Problem Set 8
Ross Enlow
11/16/2019
Problem Set 8
Problem 1: Boston
install.packages("boot", repos = "https://cran.r-project.org/web/packages/boot/boot.pdf")## Warning: unable to access index for repository https://cran.r-project.org/web/packages/boot/boot.pdf/src/contrib:
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## cannot open URL 'https://cran.r-project.org/web/packages/boot/boot.pdf/bin/macosx/el-capitan/contrib/3.6/PACKAGES'library(boot)
install.packages("data.table", repos = "https://CRAN.R-project.org/package=data.table")## Warning: unable to access index for repository https://CRAN.R-project.org/package=data.table/src/contrib:
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## cannot open URL 'https://CRAN.R-project.org/package=data.table/bin/macosx/el-capitan/contrib/3.6/PACKAGES'library(data.table)
library(splines)
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library(MASS)##
## Attaching package: 'MASS'## The following object is masked from 'package:dplyr':
##
## selectattach(Boston)
Boston=na.omit(Boston)a) Use the poly() function to fit a cubic polynomial regression to predict nox using dis. Report the regression output, and plot the resulting data and polynomial fits.
mod=lm(nox ~ poly(dis, 3), data = Boston)
summary(mod)##
## Call:
## lm(formula = nox ~ poly(dis, 3), data = Boston)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.121130 -0.040619 -0.009738 0.023385 0.194904
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.554695 0.002759 201.021 < 2e-16 ***
## poly(dis, 3)1 -2.003096 0.062071 -32.271 < 2e-16 ***
## poly(dis, 3)2 0.856330 0.062071 13.796 < 2e-16 ***
## poly(dis, 3)3 -0.318049 0.062071 -5.124 4.27e-07 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.06207 on 502 degrees of freedom
## Multiple R-squared: 0.7148, Adjusted R-squared: 0.7131
## F-statistic: 419.3 on 3 and 502 DF, p-value: < 2.2e-16dislims <- range(Boston$dis)
dis.grid <- seq(from = dislims[1], to = dislims[2], by = 0.1)
preds <- predict(mod, list(dis = dis.grid))
plot(nox ~ dis, data = Boston, col = "darkgrey")
lines(dis.grid, preds, col = "red", lwd = 2)
We may conclude that all polynomial terms are significant.
b) Plot the polynomial fits for a range of different polynomial degrees (say, from 1 to 10), and report the associated residual sum of squares.
rss <- rep(NA, 10)
for (i in 1:10) {
fit <- lm(nox ~ poly(dis, i), data = Boston)
rss[i] <- sum(fit$residuals^2)
}
plot(1:10, rss, xlab = "Degree", ylab = "RSS", type = "l")
It seems that the RSS decreases with the degree of the polynomial, and so is minimum for a polynomial of degree 10.
c) Perform cross-validation or another approach to select the optimal degree for the polynomial, and explain your results.
deltas <- rep(NA, 10)
for (i in 1:10) {
fit <- glm(nox ~ poly(dis, i), data = Boston)
deltas[i] <- cv.glm(Boston, fit, K = 10)$delta[1]
}
plot(1:10, deltas, xlab = "Degree", ylab = "Test MSE", type = "l")
It appears that the third degree polynomial has the least mean squared error and still reletively not complex, so this may be our best option.
d) Use the bs() function to fit a regression spline to predict nox using dis. Report the output for the fit using four degrees of freedom. How did you choose the knots? Plot the resulting fit.
library(splines)
attr(bs(dis,df=4), "knots")## 50%
## 3.20745modspln = fit <- lm(nox ~ bs(dis, knots = c(4, 7, 11)), data = Boston)
summary(fit)##
## Call:
## lm(formula = nox ~ bs(dis, knots = c(4, 7, 11)), data = Boston)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.124567 -0.040355 -0.008702 0.024740 0.192920
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.73926 0.01331 55.537 < 2e-16 ***
## bs(dis, knots = c(4, 7, 11))1 -0.08861 0.02504 -3.539 0.00044 ***
## bs(dis, knots = c(4, 7, 11))2 -0.31341 0.01680 -18.658 < 2e-16 ***
## bs(dis, knots = c(4, 7, 11))3 -0.26618 0.03147 -8.459 3.00e-16 ***
## bs(dis, knots = c(4, 7, 11))4 -0.39802 0.04647 -8.565 < 2e-16 ***
## bs(dis, knots = c(4, 7, 11))5 -0.25681 0.09001 -2.853 0.00451 **
## bs(dis, knots = c(4, 7, 11))6 -0.32926 0.06327 -5.204 2.85e-07 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.06185 on 499 degrees of freedom
## Multiple R-squared: 0.7185, Adjusted R-squared: 0.7151
## F-statistic: 212.3 on 6 and 499 DF, p-value: < 2.2e-16pred <- predict(fit, list(dis = dis.grid))
plot(nox ~ dis, data = Boston, col = "darkgrey")
lines(dis.grid, preds, col = "red", lwd = 2)
We see that the range of dis is from ca 1 to 13 so we split the range with knots at 4,7,10. Here we also see that all terms in spline are significant.
BONUS
e) Now fit a regression spline for a range of degrees of freedom, and plot the resulting fits and report the resulting RSS. Describe the results obtained.
rss <- rep(NA, 16)
for (i in 3:16) {
fit <- lm(nox ~ bs(dis, df = i), data = Boston)
rss[i] <- sum(fit$residuals^2)
}
plot(3:16, rss[-c(1, 2)], xlab = "Degrees of freedom", ylab = "RSS", type = "l")
We may see that RSS decreases until 14 and then slightly increases after that.
f) Perform cross-validation in order to select the best degrees of freedom for a regression spline on this data. Describe your results.
cv <- rep(NA, 16)
for (i in 3:16) {
fit <- glm(nox ~ bs(dis, df = i), data = Boston)
cv[i] <- cv.glm(Boston, fit, K = 10)$delta[1]
}## Warning in bs(dis, degree = 3L, knots = numeric(0), Boundary.knots =
## c(1.137, : some 'x' values beyond boundary knots may cause ill-conditioned
## bases
## Warning in bs(dis, degree = 3L, knots = numeric(0), Boundary.knots =
## c(1.137, : some 'x' values beyond boundary knots may cause ill-conditioned
## bases## Warning in bs(dis, degree = 3L, knots = numeric(0), Boundary.knots =
## c(1.1296, : some 'x' values beyond boundary knots may cause ill-conditioned
## bases
## Warning in bs(dis, degree = 3L, knots = numeric(0), Boundary.knots =
## c(1.1296, : some 'x' values beyond boundary knots may cause ill-conditioned
## bases## Warning in bs(dis, degree = 3L, knots = c(`50%` = 3.2157), Boundary.knots =
## c(1.1296, : some 'x' values beyond boundary knots may cause ill-conditioned
## bases
## Warning in bs(dis, degree = 3L, knots = c(`50%` = 3.2157), Boundary.knots =
## c(1.1296, : some 'x' values beyond boundary knots may cause ill-conditioned
## bases## Warning in bs(dis, degree = 3L, knots = c(`50%` = 3.2157), Boundary.knots =
## c(1.137, : some 'x' values beyond boundary knots may cause ill-conditioned
## bases
## Warning in bs(dis, degree = 3L, knots = c(`50%` = 3.2157), Boundary.knots =
## c(1.137, : some 'x' values beyond boundary knots may cause ill-conditioned
## bases## Warning in bs(dis, degree = 3L, knots = c(`33.33333%` = 2.38876666666667, :
## some 'x' values beyond boundary knots may cause ill-conditioned bases
## Warning in bs(dis, degree = 3L, knots = c(`33.33333%` = 2.38876666666667, :
## some 'x' values beyond boundary knots may cause ill-conditioned bases## Warning in bs(dis, degree = 3L, knots = c(`33.33333%` = 2.40693333333333, :
## some 'x' values beyond boundary knots may cause ill-conditioned bases
## Warning in bs(dis, degree = 3L, knots = c(`33.33333%` = 2.40693333333333, :
## some 'x' values beyond boundary knots may cause ill-conditioned bases## Warning in bs(dis, degree = 3L, knots = c(`25%` = 2.1114, `50%` = 3.2157, :
## some 'x' values beyond boundary knots may cause ill-conditioned bases
## Warning in bs(dis, degree = 3L, knots = c(`25%` = 2.1114, `50%` = 3.2157, :
## some 'x' values beyond boundary knots may cause ill-conditioned bases## Warning in bs(dis, degree = 3L, knots = c(`25%` = 2.10035, `50%` =
## 3.2721, : some 'x' values beyond boundary knots may cause ill-conditioned
## bases
## Warning in bs(dis, degree = 3L, knots = c(`25%` = 2.10035, `50%` =
## 3.2721, : some 'x' values beyond boundary knots may cause ill-conditioned
## bases## Warning in bs(dis, degree = 3L, knots = c(`20%` = 1.9512, `40%` = 2.548, :
## some 'x' values beyond boundary knots may cause ill-conditioned bases
## Warning in bs(dis, degree = 3L, knots = c(`20%` = 1.9512, `40%` = 2.548, :
## some 'x' values beyond boundary knots may cause ill-conditioned bases## Warning in bs(dis, degree = 3L, knots = c(`20%` = 1.97036, `40%` =
## 2.62694, : some 'x' values beyond boundary knots may cause ill-conditioned
## bases
## Warning in bs(dis, degree = 3L, knots = c(`20%` = 1.97036, `40%` =
## 2.62694, : some 'x' values beyond boundary knots may cause ill-conditioned
## bases## Warning in bs(dis, degree = 3L, knots = c(`16.66667%` = 1.82231666666667, :
## some 'x' values beyond boundary knots may cause ill-conditioned bases
## Warning in bs(dis, degree = 3L, knots = c(`16.66667%` = 1.82231666666667, :
## some 'x' values beyond boundary knots may cause ill-conditioned bases## Warning in bs(dis, degree = 3L, knots = c(`16.66667%` = 1.8208, `33.33333%`
## = 2.3727, : some 'x' values beyond boundary knots may cause ill-conditioned
## bases
## Warning in bs(dis, degree = 3L, knots = c(`16.66667%` = 1.8208, `33.33333%`
## = 2.3727, : some 'x' values beyond boundary knots may cause ill-conditioned
## bases## Warning in bs(dis, degree = 3L, knots = c(`14.28571%` = 1.79777142857143, :
## some 'x' values beyond boundary knots may cause ill-conditioned bases
## Warning in bs(dis, degree = 3L, knots = c(`14.28571%` = 1.79777142857143, :
## some 'x' values beyond boundary knots may cause ill-conditioned bases## Warning in bs(dis, degree = 3L, knots = c(`14.28571%` = 1.77172857142857, :
## some 'x' values beyond boundary knots may cause ill-conditioned bases
## Warning in bs(dis, degree = 3L, knots = c(`14.28571%` = 1.77172857142857, :
## some 'x' values beyond boundary knots may cause ill-conditioned bases## Warning in bs(dis, degree = 3L, knots = c(`12.5%` = 1.748425, `25%` =
## 2.10035, : some 'x' values beyond boundary knots may cause ill-conditioned
## bases
## Warning in bs(dis, degree = 3L, knots = c(`12.5%` = 1.748425, `25%` =
## 2.10035, : some 'x' values beyond boundary knots may cause ill-conditioned
## bases## Warning in bs(dis, degree = 3L, knots = c(`12.5%` = 1.751575, `25%` =
## 2.1084, : some 'x' values beyond boundary knots may cause ill-conditioned
## bases
## Warning in bs(dis, degree = 3L, knots = c(`12.5%` = 1.751575, `25%` =
## 2.1084, : some 'x' values beyond boundary knots may cause ill-conditioned
## bases## Warning in bs(dis, degree = 3L, knots = c(`11.11111%` = 1.73271111111111, :
## some 'x' values beyond boundary knots may cause ill-conditioned bases
## Warning in bs(dis, degree = 3L, knots = c(`11.11111%` = 1.73271111111111, :
## some 'x' values beyond boundary knots may cause ill-conditioned bases## Warning in bs(dis, degree = 3L, knots = c(`11.11111%` = 1.68835555555556, :
## some 'x' values beyond boundary knots may cause ill-conditioned bases
## Warning in bs(dis, degree = 3L, knots = c(`11.11111%` = 1.68835555555556, :
## some 'x' values beyond boundary knots may cause ill-conditioned bases## Warning in bs(dis, degree = 3L, knots = c(`10%` = 1.62728, `20%` =
## 1.96794, : some 'x' values beyond boundary knots may cause ill-conditioned
## bases
## Warning in bs(dis, degree = 3L, knots = c(`10%` = 1.62728, `20%` =
## 1.96794, : some 'x' values beyond boundary knots may cause ill-conditioned
## bases## Warning in bs(dis, degree = 3L, knots = c(`10%` = 1.6424, `20%` =
## 1.96794, : some 'x' values beyond boundary knots may cause ill-conditioned
## bases
## Warning in bs(dis, degree = 3L, knots = c(`10%` = 1.6424, `20%` =
## 1.96794, : some 'x' values beyond boundary knots may cause ill-conditioned
## bases## Warning in bs(dis, degree = 3L, knots = c(`9.090909%` = 1.60841818181818, :
## some 'x' values beyond boundary knots may cause ill-conditioned bases
## Warning in bs(dis, degree = 3L, knots = c(`9.090909%` = 1.60841818181818, :
## some 'x' values beyond boundary knots may cause ill-conditioned bases## Warning in bs(dis, degree = 3L, knots = c(`9.090909%` = 1.61450909090909, :
## some 'x' values beyond boundary knots may cause ill-conditioned bases
## Warning in bs(dis, degree = 3L, knots = c(`9.090909%` = 1.61450909090909, :
## some 'x' values beyond boundary knots may cause ill-conditioned bases## Warning in bs(dis, degree = 3L, knots = c(`8.333333%` = 1.5874, `16.66667%`
## = 1.83066666666667, : some 'x' values beyond boundary knots may cause ill-
## conditioned bases
## Warning in bs(dis, degree = 3L, knots = c(`8.333333%` = 1.5874, `16.66667%`
## = 1.83066666666667, : some 'x' values beyond boundary knots may cause ill-
## conditioned bases## Warning in bs(dis, degree = 3L, knots = c(`8.333333%` = 1.60476666666667, :
## some 'x' values beyond boundary knots may cause ill-conditioned bases
## Warning in bs(dis, degree = 3L, knots = c(`8.333333%` = 1.60476666666667, :
## some 'x' values beyond boundary knots may cause ill-conditioned bases## Warning in bs(dis, degree = 3L, knots = c(`7.692308%` = 1.57254615384615, :
## some 'x' values beyond boundary knots may cause ill-conditioned bases
## Warning in bs(dis, degree = 3L, knots = c(`7.692308%` = 1.57254615384615, :
## some 'x' values beyond boundary knots may cause ill-conditioned bases## Warning in bs(dis, degree = 3L, knots = c(`7.692308%` = 1.58815384615385, :
## some 'x' values beyond boundary knots may cause ill-conditioned bases
## Warning in bs(dis, degree = 3L, knots = c(`7.692308%` = 1.58815384615385, :
## some 'x' values beyond boundary knots may cause ill-conditioned bases## Warning in bs(dis, degree = 3L, knots = c(`7.142857%` = 1.56255714285714, :
## some 'x' values beyond boundary knots may cause ill-conditioned bases
## Warning in bs(dis, degree = 3L, knots = c(`7.142857%` = 1.56255714285714, :
## some 'x' values beyond boundary knots may cause ill-conditioned bases## Warning in bs(dis, degree = 3L, knots = c(`7.142857%` = 1.5299, `14.28571%`
## = 1.7996, : some 'x' values beyond boundary knots may cause ill-conditioned
## bases
## Warning in bs(dis, degree = 3L, knots = c(`7.142857%` = 1.5299, `14.28571%`
## = 1.7996, : some 'x' values beyond boundary knots may cause ill-conditioned
## basesplot(3:16, cv[-c(1, 2)], xlab = "Degrees of freedom", ylab = "Test MSE", type = "l")
It appears that our best degree of freedom may be 7.
Problem 2: College
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(ISLR)
install.packages("leaps", repos = "https://CRAN.R-project.org/package=leaps")## Warning: unable to access index for repository https://CRAN.R-project.org/package=leaps/src/contrib:
## cannot open URL 'https://CRAN.R-project.org/package=leaps/src/contrib/PACKAGES'## Warning: package 'leaps' is not available (for R version 3.6.1)## Warning: unable to access index for repository https://CRAN.R-project.org/package=leaps/bin/macosx/el-capitan/contrib/3.6:
## cannot open URL 'https://CRAN.R-project.org/package=leaps/bin/macosx/el-capitan/contrib/3.6/PACKAGES'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 Top25perc Room.Board Terminal
## -4682.0604917 3364.5887624 22.7444078 0.8485723 49.7396461
## Expend Grad.Rate
## 0.1988311 28.1908514b) 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
install.packages("gam", repos = "https://CRAN.R-project.org/package=gam")## Warning: unable to access index for repository https://CRAN.R-project.org/package=gam/src/contrib:
## cannot open URL 'https://CRAN.R-project.org/package=gam/src/contrib/PACKAGES'## Warning: package 'gam' is not available (for R version 3.6.1)## Warning: unable to access index for repository https://CRAN.R-project.org/package=gam/bin/macosx/el-capitan/contrib/3.6:
## cannot open URL 'https://CRAN.R-project.org/package=gam/bin/macosx/el-capitan/contrib/3.6/PACKAGES'library(gam)## Loading required package: foreach##
## Attaching package: 'foreach'## The following objects are masked from 'package:purrr':
##
## accumulate, when## Loaded gam 1.16.1gam.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)## Warning in model.matrix.default(mt, mf, contrasts): non-list contrasts
## argument ignoredpar(mfrow = c(2,3))
plot(gam.mod, se = TRUE, col = 'blue')
Based on the shape of the fit curves, Expend and Grad.Rate are strong non-linear with the 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] 0.7737496the R squared statitic on the test set is 0.75.
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
## -7544.60 -1071.49 -73.43 1258.11 4302.43
##
## (Dispersion Parameter for gaussian family taken to be 3207027)
##
## Null Deviance: 5856058281 on 387 degrees of freedom
## Residual Deviance: 1157736560 on 360.9999 degrees of freedom
## AIC: 6941.683
##
## Number of Local Scoring Iterations: 2
##
## Anova for Parametric Effects
## Df Sum Sq Mean Sq F value Pr(>F)
## Private 1 1568003917 1568003917 488.928 < 2.2e-16 ***
## s(Room.Board, 5) 1 1129020425 1129020425 352.046 < 2.2e-16 ***
## s(Terminal, 5) 1 411353760 411353760 128.266 < 2.2e-16 ***
## s(perc.alumni, 5) 1 153175654 153175654 47.763 2.188e-11 ***
## s(Expend, 5) 1 571017807 571017807 178.052 < 2.2e-16 ***
## s(Grad.Rate, 5) 1 58834288 58834288 18.345 2.365e-05 ***
## Residuals 361 1157736560 3207027
## ---
## 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 2.8750 0.02288 *
## s(Terminal, 5) 4 1.7835 0.13151
## s(perc.alumni, 5) 4 0.5833 0.67494
## s(Expend, 5) 4 17.3817 4.701e-13 ***
## s(Grad.Rate, 5) 4 1.2876 0.27442
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1