Approaches for Signal Detection ● Bayesian Framework Neyman-Pearson Framework
Bayesian Framework Neyman-Pearson Framework
Bayesian Framework Phenomenon e Letu= fu1 N Bayes risk: Total average cost of y y making decisions S-1 R=∑∑CPP( Decide1 Is present Fusion center where, Pr(Decide H: H, is present)=>Pr(uo=iju) Pr(u Hi) Then, the optimal fusion rule is given by the map (maximum a posteriori probability rule Pr(uHi Hi Po(c 之 10-C Pr(uHo 01
Let . Bayes Risk: Total average cost of making decisions where, Then, the optimal fusion rule is given by the MAP (maximum a posteriori probability) rule Phenomenon S-1 S-2 S-3 S-N Fusion Center y1 y2 y3 yN u0 u1 u2 u3 uN
Bayesian Framework(cont. Let c1o 11 00=0 Then, for a given set of sensor quantizers, MAP rule can be simplified as follows u: 10g +(1-,)og,M Fi <10g7 u=0 e For identical sensors, the above fusion rule simplifies to a"Kout OfN”rule If ly1,., yN is conditionally independent, then the optimal sensor decision-rules are likelihood-ratio tests A Pr(yilI
Let C10 = C01 = 1, C11 = C00 = 0. Then, for a given set of sensor quantizers, MAP rule can be simplified as follows. For identical sensors, the above fusion rule simplifies to a “K out of N” rule. If {y1 , … , yN} is conditionally independent, then the optimal sensor decision-rules are likelihood-ratio tests. log . 0 0 1 0 1 (1 )log 1 log 1 = = − + − − = u u Fi P Mi P i u Fi P Mi P i u N j
Neyman-Pearson Framework Maximize Probability of Detection under Constrained Probability of false arm max Pn s.t.Pg≤a Under the conditional independence assumption, the optimal local sensor decision rules are likelihood ratio based tests The optimal fusion rule is again likelihood ratio test A(u2x is chosen such thatP=d
Maximize Probability of Detection under Constrained Probability of False alarm Under the conditional independence assumption, the optimal local sensor decision rules are likelihood ratio based tests. The optimal fusion rule is again likelihood ratio test is chosen such that P ' F = max PD s.t. PF ≤ 𝛼′
Asymptotic results It has been shown that the use of identical thresholds is asymptotically optimal o Asymptotic performance measure: N-P Setup: Kullback-Leibler distance(KLD D1(x) D(pIllo)=pI(a)log da po( Bayesian Setup: Chernoff Information C(m,m)=max-l0g(0.,1()=/mn()2=m1(a)d 0<t<1
It has been shown that the use of identical thresholds is asymptotically optimal. Asymptotic performance measure: N-P Setup: Kullback-Leibler distance (KLD) Bayesian Setup: Chernoff Information