12.3 Boston Housing

One interesting application of discriminant analysis with respect to the Boston housing data is the classification of the districts according to the house values. The rationale behind this is that certain observables must determine the value of a district, as in Section 3.7 where the house value was regressed on the other variables. Two groups are defined according to the median value of houses $\widetilde X_{14}$: in group $\Pi_1$ the value of $\widetilde X_{14}$ is greater than or equal to the median of $\widetilde X_{14}$ and in group $\Pi_2$ the value of $\widetilde X_{14}$ is less than the median of $\widetilde X_{14}$.

The linear discriminant rule, defined on the remaining 12 variables (excluding $\widetilde X_{4}$ and $\widetilde X_{14}$) is applied. After reclassifying the 506 observations, we obtain an apparent error rate of 0.146. The details are given in Table 12.1. The more appropriate error rate, given by the AER, is 0.160 (see Table 12.2).


Table: APER for price of Boston houses. 41991 MVAdiscbh.xpl
    True
    $\Pi_1$ $\Pi_2$
  $\Pi_1$ 216 40
Predicted      
  $\Pi_2$ 34 216
       



Table: AER for price of Boston houses. 41994 MVAaerbh.xpl
    True
    $\Pi_1$ $\Pi_2$
  $\Pi_1$ 211 42
Predicted      
  $\Pi_2$ 39 214
       



Table: APER for clusters of Boston houses. 41997 MVAdiscbh.xpl
    True
    $\Pi_1$ $\Pi_2$
  $\Pi_1$ 244 13
Predicted      
  $\Pi_2$ 7 242
       


Let us now turn to a group definition suggested by the Cluster Analysis in Section 11.4. Group $\Pi_1$ was defined by higher quality of life and house. We define the linear discriminant rule using the 13 variables from $\widetilde{\data{X}}$ excluding $\widetilde X_{4}$. Then we reclassify the 506 observations and we obtain an APER of 0.0395. Details are summarized in Table 12.3. The AER turns out to be 0.0415 (see Table 12.4).


Table: AER for clusters of Boston houses. 42000 MVAaerbh.xpl
    True
    $\Pi_1$ $\Pi_2$
  $\Pi_1$ 244 14
Predicted      
  $\Pi_2$ 7 241
       


Figure 12.3 displays the values of the linear discriminant scores (see Theorem 12.2) for all of the 506 observations, colored by groups. One can clearly see the APER is derived from the 7 observations from group $\Pi_1$ with a negative score and the 13 observations from group $\Pi_2$ with positive score.

Figure: Discrimination scores for the two clusters created from the Boston housing data. 42004 MVAdiscbh.xpl
\includegraphics[width=1.2\defepswidth]{MVAdiscbh.ps}