Page 49 - Classification Parameter Estimation & State Estimation An Engg Approach Using MATLAB
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38 DETECTION AND CLASSIFICATION
probabilities are given conditionally with respect to the true classes, i.e.
def def
!
P miss ¼ P(^ ! 1 j! 2 ) and P fa ¼ P(^ ! 2 j! 1 ). In addition, we may define the prob-
!
def
ability of a detection P det ¼ P(^ ! 2 j! 2 ).
!
The overall probability of a false alarm can be derived from the prior
!
!
probability using Bayes’ theorem, e.g. P(^ ! 2 , ! 1 ) ¼ P(^ ! 2 j! 1 )P(! 1 ) ¼
P fa P(! 1 ). The probabilities P miss and P fa , as a function of the threshold
T, follow from (2.39):
Z T
P fa ðTÞ¼ PðLðzÞ < Tj! 1 Þ¼ pðLj! 1 ÞdL
1
1 ð2:42Þ
Z
P miss ðTÞ¼ PðLðzÞ > Tj! 2 Þ¼ pðLj! 2 ÞdL
T
P det ðTÞ¼ 1 P miss ðTÞ
In general, it is difficult to find analytical expressions for P miss (T) and
P fa (T). In the case of Gaussian distributed measurement vectors, with
C 1 ¼ C 2 ¼ C, expression (2.42) can be further developed. Equation
(2.41) shows that L(z) is linear in z. Since z has a normal distribution,
so has L(z); see Appendix C.3.1. The posterior distribution of L(z)is
fully specified by its conditional expectation and its variance. As L(z)is
linear in z, these parameters are obtained as:
1 1
T
E LðzÞj! 1 ¼ E½zj! 1 ðm þ m Þ C ðm m Þ
½
2
1
2
1
2
T
1 1
¼ m ðm þ m Þ C ðm m Þ ð2:43Þ
1
1
2
2
1
2
1 T 1
¼ ðm m Þ C ðm m Þ
2
1
1
2
2
Likewise:
1 T 1
E½LðzÞj! 2 ¼ ðm m Þ C ðm m Þ ð2:44Þ
2
2
1
1
2
and:
T 1
Var LðzÞj! 1 ¼ ðm m Þ C ðm m Þ¼ Var LðzÞj! 2 ð2:45Þ
½
½
1
2
1
2
With that, the signal-to-noise ratio is:
2
ðE½Lj! 2 E½Lj! 1 Þ T 1
SNR ¼ ¼ðm m Þ C ðm m Þ ð2:46Þ
1
1
2
2
Var½Lj! 2 
