Information Technology Reference
In-Depth Information
Ta b l e 7 . 1 9
Approximation based upon two-dimensional biplot given in Figure 7.14
of
√
n
times the chi-squared distances between the columns of the 2007/08 crime
data set.
Arsn
AGBH
AtMr
BNRs
BRs
CrJk
CmAs
CmRb
DrgR
InAs
Mrd
PubV
Rape
RAC
Arsn
0.000
0.354
0.078
0.045
0.038
0.292
0.037
0.103
0.646
0.117
0.064
0.072
0.072
0.469
AGBH
0.354
0.000
0.429
0.347
0.385
0.630
0.385
0.429
0.918
0.456
0.406
0.414
0.283
0.799
AtMr
0.078
0.429
0.000
0.108
0.064
0.215
0.044
0.056
0.618
0.090
0.068
0.067
0.149
0.393
BNRs
0.045
0.347
0.108
0.000
0.045
0.322
0.077
0.145
0.621
0.109
0.060
0.068
0.071
0.500
BRs
0.038
0.385
0.064
0.045
0.000
0.277
0.045
0.108
0.608
0.078
0.026
0.033
0.103
0.455
CrJk
0.292
0.630
0.215
0.322
0.277
0.000
0.256
0.202
0.633
0.254
0.271
0.266
0.360
0.178
CmAs
0.037
0.385
0.044
0.077
0.045
0.256
0.000
0.068
0.642
0.109
0.064
0.069
0.105
0.433
CmRb
0.103
0.429
0.056
0.145
0.108
0.202
0.068
0.000
0.668
0.146
0.120
0.121
0.163
0.375
DrgR
0.646
0.918
0.618
0.621
0.608
0.633
0.642
0.668
0.000
0.534
0.583
0.576
0.690
0.707
InAs
0.117
0.456
0.090
0.109
0.078
0.254
0.109
0.146
0.534
0.000
0.053
0.045
0.178
0.427
Mrd
0.064
0.406
0.068
0.060
0.026
0.271
0.064
0.120
0.583
0.053
0.000
0.008
0.126
0.448
PubV
0.072
0.414
0.067
0.068
0.033
0.266
0.069
0.121
0.576
0.045
0.008
0.000
0.134
0.443
Rape
0.072
0.283
0.149
0.071
0.103
0.360
0.105
0.163
0.690
0.178
0.126
0.134
0.000
0.536
RAC
0.469
0.799
0.393
0.500
0.455
0.178
0.433
0.375
0.707
0.427
0.448
0.443
0.536
0.000
of actual values. Dividing the first set of diagonal values by their counterparts in the
second set gives the sample predictivities for the interpolated points. Table 7.26 contains
the required predictivities. As expected, we see that the 2007/08 values are identical to
those displayed in Table 7.13.
The interpolation of selected row points in the CA biplots displayed in Figures 7.22
and 7.23 has drawn our attention to striking changes in the reported cases of drug
related crimes in the Western Cape province. Instead of relying on the positions of
interpolated points, we can investigate the crime data for the Western Cape over the
entire study period by constructing a CA biplot for the corresponding data. Table 7.27
contains the data for the Western Cape province over the entire period from 2001/02 to
2007/08.
The Pearson residuals case A biplot for the Table 7.27 data is given in Figure 7.25.
We have set the argument
markers = FALSE
in order to display only the two markers
at the ends of each biplot axis. This biplot has a very high overall quality of 98.22%.
The column and row predictivities are given in Tables 7.28 and 7.29, respectively.
The positions occupied by the row points in the Figure 7.25 biplot suggest that even a
one-dimensional CA biplot should adequately represent the distances among them. This
conclusion is supported by a one-dimensional overall quality of 92.84%.
Does the exceptionally high two-dimensional quality of 98.22% necessarily ensure
that all rows and columns are well represented in the biplot? For guidance we consider
the column and row predictivities: it is clear that the column predictivity of
Rape
is
a very low 0.1783, with that of
PubV
a moderate 0.4636. In two dimensions all row
points have very high row predictivity values, indicating that their true weighted residual
values for all the different crimes can be accurately determined from the biplot. We
also notice that in a one-dimensional biplot it is only row point
04/05
that has a very
low row predictivity but that with the addition of only one further dimension it jumps
to 0.9513.
