Exercises Unit E - Applied
Bachelor’s Degree Programme in Philosophy, International and Economic Studies, Ca’ Foscari University of Venice.
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Chapter 12 Exercise 8.
In Section 12.2.3, the Proportion of Variance Explained (PVE) was defined in Equation (12.10). We also saw that PVE can be computed using the sdev output of the prcomp() function.
Using the USArrests dataset, compute the PVE in two ways:
- Using the
sdevoutput ofprcomp(), as in Section 12.2.3
- By applying Equation (12.10) directly
For part (b), use prcomp() to obtain the principal component loadings, and then plug them into Equation (12.10) to compute the PVE.
The two methods should give the same results.
Hint: The results will match only if the same preprocessing is used in both cases.
For example, if you use centered and scaled variables in prcomp(), then you must also center and scale the data when applying Equation (12.10).
Chapter 12 Exercise 11.
Write an R function to perform matrix completion as in Algorithm 12.1, and as outlined in Section 12.5.2. In each iteration, the function should keep track of the relative error, as well as the iteration count. Iterations should continue until the relative error is small enough or until some maximum number of iterations is reached (set a default value for this maximum number). Furthermore, there should be an option to print out the progress in each iteration.
Test your function on the Boston data. First, standardize the features to have mean zero and standard deviation one using the scale() function. Run an experiment where you randomly leave out an increasing (and nested) number of observations from 5% to 30%, in steps of 5%. Apply Algorithm 12.1 with M = 1, 2, \ldots, 8. Display the approximation error as a function of the fraction of observations that are missing, and the value of M, averaged over 10 repetitions of the experiment.
Chapter 12 Exercise 12.
In Section 12.5.2, Algorithm 12.1 was implemented using the svd() function. However, given the connection between the svd() function and the prcomp() function highlighted in the lab, we could have instead implemented the algorithm using prcomp().
Write a function to implement Algorithm 12.1 that makes use of prcomp() rather than svd().