Page 234 - Classification Parameter Estimation & State Estimation An Engg Approach Using MATLAB

FEATURE REDUCTION 223

Table 7.1 Distance matrix of 13 cities in (km)

0 3290 7280 10100 10600 9540 13300 7350 7060 13900 16100 17700 4640 Bangkok
0 7460 12900 8150 8090 10100 9190 5790 11000 17300 19000 2130 Beijing
0 7390 14000 3380 12100 14000 2810 8960 9950 12800 9500 Cairo
0 18500 9670 16100 10300 10100 12600 6080 7970 14800 Capetown
0 11600 4120 8880 11200 8260 13300 11000 6220 Honolulu
0 8760 16900 2450 5550 9290 11700 9560 London
0 12700 9740 3930 10100 9010 8830 Los Angeles
0 14400 16600 13200 11200 8200 Melbourne
0 7510 11500 14100 7470 Moscow
0 7770 8260 10800 New York
0 2730 18600 Rio
0 17200 Santiago
0 Tokyo

The mapped objects in Figures 7.3 have been rotated such that New
York and Beijing lie on a horizontal line with New York on the left.
Furthermore, the vertical axis is mirrored such that London is situated
below the line New York–Beijing.
The reason for the preferred projection on the tangent plane near the
North Pole is that most cities in the list are situated in the Northern
hemisphere. The exceptions are Santiago, Rio, Capetown and Melbourne.
The distances for just these cities are least preserved. For instance, the true
geodesic distance between Capetown and Melbourne is about 10 000 (km),
but they are mapped opposite to each other with a distance of about
18 000 (km).
For specific applications, it might be fruitful to focus more on local
structure or on global structure. A way of doing so is by using the more
general stress measure, where an additional free parameter q is introduced:

N S
N S
q 1 X X q 2
E ¼ ð ij d ij Þ ð7:7Þ
S N N ij
P S P S
ðqþ2Þ i¼1 j¼iþ1
ij
i¼1 j¼iþ1
For q ¼ 0, (7.7) is equal to (7.5). However, as q is decreased, the ( ij d ij ) 2
q
term remains constant but the term will weigh small distances heavier
ij
than large ones. In other words, local distance preservation is emphasized,
whereas global distance preservation is less important. This is demon-
strated in Figure 7.3(a) to (c), where q ¼ 2, q ¼ 0and q ¼þ2have been
used, respectively. Conversely, if q is increased, the resulting stress measure
will emphasize preserving global distances over local ones. When the stress