Optimum Data Fusion Rules In Distributed Sensor Detection Systems

Ashraf Mamdouh A. Aziz and Gama1 Mabrouk A. Hamid Egyptian Anned Forces

n Abstract

The problem of decision fusion in distributed sensor systems is considered. Distributad sensors pass their decisions to a fusion center that combines the received decisions from the various sensors into a final global decision. The case where only two sensors are combined ushg an AND fusion rule was malywd by Kovattana, and the cases where two and three sensors are combined using AND and OR fusion rules were analyzed by Fe@=+ who claim€xl that OR was superior to AND. Stearns considered the case of combining two sensors using AND and OR h i o n rules and refused Fefjds claim about the superiority of OR. Stearns claimed that the receiver operating characteristics of the AND and the OR combiners must be intersected and AND was superiar to OR at low false- alarm probabilities and OR was superior to AND at high falsealarm probabilities. In this paper, a generalized fom in case of more than two sensors is derived. It also shows that the optimal fusion rule does not only depend on the desired false a h probability and the signal to noise ratio, but also depends on the probability distribution function.

1 Introduction

The problem of multiple-sensor in surveillance purposes has attracted the attention of several investigators. This interest has been sparked by the requirement of military surveillance systems to be more reliable and immune to electronic attsck than single sensor systems. This can be achieved by properly combining the infonnation obtained from H various sensors. The sensors are allowed to derive local decisions ui's, i = 1,2, ......., n and a fusion center is

responsible for combining the received decisions into a find global decision U,-,. According to Neyman-Pearson criterion, it is required to find the optimal fusion rule of the fusion center that maximizes the global detection probability PD for a desired global false alarm probability PF .

The usual fusion rule is implemented as k-out of-$n$ majority-rule voting. This means that the fusion center adopts hypothesis RI (presence of a target) as the true hypothesis when at least k-sensor favore that hypothesis. TWO

special cases are AND fusion rule, corresponding to setting k=n, and OR fusion rule, corresponding to setting k=l . The AND combiner was analyzed by Kovattana [l] for the case of two sensors and the global performance was obtained assuming event conditional independence. The AND and The OR combiners were later compared by Fefjar [2] for the cases of two and three sensors. Fefjar claimed that OR &ion rule was always superior to AND fusion rule. Stearns [3] refused FefJafs claim and claimed that depending on the choice of the global false alarm probability, either AND or OR combining can be better. Stearns claimed that the AND and the OR global performances must be intersected and AND was superior to OR at low false-alann probabilities and OR was superior to AUD at high falsealarm probabilities. Both Feqds and Steams's claims are not always me. It will be shown that, depending on the operating point on the global receiver operating charscteristic ROC and on the pamneters of the probability distributions urider hypothesis, either AND or OR can be better Bt either low or high global false alarm probabilities. Furthermore, the AND and the OR combiners may not be intersected at all,

Twenty Second National Radio Science Conference (NRSC 2005) -1 - March 15-17,2005, Cairo-Egypt

2 Optimal Data Fusion Using Two Non-Identical Sensors

Consider any two sensors with ROCs, p d l = g(Pfl) and pd2 = h i p f 2 ) . To derive the opti" threshold

setting for each sensor, we will assume that the sensofs decisions are conditionally independent given the hypothesis. h case of AND fusion d e , the global false alarm and detection probabilities are given by :

P F = P f l P f 2 3 (1)

p D = P d l P d 2 * (2) equation (2) can be rewritten as :

Substihdon for p

PD = d P f l ) N P f ; ) -

p a w = d P f 1 ) M-) 9

(3) from-(]), we obtain f 2

(4) PF

where the dependence PD(PF) is called the global ROC of the considered disiributed detection system. Clearly from (2), p f l > PF , and (3) will be

where P F - P j - 1

l-pfl a=

3 Optimal Data Fusion Using Ideatical Sensors

If the two sensors have identical ROCs, i.e. g ( P f j 1 = h(Pf* 1 9

then symmetry can be invoked to yield the following results AND: P D ( P F ) = g 2 ( f i ) ,

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where OR: fD(PF)=Zg( f l ) -g2 ( f l ) ,

f l=l-JT-PF. The extension of the two non-identical sensors case to the case of three or more non -identical sensors is complicated. However, the special case of n-identical sensors is important since the global optimization of the whole distributed detection system (the threshold's setting plus the fusion center) is straightforward. We now extend the above results to an n-sensor case [5]-[7]. The PF and the PD are given by:

where

(16) c y = - . n!

i!(n - i ) ! For k=l, the optimum fusion rule reduces to an OR fusion rule, while for k=n it becomes an AND fusion rule. For a specified value of PF, there is an optimum integer k that maximizes the PD. This is called k-out of-n fusion rule. This means that if k or more sensors decide hypothesis H I , then the global decision will be HI, i.e.

where the ui 's, i = 1,2, ......., n are the individual sensor decisions, and uo is the global decision of the fusion center.

4 Performance Optimization Examples

Stearns [3] considered the case of combining two identical sensors with Gaussian distributed observations (the case of detecting a constant in Gaussian noise). His claim about the intersection of the AND and the OR combiners and the superiority of the AND at low f a h d m probabilities and the OR at high false d m probabilities are valid in the particular example which he had considered. In general, this is not true. Here we consider the case of exponentially distributed observations; the cases of n-identical sensors and two non-identical sensors are considered:

S ( V ~ I H o ) = w ( - Y ~ ) 5 (18) f ( y j 1 H1)=AZi exp(-di y i ) , Ai > 0, yi > 0, i = 42 ,....., n ,

f ( J J i I H j ) = o , yj < 0, i = 1,2 ,.... ) n, j = 0'1 *

(19) (20)

Notice that the observations are independent when conditioned on H. The Likelihood ratio of the detectors is given by:

Twenty Second National Radio Science Conference (NRSC 2005)

March 1517,2005, Cairo-Egypt m]

which reduces to

1 4 if yj2- U-)

1 Ai

1 -d j di

1 if y i c - In(-), di > 1 , I -di dj

The corresponding false alarm and detedon probabilities m given by:

Figure 1 compares the ROCs of the OR andthe AND combiners as well FS the individual sensot's ROCs in case of two non-identical sensors with $ 4 . 4 and d2 -2.5. The global performance improvement of the OR and the AND fusion rules over the individual sensor's ROCs is obvious. Figure 1 shows an hkrsection of the OR and the

> Twenty Second National Radio Science Conference (NRSC ZOOS) W I March 15-17,2005, CabEgypt.

AND ROCs. In this figure, the AND combiner is superior M OR at low PF and the OR combiner is superior to AND at high PF . The choice between them is determined by the specified PF . Figures 2-5 compare the OR and the AND combiners for the case of two identical sensors with exponentially distributed observations for different values of coefficient d (let dl = d2 = d ). Figures 2 and 4 consider the case when d < I while Figures 3 and 5 consider the case when d > 1 ' . It is clear that when d c 1 , the OR combiner is optimum for all values of PF s, while when d > 1, the AND combiner is optimum for all values of PF s. It is also clear that there are no intersections of the OR and the AND ROCs in these figures.

Figures 6-9 compare the global ROCs of the k - out of - n h i o n niles for different values of d for the case of n- identical sensors. Figures 6 and 8 show the case when d < 1 in case of three and five identical senson, and Figures 7 and 9 show the case when d > 1. Figures 6-9 lead to the same conclusions as above.

From Figures 2-9, it is clear that the optimum fusion rule depends only on the coefficient of the probability distributions d . It is also clear that depending on the coefficient d , either the AND or the OR combiner can be superior for all values of PF s. Furthermore, it is clear that the ROCs of the OR and the AND combhers may not be intersected. From the above results, we remark that (1) the AND combiner is not always optimum at low PF s, (2) the OR combiner is not always optimum at high PF, (3) the choice between the AND and the OR combiners depends on the desired values of PF s as well as the coefficient of the probability distributions ( d in OUT examples), and (4) the ROCs of the OR and the AND combiners may not be intersected at all.

5 Conclusion

Optimum fusion rules for multiple sensor distributed detection systems have been considered. For the case of exponentially distributed observations, the optimal fusion rule is based on the value of d . For d el, the OR fusion rule is found to be optimal whereas the AND fusion rule is optimal for d >1. The choice between them depends on the desired global false alarm probability as well as the parameters of the probability distributions under both hypotheses. Furthermore, the global ROCs of the OR and the AND combhers may or may not be intersected. This again depends on the coefficients of the probability distribution fimction.

References

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Twenty Second National Radio Science Conference (NRSC 2005) Markh 1547,2005, CairolEgypt

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Twenty Second National Radio Seienoe Conference (NRSC 2005) -1 T

March 1547,2005, Cairo-Egypt

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