HW 5
Michal Greenwood
March 27, 2019
Q2 For parts (a) through (c), indicate which of i. through iv. is correct. Justify your answer.
(a) The lasso, relative to least squares, is: iii. Less flexible and hence will give improved prediction accuracy when its increase in bias is less than its decrease in variance.
- is less flexible and puts a constraint on the least squares. (b) Repeat (a) for ridge regression relative to least squares.
- is also accurate the ridge puts a budget constraint on the least squares.
(c) Repeat (a) for non-linear methods relative to least squares.
For more flexible, less bias and higher variance a non-linear model would be best.
Q9. In this exercise, we will predict the number of applications received using the other variables in the College data set.
(a) Split the data set into a training set and a test set.
library(ISLR)## Warning: package 'ISLR' was built under R version 3.5.2data(College)
set.seed(1)
sum(is.na(College))## [1] 0train.size = dim(College)[1] / 2
train = sample(1:dim(College)[1], train.size)
test = -train
College.train = College[train, ]
College.test = College[test, ](b) Fit a linear model using least squares on the training set, and report the test error obtained.
lm.fit = lm(Apps~., data=College.train)
lm.pred = predict(lm.fit, College.test)
mean((College.test[, "Apps"] - lm.pred)^2)## [1] 1108531(c) Fit a ridge regression model on the training set, with ?? chosen by cross-validation. Report the test error obtained.
library(glmnet)## Warning: package 'glmnet' was built under R version 3.5.3## Loading required package: Matrix## Loading required package: foreach## Warning: package 'foreach' was built under R version 3.5.3## Loaded glmnet 2.0-16train.mat = model.matrix(Apps~., data=College.train)
test.mat = model.matrix(Apps~., data=College.test)
grid = 10 ^ seq(4, -2, length=100)
mod.ridge = cv.glmnet(train.mat, College.train[, "Apps"], alpha=0, lambda=grid, thresh=1e-12)
lambda.best = mod.ridge$lambda.min
lambda.best## [1] 0.01149757ridge.pred = predict(mod.ridge, newx=test.mat, s=lambda.best)
mean((College.test[, "Apps"] - ridge.pred)^2)## [1] 1108512(d) Fit a lasso model on the training set, with ?? chosen by crossvalidation. Report the test error obtained, along with the number of non-zero coefficient estimates.
mod.lasso = cv.glmnet(train.mat, College.train[, "Apps"], alpha=1, lambda=grid, thresh=1e-12)
lambda.best = mod.lasso$lambda.min
lambda.best## [1] 28.48036lasso.pred = predict(mod.lasso, newx=test.mat, s=lambda.best)
mean((College.test[, "Apps"] - lasso.pred)^2)## [1] 1028718mod.lasso = glmnet(model.matrix(Apps~., data=College), College[, "Apps"], alpha=1)
predict(mod.lasso, s=lambda.best, type="coefficients")## 19 x 1 sparse Matrix of class "dgCMatrix"
## 1
## (Intercept) -6.491800e+02
## (Intercept) .
## PrivateYes -3.910803e+02
## Accept 1.414561e+00
## Enroll -6.880133e-02
## Top10perc 2.975286e+01
## Top25perc -5.898729e-03
## F.Undergrad .
## P.Undergrad 8.118367e-03
## Outstate -4.802638e-02
## Room.Board 1.154551e-01
## Books .
## Personal .
## PhD -4.573050e+00
## Terminal -3.263399e+00
## S.F.Ratio 5.152699e-01
## perc.alumni -1.066814e+00
## Expend 6.615233e-02
## Grad.Rate 4.204566e+00(e) Fit a PCR model on the training set, with M chosen by crossvalidation. Report the test error obtained, along with the value of M selected by cross-validation.
library(pls)## Warning: package 'pls' was built under R version 3.5.3##
## Attaching package: 'pls'## The following object is masked from 'package:stats':
##
## loadingspcr.fit = pcr(Apps~., data=College.train, scale=T, validation="CV")
validationplot(pcr.fit, val.type="MSEP")
pcr.pred = predict(pcr.fit, College.test, ncomp=10)
mean((College.test[, "Apps"] - data.frame(pcr.pred))^2)## Warning in mean.default((College.test[, "Apps"] - data.frame(pcr.pred))^2):
## argument is not numeric or logical: returning NA## [1] NA(f) Fit a PLS model on the training set, with M chosen by crossvalidation. Report the test error obtained, along with the value of M selected by cross-validation.
pls.fit = plsr(Apps~., data=College.train, scale=T, validation="CV")
validationplot(pls.fit, val.type="MSEP")
pls.pred = predict(pls.fit, College.test, ncomp=10)
mean((College.test[, "Apps"] - data.frame(pls.pred))^2)## Warning in mean.default((College.test[, "Apps"] - data.frame(pls.pred))^2):
## argument is not numeric or logical: returning NA## [1] NA(g) Comment on the results obtained. How accurately can we predict the number of college applications received? Is there much difference among the test errors resulting from these five approaches?
test.avg = mean(College.test[, "Apps"])
lm.test.r2 = 1 - mean((College.test[, "Apps"] - lm.pred)^2) /mean((College.test[, "Apps"] - test.avg)^2)
ridge.test.r2 = 1 - mean((College.test[, "Apps"] - ridge.pred)^2) /mean((College.test[, "Apps"] - test.avg)^2)
lasso.test.r2 = 1 - mean((College.test[, "Apps"] - lasso.pred)^2) /mean((College.test[, "Apps"] - test.avg)^2)
pcr.test.r2 = 1 - mean((College.test[, "Apps"] - data.frame(pcr.pred))^2) /mean((College.test[, "Apps"] - test.avg)^2)## Warning in mean.default((College.test[, "Apps"] - data.frame(pcr.pred))^2):
## argument is not numeric or logical: returning NApls.test.r2 = 1 - mean((College.test[, "Apps"] - data.frame(pls.pred))^2) /mean((College.test[, "Apps"] - test.avg)^2)## Warning in mean.default((College.test[, "Apps"] - data.frame(pls.pred))^2):
## argument is not numeric or logical: returning NAbarplot(c(lm.test.r2, ridge.test.r2, lasso.test.r2, pcr.test.r2, pls.test.r2), col="red", names.arg=c("OLS", "Ridge", "Lasso", "PCR", "PLS"), main="R-squared")
Q11 We will now try to predict per capita crime rate in the Boston data set.
(a) Try out some of the regression methods explored in this chapter, such as best subset selection, the lasso, ridge regression, and PCR. Present and discuss results for the approaches that you consider.
set.seed(1)
library(glmnet)
library(MASS)
library(leaps)## Warning: package 'leaps' was built under R version 3.5.3data(Boston)trainid <- sample(1:nrow(Boston), nrow(Boston)/2)
train <- Boston[trainid,]
test <- Boston[-trainid,]
xmat.train <- model.matrix(crim~., data=train)[,-1]
xmat.test <- model.matrix(crim~., data=test)[,-1]
str(Boston)## 'data.frame': 506 obs. of 14 variables:
## $ crim : num 0.00632 0.02731 0.02729 0.03237 0.06905 ...
## $ zn : num 18 0 0 0 0 0 12.5 12.5 12.5 12.5 ...
## $ indus : num 2.31 7.07 7.07 2.18 2.18 2.18 7.87 7.87 7.87 7.87 ...
## $ chas : int 0 0 0 0 0 0 0 0 0 0 ...
## $ nox : num 0.538 0.469 0.469 0.458 0.458 0.458 0.524 0.524 0.524 0.524 ...
## $ rm : num 6.58 6.42 7.18 7 7.15 ...
## $ age : num 65.2 78.9 61.1 45.8 54.2 58.7 66.6 96.1 100 85.9 ...
## $ dis : num 4.09 4.97 4.97 6.06 6.06 ...
## $ rad : int 1 2 2 3 3 3 5 5 5 5 ...
## $ tax : num 296 242 242 222 222 222 311 311 311 311 ...
## $ ptratio: num 15.3 17.8 17.8 18.7 18.7 18.7 15.2 15.2 15.2 15.2 ...
## $ black : num 397 397 393 395 397 ...
## $ lstat : num 4.98 9.14 4.03 2.94 5.33 ...
## $ medv : num 24 21.6 34.7 33.4 36.2 28.7 22.9 27.1 16.5 18.9 ...fit.ridge <- cv.glmnet(xmat.train, train$crim, alpha=0)
(lambda <- fit.ridge$lambda.min) ## [1] 0.5982585pred.ridge <- predict(fit.ridge, s=lambda, newx=xmat.test)
(err.ridge <- mean((test$crim - pred.ridge)^2)) ## [1] 38.36353predict(fit.ridge, s=lambda, type="coefficients")## 14 x 1 sparse Matrix of class "dgCMatrix"
## 1
## (Intercept) 4.429222237
## zn 0.036521710
## indus -0.061214283
## chas -0.775621731
## nox -8.045252028
## rm 1.179585528
## age 0.005683303
## dis -0.884805548
## rad 0.429755275
## tax 0.003261068
## ptratio -0.169564879
## black -0.004036776
## lstat 0.193028502
## medv -0.171441089fit.lasso <- cv.glmnet(xmat.train, train$crim, alpha=1)
(lambda <- fit.lasso$lambda.min) # optimal lambda## [1] 0.2530181pred.lasso <- predict(fit.lasso, s=lambda, newx=xmat.test)
(err.lasso <- mean((test$crim - pred.lasso)^2)) # test error## [1] 38.4593predict(fit.lasso, s=lambda, type="coefficients")## 14 x 1 sparse Matrix of class "dgCMatrix"
## 1
## (Intercept) -1.803083089
## zn 0.017767800
## indus .
## chas -0.266407050
## nox .
## rm 0.510067271
## age .
## dis -0.377582056
## rad 0.457627436
## tax .
## ptratio .
## black -0.002413684
## lstat 0.159515883
## medv -0.093917991predict.regsubsets <- function(object, newdata, id, ...){
form <- as.formula(object$call[[2]])
mat <- model.matrix(form, newdata)
coefi <- coef(object, id=id)
xvars <- names(coefi)
mat[,xvars]%*%coefi
}fit.fwd <- regsubsets(crim~., data=train, nvmax=ncol(Boston)-1)
(fwd.summary <- summary(fit.fwd))## Subset selection object
## Call: regsubsets.formula(crim ~ ., data = train, nvmax = ncol(Boston) -
## 1)
## 13 Variables (and intercept)
## Forced in Forced out
## zn FALSE FALSE
## indus FALSE FALSE
## chas FALSE FALSE
## nox FALSE FALSE
## rm FALSE FALSE
## age FALSE FALSE
## dis FALSE FALSE
## rad FALSE FALSE
## tax FALSE FALSE
## ptratio FALSE FALSE
## black FALSE FALSE
## lstat FALSE FALSE
## medv FALSE FALSE
## 1 subsets of each size up to 13
## Selection Algorithm: exhaustive
## zn indus chas nox rm age dis rad tax ptratio black lstat medv
## 1 ( 1 ) " " " " " " " " " " " " " " "*" " " " " " " " " " "
## 2 ( 1 ) " " " " " " " " " " " " " " "*" " " " " " " "*" " "
## 3 ( 1 ) " " " " " " " " "*" " " " " "*" " " " " " " "*" " "
## 4 ( 1 ) "*" " " " " " " " " " " "*" "*" " " " " " " " " "*"
## 5 ( 1 ) "*" " " " " "*" " " " " "*" "*" " " " " " " " " "*"
## 6 ( 1 ) "*" " " " " "*" "*" " " "*" "*" " " " " " " " " "*"
## 7 ( 1 ) "*" " " " " "*" "*" " " "*" "*" " " " " " " "*" "*"
## 8 ( 1 ) "*" " " " " "*" "*" " " "*" "*" " " "*" " " "*" "*"
## 9 ( 1 ) "*" " " " " "*" "*" " " "*" "*" "*" "*" " " "*" "*"
## 10 ( 1 ) "*" " " "*" "*" "*" " " "*" "*" "*" "*" " " "*" "*"
## 11 ( 1 ) "*" " " "*" "*" "*" " " "*" "*" "*" "*" "*" "*" "*"
## 12 ( 1 ) "*" " " "*" "*" "*" "*" "*" "*" "*" "*" "*" "*" "*"
## 13 ( 1 ) "*" "*" "*" "*" "*" "*" "*" "*" "*" "*" "*" "*" "*"err.fwd <- rep(NA, ncol(Boston)-1)
for(i in 1:(ncol(Boston)-1)) {
pred.fwd <- predict(fit.fwd, test, id=i)
err.fwd[i] <- mean((test$crim - pred.fwd)^2)
}
plot(err.fwd, type="b", main="Test MSE for Forward Selection", xlab="Number of Predictors")
which.min(err.fwd)## [1] 4fit.bwd <- regsubsets(crim~., data=train, nvmax=ncol(Boston)-1)
(bwd.summary <- summary(fit.bwd))## Subset selection object
## Call: regsubsets.formula(crim ~ ., data = train, nvmax = ncol(Boston) -
## 1)
## 13 Variables (and intercept)
## Forced in Forced out
## zn FALSE FALSE
## indus FALSE FALSE
## chas FALSE FALSE
## nox FALSE FALSE
## rm FALSE FALSE
## age FALSE FALSE
## dis FALSE FALSE
## rad FALSE FALSE
## tax FALSE FALSE
## ptratio FALSE FALSE
## black FALSE FALSE
## lstat FALSE FALSE
## medv FALSE FALSE
## 1 subsets of each size up to 13
## Selection Algorithm: exhaustive
## zn indus chas nox rm age dis rad tax ptratio black lstat medv
## 1 ( 1 ) " " " " " " " " " " " " " " "*" " " " " " " " " " "
## 2 ( 1 ) " " " " " " " " " " " " " " "*" " " " " " " "*" " "
## 3 ( 1 ) " " " " " " " " "*" " " " " "*" " " " " " " "*" " "
## 4 ( 1 ) "*" " " " " " " " " " " "*" "*" " " " " " " " " "*"
## 5 ( 1 ) "*" " " " " "*" " " " " "*" "*" " " " " " " " " "*"
## 6 ( 1 ) "*" " " " " "*" "*" " " "*" "*" " " " " " " " " "*"
## 7 ( 1 ) "*" " " " " "*" "*" " " "*" "*" " " " " " " "*" "*"
## 8 ( 1 ) "*" " " " " "*" "*" " " "*" "*" " " "*" " " "*" "*"
## 9 ( 1 ) "*" " " " " "*" "*" " " "*" "*" "*" "*" " " "*" "*"
## 10 ( 1 ) "*" " " "*" "*" "*" " " "*" "*" "*" "*" " " "*" "*"
## 11 ( 1 ) "*" " " "*" "*" "*" " " "*" "*" "*" "*" "*" "*" "*"
## 12 ( 1 ) "*" " " "*" "*" "*" "*" "*" "*" "*" "*" "*" "*" "*"
## 13 ( 1 ) "*" "*" "*" "*" "*" "*" "*" "*" "*" "*" "*" "*" "*"err.bwd <- rep(NA, ncol(Boston)-1)
for(i in 1:(ncol(Boston)-1)) {
pred.bwd <- predict(fit.bwd, test, id=i)
err.bwd[i] <- mean((test$crim - pred.bwd)^2)
}
plot(err.bwd, type="b", main="Test MSE for Backward Selection", xlab="Number of Predictors")
which.min(err.bwd)## [1] 4par(mfrow=c(3,2))min.cp <- which.min(fwd.summary$cp)
plot(fwd.summary$cp, xlab="Number of Poly(X)", ylab="Forward Selection Cp", type="l")
points(min.cp, fwd.summary$cp[min.cp], col="red", pch=19, lwd=5)
min.cp <- which.min(bwd.summary$cp)
plot(bwd.summary$cp, xlab="Number of Poly(X)", ylab="Backward Selection Cp", type="l")
points(min.cp, bwd.summary$cp[min.cp], col="red", pch=19, lwd=5)
min.bic <- which.min(fwd.summary$bic)
plot(fwd.summary$bic, xlab="Number of Poly(X)", ylab="Forward Selection BIC", type="l")
points(min.bic, fwd.summary$bic[min.bic], col="red", pch=4, lwd=5)
min.bic <- which.min(bwd.summary$bic)
plot(bwd.summary$bic, xlab="Number of Poly(X)", ylab="Backward Selection BIC", type="l")
points(min.bic, bwd.summary$bic[min.bic], col="red", pch=4, lwd=5)
min.adjr2 <- which.max(fwd.summary$adjr2)
plot(fwd.summary$adjr2, xlab="Number of Poly(X)", ylab="Forward Selection Adjusted R^2", type="l")
points(min.adjr2, fwd.summary$adjr2[min.adjr2], col="red", pch=4, lwd=5)
min.adjr2 <- which.max(bwd.summary$adjr2)
plot(bwd.summary$adjr2, xlab="Number of Poly(X)", ylab="Backward Selection Adjusted R^2", type="l")
points(min.adjr2, bwd.summary$adjr2[min.adjr2], col="red", pch=4, lwd=5)
(b) Propose a model (or set of models) that seem to perform well on this data set, and justify your answer. Make sure that you are evaluating model performance using validation set error, crossvalidation, or some other reasonable alternative, as opposed to using training error.
err.ridge## [1] 38.36353err.lasso## [1] 38.4593err.fwd## [1] 41.20712 39.46705 40.40271 38.96427 39.28537 40.02194 39.63462
## [8] 39.56733 39.64198 39.59028 39.27845 39.33711 39.27592err.bwd## [1] 41.20712 39.46705 40.40271 38.96427 39.28537 40.02194 39.63462
## [8] 39.56733 39.64198 39.59028 39.27845 39.33711 39.27592(c) Does your chosen model involve all of the features in the data set? Why or why not? No the model indicated above does not have all the predictors and doesn’t change the value of the model.