Page 293 - Introduction to Statistical Pattern Recognition

6 Nonparametric Density Estimation 275

Solving alMSEIak = 0 generates [5]

(6.100)

-- 4
xN ’lM . (6.101)
It should be pointed out that Ex (MSE { p(X)} } can be minimized by a similar
procedure to obtain the globally optimal k. The resulting k* is similar but
slightly smaller than k* of (6.100).

Optimal metric: The optimal metric also can be computed as in the Par-
zen case. Again, a family of density functions with the form of (6.51) is stu-
died with the metric of (6.24). In order to diagonalize both B and A to I and A
respectively, X is linearly transformed to Z. In the transformed Z-space,
IMSE; becomes, from (6.101) and (6.13),

IMSE; = c I I c2jpP’”(Z)t? [ v2p~(Z)- A Idz]& , (6.102)

where cI and c2 are positive constants. IMSE; can be minimized with respect
to A by minimizing
n
J = tr2(V2pZ(Z)A) - p( nh,-l) , (6.103)
i=l
which is identical to (6.59).
Therefore, the optimal metric A for the kNN density estimate is identical