ANLY512: Homework 2
Due Feb 24 11:59pm
Problem 1: Use best subset selection to determine best heating values
For bioenergy production, the heating value is a measure of the amount of heat released during combustion.
The Higher heating value (HHV) is a particular method for determining the heat released during combustion.
The higher HHV the more energy released for a given amount of material. You will use the biomass dataset
from the model_data package. Run a ?biomass after importing the data to read about the domain. The
response variable is HHV and the predictor variables are the percentages of different elements. Do not include
the sample and dataset variables in your analysis.
a: Create scatterplots of the response and predictor variables and comment on your findings.
b: Split the dataset into train and test datasets using train-test split.
c: Use regsubsets() to perform best subset selection to pick the best model according to Cp,
BIC, and adjusted R2.
d: Repeat this procedure for forward stepwise selection and backward stepwise selection,
compare the best models from each selection method.
e: Use the predict() function to investigate the test performance in RMSE using your “best
model”.
Problem 2: Create your own cross validation algorithm to predict the interest
rate for loan applications
Lending club gained fame for being one of the first major players in retail lending. The dataset lending_club
in the model_data package includes 9,857 loans that were provided. Interest rates of loans are often a good
indicator of the level of risk associated with lending. If you are likely to pay back a loan, then you will likely
be charged lower interest than someone who has a higher chance of default. Your goal is to determine the
best model for predicting the interest rate charged to borrowers using best, forward, and backward subset
selection within a five-fold cross-validation framework.
Prep steps: - drop all rows with missing data in the following columns
a: Create a correlation plot of all the numeric variables in the dataset using the corrplot
package to create a high quality graph, then comment on your findings
b: Run best, forward, and backward subset selection on the entire dataset comment on the
findings
c: Create a five-fold cross-validation algorithm usng for loops to compare the CV mse perfor-
mance of your best two models
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Problem 3: Properties of k-fold cross validation
Suppose we are given a training set with n observations and want to conduct k-fold cross-validation. Assume
always that n = km where m is an integer.
a: Let k = 2. Explain carefully why there are 12
(
n
m
)
ways to partition the data into 2 folds.
b: Let k = 3. Explain carefully why there are n!3!m!m!m! ways to partition the data into 3 folds.
c: Guess a formula for the number of ways to partition the data into k folds for general k.
Check if your formula gives the correct answer for k=n (leave-one-out c.v.).
Problem 4: Using cross-validation to select best advertising budget
In this problem, we use the Advertising data download here. We want to predict Sales from TV, Radio and
Newspaper, using multiple regression with all three predictors plus up to one interaction term of these three
predictors, e.g. TV * Radio or Radio * Newspaper.
a: Should such an interaction term be included? Which one? Try to answer this question
by estimating the residual standard error using 10-fold cross validation for all four possible
models.
b: Create a single plot showing the return on investment of each advertsing method where
the y-axis is Sales and the x-axis is advertsing dollars. There should be three lines, one for
each method. The slope is the coefficient from you regression. What is the best advertising
method to invest in based on return on investment?
Problem 5: ISLR 6.8 #8(a-d)
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