You will examine the Carseats data, which is part of the ISLR library. You will attempt to predict Sales (child car seat sales) in 400 locations based on a number of predictors.
A data frame with 400 observations on the following 11 variables.
Sales Unit sales (in thousands) at each location
CompPrice Price charged by competitor at each location
Income Community income level (in thousands of dollars)
Advertising Local advertising budget for company at each location (in thousands of dollars)
Population Population size in region (in thousands)
Price Price company charges for car seats at each site
ShelveLoc A factor with levels Bad, Good and Medium indicating the quality of the shelving
location for the car seats at each site
Age Average age of the local population
Education Education level at each location
Urban A factor with levels No and Yes to indicate whether the store is in an urban or rural location
US A factor with levels No and Yes to indicate whether the store is in the US or not
(a) Split the data into a training set (70%) and a test set (30%).
(b) Fit a multiple linear regression model on the training data to predict Sales using all of the variables. Use the summary() function to print the results. Comment on the output. What test MSE do you obtain using the validation set approach?
(c) Apply regsubsets() to the training set in order to perform forward stepwise selection. What is the best model obtained according to Cp, BIC, and adjusted R^2? Show some plots to provide evidence for your answer, and report the coefficients of the best model obtained according to Cp, BIC, and adjusted R^2.
(d) Fit a ridge regression model on the training set, with λ chosen by cross-validation. Report the test MSE obtained.
(e) Fit a lasso model on the training set, with λ chosen by cross-validation. Report the test MSE obtained, along with the number of non-zero coefficient estimates.
(f) Fit a PCR model on the training set, with M chosen by cross-validation. Report the test MSE obtained, along with the value of M selected by cross-validation.
(g) Fit a PLS model on the training set, with M chosen by cross-validation. Report the test error obtained, along with the value of M selected by cross-validation.
(h) Fit a regression tree to the training set. Plot the tree and interpret the results. What test MSE do you obtain?
(i) Use cross-validation in order to determine the optimal level of tree complexity. Does pruning the tree improve the test MSE?