Page 196 - Introduction to Statistical Pattern Recognition

178 Introduction to Statistical Pattern Recognition

(b) Assuming that the density functions of h(X) for o1 and 02 can be
approximated by normal densities, compute the approximated value
of the Bayes error for n = 8, h2A, = 2.5, and PI = P2 = 0.5.

2. Two normal distributions are characterized by

Calculate the errors due to the Bayes classifier and the bisector.

3. Using the same data as in Problem 2 except

find the linear discriminant function which maximizes the Fisher criterion,
and minimize the error by adjusting the threshold.

4. Using the same data as in Problem 3, find the optimum linear discriminant
function which minimizes the probability of error. Show that the error is
smaller than the one of Problem 3. (Check the errors for s = 0, 0.02 and
0.25.)

5. Design the optimum linear classifier by minimizing the mean-square error
of
-2
E = E { ( vTx + Yo - y(X))2 )
where y(X) = +1 for X E o2 and -1 for X E ol, Without using the pro-
cedure discussed in this chapter, take the derivative of E2 with respect to
V and 1'0, equate the derivative to zero, and solve the equation for V. Set-
ting the mixture mean, Mo = PIMI + P2M2, as the coordinate origin,
confirm that the resulting optimum V is