13.4 Exercises

EXERCISE 13.1   Show that the matrices $\data{A}^{-1}\data{X}\data{B}^{-1}\data{X}^{\top}$ and $\data{B}^{-1}\data{X}^{\top}\data{A}^{-1}\data{X}$ have an eigenvalue equal to $1$ and that the corresponding eigenvectors are proportional to $(1,\ldots,1)^{\top}$.

EXERCISE 13.2   Verify the relations in (13.8), (13.14) and (13.17).

EXERCISE 13.3   Do a correspondence analysis for the car marks data (Table B.7)! Explain how this table can be considered as a contingency table.

EXERCISE 13.4   Compute the $\chi^2$-statistic of independence for the French baccalauréat data.

EXERCISE 13.5   Prove that $\data{C} = \data{A}^{-1/2} (\data{X} - E) \data{B}^{-1/2}
\sqrt{x_{\bullet \bullet}}$ and $E= \frac{ab^{\top}}{x_{\bullet \bullet}}$ and verify (13.20).

EXERCISE 13.6   Do the full correspondence analysis of the U.S. crime data (Table B.10), and determine the absolute contributions for the first three axes. How can you interpret the third axis? Try to identify the states with one of the four regions to which it belongs. Do you think the four regions have a different behavior with respect to crime?

EXERCISE 13.7   Repeat Exercise 13.6 with the U.S. health data (Table B.16). Only analyze the columns indicating the number of deaths per state.

EXERCISE 13.8   Consider a $(n \times n)$ contingency table being a diagonal matrix $\data X$. What do you expect the factors $r_k, s_k$ to be like?

EXERCISE 13.9   Assume that after some reordering of the rows and the columns, the contingency table has the following structure:

\begin{displaymath}
\data X = \begin{array}{\vert l\vert c\vert c\vert} \hline
...
...ine
I_1 & * & 0 \\ \hline
I_2 & 0 & * \\ \hline
\end{array}\end{displaymath}

That is, the rows $I_i$ only have weights in the columns $J_i$, for $i=1,2$. What do you expect the graph of the first two factors to look like?

EXERCISE 13.10   Redo Exercise 13.9 using the following contingency table:

\begin{displaymath}
\data X = \begin{array}{\vert l\vert c\vert c\vert c\vert} \...
..._2 & 0 & * & 0\\ \hline
I_3 & 0 & 0 & *\\ \hline
\end{array}\end{displaymath}

EXERCISE 13.11   Consider the French food data (Table B.6). Given that all of the variables are measured in the same units (Francs), explain how this table can be considered as a contingency table. Perform a correspondence analysis and compare the results to those obtained in the NPCA analysis in Chapter 9.