## Description

## Problem 1 (R exercise for linear Regression. 50 points)

Consider the data set “fat” in the “faraway”

library of R. The data is also available at T-square or at

fat <- read.table(file = “http://www.isye.gatech.edu/~ymei/7406/Handouts/fat.csv”,

sep=”,”, header=TRUE);

The dataset fat has 252 observations and 18 variables, and, for more detailed description, see the link

(http://artax.karlin.mff.cuni.cz/r-help/library/faraway/html/fat.html) or

(http://cran.r-project.org/web/packages/faraway/faraway.pdf) (page #37).

For more background information, you can also see

http://en.wikipedia.org/wiki/Body_fat_percentage

The purpose of this homework is to help you better understand linear regression and R. Here we assume

that the percentage of body fat using Brozek’s equation (brozek, the first column) as the response variable,

and the other 17 variables as potential predictors. We will use several different statistical methods to fit this

dataset in the problem of predicting brozek using the other 17 potential predictors. For that purpose, it is

useful to split it into the following sub-tasks.

(a) First, we should split the original data set into disjoint training and testing data sets, so that we

can better evaluate and compare different models. One possible simple way is to random select a

proportion, say, 10% of observations from the data for use as a test sample, and use the remaining

data as a training sample building different models.

Note that in practice, it is more reasonable

to select much larger proportion, say 30% or 20%, as testing sample. Here we chose only 10%

as the testing sample, so that we can list those testing observations explicitly below. You can do so by

the following R code.

n = dim(fat)[1]; ### total number of observations

n1 = round(n/10); ### number of observations randomly selected for testing data

set.seed(7406); ### set the seed for randomization

flag = sort(sample(1:n, n1));

## If you are using other software, the 25 rows of testing observations are:

flag = c(1, 21, 22, 57, 70, 88, 91, 94, 121, 127, 149, 151, 159, 162,

164, 177, 179, 194, 206, 214, 215, 221, 240, 241, 243);

fat1train = fat[-flag,]; fat1test = fat[flag,];

(b) Second, for the training data “fat1train,” do some exploratory (or preliminary) data analysis

such as scatter plots or summary statistics of some variables that you feel are important (e.g., explain

the unusual pattern).

(c) Based on the training data “fat1train,” build the following models

(i) Linear regression with all predictors.

(ii) Linear regression with the best subset of k = 5 predictors variables;

(iii) Linear regression with variables (stepwise) selected using AIC;

(iv) Ridge regression;

(v) LASSO;

(vi) Principal component regression;

(vii) Partial least squares.

(d) Use the models you find in part (c) to predict the response in the testing data “fat1test” in part (a).

Report the performance of each model ˆf on the testing data, say, {(Y

test

i

, x

test

i

)}

n1

i=1. Here n1 = 25 and

we assume that the performance of each model is evaluated by the following testing error

T E =

1

n1

Xn1

i=1

[Y

test

i − ˆf(x

test

i

)]2

.

(e) The above steps are sufficient when one has a large data set. However, for a relatively small data, one

may want to do further to assess the robustness of each method. One general approach is Monte

Carlo Cross-Validation algorithm that repeats the above computation B times (B = 100 say).

That is, for each loop b = 1, . . . , B, we randomly select, say n1 = 25, observations from the original data

as the testing data, and use the remaining data as a training sample. Within each loop, we first build

different models from “the training data of that specific loop”, and then evaluate their performances

on “the corresponding testing data.” Therefore, for each model or method in part (c), we will obtain

B values of testing errors on B different subsets of testing data, denote by T Eb for b = 1, 2, . . . , B.

Then the “average” performances of each model can be summarized by the sample mean and sample

variables of these B TE values:

T E∗ =

1

B

X

B

b=1

T Eb and V ar ˆ (T E) = 1

B − 1

X

B

b=1

T Eb − T E∗

2

.

Compute and compare the “average” performances of each model mentioned in part (c).

Write a report to summarize your findings. The report should include (i) Introduction, (ii) Exploratory (or preliminary) Data Analysis of training data in part (a), (iii) Methods, (iv) Results

and (v) Findings. Also see the guidelines on the final report of our course project. Please attach your R

code (without, or with limited, output) in the appendix of your report, and please do not just dump the R

output in the body of the report.

Remark: In Part (c) and (e), please see the update R code for linear regression at Canvas. Note that in

part (e), the same original data is repeatedly used B times as a whole, but it is used differently at different

loops due to the different split of training and testing data. The idea of repeating the similar data analysis

process B times is essential in many well-known statistical tools such as bootstrapping and Random

Forest, and has been widely used in other fields such as bioinformatics or computational biology.

For your convenience, I also post some R codes at the pdf file of this homework at Canvas that might

be useful. Please feel free to modify those R codes if you want. To encourage everyone learn the materials,

each student must write their R or any other software codes by themselves, and no collaborations allowed!

It is cheating if you copy and paste your classmates’ computing codes.

Appendix: the following R code might be useful for Problem 1 of Homework #2:

### Read the data

fat <- read.table(file = “http://www.isye.gatech.edu/~ymei/7406/Handouts/fat.csv”,

sep=”,”, header=TRUE);

### Split the data as in Part (a)

n = dim(fat)[1]; ### total number of observations

n1 = round(n/10); ### number of observations randomly selected for testing data

set.seed(7406); ### set the seed for randomization

flag = sort(sample(1:n, n1));

## If you are using other software, the 25 rows of testing observations are:

flag = c(1, 21, 22, 57, 70, 88, 91, 94, 121, 127, 149, 151, 159, 162,

164, 177, 179, 194, 206, 214, 215, 221, 240, 241, 243);

fat1train = fat[-flag,]; fat1test = fat[flag,];

###In Part (b)-(d), Please see the update R code for linear regression at T-square.

### Please write your own R or other software code to analyze the training data “fat1train”

### and evaluate different models on the testing data “fat1test”.

### Part (e): the following R code might be useful, and feel free to modify it.

### save the TE values for all models in all $B=100$ loops

B= 100; ### number of loops

TEALL = NULL; ### Final TE values

for (b in 1:B){

### randomly select 25 observations as testing data in each loop

flag <- sort(sample(1:n, n1));

fattrain <- fat[-flag,];

fattest <- fat[flag,];

### you can write your own R code here to first fit each model to “fattrain”

### then get the testing error (TE) values on the testing data “fattest”

### Suppose that you save the TE values for these five models as

### te1, te2, te3, te4, te5, te6, te7, respectively, within this loop

### Then you can save these 5 Testing Error values by using the R code

###

TEALL = rbind( TEALL, cbind(te1, te2, te3, te4, te5, te6, te7) );

}

dim(TEALL); ### This should be a Bx7 matrices

### if you want, you can change the column name of TEALL

colnames(TEALL) <- c(“mod1”, “mod2”, “mod3”, “mod4”, “mod5”, “mod6”, “mod7”);

## You can report the sample mean and sample variances for the five models

apply(TEALL, 2, mean);

apply(TEALL, 2, var);

### END ###