Page 170 - Introduction to Statistical Pattern Recognition
P. 170
152 Introduction to Statistical Pattern Recognition
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llc1I2=o -+ uTw=r -+ E =o. (4.95)
As U, increases, the change of IlC(1)112 can be calculated by substituting (4.92)
and (4.94) into (4.91). The result is
IIC(t+ 1)112 - Itc(e)1I2
UTU
= -2pCT(e)[!-,~]AI'(t)+p2ArT(P$[l- -]Ar(t) . (4.96)
N N
On the other hand, from (4.91), (4.93), and (4.86),
CTUTU = (wTu - rT)uTu (wT rTuT)u o (4.97)
-
=
=
and
2CTAT=((C+ ICI)'+(C- ICI)')(C+ ICI)
=(C+ lCl)'(C+ lCl)=ArTAr. (4.98)
Therefore, (4.96) can be simplified as
forOcpcl. (4.99)
The equality holds only when llAr1I2 = 0. Thus, as t increases, )lC(U,)112
decreases monotonically, until llAr112 equals zero. It means either IIC 112 = 0 or
C =-IC I from (4.90). When llC11* = 0, we can achieve E2 = 0 [see (4.931.
On the other hand, when C = - IC I, all components of C become negative or
zero and the iteration stops with UTW I r satisfied from (4.91).
