Homework 2

There are two exercises for Homework 2. The first is about bootstrapping, and the second is about classification. Specifically:

• The first exercise is a replication exercise. Your task is as follows: when you find the R scripts missing for the numeric/figure outputs, provide the R scripts yourself. Your output should match my outputs exactly if you have followed the procedures described in the exercise.

• The second exercise is essentially a mini-report of classification analysis, which leaves you lots of room for creativity.

Exercise 1: Bootstrapping

Bootstrapping is a resampling method commonly used to calculate the standard error (and confidence intervals) of an estimator. This assignment uses an artifical example of “weighing panda Yaya” to illustrate the use of bootstrapping.

Set the random seed:

set.seed(2048)

Description

The panda keeper wants to measure the weight of panda Yaya. She lures Yaya onto the scale multiple times, but because Yaya keeps moving on the scale, each measurement may have a large error.

Assume the keeper weighs Yaya ten times, and the recorded weights are:

k_weights <- c(93, 77, 62, 78, 75, 85, 66, 83, 91, 72)

A simple way to reduce noise is to take the mean:

#> 78.2

The sample mean we obtain, 78.2, is just “one” statistic. Without additional assumptions to specify the “model” of panda Yaya’s weight, we cannot perform statistical inference on this result nor assess its accuracy.

Bootstrapping: Resampling

We use the sample() function for resampling. Unlike cross‑validation, bootstrapping uses sampling with replacement, so we set replace=TRUE:

sample(k_weights, size=10, replace=TRUE)
#>  [1] 66 93 83 91 62 91 62 62 72 75

We perform K = 100 resamplings:

K = 100
panda_boots <- replicate(K, sample(k_weights, size=10, replace=TRUE))

Check that the resulting resampled data panda_boots is a matrix with 10 rows and 100 columns:

#> [1]  10 100

Calculating the Standard Error

We compute the mean of each column, obtaining a vector of length 100:

panda_boot_means <- apply(panda_boots, 2, mean)
length(panda_boot_means)
#> [1] 100

panda_boot_means contains more information than mean(k_weights). Intuitively, panda_boot_means can be seen as 100 realizations of the mean statistic.

The (sample) variance of panda_boot_means can be used as an estimate of the standard error of the mean statistic (the bootstrap standard error).

#> [1] 3.278438

plot of chunk unnamed-chunk-10

Calculating Confidence Intervals

In the analysis above, we performed K = 100 resamplings. With modern computers, we can perform K = 1000 or K = 10000 resamplings to obtain a more accurate estimate of the standard error. Moreover, a larger K helps us construct more accurate confidence intervals for the mean statistic.

In this simple example, by the Central Limit Theorem, when K is large, panda_boot_means should be approximately normally distributed. We can directly construct a 95% confidence interval using the bootstrap standard error:

qnorm(0.025)

[1] -1.959964

K = 10000
panda_boots <- replicate(K, sample(k_weights, size=10, replace=TRUE))
panda_boot_means <- apply(panda_boots, 2, mean)

plot of chunk unnamed-chunk-12

Below is the confidence interval. The formula is $\mathrm{mean} \pm 1.96 \times \mathrm{sd}$, where the value 1.96 is obtained from the R expression qnorm(0.025).

#> [1] 72.22867 84.17133

Summary

This exercise illustrates the bootstrap method with a simple example. We have used the sample() function for resampling.

Exercise 2: Classsification

You should first install the R package ISLR2 if you have not done it yet:

install.packages("ISLR2")

This is the accompanying R package for our textbook, Intro to Statistical Learning. This package provides many useful data sets:

When you run library(ISLR2), these data sets will be loaded into your current R session. You can use the familiar ? syntax to read the manual for these data sets. For example, by running ?Wage, the documentation for the Wage dataset will be opened.

Exercise description

Choose one dataset from the 21 datasets above, perform some exploratory analysis, and then conduct some classification analysis.

You may refer to our course website to see some classification analyses of the Default dataset and the stock market dataset.

Colophon. This document is built with Pandoc and Knitr, with the following knitr setup:

knitr::opts_chunk$set(collapse = TRUE, comment = "#>")