Page 197 - Introduction to Statistical Pattern Recognition

4 Parametric Classifiers 179

v = [PIC, + P,C2]-1(M2 -M,)

6. Prove that E( Fowl)F*ow2)) = 0 for wI # o2 where Fuw) is the Fourier
transform of a stationary random process, x(r), as

7. Two stationary normal distributions are characterized by P, = P2 = 0.5,
MI =0, M2 =A[l.. . 1lT, and C=C, =C2 =02R where R is given in
(4.126).
(a) Compute the Bayes error for n = 10, A = 2, o2 = 1, and p = 0.5.
(b) Using the same numbers as in (a), compute the error when
NX(Ml,02Z) NX(M2,021) are used to design the classifier and
and
NX(Ml,C) and Nx(Mz,C) used to test the classifier.
are
8. Repeat Problem 7 for a two-dimensional random field of nxn. The verti-
cal and horizontal correlation matrices are the same and specified by
(4.126).

9. Design a linear classifier by minimizing the mean-square error for the data
given in the following Table, assuming P I = P, = 0.5.

XI x2 X3 PlW) P2W
-1 -1 -1 1/3 0
+1 -1 -1 1/24 1/8
-1 +1 -1 1/24 1/8
+1 +1 -1 0 1/3
-1 -1 +1 1 /3 0
+1 -1 +1 1/24 1/8
-1 +I +1 1/24 1/8
+1 +1 +1 0 1/3