Combining Pattern Classifiers

28 FUNDAMENTALS OF PATTERN RECOGNITION
Fig. 1.9
(a) The skeletons of the two classes to be generated. (b) The generated data set.
1.5.3.1 Noisy Geometric Figures. The following example suggests one possible
way for achieving this. (The Matlab code is shown in Appendix 1E.)
Suppose we want to generate two classes in R 2 with prior probabilities 0.6 and
0.4, respectively. Each class will be distributed along a piece of a parametric
curve. Let class v 1 have a skeleton of a Lissajous figure with a parameter t, such that
x ¼ a sin nt, y ¼ b cos t, t [ ½ p, pŠ (1:36)
Pick a ¼ b ¼ 1 and n ¼ 2. The Lissajous figure is shown in Figure 1.9a.
Let class v 2 be shaped as a segment of a line with a parametric equation
x ¼ t, y ¼ at þ b, for t [ ½ 0:3, 1:5Š (1:37)
Let us pick a ¼ 1:4 and b ¼ 1:5. The segment is depicted in Figure 1.9a.
We shall draw random samples with uniform distributions along the skeletons
with overlaid normal distributions of specified variances. For v 1 we shall use s 2 ¼
0:005 on both axes and a diagonal covariance matrix. For v 2 , we shall use s 2 1 ¼
0:01 (1:5 x) 2 and s 2 2 ¼ 0:001. We chose s 1 to vary with x so that smaller x
values exhibit larger variance. To design the data set, select the total number of
data points T, and follow the list of steps below. The normal distributions for the
example are generated within the code. Only the standard (uniform) random generator
of Matlab will be used.
1. Generate a random number r [ ½0;1Š.
2. If r , 0:6, then proceed to generate a point from v 1 .
(a) Generate randomly t in the interval ½ p, pŠ.
(b) Find the point (x, y) on the curve using Eq. (1.36).
(c) To superimpose the noise generate a series of triples of random numbers
u, v within ½ 3s, 3sŠ, and w [ ½0, 1Š, until the following condition,