors
221
Regularized Discriminant Analysis
• The method tries to split the difference between LDA and
QDA.
• It uses two tuning parameters, gamma and lambda:
– gamma controls the correlation assumption for the predictors
• as gamma  1 the model assumes less predictor correlations
– lambda toggles between linear and quadratic boundaries
• gamma = 0 & lambda = 1  LDA
• gamma = 0 & lambda = 0  QDA
• Other combinations of gamma and lambda produce
models that are compromises between LDA and QDA
222
Regularized Discriminant Analysis
• To see the effect of changing gamma:
– RdaMovieA.gif
• To see the effect of changing lambda:
– RdaMovieB.gif
• We can find the optimal gamma and lambda by
cross-validation
223
Flexible Discriminant Analysis
• FDA generalizes LDA to highly nonlinear boundaries
• In addition to the original predictors, nonlinear functions of
the predictors are added to the data
– This is known as a “basis expansion” of the original data
• This procedure essentially builds a set of “one versus all”
classification models
– a 0/1 outcome is used for each model
– the softmax function is used to convert the model output to class
probabilities
224
Flexible Discriminant Analysis
• For example, the MARS “hinge functions” can be
used
• For each 0/1 outcome, the best predictor/split of
the data is determined and two hinge functions
are added
• Hinge functions are added until a pre-specified
number of terms is reached
• Like the MARS model, the number of features is
reduced until the fit begins to suffer
225
FDA Example
• FDA uses the MARS procedure to determine new
hinge features
– for these data, 3 sets of features were used in to
discriminate the classes
226
Modern Classification Methods
227
Classification Trees
• Like regression trees, classification trees search
through each predictor to find a value of single
predictor that splits the data into two (or more)
groups that are more pure than the original
group.
• For each partition, each predictor is evaluated at
all possible split points and the best predictor
and split are selected.
– Process continues until some criterion for stopping is
met (like minimum number of observations in a node)
228
Splitting Example
Pred A
A > Thresh 1
A  Thresh 1
Pred B
B > Thresh 2
Pred D
B  Thresh 2
Pred A
A > Thresh 3
229
1
2
D > Thresh 4
D  Thresh 4
A  Thresh 3
1
2
1
2
1
2
1
2
Impurity Measures
• There are several measures for determining the
purity of the split. For a two-class, two common
measures are
– Misclassification error
– Gini index
230
Impurity Measure Definitions
x<k
Class 1
a
Class 2
c
x≥k
b
d
c 
 a
p1  min
,

ac ac
d 
 b
p2  min
,

bd bd 
ac
bd
w1 
, w2 
n
n
• Misclassification error: w1p1 + w2p2
– When w1 = w2= 0.5, ME = 0.5*(p1 + p2)
• Gini index: w1p1(1-p1) + w2p2(1-p2)
– When w1 = w2= 0.5, GI = 0.5*(p1(1-p1) + p2(1-p2))
231
Impurity Measure Comparison
232
Simple Example
10
• In this example a few
possible partitions clearly
stand out:
8
– x1 = 5,
0
• How does each impurity
measure rank these
partitions?
2
4
x2
– x2 = 1.5
6
– x2 = 7.5, or
0
2
4
6
x1
233
8
10
Classification Results
Black
Red
Total
x1 ≥ 5
40
7
47
x1 < 5
11
42
53
Black
Red
Total
x2 < 7.5 x2 ≥ 7.5
51
0
32
17
83
17
Black
Red
Total
x2 < 1.5
14
0
14
Partition Misclassification Error
Gini Index
x1 ≥ 5
0.15
0.25
x1 < 5
0.21
0.33
(0.47)(0.15)+(0.53)(0.21) (0.47)(0.25)+(0.53)(0.33)
Total
= 0.18
= 0.29
x2 < 7.5
x2 ≥ 7.5
Total
x2 < 1.5
x2 ≥ 1.5
Total
234
0.39
0
(0.83)(0.39) + (0.17)(0)
= 0.32
0.47
0
(0.83)(0.47) + (0.17)(0)
= 0.39
0.43
0
(0.14)(0) + (0.86)(0.43)
= 0.37
0.49
0
(0.14)(0) + (0.86)(0.49)
= 0.42
x2 ≥ 1.5
37
49
86
Ensemble Methods
• Like individual regression trees, single
classification trees
– are not optimal classification methods.
– have high variability—small changes in the data can
drastically affect the structure of the tree.
• Bagging, random forests, and boosting can also
be implemented for classification problems
235
Bagging, Random Forests, and Boosting
• Each of these ensemble methods are
implemented in the same way as in regression.
• The objective is to minimize misclassification
error
– The loss function changes to exponential loss rather
than squared error loss.
• Tuning parameters for these methods are the
same as in regression
236
Neural Networks
• Like PLS, neural networks for classification
translate the classes to a set of binary (zero/one)
variables.
• The binary variables are modeled using the
predictors and the softmax technique is used to
make sure that the model outputs behave like
probabilities
237
Fitting Neural Networks
• As in regression models, there are two
complexity parameters:
– The number of hidden units
– The amount of weight decay
• The second parameter helps determine the
smoothness of the classification boundaries
• For the example data:
– nnetMovie.gif
238
Support Vector Machines (SVMs)
• SVMs for classification use
a completely different
objective function:
– the margin
• Suppose we have two
predictors and a bunch of
compounds
• We may want to classify
compounds as active or
inactive
• Let’s further suppose that
these two predictors
completely separate these
classes
239
The Margin
• There are an infinite
number of straight lines
that we can use to
separate these two
groups
– some must be better
than others
• The margin is a defined
by equally spaced
boundaries on each
side of the line
240
The Margin
• To maximize the
margin, we try to make
it as large as possible
– without capturing any
compounds
• As the margin
increases, the solution
becomes more robust
• SVMs maximize the
margin to estimate
parameters
241
Support Vectors and Data Reduction
• When the classes overlap, points are allowed
within the margin
– the number of points is controlled by a cost parameter
• The points that are within the margin (or on it’s
boundary) are the support vectors
• It turns out that the prediction function only uses
the support vectors
– the prediction equation is more compact and efficient
– the model may be more robust to outliers
242
Nonlinear Boundaries
• Similar to regression models, the “kernel trick”
can be used to generate highly nonlinear class
boundaries
• For classification, there are two common kernel
functions
– polynomial (3 tuning variables)
– radial basis functions (2 parameters)
243
SVM Example Class Boundary
RBF Kernel
244
79 SVs (31.6%)
The Effect of the Cost Parameter
• As the cost parameter is increased, the model will
work very hard to correctly classify the
compounds
– This can lead to over-fitting
• To see the effect of the cost parameter, the link
below shows an animation for a radial basis
function SVM
– SvmMovieB.gif
• Note that, as the boundary becomes more
complicated, the #SV decreases
– The margin is becoming very small
245
Nearest Neighbor Classifiers
• To predict the class of a new compound, this
procedure uses the most frequent class of the
closest k neighbors
– if a tie, randomly pick from the most frequent classes
• k, the number of neighbors, is the tuning
parameter
• Since distance is used to define the nearest
points, the predictors should be centered and
scaled
246
Nearest Neighbor Classifiers
• For the simulated data,
the model was tuned
across k values from 1 to
20
– 7 neighbors was found to
be optimal
• k-NN class boundaries
tend to be somewhat
jagged but smooth out as
k increases
247
Naïve Bayes
• Recall Bayes theorem:
• Of course, the predictor distributions are usually
multivariate and these probabilities would involve
multidimensional integration
248
Naïve Bayes
• In “naïve Bayes,” aka “Idiot’s Bayes,” the
relationships between predictors are ignored
– i.e all predictors are treated as uncorrelated
249
Naïve Bayes
• Despite this assumption, this model usually is
very competitive, even with strong correlations
• How do we estimate continuous predictor
distributions?
– parametrically: assume normality and use the sample
mean and variance
– non-parametrically: use a nonparametric density
estimator
250
Naïve Bayes
• For example, looking at only the distribution of
predictor A in our example, we see a slight shift
between the distributions of the predictor for
each class:
251
Naïve Bayes
• If a new sample has a
value of predictor A = 1, it is more likely to be
active
– active density ~ 0.40
– inactive density ~ 0.17
252
Naïve Bayes
• For predictor B, the
inactive probability is much
larger for values between
-0.5 and 0.5
• For each predictor, the
distributions are modeled
– class probabilities can be
computed for each predictor
• The final class probability
is calculated by multiplying
all the probabilities
together
253
A Tale of Two Samples
Sample 1
254
Sample 2
Pred A
Pred B
Pred B Pred A
-1
0
Total
-1
-1
Total
Active
0.40
0.14
0.06
0.40
0.30
0.12
Inactive
0.17
0.62
0.10
0.17
0.08
0.01
Naïve Bayes and Many Predictors
• Like LDA, naïve Bayes
models can overfit when
many noisy predictors are
included in the model
• As with LDA, we simulated
noise data and were able
to see class separation
increase as the number of
predictors went up
255
Naïve Bayes Classifiers
• Class boundaries for
naïve Bayes models
can show circular or
elliptical islands
• Since the predictors
are treated as
uncorrelated, there
cannot be any
diagonal ellipses
256
Example: Prediction of Spam
• These data were collect by HP. 4,601 e-mails were
classified as spam or not spam.
• Predictor variables are derived form the emails related to
the frequency of words or characters in the e-mail.
Variables include:
– A set of word frequency variables. For example, the variable make
measures the relative frequency of that word in the email
– Variables related to numbers: words that start with numbers are
also measured. For example, the variable num415 measures how
often the number 415 appears
– Other variables relate to special characters (e.g. the variable
charExclamation) or capital letters (capitalAve)
257
Example: Prediction of Spam
• We would like to classify emails as being spam with an
emphasis on high specificity, i.e. a low probability of nonspam being labeled as spam
• For training, an 80% split was used via stratified random
sampling
258
Method Comparison
259
Method Comparison
260
ROC Comparison
261
Classification Datasets
262
Glaucoma Data
• 62 variables are derived from a confocal laser scanning
image of the optic nerve head, describing its morphology.
Observations are from normal and glaucomatous eyes,
respectively. Examples of variables are:
– as: superior area
– vbss: volume below surface temporal
– mhcn: mean height contour nasal
– vari: volume above reference inferior, etc
• We would like to predict whether a subject has glaucoma
given their imaging data
263
Predicting Diabetes in Pima Indians
• These data are from Pima Indian women living in Arizona.
Several variables were collected, such as:
– pregnant: number of
pregnancies
– glucose: plasma glucose
levels
– pressure: diastolic BP
– mass: body mass index
– pedigree: diabetic pedigree
function,
– age
– diabetes: negative or positive
– triceps: skin fold thickness
– insulin: serum insulin
• We would like to predict a new Indian woman's diabetic
status given their other information.
264
Classification Backup Slides
265
FDA Pre-Processing
• FDA models often use the MARS hinge functions, so they
share similar properties.
• FDA models are resistant to zero- and near-zero variance
predictors
• Highly correlated predictors are allowed, but this can lead
to significant amount of randomness during the predictor
selection process
– The split choice between two highly correlated predictors becomes
a toss-up
• Centering and scaling are not required but are suggested
266
Tree Pre-Processing
• Same as for regression…
– missing predictor values are allowed
– centering and scaling are not required
• centering and scaling do not affect results
– highly correlated predictors are allowed
• Including highly correlated predictors can cause
instability and make predictor importance rankings
somewhat random
– zero- and near-zero variance predictors are
allowed
267
RDA Pre-Processing
• RDA models are cannot deal with zero- and near-zero
variance predictors
– they must be removed
• Highly correlated predictors are allowed, but not
suggested
– However, perfectly correlated predictors will cause the model to fail
• Centering and scaling are not required but are suggested
• Additionally, there cannot be linear dependencies between
predictors
268
Neural Network Pre-Processing
• Neural network models will not fail with zero-variance
predictors
• However, these models use a large number of parameters
and near-zero variance predictors may lead to numerical
issues such as a failure to converge
• Highly correlated predictors should be removed.
• Centering and scaling are required
269
Nearest Neighbor Pre-Processing
• These models are resistant to zero- and near-zero
variance predictors as well as highly correlated predictors
• Centering and scaling are required
270
Naïve Bayes Pre-Processing
• These model will not fail with zero-variance predictors
• Highly correlated predictors are also allowed.
• Centering and scaling are not required
271
Model Building Training
Other Considerations
272
Variables to Select
• Variables thought to be related to the response
should be included in the model
• Sometimes we don’t know if a set of variables
are related to the response
• Should these be included in the analysis?
• If the variables are not related to the response,
then we are including noise into our predictor set
• What happens to the performance of the
techniques when noise is added?
– Can we still find signal?
273
Illustration
• To the blood brain barrier data of Mente and Lombardo
(2005), we have added 10, 50, 100, and 200 random
predictors
• For each of these new data sets, we have built each
regression model, using cross-validation to determine the
optimal parameter settings
• The results are on the following slides
– Keep in mind that these results are for one example
– Methods may have different rankings for other examples
274
Performance Comparison
R2: CV for Training Set
0.5
0.4
0.3
0.2
0.1
0
0
50
Noise
275
100
150
200
Performance Comparison
R2: Test Set
0.5
0.4
0.3
0.2
0.1
0
50
Noise
276
100
150
200
Variables to Select
• Hopefully, we’ve demonstrated that resampling is
a good way to avoid over-fitting
• Realize that predictor selection is part of the
modeling process
• Doing predictor selection outside of crossvalidation can lead to sever predictor selection
bias
– and potential over-fitting (but you won’t know until a
test set)
277
Effects of Categorizing a Continuous Response
• A majority of responses are measured on a continuous
scale
• The continuous scale allows us to compare observations
on their original scale
• Sometimes the continuous response naturally falls into
two or more modes
– If the relative distance between these modes is not relevant, then
the response can be binned
– However, if the distance between modes is relevant, then we lose
information by binning the response
• Binning a continuous response that does not have
natural modes will make us lose even more information
and will degrade model
278
Thanks
• Thanks for sitting through all this
• More thanks to:
– Benevolent overlords David Potter and Ed
Kadyszewski
– Nathan Coulter and Gautam Bhola for computing
support
– Pfizer Chemistry for feedback on earlier versions of
this training
279