Page 37 - Classification Parameter Estimation & State Estimation An Engg Approach Using MATLAB

26 DETECTION AND CLASSIFICATION

section discusses the case in which these densities are modelled as
normal. Suppose that the measurement vectors coming from an object
with class ! k are normally distributed with expectation vector m and
k
covariance matrix C k (see Appendix C.3):

T
1
1 ðz m Þ C ðz m Þ !
k
k
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp k ð2:17Þ
pðzj! k Þ¼ q
N 2
ð2 Þ jC k j
where N is the dimension of the measurement vector.
Substitution of (2.17) in (2.12) gives the following minimum error rate
classification:

^ ! !ðzÞ¼ ! i with
8 9
!
> T 1 >
1 ðz m Þ C ðz m Þ
< =
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp k k k
i ¼ argmax q Pð! k Þ
N 2
> >
ð2 Þ jC k j
k¼1;...;K : ;
ð2:18Þ
We can take the logarithm of the function between braces without
changing the result of the argmaxfg function. Furthermore, all terms
not containing k are irrelevant. Therefore (2.18) is equivalent to

^ ! !ðzÞ¼! i with
1 1 T 1

i ¼ argmax ln jC k jþ ln Pð! k Þ ðz m Þ C ðz m Þ
k
k
k
k¼1;...;K 2 2
T
1
1
T
T
1

¼ argmax ln jC k jþ 2ln Pð! k Þ m C m þ 2z C m z C z
k k k k k k
k¼1;...;K
ð2:19Þ
Hence, the expression of a minimum error rate classification with nor-
mally distributed measurement vectors takes the form of:
T
T
^ ! !ðzÞ¼ ! i with i ¼ argmaxfw k þ z w k þ z W k zg ð2:20Þ
k¼1;...;K