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Penalized Regression, Part 2

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Penalized Regression, Part 2

Recall in penalized regression, we re-write our loss function to include not only the squared error loss but a penalty term

Our goal then becomes to minimize our a loss function (i.e. SS)

In the regression setting we can write M(q ) in terms of our regression parameters b as follows

The penalty function takes the form

Last class we discussed ridge regression as an alternative to OLS when covariates are collinear

Ridge regression can reduce the variability and improve accuracy of a regression model

However, there is not a means of variable selection in ridge regression

Ideally we want to be able to reduce the variability in a model but also be able to select which variables are most strongly associated with our outcome

In ridge regression, the new function is

Consider instead the estimator which minimizes

The only change is to the penalty function and while the change is subtle, is has a big impact on our regression estimator

The name lasso stands for “Least Absolute Shrinkage and Selection Operator”

Like ridge regression, penalizing the absolute values of the coefficients shrinks them towards zero

But in the lasso, some coefficients are shrunk completely to zero

Solutions where multiple coefficient estimates are identically zero are called sparse

Thus the penalty performs a continuous variable selection, hence the name

Solid areas represent the constraint regions

The ellipses represent the contours of the least square error function

Because the lasso penalty has an absolute value operation, the objective function is not differentiable and therefore lacks a closed form

As a result, we must use optimization algorithms to find the minimum

Examples of these algorithms include

-Quadratic programming (limit ~100 predictors)

-Least Angle Regression/LAR (limit ~10,000 predictors)

Since lasso is not a linear estimator, we have no H matrix such that

Thus determining the degrees of freedom are more difficult to estimate

One means is to estimate the degrees of freedom based on the number of non-zero parameters in the model and then use AIC, BIC or Cp to select the best l

Alternatively (and often more preferred) we could select l via cross-validation

Alternative method for variable subset selection designed to handle correlated predictors

Iterative process that begins with all coefficients equal to zero and build regression function in successive small steps

Similar algorithm to forward selection in that predictors added successively to the model

However, it is much more cautious than forward stepwise model selection

-e.g. for a model with 10 possible predictors stepwise takes 10 steps at most, stagewise may take 5000+

Stagewise algorithm:

(1) Initialize model such that

(2) Find the predictor Xj1 that is most correlated with r and add it to the model (here )

(3) Update

-Note, h is a small constant controlling step-length

(4) Update

(5) Repeat steps 2 thru 4 until

Although the algorithms look entirely different, their results are very similar!

They will trace very similar paths for addition of predictors to the model

They both represent special cases of a method called least angle regression (LAR)

LAR algorithm:

(1) Initialize model such that

Also initialize an empty “active set” A (subset of indices)

(2) Find the predictor that is most correlated with r where

; update the active set to include

(3) Move toward until some other covariate has the same correlation with r that does. Update the active set to include

(4) Update rand move along towards the joint OLS direction for the regression of r on until a third covariate

is as correlated with r as the first two predictors.Update the active set to include

(5) Continue until all k covariates have been added to the model

Consider a case where we have 2 predictors…

Efron et al. 2004

LAR is a more general method than lasso

A modification of the LAR algorithm produces the entire lasso path for l varied from 0 to infinity

Modification occurs if a previously non-zero coefficient estimated to be zero at some point in the algorithm

If this occurs, the LAR algorithm is modified such that the coefficient is removed from the active set and the joint direction is recomputed

This modification is the most frequently implements version of LAR

LAR is also a more general method than stagewise selection

Can also reproduce stagewise results using modified LAR

Start with the LAR algorithm and determine the best direction at each stage

If the direction for any predictor in the active set doesn’t agree in sign with the correlation between r and Xj, adjust to move in the direction of corr(r, Xj)

As step sizes go to 0, we get a modified version of the LAR algorithm

- LARS
- Uses least square directions in the active set of variables

- Lasso
- Uses the least square directions
- If the variable crosses 0, it is removed from the active set

- Forward stagewise
- Uses non-negative least squares directions in the active set

Consider fitting a LAR model with k < pparameters

Equivalently use a lasso bound t that constrains the full regression fit

General definition for the effective degrees of freedom (edf) for an adaptively fit model:

For LARS at the kth step, the edf for the fit vector is exactly k

For lasso, at any stage in the fit the effective degrees of freedom is approximately the number of predictors in the model

What if we consider lasso, forward stagewise, or LAR as alternatives?

There are 2 packages in R that will allow us to do this

-lars

-glmnet

The lars package has the advantage of being able to fit all three model types (plus a typical forward stepwise selection algorithm)

However, the glmnet package can fit lasso regression models for different types of regression

-linear, logistic, cox-proportional hazards, multinomial, and poisson

Recall our regression model

> summary(mod13)

Call:

lm(formula = PBF ~ Age + Wt + Ht + Neck + Chest + Abd + Hip + Thigh + Knee + Ankle + Bicep + Arm + Wrist, data = bodyfat, x = T)

Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) -18.18849 17.34857 -1.048 0.29551

Age 0.06208 0.03235 1.919 0.05618 .

Wt -0.08844 0.05353 -1.652 0.09978 .

Ht -0.06959 0.09601 -0.725 0.46925

Neck -0.47060 0.23247 -2.024 0.04405 *

Chest -0.02386 0.09915 -0.241 0.81000

Abd 0.95477 0.08645 11.04 < 2e-16 ***

Hip -0.20754 0.14591 -1.422 0.15622

Thigh 0.23610 0.14436 1.636 0.10326

Knee 0.01528 0.24198 0.063 0.94970

Ankle 0.17400 0.22147 0.786 0.43285

Bicep 0.18160 0.17113 1.061 0.28966

Arm 0.45202 0.19913 2.270 0.02410 *

Wrist -1.62064 0.53495 -3.030 0.00272 **

Residual standard error: 4.305 on 238 degrees of freedom. Multiple R-squared: 0.749,

Adjusted R-squared: 0.7353 . F-statistic: 54.65 on 13 and 238 DF, p-value: < 2.2e-16

LAR:

>library(lars)

>par(mfrow=c(2,2))

>object <- lars(x=as.matrix(bodyfat[,3:15]),y=as.vector(bodyfat[,2]), type="lasso")

>plot(object, breaks=F)

>object2 <- lars(x=as.matrix(bodyfat[,3:15]),y=as.vector(bodyfat[,2]), type="lar")

>plot(object2, breaks=F)

>object3 <- lars(x=as.matrix(bodyfat[,3:15]),y=as.vector(bodyfat[,2]), type=“for")

>plot(object3, breaks=F)

>object4 <- lars(x=as.matrix(bodyfat[,3:15]),y=as.vector(bodyfat[,2]), type=“stepwise")

>plot(object4, breaks=F)

A closer look at the model:

>object <- lars(x=as.matrix(bodyfat[,3:15]),y=as.vector(bodyfat[,2]), type="lasso")

> names(object)

[1] "call" "type" "df" "lambda" "R2" "RSS" "Cp" "actions"

[9] "entry" "Gamrat" "arc.length" "Gram" "beta" "mu" "normx" "meanx"

> object$beta

Age Wt Ht Neck Chest Abd Hip Thigh Knee Ankle

0 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.0000000 0.000000000 0.00000000 0.00000000 0.0000000

1 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.5164924 0.000000000 0.00000000 0.00000000 0.0000000

2 0.00000000 0.00000000 -0.04395065 0.00000000 0.00000000 0.5314218 0.000000000 0.00000000 0.00000000 0.0000000

3 0.01710504 0.00000000 -0.13752803 0.00000000 0.00000000 0.5621288 0.000000000 0.00000000 0.00000000 0.0000000

4 0.04880181 0.00000000 -0.15894236 0.00000000 0.00000000 0.6550929 0.000000000 0.00000000 0.00000000 0.0000000

5 0.04994577 0.00000000 -0.15905246 -0.02624509 0.00000000 0.6626603 0.000000000 0.00000000 0.00000000 0.0000000

6 0.06499276 0.00000000 -0.15911969 -0.25799496 0.00000000 0.7079872 0.000000000 0.00000000 0.00000000 0.0000000

7 0.06467180 0.00000000 -0.15921694 -0.26404701 0.00000000 0.7118167 -0.004720494 0.00000000 0.00000000 0.0000000

8 0.06022586 -0.01117359 -0.14998300 -0.29599536 0.00000000 0.7527298 -0.022557736 0.00000000 0.00000000 0.0000000

9 0.05710956 -0.02219531 -0.14039586 -0.32675736 0.00000000 0.7842966 -0.035675017 0.00000000 0.00000000 0.0000000

10 0.05853733 -0.04577935 -0.11203059 -0.39386199 0.00000000 0.8425758 -0.101022340 0.09657784 0.00000000 0.0000000

11 0.06132775 -0.07889636 -0.07798153 -0.45141574 0.00000000 0.9142944 -0.171178163 0.20141924 0.00000000 0.1259630

12 0.06214695 -0.08452690 -0.07220347 -0.46528070 -0.01582661 0.9402896 -0.194491760 0.22553958 0.00000000 0.1586161

13 0.06207865 -0.08844468 -0.06959043 -0.47060001 -0.02386415 0.9547735 -0.207541123 0.23609984 0.01528121 0.1739954

Bicep Arm Wrist

0 0.00000000 0.0000000 0.000000

1 0.00000000 0.0000000 0.000000

2 0.00000000 0.0000000 0.000000

3 0.00000000 0.0000000 0.000000

4 0.00000000 0.0000000 -1.169755

5 0.00000000 0.0000000 -1.198047

6 0.00000000 0.2175660 -1.535349

7 0.00000000 0.2236663 -1.538953

8 0.00000000 0.2834326 -1.535810

9 0.04157133 0.3117864 -1.534938

10 0.09096070 0.3635421 -1.522325

11 0.15173471 0.4229317 -1.587661

12 0.17055965 0.4425212 -1.607395

13 0.18160242 0.4520249 -1.620639

A closer look at the model:

> names(object)

[1] "call" "type" "df" "lambda" "R2" "RSS" "Cp" "actions"

[9] "entry" "Gamrat" "arc.length" "Gram" "beta" "mu" "normx" "meanx"

> object$df

Intercept

1 2 3 4 5 6 7 8 9 10 11

12 13 14

> object$Cp

0 1 2 3 4 5 6 7 8 9 10

698.4 93.62 85.47 65.41 30.12 30.51 19.39 20.91 18.68 17.41 12.76

11 12 13

10.47 12.06 14.00

Glmnet:

>fit<-glmnet(x=as.matrix(bodyfat[,3:15]),y=as.vector(bodyfat[,2]), alpha=1)

>fit.cv<-cv.glmnet(x=as.matrix(bodyfat[,3:15]), y=as.vector(bodyfat[,2]), alpha=1)

>plot(fit.cv, sign.lambda=-1)

>fit<-glmnet(x=as.matrix(bodyfat[,3:15]),y=as.vector(bodyfat[,2]), alpha=1, 0.02123575)

Glmnet:

>fit<-glmnet(x=as.matrix(bodyfat[,3:15]),y=as.vector(bodyfat[,2]), alpha=1)

>names(fit)

[1] "a0" "beta" "df" "dim" "lambda" "dev.ratio" "nulldev" "npasses" "jerr"

[10] "offset" "call" "nobs"

> fit$lambda

[1] 6.793883455 6.190333574 5.640401401 5.139323686 4.682760334 4.266756812 3.887709897 3.542336464 3.227645056

[10] 2.940909965 2.679647629 2.441595119 2.224690538 2.027055162 1.846977168 1.682896807 1.533392893 1.397170495

[19] 1.273049719 1.159955490 1.056908242 0.963015426 0.877463790 0.799512325 0.728485854 0.663769178 0.604801754

[28] 0.551072833 0.502117041 0.457510347 0.416866389 0.379833128 0.346089800 0.315344136 0.287329832 0.261804242

[37] 0.238546274 0.217354481 0.198045308 0.180451508 0.164420694 0.149814013 0.136504949 0.124378225 0.113328806

[46] 0.103260988 0.094087566 0.085729086 0.078113150 0.071173793 0.064850910 0.059089734 0.053840365 0.049057335

[55] 0.044699216 0.040728261 0.037110075 0.033813318 0.030809436 0.028072411 0.025578535 0.023306209 0.021235749

[64] 0.019349224 0.017630292 0.016064066 0.014636978 0.013336669 0.012151876 0.011072337 0.010088701 0.009192449

[73] 0.008375817 0.007631733 0.006953750 0.006335998 0.005773126 0.005260257

Glmnet:

>fit.cv<-cv.glmnet(x=as.matrix(bodyfat[,3:15]), y=as.vector(bodyfat[,2]), alpha=1)

> names(fit.cv)

[1] "lambda" "cvm" "cvsd" "cvup" "cvlo" "nzero" "name" "glmnet.fit"

[9] "lambda.min" "lambda.1se"

> fit.cv$lambda.min

[1] 0.02123575

Ridge

Lasso

LARS

Stagewise

Stepwise

What can we do in SAS?

SAS can also do cross-validation

However, it only fits linear regression

Here’s the basic SAS code

ods graphics on;

proc glmselect data=bf plots=all;

model pbf=age wt ht neck chest abd hip thigh knee ankle bicep arm wrist/selection=lasso(stop=none choose=AIC);

run;

ods graphics off;

The GLMSELECT Procedure

LASSO Selection Summary

Effect Effect Number

Step Entered Removed Effects In AIC

0 Intercept1 11325.7477

-----------------------------------------------------------------------------------------------

Abd2 21070.4404

Ht 3 31064.8357

Age4 41049.4793

Wrist5 51019.1226

Neck6 61019.6222

Arm7 71009.0982

Hip8 81010.6285

Wt9 91008.4396

Bicep10 101007.1631

Thigh11 111002.3524

Ankle12 12999.8569*

Chest13 131001.4229

13 Knee14 141003.3574

Penalized regression methods are most useful when

-high collinearity exists

-when p >> n

Keep in mind you still need to look at the data first

Could also consider other forms of penalized regression, though in practice alternatives are not used