Distributed Detection of a Nuclear Radioactive Source using Fusion of ...

Distributed Detection of a Nuclear Radioactive Source using Fusion of Correlated Decisions Ashok Sundaresan and Pramod K. Varshney

Nageswara S.V. Rao

Department of Electrical Engineering and Computer Science Syracuse University Syracuse, U.S.A. Email: asundare,[email protected]

Computer Science and Mathematics Division Oak Ridge National Laboratory Oak Ridge, U.S.A. Email: [email protected]

Abstract—A distributed detection method is developed for the detection of a nuclear radioactive source using a small number of radiation counters. Local one bit decisions are made at each sensor over a period of time and a fusion center makes the global decision. A novel test for the fusion of correlated decisions is derived using the theory of copulas and optimal sensor thresholds are obtained using the Normal copula function. The performance of the derived fusion rule is compared with that of the Chair-Varshney rule. An increase in detection performance is observed. A method to estimate the correlation between the sensor observations using only the vector of sensor decisions is also proposed.

I. I NTRODUCTION Detection of radiation from nuclear materials has become an important task due to the increasing threats from potential terrorist activities. One possible scenario is the dispersion of radioactive material using a conventional explosive device, namely the so called dirty bomb, in a densely populated area. The radioactive materials used in creating a dirty bomb are usually isotopes like Cs-137 that are widely used in industries and in hospitals for medical purposes and can be obtained with considerable ease. The radioactive materials for clandestine activities will need to be transported to the destination place. The task is to detect the low level radiations from the vehicles carrying these sources before they reach their destination. We propose a system comprising of a network of radiation counters operating collaboratively to detect the presence of a radioactive source. Such a network of sensors could be deployed at suitable places along the road side or at places like weigh stations, inspection stations, etc. Detection of radioactive sources using sensor networks has received some attention off-late. In [1], the authors examine the increase in signal-to-noise ratio obtained by a simple combination of data from networked sensors compared to a single sensor. The costs and benefits of using a network of radiation detectors for radioactive source detection are analyzed and evaluated in [2]. In [3], the authors propose a Bayesian methodology by assuming independence of sensor observations. While this work is rigorous, the independence property is not satisfied in practice since the sensor measurements are correlated based on the relative sensor locations and with respect to the source. In this work, we propose a novel distributed framework for radiation source detection and

exploit the correlation of sensor observations for improved detection performance. Due to the nature of the problem, the radioactive sensors used are expensive high precision devices requiring more battery power unlike low cost, low power sensors used in most sensor network applications [4]. This makes deployment of a large number of sensors infeasible. In this work, we assume the presence of a few sensors (typically ranging from 1-5) monitoring a region for the possible presence of a radiation source, and propose a distributed bandwidth-constrained scheme for radioactive source detection. The sensors provide radiation counts based on the intensity of the radiation, and a decision regarding the presence or absence of a radioactive source may be made based on a local threshold. The radiation counts typically follow a Poisson distribution with the parameter proportional to inverse square distance [5]. Thus, sensors that are closer to the source register higher counts compared to farther ones, and such a relationship makes the measurements correlated, and hence non-independent. But the individual sensor measurements that correspond to the margins of the joint measurement distribution are fairly well known for this problem [6]. However, in general the correlations are not a priori known and must be explicitly accounted for in combining the information from multiple sensors. The radiation source or target to be detected is assumed to be stationary over a period of time. A sequence of binary decisions (over a length of time) are made at the individual sensors and sent to a fusion center which combines them to make a final decision on the presence or absence of the radioactive source. As noted previously, if a radioactive source is indeed present, the sensor decisions at any instant of time would be correlated since all sensors observe a common random phenomenon. Fusion of correlated decisions has been studied in [7], [8]. These approaches assumed a complete knowledge of the joint distribution of the sensor observations. Such methods are feasible in special cases such as when sensor observations are realizations of multivariate Gaussian random variables. We present a novel approach using copulas to fuse correlated decisions and obtain optimal thresholds for sensor quantization. Using the copula theory, joint distribution functions can be constructed from the marginal distributions even when the observations are correlated non-Gaussian random

variables. Hence the fusion strategy described in this paper is particularly attractive in practical cases where the underlying distributions are non-Gaussian. The rest of the paper is arranged as follows. The problem is described in Section II. The design of the optimal fusion rule and individual sensor tests is considered in Section III. Experimental results are shown in Section IV. Some concluding remarks are drawn in Section V. II. P ROBLEM F ORMULATION The problem is formulated as a binary hypothesis testing problem with the H0 hypothesis indicating the absence of any radioactive source and the H1 hypothesis indicating the presence of a radioactive source. The observations received by the sensors under both hypotheses are as follows. H0 : zij = bij + wij i = 1, 2; j = 1, . . . , N H1 : zij = cij + bij + wij i = 1, 2; j = 1, . . . , N

(1)

where bij , cij and wij are the background radiation count, source radiation count and measurement noise respectively, at sensor i located at (xi , yi ) during the j th time interval. The background radiation count received during the time interval (0, t] is assumed to be Poisson distributed with known rate λb . The source radiation count at sensor i located at (xi , yi ) is assumed to be Poisson distributed with rate λci . We assume an isotropic behavior of radiation in the presence of the source so that the rate λci is a function of the source intensity A, and distance of the ith sensor from the source given by λci =

A (x0 − xi )2 + (y0 − yi )2

(2)

where (x0 , y0 ) represent the source coordinates. The measurement noise wij is Gaussian distributed with a known variance 2 . The background radiation count bij and measurement σw noise wij are assumed to be spatially and temporally independent. This implies that under the H0 hypothesis, sensor observations are independent over space and time. Under the H1 hypothesis, the sensors observe a spatio-temporal phenomenon giving rise to spatial and temporal correlation. The overall problem is solved in a distributed fashion. It consists of determining individual sensor thresholds to form sensor decisions and the fusion test to declare the global decision using the vector of sensor decisions. In this work, we assume temporal independence and while designing the system, focus on exploiting only spatial correlation between the sensors for improved detection performance. Also, in this paper, the problem is solved for a known signal case, i.e., values of source intensity A and source coordinates (x0 , y0 ) are assumed to be known. III. S YSTEM D ESIGN

Fig. 1.

A two sensor distributed detection scheme

assume that τ1 and τ2 are individual sensor thresholds used for making the one-bit decisions. Then the sensor decisions, at any time interval 1 ≤ i ≤ N , are quantized versions of sensor observations defined as ( 0 if −∞ < z1i ≤ τ1 , u1i = Q(z1i ) = (3) 1 if τ1 ≤ z1i < ∞ ( 0 if −∞ < z2i < τ2 , u2i = Q(z2i ) = (4) 1 if τ2 < z2i < ∞ Also let P r(u1i = 1|H1 ) = p1 , P r(u1i = 1|H0 ) = q1 P r(u2i = 1|H1 ) = p2 , P r(u2i = 1|H0 ) = q2 If f (z1i |H1 ) and f (z1i |H0 ) are the conditional density functions (under H1 and H0 respectively) of the ith observation received at sensor 1 (z1i ), then it can be readily seen that Z∞ p1 = f (z1i |H1 )dz1i τ1 Z∞

f (z1i |H0 )dz1i

q1 = τ1

We can define p2 and q2 in a similar manner. Under H0 , the observation received by any sensor during a particular time interval i, 1 ≤ i ≤ N is given by

A. Decision Fusion For the sake of simplicity, in this paper we will assume that two sensors are observing the common phenomenon over N time intervals each of length (0, t] (see Figure 1). Let us

zi = bi + wi where the sensor subscript has been omitted for notational convenience. It is obvious that zi follows the hierarchical

distribution [9] Λ(u) = zi ∼

2 N (bi , σw )

bi ∼ P oisson(λb )

Hence under H0 , the marginal probability density function (pdf) of zi is given by f (zi )

= =

∞ X kb =0 ∞ X

f (b = kb , zi ) f (zi |b = kb )P (b = kb )

kb =0 ∞ X

" # (zi − kb )2 exp(−λb )λkb b 1 p exp − = 2 2 2σw kb ! 2πσw kb =0 From the above equation it is obvious that f (zi ) is an infinite sum of scaled Gaussian densities. Hence f (zi ) under H0 is a Gaussian mixture distribution with the following mean and variance. E(zi ) = E(E(zi |b)) = E(b) = λb 2 var(zi ) = E(var(zi |b)) + var(E(zi |b)) = σw + λb Note that the Gaussian mixture has its components centered around the Poisson counts and weighted by the Poisson count probabilities. Components centered around count values close to the Poisson rate λb are more heavily weighted. In this paper, we approximate the Gaussian mixture distribution by a Gaussian distribution with the same mean and the variance. Similar approximations have been employed in the literature [10]. Thus, 2 f (zi |H0 ) ∼ N (λb , σw + λb ) Similarly, under the H1 hypothesis, 2 f (zi |H1 ) ∼ N (λb + λc , σw + λb + λ c )

where λc is a function of the sensor’s position relative to the source and hence may be different for sensor 1 and sensor 2. Once the pdfs of individual sensor’s observations under both hypotheses are known, we can readily evaluate   τ −λ −λ 1 b c1 (5) p1 = Q p 2 +λ +λ σw b c1  τ −λ −λ  2 b c2 p2 = Q p (6) 2 +λ +λ σw b c2  τ −λ  1 b q1 = Q p (7) 2 σw + λb  τ −λ  2 b q2 = Q p (8) 2 σw + λb where, Q(.) is the complementary cumulative distribution function of the standard Normal. Let u1 and u2 be the vector of sensor decisions, then the optimal test at the fusion center is the likelihood ratio test (LRT) given by [11]

P (u1 , u2 |H1 ) H1 ≷ γ P (u1 , u2 |H0 ) H0

(9)

Assuming temporal independence of sensor decisions, the optimal fusion statistic becomes QN P (u1i , u2i |H1 ) T (u) = Qi=1 (10) N i=1 P (u1i , u2i |H0 ) Let P (u1i P (u1i P (u1i P (u1i

= 0, u2i = 1, u2i = 0, u2i = 1, u2i

= 0|H1 ) = P00 , P (u1i = 0, u2i = 1|H1 ) = P01 = 0|H1 ) = P10 , P (u1i = 1, u2i = 1|H1 ) = P11 = 0|H0 ) = Q00 , P (u1i = 0, u2i = 1|H0 ) = Q01 = 0|H0 ) = Q10 , P (u1i = 1, u2i = 1|H0 ) = Q11

Then the joint probability mass function (pmf) of u1 and u2 under H1 and H0 is given by (1−u1i )(1−u2i )

P01

(1−u1i )(1−u2i )

Q01

P (u1i , u2i |H1 ) = P00

(1−u1i )u2i

P101i

u (1−u2i )

(1−u1i )u2i

Q101i

u1i u2i P11 (11)

u (1−u2i )

Qu111i u2i (12) Using Eq. (11) and Eq. (12) in Eq. (10), taking log on both sides and simplifying, we get P (u1i , u2i |H0 ) = Q00

log Λ(u) = C1

N X

u1i + C2

i=1

N X

u2i + C3

i=1

N X

u1i u2i

(13)

i=1

where, C1 = log

P10 Q00 P00 Q10

C2 = log

P01 Q00 P00 Q01

C3 = log

P00 P11 Q01 Q10 P01 P10 Q00 Q11

It is known that u1i , u2i and u3i = u1i u2i are each Bernoulli random variables. The success probabilities of u1i and u2i are p1 and p2 respectively under H1 and q1 and q2 respectively under H0 . Let the success probability of u3i be p3 = P11 under H1 and q3 = Q 11 under P P H0 . Under P the assumption of time independence, u1i , u2i and u3i are each Binomial distributed. Using Laplace-DeMoivre approximation [12], the optimal fusion test statistic is Gaussian distributed under both hypotheses. Let µ1 and σ12 be the mean and the variance of log Λ(u) under H1 and µ0 and σ02 be the mean and the variance of log Λ(u) under H0 . Then it can be shown that µ0 = N [C1 q1 + C2 q2 + C3 q3 ]

(14)

σ02 = N [C12 q1 (1 − q1 ) + C22 q2 (1 − q2 ) + C32 q3 (1 − q3 )] (15) µ1 = N [C1 p1 + C2 p2 + C3 p3 ]

(16)

σ12 = N [C12 q1 (1 − q1 ) + C22 q2 (1 − q2 ) + C32 q3 (1 − q3 )] (17)

The system probability of false alarm (PF A ) and system probability of detection (PD ) are now given by  γ0 − µ  1 PD = Q σ1  γ0 − µ  0 PF A = Q σ0

(18) (19)

where, γ 0 is the threshold for the fusion test. Under the Neyman-Pearson criterion, γ 0 can be obtained by constraining PF A = α as below γ 0 = σ0 Q−1 (PF A ) + µ0

(20)

For performing the test at the fusion center, we require the quantities P00 , P01 , P10 , P11 , Q00 , Q01 , Q10 , Q11 that completely specify the joint conditional pmfs of the sensor decisions u1i and u2i under both hypotheses. Because of the independence of sensor observations and hence sensor decisions under H0 , we get Q00 Q01 Q10 Q11

= (1 − q1 )(1 − q2 ) = (1 − q1 )q2 = q1 (1 − q2 ) = q1 q2

Zτ2 f (z1i , z2i |H1 )dz1i dz2i

P00 =

(21)

z1i =−∞ z2i =−∞

Zτ1

Z∞

P01 =

f (z1i , z2i |H1 )dz1i dz2i

(22)

f (z1i , z2i |H1 )dz1i dz2i

(23)

z1i =−∞ z2i =τ2

Z∞

Zτ2

P10 = z1i =τ1 z2i =−∞

Z∞

Z∞ f (z1i , z2i |H1 )dz1i dz2i

P11 =

Recently a lot of progress has been made in the study of copulas and their applications in statistics. Copulas are basically functions that join or couple multivariate distribution functions to their one-dimensional marginal distribution functions [13]. Another definition of copulas states that they are joint distribution functions of uniform distributed random variables. The role of copulas in relating multivariate distribution functions and their univariate marginals is explained by Sklar’s theorem [13], [14], which is stated as follows. Sklar’s Theorem: Consider an m-dimensional continuous distribution function F with continuous marginal distribution functions F1 , . . . , Fm . Then there exists a unique copula C, such that for all x1 , . . . , xm in [−∞, ∞] F (x1 , x2 , . . . , xm ) = C(F1 (x1 ), F2 (x2 ), . . . , Fm (xm )) (25) Conversely, consider a copula C and univariate cdfs F1 , . . . , Fm , then F as defined in Eq.(25) is a multivariate cdf with marginals F1 , . . . , Fm . As a direct consequence of the above theorem ,we obtain by differentiating both sides of Eq.(25), m Y  f (x1 , . . . , xm ) = f (xi ) c(F1 (x1 ), . . . , Fm (xm )) (26) i=1

However, under H1 , the observations are correlated and the joint pmf of the sensor decisions cannot be evaluated in a straightforward manner. Under H1 , the probabilities P00 , P01 , P10 , P11 need to be calculated as follows. Zτ1

B. Copula Theory

(24)

z1i =τ1 z2i =τ2

Notice that the joint distribution of the sensor observations under H1 (f (z1i , z2i |H1 )) is required to calculate the probabilities P00 , P01 , P10 , P11 . It is known that the marginals f (z1i |H1 ) and f (z2i |H1 ) are Gaussian. However, a conclusion about the joint density of z1i and z2i under H1 cannot be made directly since z1i and z2i do not originate from a bi-variate Gaussian distribution. Here we employ the copula theory to construct the joint distribution of z1i and z2i under the H1 hypothesis.

where, c is termed as the copula density given by c(k) =

∂ m (C(k1 , . . . , km )) ∂k1 , . . . , ∂km

(27)

where, ki = Fi (xi ). Thus, we can construct a joint density function with specified marginal densities by employing Eq.(26). The choice of a copula function to construct the joint density is an important consideration here. Various families of copula functions exist in the literature [13], [14]. However, it is not very clear as to which copula function should be used in which case. It is conjectured by many authors that the use of different copula functions may exhibit different dependence behavior among the random variables. In this work, we make use of the Normal (or Gaussian) copula to construct the joint density function of the sensor observations under the H1 hypothesis. The use of other copula functions is under investigation. The copula density for a Normal copula can be obtained easily from Eq.(26) as below. c(φ(x1 ), . . . , φ(xm )) =

Φ(x1 , . . . , xm ) φ(x1 ), . . . , φ(xm )

(28)

where, Φ is the multivariate Normal density function and φ is the univariate normal density function. The Normal copula incorporates the dependencies among the random variables in a manner exactly similar to the way a multivariate Normal distribution does by using the covariance matrix. Hence, to use the Normal copula in constructing a joint distribution, the linear correlation coefficients between the random variables are needed. In some cases, the correlation between the random

variables may be available in advance or evaluated from a correlation model. Otherwise, they need to be estimated from the data. This will be elucidated in more detail in Section IV. Making use of the Normal copula density(see Eq.(28)) to construct the joint distribution of the sensor observations under H1 , we get f (z1i , z2i |H1 ) =f (z1i |H1 )f (z2i |H1 ) cg (FZ1i |H1 (z1i ), FZ2i |H1 (z2i ))

Starting from bi-variate Normal, bi-variate Poisson random variables with mean equal to λc are generated (using the probability integral transform and then applying the inverse Poisson distribution function) with varying values of correlation (ρs ). Due to the effect of background radiation and measurement noise, the correlation between the sensor observations (ρz ) at any time instant i is reduced and is given by

(29) ρz =

where, cg (u, v) is the Normal copula density evaluated from Eq.(28). On simplifying Eq.(29), f (z1i , z2i |H1 ) is found to be nothing but the bi-variate Gaussian density as expected. It is important to note here that the use of a different copula function would not have resulted in the bi-variate Gaussian density but might have still served our purpose of determining the joint probabilities. Using the determined joint density function of the observations under H1 , expressions for the probabilities P00 , P01 , P10 , P11 can be obtained by using Eqs.(21)-(24). C. Optimal Threshold for Local Sensors In Section III-A, we assumed that τ1 and τ2 are individual sensor thresholds. It can be seen that PD and PF A given by Eq.(18) and Eq.(19) are functions of τ1 and τ2 . Constraining PF A = α, PD can be written as

ρs λ c λc + λ b + σ w 2

For each value of ρs and consequently ρz , the expression for PD as a function of τ1 and τ2 by constraining the value of PF A is obtained from Eq.(30) and the same is maximized w.r.t τ1 and τ2 to obtain the optimal sensor thresholds. The maximum value of PD is noted. The same is repeated for various values of PF A and the ROC is generated. The experiment is repeated for different values of ρs and the results are plotted as shown in Figure 2. For comparison purposes, we also evaluated the detection performance of the system that assumes independence of sensor decisions under the H1 hypothesis. Under the conditional independence assumption, the term C3 in Eq.(13) becomes zero and the optimal test statistic reduces to log Λ2 (u) = C1

N X i=1

σ0 (τ1 , τ2 )Q−1 (α) + µ0 (τ1 , τ2 ) − µ1 (τ1 , τ2 ) ) σ1 (τ1 , τ2 ) (30) The sensor thresholds are chosen to maximize PD at a particular value of PF A . Hence the optimal sensor thresholds are given by

PD (τ1 , τ2 ) = Q(

(τ1∗ , τ2∗ ) = arg maxPD (τ1 , τ2 )

(31)

τ1 ,τ2

For the results shown in this paper, a search algorithm is used to perform the above optimization. IV. E XPERIMENTAL I NVESTIGATIONS As mentioned previously, in this work we present results for the case when the count rate of the source at each sensor (λci ) is known. The count rate is determined by using the following source parameter values: A = 10, (x0 , y0 ) = (10, 10). It is assumed that both sensors are located equidistant from the source at a distance 4 units from the source resulting in equal values of λci (λc1 = λc2 = λc = 0.625). A mean background radiation with count λB = 10 and measurement noise with 2 variance σw = 10 is considered. It is assumed that the sensors are observing the phenomena over N = 100 time intervals each of length one second. A. Known Correlation Case In this case, we assume that the correlation between the random variables z1i and z2i is known. Spatial correlation functions modeling the correlation between two sensors as a function of the distance between them exist in literature [15] and may be used in this case.

(32)

u1i + C2

N X

u2i

(33)

i=1

which is nothing but the Chair-Varshney test statistic [16]. Also, the joint probabilities of the sensor decisions under the H1 hypothesis are now given by, P00 P01 P10 P11

= (1 − p1 )(1 − p2 ) = (1 − p1 )p2 = p1 (1 − p2 ) = p 1 p2

Using the Laplace-DeMoivre approximation [12], the ChairVarshney test statistic is also Gaussian distributed (assuming time independence) whose mean and variance under either hypothesis can be calculated from Eqs.(14)-(17) by substituting C3 = 0. PD and PF A for the Chair-Varshney statistic, as functions of τ1 and τ2 , are obtained and the detection performance of the Chair-Varshney statistic is also evaluated in the same manner as described afore (see Figure 2). From Figure 2, it is clear that the detection performance is improved by taking correlation into account. The proposed approach is able to do much better than the Chair-Varshney fusion rule even at low values of correlation. The increase in PD is noticeable especially at lower values of PF A which is desirable. As expected, the detection performance increases with increase in signal correlation. B. Unknown Correlation Case In this case, the correlation between z1 and z2 needs to be estimated first before the pdf of the test statistic given by Eq.(13) can be evaluated under either hypotheses. A two step procedure is adopted here and is described as follows.

Fig. 2.

Detection Performance for Known Correlation Case

First the sensor thresholds are obtained by assuming the sensor decisions to be independent under H1 and using the Chair-Varshney test statistic as detailed in the previous section. The dependence between the sensor observations is evaluated (estimated from the decision vectors (u1 and u2 ) using a nonparametric rank correlation measure, Kendall’s τ [13]. Given a vector (a, b) of observations from a bi-variate random vector (A, B), Kendall’s τ is defined ([13]) as the ratio of the difference in the number of concordant pairs (c) and discordant pairs (d) to the total number of pairs of observations, i.e., c−d kτ = (34) c+d (ai , bi ) and (aj , bj ) are said to be concordant if (ai − aj )(bi − bj ) > 0 and discordant if (ai −aj )(bi −bj ) < 0. An interesting property of the Kendall’s τ correlation measure is that it remains invariant under non-decreasing transformations of the original data. Because of this property and from Eq.(3) and Eq.(4) we can infer that Kendall’s τ between u1 and u2 is equal to that between z1 and z2 . Once Kendall’s τ (kτ )between z1 and z2 is known, the linear correlation coefficient between z1 and z2 is given by [14]  πk  τ ρ(z1 , z2 ) = sin (35) 2 In the second step, using the correlation estimated in the first step the pdf of the optimal test statistic (see Eq.(13)) is obtained under both hypotheses. The same procedure described in Section IV-A is then carried out to determine optimum sensor thresholds that maximize PD at a given value of PF A . The results of our simulation are shown in Table 1. The results are shown for the case when A = 10, λc1 = λc2 = 0.625, 2 λB = 10 and σw = 10. The correlation between the source signal received at the two sensors ρs = 0.9. A degradation in performance is noticeable compared to the known signal case. This is expected since the correlation is being estimated from a sequence of quantized data. However, the use of correlation

still has increased the detection performance compared to the Chair-Varshney test especially for low PF a values. Note that the length of time over which decisions are obtained plays an important role here. The longer the observation time better will the estimate of the correlation between the sensor observations. V. C ONCLUSION A distributed scheme using a network of two sensors for detection of a nuclear radioactive source was developed. A new fusion test taking correlation of sensor decisions was developed and used to determine optimal sensor quantization thresholds. The performance of the proposed scheme was compared to the Chair-Varshney test which assumes conditional independence of sensor decisions. The proposed scheme is able to achieve a better detection performance than the ChairVarshney fusion rule. Our future work will consist of investigating methods to estimate the covariance matrix of the Normal copula function by utilizing training data obtained by generating a set of measurements under both hypotheses. In our case, these measurements can be generated in two different ways: a) Poisson and Gaussian distributions can be used to generate cij , bij and wij , which will be added to provide zij as in Eq. (1), or b) the intensity level λci can be computed based on the source parameters, and radiation sensors can be simultaneously subjected to radiation levels λci and λb in a controlled laboratory environment. The output of the sensors can taken as random samples of (z1 , z2 ). In future work, we will also consider the use of other copula functions to construct the joint density of sensor observations. Also, a more general detection problem in the event of unknown source location parameters will be considered.

ACKNOWLEDGEMENT This material is based on work supported by UT Battelle, LLC Subcontract number 4000053980, with funding originating from Department of Energy Contract Number DE-AC0500OR22725. R EFERENCES [1] R.J. Nemzek, J.S. Dreicer, D.C. Torney and T.T. Warnock, “Distributed sensor networks for detection of mobile radioactive sourcesr,” IEEE Trans. on Nuclear Science, vol. 15, no. 4, pp. 1693–1700, Aug. 2004. [2] D.L. Stephens,Jr. and A.J. Peurrung, “Detection of moving radioactive sources using sensor networks,” IEEE Trans. on Nuclear Science, vol. 51, no. 5, pp. 2273–2278, Oct. 2004. [3] S.M. Brennan, A.M. Mielke and D.C. Torney, “Radioactive source detection by sensor networks,” IEEE Trans. on Nuclear Science, vol. 52, no. 3, pp. 813–819, June 2005. [4] G.F. Knoll, Radiation Detection and Measurement. John-Wiley, 2000. [5] R.E. Lapp and H.L. Andrews, Nuclear Radiation Physics. PrenticeHall, New Jersey, 1948. [6] D. Mihalas and B.W. Mihalas, Foundations of Radiation Hydrodynamics. Courier Dover Publications, 2000. [7] E. Drakopoulos and C.C. Lee, “Optimum multisensor fusion of correlated local decisions,” IEEE Trans. on Aerospace and Elect. Syst., vol. 27, no. 4, pp. 593–605, July 1991. [8] M. Kam, Q. Zhu and W.S. Gray, “Optimum data fusion of correlated local decisions in multiple sensor detection systems,” IEEE Trans. on Aerospace and Elect. Syst., vol. 28, pp. 916–920, July 1992. [9] G. Casella and R.L. Berger, Statistical Inference. Duxbury Press, Belmont, CA, 1990. [10] Y. Bar-Shalom and X.R. Li, Multitarget-Multisensor Tracking: Princliples and Techniques. YBS, Storrs, CT, 1995. [11] P.K. Varshney, Distributed Detection and Data Fusion. Springer-Verlag, New York, 1997. [12] A. Papoulis, Probability, Random Variables and Stochastic Processes. McGraw-Hill, New York, 1984. [13] R.B. Nelsen, An Introduction to Copulas. Springer-Verlag, New York, 1999. [14] T. Schmidt, Coping with Copulas. Risk Books, 2007. [15] M.C. Vuran, O.B. Akan, I.F. Akyildiz, “Spatio-temporal correlation: theory and applications for wireless sensor networks,” Computer Networks, vol. 45, Issue 3, pp. 245–259, June 2004. [16] Z. Chair and P.K. Varshney, “Optimal data fusion in multiple sensor detection systems,” IEEE Trans. on Aerospace and Elect. Syst., vol. 22, pp. 98–101, Jan 1986.