Page 185 - Introduction to Statistical Pattern Recognition

4 Parametric Classifiers 167

Experiment 3: Computation of the Bhattacharyya distance
Data: RADAR
Dimension: n = 66
Sample size: N I = N2 = 8800, 720, 360
n
Approximation: Toeplitz approximation for Cj
No. of trials: z = 1
Results: Table 4-1

TABLE 4-1
EFFECT OF TOEPLITZ APPROXIMATION
NI =N2 Without Approx. Toeplitz Approx.
8,800 0.64 0.73
720 1.57 0.77
~~
360 2.52 0.8 1
4,400 (Design)
20.2 26.3
4,400 (Test)
720 (Design)
25.9 26.6
4,400 (Test)
360 (Design)
30.1 26.8
4,400 (Test)

n n
In this experiment, the sample mean M, and sample covariance matrix C,
n
were estimated from N, samples, and the correlation matrix of C, was approxi-
n n
mated by the toeplitz form of (4.143). Using M, and the approximated C,, the
Bhattacharyya distance was computed and was compared with the one com-
puted from h, and (without the approximation). Both are fairly close for
N, = 8800, indicating the validity of the approximation. Furthermore, since the
approximated covariance matrices depend on a smaller number of parameters,
its estimates are less sensitive to the sample size. Without approximation, the
effect of the sample size is evident. That is, p(1/2) increases as N, decreases.