Model selection

Linear Model Selection and Regularization
• Recall the linear model
Y = β0 + β1X1 + · · · + βpXp + .
• In the lectures that follow, we consider some approaches for
extending the linear model framework. In the lectures
covering Chapter 7 of the text, we generalize the linear
model in order to accommodate non-linear, but still
additive, relationships.
• In the lectures covering Chapter 8 we consider even more
general non-linear models.
1 / 57

In praise of linear models!
• Despite its simplicity, the linear model has distinct
advantages in terms of its interpretability and often shows
good predictive performance.
• Hence we discuss in this lecture some ways in which the
simple linear model can be improved, by replacing ordinary
least squares ﬁtting with some alternative ﬁtting
procedures.
2 / 57

Why consider alternatives to least squares?
• Prediction Accuracy: especially when p > n, to control the
variance.
• Model Interpretability: By removing irrelevant features —
that is, by setting the corresponding coeﬃcient estimates
to zero — we can obtain a model that is more easily
interpreted. We will present some approaches for
automatically performing feature selection.
3 / 57

Three classes of methods
• Subset Selection. We identify a subset of the p predictors
that we believe to be related to the response. We then fit a
model using least squares on the reduced set of variables.
• Shrinkage. We fit a model involving all p predictors, but
the estimated coefficients are shrunken towards zero
relative to the least squares estimates. This shrinkage (also
known as regularization) has the effect of reducing variance
and can also perform variable selection.
• Dimension Reduction. We project the p predictors into a
M-dimensional subspace, where M < p. This is achieved by
computing M different linear combinations, or projections,
of the variables. Then these M projections are used as
predictors to fit a linear regression model by least squares.
4 / 57

Subset Selection
Best subset and stepwise model selection procedures
Best Subset Selection
1. Let M0 denote the null model, which contains no
predictors. This model simply predicts the sample mean
for each observation.
2. For k = 1, 2, . . . p:
(a) Fit all p
k models that contain exactly k predictors.
(b) Pick the best among these p
k models, and call it Mk. Here
best is deﬁned as having the smallest RSS, or equivalently
largest R2
.
3. Select a single best model from among M0, . . . , Mp using
cross-validated prediction error, Cp (AIC), BIC, or
adjusted R2.
5 / 57

Example- Credit data set
2 4 6 8 10
2e+074e+076e+078e+07
Number of Predictors
ResidualSumofSquares
2 4 6 8 10
0.00.20.40.60.81.0
R2
For each possible model containing a subset of the ten predictors
in the Credit data set, the RSS and R2 are displayed. The red
frontier tracks the best model for a given number of predictors,
according to RSS and R2. Though the data set contains only
ten predictors, the x-axis ranges from 1 to 11, since one of the
variables is categorical and takes on three values, leading to the
creation of two dummy variables
6 / 57

Extensions to other models
• Although we have presented best subset selection here for
least squares regression, the same ideas apply to other
types of models, such as logistic regression.
• The deviance— negative two times the maximized
log-likelihood— plays the role of RSS for a broader class of
models.
7 / 57

Stepwise Selection
• For computational reasons, best subset selection cannot be
applied with very large p. Why not?
• Best subset selection may also suffer from statistical
problems when p is large: larger the search space, the
higher the chance of finding models that look good on the
training data, even though they might not have any
predictive power on future data.
• Thus an enormous search space can lead to overfitting and
high variance of the coefficient estimates.
• For both of these reasons, stepwise methods, which explore
a far more restricted set of models, are attractive
alternatives to best subset selection.
8 / 57

Forward Stepwise Selection
• Forward stepwise selection begins with a model containing
no predictors, and then adds predictors to the model,
one-at-a-time, until all of the predictors are in the model.
• In particular, at each step the variable that gives the
greatest additional improvement to the ﬁt is added to the
model.
9 / 57

In Detail
Forward Stepwise Selection
1. Let M0 denote the null model, which contains no
predictors.
2. For k = 0, . . . , p − 1:
2.1 Consider all p − k models that augment the predictors in
Mk with one additional predictor.
2.2 Choose the best among these p − k models, and call it
Mk+1. Here best is deﬁned as having smallest RSS or
highest R2
.
adjusted R2.
10 / 57

More on Forward Stepwise Selection
• Computational advantage over best subset selection is
clear.
• It is not guaranteed to ﬁnd the best possible model out of
all 2p models containing subsets of the p predictors. Why
not? Give an example.
11 / 57

Credit data example
# Variables Best subset Forward stepwise
One rating rating
Two rating, income rating, income
Three rating, income, student rating, income, student
Four cards, income rating, income,
student, limit student, limit
The first four selected models for best subset selection and
forward stepwise selection on the Credit data set. The first
three models are identical but the fourth models differ.
12 / 57

Backward Stepwise Selection
• Like forward stepwise selection, backward stepwise selection
provides an eﬃcient alternative to best subset selection.
• However, unlike forward stepwise selection, it begins with
the full least squares model containing all p predictors, and
then iteratively removes the least useful predictor,
one-at-a-time.
13 / 57

Backward Stepwise Selection: details
Backward Stepwise Selection
1. Let Mp denote the full model, which contains all p
predictors.
2. For k = p, p − 1, . . . , 1:
2.1 Consider all k models that contain all but one of the
predictors in Mk, for a total of k − 1 predictors.
2.2 Choose the best among these k models, and call it Mk−1.
Here best is deﬁned as having smallest RSS or highest R2
.
adjusted R2.
14 / 57

More on Backward Stepwise Selection
• Like forward stepwise selection, the backward selection
approach searches through only 1 + p(p + 1)/2 models, and
so can be applied in settings where p is too large to apply
best subset selection
• Like forward stepwise selection, backward stepwise
selection is not guaranteed to yield the best model
containing a subset of the p predictors.
• Backward selection requires that the number of samples n
is larger than the number of variables p (so that the full
model can be ﬁt). In contrast, forward stepwise can be
used even when n < p, and so is the only viable subset
method when p is very large.
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Choosing the Optimal Model
• The model containing all of the predictors will always have
the smallest RSS and the largest R2, since these quantities
are related to the training error.
• We wish to choose a model with low test error, not a model
with low training error. Recall that training error is usually
a poor estimate of test error.
• Therefore, RSS and R2 are not suitable for selecting the
best model among a collection of models with diﬀerent
numbers of predictors.
16 / 57

Estimating test error: two approaches
• We can indirectly estimate test error by making an
adjustment to the training error to account for the bias due
to overﬁtting.
• We can directly estimate the test error, using either a
validation set approach or a cross-validation approach, as
discussed in previous lectures.
• We illustrate both approaches next.
17 / 57

Cp, AIC, BIC, and Adjusted R2
• These techniques adjust the training error for the model
size, and can be used to select among a set of models with
diﬀerent numbers of variables.
• The next ﬁgure displays Cp, BIC, and adjusted R2 for the
best model of each size produced by best subset selection
on the Credit data set.
18 / 57

Credit data example
2 4 6 8 10
1000015000200002500030000
Cp
2 4 6 8 10
1000015000200002500030000
BIC
2 4 6 8 10
0.860.880.900.920.940.96
AdjustedR2
19 / 57

Now for some details
• Mallow’s Cp:
Cp =
1
n
RSS + 2dˆσ2
,
where d is the total # of parameters used and ˆσ2 is an
estimate of the variance of the error associated with each
response measurement.
• The AIC criterion is deﬁned for a large class of models ﬁt
by maximum likelihood:
AIC = −2 log L + 2 · d
where L is the maximized value of the likelihood function
for the estimated model.
• In the case of the linear model with Gaussian errors,
maximum likelihood and least squares are the same thing,
and Cp and AIC are equivalent. Prove this.
20 / 57

Details on BIC
BIC =
1
n
RSS + log(n)dˆσ2
.
• Like Cp, the BIC will tend to take on a small value for a
model with a low test error, and so generally we select the
model that has the lowest BIC value.
• Notice that BIC replaces the 2dˆσ2 used by Cp with a
log(n)dˆσ2 term, where n is the number of observations.
• Since log n > 2 for any n > 7, the BIC statistic generally
places a heavier penalty on models with many variables,
and hence results in the selection of smaller models than
Cp. See Figure on slide 19.
21 / 57

Adjusted R2
• For a least squares model with d variables, the adjusted R2
statistic is calculated as
Adjusted R2
= 1 −
RSS/(n − d − 1)
TSS/(n − 1)
.
where TSS is the total sum of squares.
• Unlike Cp, AIC, and BIC, for which a small value indicates
a model with a low test error, a large value of adjusted R2
indicates a model with a small test error.
• Maximizing the adjusted R2 is equivalent to minimizing
RSS
n−d−1. While RSS always decreases as the number of
variables in the model increases, RSS
n−d−1 may increase or
decrease, due to the presence of d in the denominator.
• Unlike the R2 statistic, the adjusted R2 statistic pays a
price for the inclusion of unnecessary variables in the
model. See Figure on slide 19.
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Validation and Cross-Validation
• Each of the procedures returns a sequence of models Mk
indexed by model size k = 0, 1, 2, . . .. Our job here is to
select ˆk. Once selected, we will return model Mˆk
23 / 57

Validation and Cross-Validation
• Each of the procedures returns a sequence of models Mk
indexed by model size k = 0, 1, 2, . . .. Our job here is to
select ˆk. Once selected, we will return model Mˆk
• We compute the validation set error or the cross-validation
error for each model Mk under consideration, and then
select the k for which the resulting estimated test error is
smallest.
• This procedure has an advantage relative to AIC, BIC, Cp,
and adjusted R2, in that it provides a direct estimate of
the test error, and doesn’t require an estimate of the error
variance σ2.
• It can also be used in a wider range of model selection
tasks, even in cases where it is hard to pinpoint the model
degrees of freedom (e.g. the number of predictors in the
model) or hard to estimate the error variance σ2.
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Credit data example
2 4 6 8 10
100120140160180200220
SquareRootofBIC
2 4 6 8 10
100120140160180200220
ValidationSetError
2 4 6 8 10
100120140160180200220
Cross−ValidationError
24 / 57

Details of Previous Figure
• The validation errors were calculated by randomly selecting
three-quarters of the observations as the training set, and
the remainder as the validation set.
• The cross-validation errors were computed using k = 10
folds. In this case, the validation and cross-validation
methods both result in a six-variable model.
• However, all three approaches suggest that the four-, ﬁve-,
and six-variable models are roughly equivalent in terms of
their test errors.
• In this setting, we can select a model using the
one-standard-error rule. We ﬁrst calculate the standard
error of the estimated test MSE for each model size, and
then select the smallest model for which the estimated test
error is within one standard error of the lowest point on
the curve. What is the rationale for this?
25 / 57

Shrinkage Methods
Ridge regression and Lasso
• The subset selection methods use least squares to fit a
linear model that contains a subset of the predictors.
• As an alternative, we can fit a model containing all p
predictors using a technique that constrains or regularizes
the coefficient estimates, or equivalently, that shrinks the
coefficient estimates towards zero.
• It may not be immediately obvious why such a constraint
should improve the fit, but it turns out that shrinking the
coefficient estimates can significantly reduce their variance.
26 / 57

Ridge regression
• Recall that the least squares ﬁtting procedure estimates
β0, β1, . . . , βp using the values that minimize
RSS =
n
i=1

yi − β0 −
p
j=1
βjxij


2
.
• In contrast, the ridge regression coeﬃcient estimates ˆβR
are the values that minimize
n
i=1

yi − β0 −
p
j=1
βjxij


2
+ λ
p
j=1
β2
j = RSS + λ
p
j=1
β2
j ,
where λ ≥ 0 is a tuning parameter, to be determined
separately.
27 / 57

Ridge regression: continued
• As with least squares, ridge regression seeks coefficient
estimates that fit the data well, by making the RSS small.
• However, the second term, λ j β2
j , called a shrinkage
penalty, is small when β1, . . . , βp are close to zero, and so it
has the effect of shrinking the estimates of βj towards zero.
• The tuning parameter λ serves to control the relative
impact of these two terms on the regression coefficient
estimates.
• Selecting a good value for λ is critical; cross-validation is
used for this.
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Credit data example
1e−02 1e+00 1e+02 1e+04
−300−1000100200300400
StandardizedCoefficients
Income
Limit
Rating
Student
0.0 0.2 0.4 0.6 0.8 1.0−300−1000100200300400
λ ˆβR
λ 2/ ˆβ 2
29 / 57

Details of Previous Figure
• In the left-hand panel, each curve corresponds to the ridge
regression coefficient estimate for one of the ten variables,
plotted as a function of λ.
• The right-hand panel displays the same ridge coefficient
estimates as the left-hand panel, but instead of displaying
λ on the x-axis, we now display ˆβR
λ 2/ ˆβ 2, where ˆβ
denotes the vector of least squares coefficient estimates.
• The notation β 2 denotes the 2 norm (pronounced “ell
2”) of a vector, and is defined as β 2 = p
j=1 βj
2
.
30 / 57

Ridge regression: scaling of predictors
• The standard least squares coefficient estimates are scale
equivariant: multiplying Xj by a constant c simply leads to
a scaling of the least squares coefficient estimates by a
factor of 1/c. In other words, regardless of how the jth
predictor is scaled, Xj
ˆβj will remain the same.
• In contrast, the ridge regression coefficient estimates can
change substantially when multiplying a given predictor by
a constant, due to the sum of squared coefficients term in
the penalty part of the ridge regression objective function.
• Therefore, it is best to apply ridge regression after
standardizing the predictors, using the formula
˜xij =
xij
1
n
n
i=1(xij − xj)2
31 / 57

Why Does Ridge Regression Improve Over Least
Squares?
The Bias-Variance tradeoﬀ
1e−01 1e+01 1e+03
0102030405060
MeanSquaredError
0.0 0.2 0.4 0.6 0.8 1.0
0102030405060
MeanSquaredError
λ ˆβR
λ 2/ ˆβ 2
Simulated data with n = 50 observations, p = 45 predictors, all having
nonzero coeﬃcients. Squared bias (black), variance (green), and test
mean squared error (purple) for the ridge regression predictions on a
simulated data set, as a function of λ and ˆβR
λ 2/ ˆβ 2. The
horizontal dashed lines indicate the minimum possible MSE. The
purple crosses indicate the ridge regression models for which the MSE
is smallest. 32 / 57

The Lasso
• Ridge regression does have one obvious disadvantage:
unlike subset selection, which will generally select models
that involve just a subset of the variables, ridge regression
will include all p predictors in the final model
• The Lasso is a relatively recent alternative to ridge
regression that overcomes this disadvantage. The lasso
coefficients, ˆβL
λ , minimize the quantity
n
i=1

yi − β0 −
p
j=1
βjxij


2
+ λ
p
j=1
|βj| = RSS + λ
p
j=1
|βj|.
• In statistical parlance, the lasso uses an 1 (pronounced
“ell 1”) penalty instead of an 2 penalty. The 1 norm of a
coefficient vector β is given by β 1 = |βj|.
33 / 57

The Lasso: continued
• As with ridge regression, the lasso shrinks the coefficient
estimates towards zero.
• However, in the case of the lasso, the 1 penalty has the
effect of forcing some of the coefficient estimates to be
exactly equal to zero when the tuning parameter λ is
sufficiently large.
• Hence, much like best subset selection, the lasso performs
variable selection.
• We say that the lasso yields sparse models — that is,
models that involve only a subset of the variables.
• As in ridge regression, selecting a good value of λ for the
lasso is critical; cross-validation is again the method of
choice.
34 / 57

Example: Credit dataset
20 50 100 200 500 2000 5000
−2000100200300400
0.0 0.2 0.4 0.6 0.8 1.0−300−1000100200300400
Income
Limit
Rating
Student
λ ˆβL
λ 1/ ˆβ 1
35 / 57

The Variable Selection Property of the Lasso
Why is it that the lasso, unlike ridge regression, results in
coeﬃcient estimates that are exactly equal to zero?
36 / 57

The Variable Selection Property of the Lasso
Why is it that the lasso, unlike ridge regression, results in
coeﬃcient estimates that are exactly equal to zero?
One can show that the lasso and ridge regression coeﬃcient
estimates solve the problems
minimize
β
n
i=1

yi − β0 −
p
j=1
βjxij


2
subject to
p
j=1
|βj| ≤ s
and
minimize
β
n
i=1

yi − β0 −
p
j=1
βjxij


2
subject to
p
j=1
β2
j ≤ s,
respectively.
36 / 57

Comparing the Lasso and Ridge Regression
0.02 0.10 0.50 2.00 10.00 50.00
0102030405060
MeanSquaredError
0.0 0.2 0.4 0.6 0.8 1.0
0102030405060
R2
on Training Data
MeanSquaredError
λ
Left: Plots of squared bias (black), variance (green), and test
MSE (purple) for the lasso on simulated data set of Slide 32.
Right: Comparison of squared bias, variance and test MSE
between lasso (solid) and ridge (dashed). Both are plotted
against their R2 on the training data, as a common form of
indexing. The crosses in both plots indicate the lasso model for
which the MSE is smallest.
38 / 57

Comparing the Lasso and Ridge Regression: continued
0.02 0.10 0.50 2.00 10.00 50.00
020406080100
MeanSquaredError
0.4 0.5 0.6 0.7 0.8 0.9 1.0
020406080100
R2
on Training Data
MeanSquaredError
λ
Left: Plots of squared bias (black), variance (green), and test
MSE (purple) for the lasso. The simulated data is similar to
that in Slide 38, except that now only two predictors are related
to the response. Right: Comparison of squared bias, variance
and test MSE between lasso (solid) and ridge (dashed). Both
are plotted against their R2 on the training data, as a common
form of indexing. The crosses in both plots indicate the lasso
model for which the MSE is smallest.
39 / 57

Conclusions
• These two examples illustrate that neither ridge regression
nor the lasso will universally dominate the other.
• In general, one might expect the lasso to perform better
when the response is a function of only a relatively small
number of predictors.
• However, the number of predictors that is related to the
response is never known a priori for real data sets.
• A technique such as cross-validation can be used in order
to determine which approach is better on a particular data
set.
40 / 57

Selecting the Tuning Parameter for Ridge Regression
and Lasso
• As for subset selection, for ridge regression and lasso we
require a method to determine which of the models under
consideration is best.
• That is, we require a method selecting a value for the
tuning parameter λ or equivalently, the value of the
constraint s.
• Cross-validation provides a simple way to tackle this
problem. We choose a grid of λ values, and compute the
cross-validation error rate for each value of λ.
• We then select the tuning parameter value for which the
cross-validation error is smallest.
• Finally, the model is re-ﬁt using all of the available
observations and the selected value of the tuning
parameter.
41 / 57

Credit data example
5e−03 5e−02 5e−01 5e+00
25.025.225.425.6
5e−03 5e−02 5e−01 5e+00
−300−1000100300
λλ
Left: Cross-validation errors that result from applying ridge
regression to the Credit data set with various values of λ.
Right: The coeﬃcient estimates as a function of λ. The vertical
dashed lines indicates the value of λ selected by cross-validation.
42 / 57

Simulated data example
0.0 0.2 0.4 0.6 0.8 1.0
020060010001400
0.0 0.2 0.4 0.6 0.8 1.0
−5051015
ˆβL
λ 1/ ˆβ 1
ˆβL
λ 1/ ˆβ 1
Left: Ten-fold cross-validation MSE for the lasso, applied to the
sparse simulated data set from Slide 39. Right: The
corresponding lasso coeﬃcient estimates are displayed. The
vertical dashed lines indicate the lasso ﬁt for which the
cross-validation error is smallest.
43 / 57

Dimension Reduction Methods
• The methods that we have discussed so far in this chapter
have involved ﬁtting linear regression models, via least
squares or a shrunken approach, using the original
predictors, X1, X2, . . . , Xp.
• We now explore a class of approaches that transform the
predictors and then ﬁt a least squares model using the
transformed variables. We will refer to these techniques as
dimension reduction methods.
44 / 57

Dimension Reduction Methods: details
• Let Z1, Z2, . . . , ZM represent M < p linear combinations of
our original p predictors. That is,
Zm =
p
j=1
φmjXj (1)
for some constants φm1, . . . , φmp.
• We can then ﬁt the linear regression model,
yi = θ0 +
M
m=1
θmzim + i, i = 1, . . . , n, (2)
using ordinary least squares.
• Note that in model (2), the regression coeﬃcients are given
by θ0, θ1, . . . , θM . If the constants φm1, . . . , φmp are chosen
wisely, then such dimension reduction approaches can often
outperform OLS regression.
45 / 57

• Notice that from definition (1),
M
m=1
θmzim =
M
m=1
θm
p
j=1
φmjxij =
p
j=1
M
m=1
θmφmjxij =
p
j=1
βjxij,
where
βj =
M
m=1
θmφmj. (3)
• Hence model (2) can be thought of as a special case of the
original linear regression model.
• Dimension reduction serves to constrain the estimated βj
coefficients, since now they must take the form (3).
• Can win in the bias-variance tradeoff.
46 / 57

Principal Components Regression
• Here we apply principal components analysis (PCA)
(discussed in Chapter 10 of the text) to define the linear
combinations of the predictors, for use in our regression.
• The first principal component is that (normalized) linear
combination of the variables with the largest variance.
• The second principal component has largest variance,
subject to being uncorrelated with the first.
• And so on.
• Hence with many correlated original variables, we replace
them with a small set of principal components that capture
their joint variation.
47 / 57

Pictures of PCA
10 20 30 40 50 60 70
05101520253035
Population
AdSpending
The population size (pop) and ad spending (ad) for 100
diﬀerent cities are shown as purple circles. The green solid line
indicates the ﬁrst principal component, and the blue dashed line
indicates the second principal component.
48 / 57

Pictures of PCA: continued
20 30 40 50
51015202530
Population
AdSpending
−20 −10 0 10 20
−10−50510
1st Principal Component
2ndPrincipalComponent
A subset of the advertising data. Left: The ﬁrst principal
component, chosen to minimize the sum of the squared
perpendicular distances to each point, is shown in green. These
distances are represented using the black dashed line segments.
Right: The left-hand panel has been rotated so that the ﬁrst
principal component lies on the x-axis.
49 / 57

−3 −2 −1 0 1 2 3
2030405060
1st Principal Component
Population
−3 −2 −1 0 1 2 3
51015202530
1st Principal ComponentAdSpending
Plots of the ﬁrst principal component scores zi1 versus pop and
ad. The relationships are strong.
50 / 57

−1.0 −0.5 0.0 0.5 1.0
2030405060
2nd Principal Component
Population
−1.0 −0.5 0.0 0.5 1.0
51015202530
2nd Principal ComponentAdSpending
Plots of the second principal component scores zi2 versus pop
and ad. The relationships are weak.
51 / 57

Application to Principal Components Regression
0 10 20 30 40
010203040506070
Number of Components
MeanSquaredError
0 10 20 30 40
050100150
MeanSquaredError
Squared Bias
Test MSE
Variance
PCR was applied to two simulated data sets. The black, green,
and purple lines correspond to squared bias, variance, and test
mean squared error, respectively. Left: Simulated data from
slide 32. Right: Simulated data from slide 39.
52 / 57

Choosing the number of directions M
2 4 6 8 10
−300−1000100200300400
Income
Limit
Rating
Student
2 4 6 8 10
20000400006000080000
Cross−ValidationMSE
Left: PCR standardized coeﬃcient estimates on the Credit
data set for diﬀerent values of M. Right: The 10-fold cross
validation MSE obtained using PCR, as a function of M.
53 / 57

Partial Least Squares
• PCR identifies linear combinations, or directions, that best
represent the predictors X1, . . . , Xp.
• These directions are identified in an unsupervised way, since
the response Y is not used to help determine the principal
component directions.
• That is, the response does not supervise the identification
of the principal components.
• Consequently, PCR suffers from a potentially serious
drawback: there is no guarantee that the directions that
best explain the predictors will also be the best directions
to use for predicting the response.
54 / 57

Partial Least Squares: continued
• Like PCR, PLS is a dimension reduction method, which
first identifies a new set of features Z1, . . . , ZM that are
linear combinations of the original features, and then fits a
linear model via OLS using these M new features.
• But unlike PCR, PLS identifies these new features in a
supervised way – that is, it makes use of the response Y in
order to identify new features that not only approximate
the old features well, but also that are related to the
response.
• Roughly speaking, the PLS approach attempts to find
directions that help explain both the response and the
predictors.
55 / 57

Details of Partial Least Squares
• After standardizing the p predictors, PLS computes the
first direction Z1 by setting each φ1j in (1) equal to the
coefficient from the simple linear regression of Y onto Xj.
• One can show that this coefficient is proportional to the
correlation between Y and Xj.
• Hence, in computing Z1 = p
j=1 φ1jXj, PLS places the
highest weight on the variables that are most strongly
related to the response.
• Subsequent directions are found by taking residuals and
then repeating the above prescription.
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Summary
• Model selection methods are an essential tool for data
analysis, especially for big datasets involving many
predictors.
• Research into methods that give sparsity, such as the lasso
is an especially hot area.
• Later, we will return to sparsity in more detail, and will
describe related approaches such as the elastic net.
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Model selection

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Model selection