Page 47 - Classification Parameter Estimation & State Estimation An Engg Approach Using MATLAB

36 DETECTION AND CLASSIFICATION

If the test fails, it is decided for ! 2 ,otherwise for ! 1 . We write symbolically:

! 1
>
pðzj! 1 ÞPð! 1 Þ pðzj! 2 ÞPð! 2 Þ ð2:35Þ
<
! 2
Rearrangement gives:

! 1
pðzj! 1 Þ > Pð! 2 Þ
ð2:36Þ
pðzj! 2 Þ < Pð! 1 Þ
! 2
Regarded as a function of ! k the conditional probability density p(zj! k )
is called the likelihood function of ! k . Therefore, the ratio:

pðzj! 1 Þ
LðzÞ¼ ð2:37Þ
pðzj! 2 Þ
is called the likelihood ratio. With this definition the classification
becomes a simple likelihood ratio test:

! 1
> Pð! 2 Þ
LðzÞ ð2:38Þ
< Pð! 1 Þ
! 2

The test is equivalent to a threshold operation applied to L(z) with
threshold P(! 2 )/P(! 1 ).
Even if the cost function is not uniform, the Bayes detector retains the
structure of (2.38), only the threshold should be adapted so as to reflect
the change of cost. The proof of this is left as an exercise for the reader.
In case of measurement vectors with normal distributions, it is con-
venient to replace the likelihood ratio test with a so-called log-likelihood
ratio test:

! 1
>
Pð! 2 Þ
LðzÞ T with LðzÞ¼ ln LðzÞ and T ¼ ln ð2:39Þ
Pð! 1 Þ
<
! 2