Assignment 6
ikq392
2022-11-06
Problem 6.
In this exercise, you will further analyze the Wage data set considered throughout this chapter.
(a) Perform polynomial regression to predict wage using age. Use cross-validation to select the optimal degree d for the polynomial. What degree was chosen, and how does this compare to the results of hypothesis testing using ANOVA? Make a plot of the resulting polynomial fit to the data.
library(ISLR2)
library(boot)
set.seed(1)
degree <- 10
cv.errs <- rep(NA, degree)
for (i in 1:degree) {
fit <- glm(wage ~ poly(age, i), data = Wage)
cv.errs[i] <- cv.glm(Wage, fit)$delta[1]
}
plot(1:degree, cv.errs, xlab = 'Degree', ylab = 'Test MSE', type = 'l')
deg.min <- which.min(cv.errs)
points(deg.min, cv.errs[deg.min], col = 'red', cex = 2, pch = 19)
The minimum test MSE is at the 9th degree. The ANOVA test suggested degree 4 (section 7.8.1).
plot(wage ~ age, data = Wage, col = "darkgrey")
age.range <- range(Wage$age)
age.grid <- seq(from = age.range[1], to = age.range[2])
fit <- lm(wage ~ poly(age, 3), data = Wage)
preds <- predict(fit, newdata = list(age = age.grid))
lines(age.grid, preds, col = "red", lwd = 2)
(b) Fit a step function to predict wage using age, and perform crossvalidation to choose the optimal number of cuts. Make a plot of the fit obtained.
plot(wage ~ age, data = Wage, col = "darkgrey")
age.range <- range(Wage$age)
age.grid <- seq(from = age.range[1], to = age.range[2])
fit <- lm(wage ~ poly(age, 3), data = Wage)
preds <- predict(fit, newdata = list(age = age.grid))
lines(age.grid, preds, col = "red", lwd = 2)
The optimal number of cuts is 8.
plot(wage ~ age, data = Wage, col = "darkgrey")
fit <- glm(wage ~ cut(age, 8), data = Wage)
preds <- predict(fit, list(age = age.grid))
lines(age.grid, preds, col = "red", lwd = 2)
Problem 10.
This question relates to the College data set.
(a) Split the data into a training set and a test set. Using out-of-state tuition as the response and the other variables as the predictors, perform forward stepwise selection on the training set in order to identify a satisfactory model that uses just a subset of the predictors.
library(leaps)
train <- sample(1: nrow(College), nrow(College)/2)
test <- -train
fit <- regsubsets(Outstate ~ ., data = College, subset = train, method = 'forward')
fit.summary <- summary(fit)
fit.summary## Subset selection object
## Call: regsubsets.formula(Outstate ~ ., data = College, subset = train,
## method = "forward")
## 17 Variables (and intercept)
## Forced in Forced out
## PrivateYes FALSE FALSE
## Apps FALSE FALSE
## Accept FALSE FALSE
## Enroll FALSE FALSE
## Top10perc FALSE FALSE
## Top25perc FALSE FALSE
## F.Undergrad FALSE FALSE
## P.Undergrad FALSE FALSE
## Room.Board FALSE FALSE
## Books FALSE FALSE
## Personal FALSE FALSE
## PhD FALSE FALSE
## Terminal FALSE FALSE
## S.F.Ratio FALSE FALSE
## perc.alumni FALSE FALSE
## Expend FALSE FALSE
## Grad.Rate FALSE FALSE
## 1 subsets of each size up to 8
## Selection Algorithm: forward
## PrivateYes Apps Accept Enroll Top10perc Top25perc F.Undergrad
## 1 ( 1 ) " " " " " " " " " " " " " "
## 2 ( 1 ) "*" " " " " " " " " " " " "
## 3 ( 1 ) "*" " " " " " " " " " " " "
## 4 ( 1 ) "*" " " " " " " " " " " " "
## 5 ( 1 ) "*" " " " " " " " " " " " "
## 6 ( 1 ) "*" " " " " " " " " " " " "
## 7 ( 1 ) "*" " " "*" " " " " " " " "
## 8 ( 1 ) "*" " " "*" "*" " " " " " "
## P.Undergrad Room.Board Books Personal PhD Terminal S.F.Ratio
## 1 ( 1 ) " " " " " " " " " " " " " "
## 2 ( 1 ) " " " " " " " " " " " " " "
## 3 ( 1 ) " " "*" " " " " " " " " " "
## 4 ( 1 ) " " "*" " " " " " " " " " "
## 5 ( 1 ) " " "*" " " " " " " " " " "
## 6 ( 1 ) " " "*" " " " " " " "*" " "
## 7 ( 1 ) " " "*" " " " " " " "*" " "
## 8 ( 1 ) " " "*" " " " " " " "*" " "
## perc.alumni Expend Grad.Rate
## 1 ( 1 ) " " "*" " "
## 2 ( 1 ) " " "*" " "
## 3 ( 1 ) " " "*" " "
## 4 ( 1 ) "*" "*" " "
## 5 ( 1 ) "*" "*" "*"
## 6 ( 1 ) "*" "*" "*"
## 7 ( 1 ) "*" "*" "*"
## 8 ( 1 ) "*" "*" "*"coef(fit, id = 6)## (Intercept) PrivateYes Room.Board Terminal perc.alumni
## -3687.2054828 2626.9333231 0.9736420 31.3139389 47.9371829
## Expend Grad.Rate
## 0.2491109 30.3780593(b) Fit a GAM on the training data, using out-of-state tuition as the response and the features selected in the previous step as the predictors. Plot the results, and explain your findings.
library(gam)
gam.mod <- gam(Outstate ~ Private + s(Room.Board, 5) + s(Terminal, 5) + s(perc.alumni, 5) + s(Expend, 5) + s(Grad.Rate, 5), data = College, subset = train)
par(mfrow = c(2,3))
plot(gam.mod, se = TRUE, col = 'blue')
The fit curves of Expend and Grad.Rate are strong non-linear with Outstate.
(c) Evaluate the model obtained on the test set, and explain the results obtained.
preds <- predict(gam.mod, College[test, ])
RSS <- sum((College[test, ]$Outstate - preds)^2)
TSS <- sum((College[test, ]$Outstate - mean(College[test, ]$Outstate)) ^ 2)
1 - (RSS / TSS)## [1] 0.7900434The r-squared statistic is 0.79.
(d) For which variables, if any, is there evidence of a non-linear relationship with the response?
summary(gam.mod)##
## Call: gam(formula = Outstate ~ Private + s(Room.Board, 5) + s(Terminal,
## 5) + s(perc.alumni, 5) + s(Expend, 5) + s(Grad.Rate, 5),
## data = College, subset = train)
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -6542.859 -1026.898 3.791 1199.213 6862.706
##
## (Dispersion Parameter for gaussian family taken to be 3497059)
##
## Null Deviance: 6077526933 on 387 degrees of freedom
## Residual Deviance: 1262438051 on 360.9999 degrees of freedom
## AIC: 6975.275
##
## Number of Local Scoring Iterations: NA
##
## Anova for Parametric Effects
## Df Sum Sq Mean Sq F value Pr(>F)
## Private 1 1487299914 1487299914 425.300 < 2.2e-16 ***
## s(Room.Board, 5) 1 1127309685 1127309685 322.359 < 2.2e-16 ***
## s(Terminal, 5) 1 319183857 319183857 91.272 < 2.2e-16 ***
## s(perc.alumni, 5) 1 266615907 266615907 76.240 < 2.2e-16 ***
## s(Expend, 5) 1 684447052 684447052 195.721 < 2.2e-16 ***
## s(Grad.Rate, 5) 1 60429296 60429296 17.280 4.033e-05 ***
## Residuals 361 1262438051 3497059
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Anova for Nonparametric Effects
## Npar Df Npar F Pr(F)
## (Intercept)
## Private
## s(Room.Board, 5) 4 1.7686 0.1345
## s(Terminal, 5) 4 1.8852 0.1124
## s(perc.alumni, 5) 4 1.2801 0.2773
## s(Expend, 5) 4 21.1148 1.11e-15 ***
## s(Grad.Rate, 5) 4 1.1452 0.3350
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1The data under the “Anova for Nonparametric Effects” section show that Expend has a strong non-linear relationship with Outstate.