Page 269 - Introduction to Statistical Pattern Recognition

5 Parameter Estimation 25 1

5. Derive v, of (5.92) for Data 1-1.
6. A quadratic classifier for zero-mean distributions is given as

01
h(X) =XT[C;I - ZC,']X >< t .
02
In the design phase, C1 and C2 are estimated by
N
I
N
I
T
T
*
CI = - XiX; and C2 = - Y,Y, ,
N ;=I N ;=I
where XI and Y, are samples from wl and w2 respectively. For testing
n
Xk, Xk is excluded from design to get Elk. Prove

7. Modify the Procedure 111 of Section 3.2 to the leave-one-out method.
8. Assuming MI = 0, M2 = M = [m 0. . . O]', and XI = C2 = I, compute
the integral of (5.153) along the Bayes boundary (xI = -/2) to
obtain PI. Use h(X)==x, -MTM/2, d':(X)=(C,:'=lx-f)2, and
1
p (x) = (2~)-"'~exp[- -E ,:'= I x,' 1.
2
9. N boxes have equal probability of getting a sample. When N samples
are thrown, the ith box receives k, samples. Defining w, = k,/N, prove
that
(1) E{w,) = 1/N,
(2) E((w, - l/N)(w, - l/N)} =6,/N2 - l/N3.