9.10 Exercises

EXERCISE 9.1   Prove Theorem 9.1. (Hint: use (4.23).)

EXERCISE 9.2   Interpret the results of the PCA of the U.S. companies. Use the analysis of the bank notes in Section 9.3 as a guide. Compare your results with those in Example 9.9.

EXERCISE 9.3   Test the hypothesis that the proportion of variance explained by the first two PCs for the U.S. companies is $\psi=0.75$.

EXERCISE 9.4   Apply the PCA to the car data (Table B.7). Interpret the first two PCs. Would it be necessary to look at the third PC?

EXERCISE 9.5   Take the athletic records for 55 countries (Appendix B.18) and apply the NPCA. Interpret your results.

EXERCISE 9.6   Apply a PCA to $\Sigma=\left(\begin{array}{rr} 1 & \rho\\ \rho & 1 \end{array}\right)$, where $\rho > 0$. Now change the scale of $X_1$, i.e., consider the covariance of $cX_1$ and $X_2$. How do the PC directions change with the screeplot?

EXERCISE 9.7   Suppose that we have standardized some data using the Mahalanobis transformation. Would it be reasonable to apply a PCA?

EXERCISE 9.8   Apply a NPCA to the U.S. CRIME data set (Table B.10). Interpret the results. Would it be necessary to look at the third PC? Can you see any difference between the four regions? Redo the analysis excluding the variable ``area of the state.''

EXERCISE 9.9   Repeat Exercise 9.8 using the U.S. HEALTH data set (Table B.16).

EXERCISE 9.10   Do a NPCA on the GEOPOL data set (see Table B.15) which compares 41 countries w.r.t. different aspects of their development. Why or why not would a PCA be reasonable here?

EXERCISE 9.11   Let $U$ be an uniform r.v. on $[0,1]$. Let $a \in \mathbb{R}^3$ be a vector of constants. Suppose that $X=Ua^{\top}=(X_1, X_2, X_3)$. What do you expect the NPCs of $X$ to be?

EXERCISE 9.12   Let $U_1$ and $U_2$ be two independent uniform random variables on $[0,1]$. Suppose that $X=(X_1, X_2, X_3,X_4)^{\top}$ where $X_1=U_1$, $X_2=U_2$, $X_3=U_1+U_2$ and $X_4=U_1-U_2$. Compute the correlation matrix $P$ of $X$. How many PCs are of interest? Show that $\gamma_1 =\left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}, 1, 0\right)^{\top}$ and $\gamma_2 =\left(\frac{1}{\sqrt{2}}, \frac{-1}{\sqrt{2}}, 0, 1\right)^{\top}$ are eigenvectors of $P$ corresponding to the non trivial $\lambda$`s. Interpret the first two NPCs obtained.

EXERCISE 9.13   Simulate a sample of size $n=50$ for the r.v. $X$ in Exercise 9.12 and analyze the results of a NPCA.

EXERCISE 9.14   Bouroche (1980) reported the data on the state expenses of France from the period 1872 to 1971 (24 selected years) by noting the percentage of 11 categories of expenses. Do a NPCA of this data set. Do the three main periods (before WWI, between WWI and WWII, and after WWII) indicate a change in behavior w.r.t. to state expenses?