Distributed Detection

Module by: Don Johnson

So far, we have derived a detector's computational structure for a given problem from the likelihood ratio. In some cases, we need to structure the detector to meet real-world constraints of how we make observations and where decisions must be made. One particular structure that has received much attention is the distributed structure shown in Figure 1.

Here, observations that reflect the same model are made at several locations, what we call sensors. Let the number of sensors be NN and let each sensor's observation be of the form ℳ i : r n = s i + n n , n∈0…N-1 ℳ i : r n s i n n , n 0 … N 1 with the noise at each sensor statistically independent of the others. Each sensor makes a decision about its local observations and sends these hard decisions to "headquarters," the fusion center, which assimilates all the decisions and makes a final decision as to which model applies best. This decision structure is optimal: examples can be easily found where a smaller probability of error would result if each sensor sent its likelihood ratio rather than its decision. We want to understand the consequences and design of this simpler but suboptimal structure.

Let μ n ∈01 μ n 0 1 , n∈1…N n 1 … N denote each sensor's decision (however it is made) as to which model described its observations best. The fusion center makes its decision based on the likelihood ratio formed from the sensor decision variables. Because each sensor's observations and their decision processes are statistically independent, the log likelihood ratio takes on a simple form. logΛμ= ∑nlogP μ n | ℳ 1 P μ n | ℳ 0 ≷ ℳ 0 ℳ 1 logη Λ μ n n P μ n | ℳ 1 P μ n | ℳ 0 ≷ ℳ 0 ℳ 1 η Here P μ n | ℳ 1 P μ n | ℳ 0 = P D ( n ) P F ( n ) if μ n =11- P D ( n ) 1- P F ( n ) if μ n =0 P μ n | ℳ 1 P μ n | ℳ 0 P D ( n ) P F ( n ) μ n 1 1 P D ( n ) 1 P F ( n ) μ n 0 where P D ( n ) P D ( n ) , P F ( n ) P F ( n ) are the n th n th sensor's detection and false-alarm probabilities, respectively. Assume for the moment that all sensors have identical performance: P D ( n ) = P D P D ( n ) P D and P F ( n ) = P F P F ( n ) P F . If kk denotes the number of sensors choosing ℳ 1 ℳ 1 , which equals ∑n μ n n n μ n , the log likelihood ratio test becomes klog P D P F +N-klog1- P D 1- P F ≷ ℳ 0 ℳ 1 logη k P D P F N k 1 P D 1 P F ≷ ℳ 0 ℳ 1 η klog P D 1- P F P F 1- P D ≷ ℳ 0 ℳ 1 logη+Nlog1- P F 1- P D k P D 1 P F P F 1 P D ≷ ℳ 0 ℳ 1 η N 1 P F 1 P D Hopefully, P D > P F P D P F , which means that P D 1- P F P F 1- P D >1 P D 1 P F P F 1 P D 1 ; in this case, we can bring the logarithm of this term to the right side without changing the sense of the inequality. We find the optimal decision rule to be k ≷ ℳ 0 ℳ 1 γ k ≷ ℳ 0 ℳ 1 γ with kk, the number of sensors pronouncing ℳ 1 ℳ 1 , as the sufficient statistic. To calculate performance probabilities and to find γγ, we only need that kk has a binomial distribution. Prk=NkPk1-PN-k k N k P k 1 P N k where according to ℳ 0 ℳ 0 , P= P F P P F , and according to ℳ 1 ℳ 1 , P= P D P P D .

An issue that must be addressed is whether any threshold setting can result in better performance than can be obtained with a single sensor. Consider the false-alarm probability. Prk>γ| ℳ 0 =∑k=⌈γ⌉NkPk1-PN-k ℳ 0 k γ k k γ N k P k 1 P N k If N=2 N 2 , this false-alarm probability is either 11, 1-1- P F 2 1 1 P F 2 , or P F 2 P F 2 . Only in the latter case is the distributed system's false-alarm probability smaller. With this threshold, however, its detection probability is P D 2 P D 2 , which is smaller than that for one sensor. If one plotted the entire ROC curve for the two-sensor case, the point P F P D P F P D would not lie under it. For N≥3 N 3 , the situation improves; superior performance can be obtained using distributed detection.

A sublty is choosing each sensor's decision rule. So far, we have assumed that with identically distributed observations, each sensor uses the same decision rule. Presumably we should use a likelihood ratio test; but should every sensor uses the same threshold? If not, sensors yield different false-alarm and detection probabilities, which complicates the analysis. Because of the following example, non-uniform choices for detection thresholds can yield superior performance.

This example illustrates two points. First of all, choosing the sensor's detection threshold to optimize system performance depends on choices made at the fusion center. Secondly, different as well as identical sensor thresholds can be optimal. Consequently, making sweeping assumptions about distributed decision rules can lead to suboptimal performance. What has been shown is that as the number of sensors increases, setting identical sensor thresholds will yield optimal performance.

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This work is licensed by Don Johnson under a Creative Commons Attribution License (CC-BY 1.0).

Last edited by Charlet Reedstrom on Aug 22, 2003 11:14 am GMT-5.