Week 4 Tutorial: Principal Component Analysis Solution

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Description

Instructions for assessment

Please submit your answers to Question 1 as a pdf document via Quercus. It will be Due Friday 11 October, 2018 at 12pm (Midday). Late submissions will be heavily penalized.

The document should contain answers to the four subquestions (i)–(iv) of Question 1.

Question 1 (For Assessment)

A B

0

y

4

8

3 −2 −1 0 1 2 −3 −2 −1 0 1 2

x

Consider the two data sets shown in the above plot.

  1. Sketch (on each panel) the pair of eigenvectors you would expect ot obtain from a principal component analysis of each data set.

  1. For each data set, indicate which eigenvector (if any) you’d expect to have the larger eigenvalue.

  1. What do the numerical values of the eigenvalues tell you about the data?

1

  1. Do you think it is appropriate to use PCA to reduce the dimensionality of the data shown in panel B? Why or why not?

Question 2

Do Labs 1 and 3 in Chapter 10 of the Textbook (Sections 10.4 and 10.6)

Question 3

By modifying the argument used in the lecture, argue that the second principal component of a (centred) feature matrix X should maximise the Rayleigh quotent of XT X over all vectors that are orthogonal to the first principal component and show that this implies that the second prinicipal component is the eigenvector of XT X that corresponds to the second largest eigenvalue.

Question 4

An experiment was undertaken to exmaine differences in burritos made across Toronto. 400 burritos were purchased at commercial venues and four measurements were taken of each burrito: mass (in grams), length (in centimetres), sturdiness (scored from 1-10), and taste (scored from 1-10). The scaled sample covariance matrix was calculated and its four eigenvalues were found to be 14.1, 4.3, 1.2 and 0.4. The corresponding first and second eigenvectors were

vT = [0.39, 0.42, 0.44, 0.69]

1

  1. What is the proportion of variance explained by the first two prinicple components?

  1. Give an interpretation of the first two principal components

  1. Suppose that the data were stored by recording the eigenvalues and eigenvectors together with the 400 values of the first and second principal components and the mean values for the original variables. How would you use this information reconstruct approximations to the original covariance matrix and the original data?

2