Page 162 - Introduction to Statistical Pattern Recognition

144 Introduction to Statistical Pattern Recognition

6y(X) = 6V'X . (4.57)
Inserting (4.57) into (4.53), and using y (X) = V'X = X'V,

1

where Si = E(XXT lo,) is the autocorrelation matrix of 0,. Since 6f = 0
regardless of 6V', [.] must be zero. Thus,

where af/asy is replaced by af /do: in order to maintain the uniformity of
expressions. Note that (4.59) is the same as (4.25), except that Si is used in
(4.59) while Ci is used in (4.25). This is due to the difference in criteria we
used; f (q1,q2,s:,s$) for (4.59) and f(q1,q2,0:,o~) for (4.25). Since
s' =qy+o?, f (q1,q2,s:,s$) is also a function of ql, q2, o:, and 0:. There-
fore, both (4.59) and (4.25) must give the same optimum V.
In order to confirm the above argument, let us prove that
V = [sS I+(l-s)S2]-1 [(af/&ll)Ml+(df/&12)M2] same vector as
the
is
V = [sZI+(1-s)C2]-' [@f /&ll)Ml+(af /&12)M2] except for its length. Since
the result must be independent of where the coordinate origin is, let us choose
M as the coordinate origin for simplicity. Then, MI and M2 are replaced by
0 and M = M2-M I. Ignoring the constant (af/&12), we start from

v = [SS I + (l-s)S2]-IM . (4.60)
Since SI = C and S2 = C2+MMT,
I