Page 275 - Introduction to Statistical Pattern Recognition
P. 275
6 Nonparametric Density Estimation 257
j~(x)dX = 1 and Z, = r’A where C, is the covariance matrix of K(X).
Convolution expression: Equation (6.2) can be rewritten in convolution
form as
where p,T is an impulsive density function with impulses at the locations of
existing N samples.
That is, the estimated density p(X) is obtained by feeding p,(X) through a
linear (noncausal) filter whose impulse response is given by K(X). Therefore,
n
p(X) is a smoothed version of p,(X).
Moments of p(X): The first and second order moments of (6.4) can be
n
easily computed. First, let us compute the expected value of p,(X) as
IN l N
E(P,(X)J = -Zj6(X-Z)P(Z)dZ = -ZP(X) =p(X). (6.6)
N,=I N ;=,
That is, p,(X) is an unbiased estimate of p(X). Then, the expected value of
p(X) of (6.4) may be computed as
Also,
