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Networked Decision Making for Poisson Processes: Application to nuclear detection

This paper addresses a detection problem where several spatially distributed sensors independently observe a time-inhomogeneous stochastic process. The task is to decide between two hypotheses regarding the statistics of the observed process at the end of a fixed time interval. In the proposed method, each of the sensors transmits once to a fusion center a locally processed summary of its information in the form of a likelihood ratio. The fusion center then combines these messages to arrive at an optimal decision in the Neyman-Pearson framework. The approach is motivated by applications arising in the detection of mobile radioactive sources, and offers a pathway toward the development of novel fixed- interval detection algorithms that combine decentralized processing with optimal centralized decision making.


I. INTRODUCTION
Decision making is crucial in translating information to action. Human decision makers can of point processes [28]- [31]. A realization of a point process is a random sequence of points, each representing the time and/or spatial location of an event. A point process can be characterized in terms of its intensity which corresponds to the rate at which events occur. Beyond nuclear measurement, typical examples of such processes include customers to and from a service facility in queueing theory [28], [32], electron emission from a photodetector in optical communications systems [33], generation of electrical pulses in neurons [34], and others. Of special interest in nuclear detection are Poisson processes which provide the natural models describing the emission and measurement of radiation [24], [25], [35].
In regards to point processes, the problem of decision making between two alternative hypotheses ("all clear" versus "alarm") has been addressed in [28], [36], [37]; the solution typically involves the computation of a likelihood ratio, whose comparison against a threshold provides the decision. Error probability bounds for such decision problems are studied in [38], while robust decision making (in the presence of modeling uncertainty) is explored in [39]. Decision problems with time-inhomogeneous point processes also arise in optical communications [33], [40]. In sum, for the classical (single observer) case, decision theory for general point processes is wellunderstood. However, the realization of these results in a network setting requires care. Indeed, the likelihood ratios in [28], [36] involve intensities computed on the basis of all accumulated information, 1 necessitating a modicum of caution in a setting such as ours where much of the computation and most of the raw data are decentralized (See Remark 2).
The problem of detecting (moving and stationary) radioactive sources using networks of sensors has received a fair bit of attention in the literature. In situations where the parameters (location, trajectory, activity) of the source are unknown, Bayesian methods are frequently used [13], [24], [25], [41], embedding the issue of detection in a parameter estimation problem. While powerful, Bayesian methods for source parameter estimation exhibit computational complexity exponential in the number of parameters estimated, posing challenges for their implementation in real time for networks with more than ten nodes [25], [41]. An important insight-and one that serves as the starting point for our analysis-is that in many cases of interest, the problem of source localization can be decoupled from the problem of source detection. Indeed, there are improved methods [42]- [44] for tracking the carrier of a potential radioactive source using 1 Strictly speaking, the intensity is a conditional rate at which events occur (conditioned on available information). sensor modalities other than a Geiger counter. Armed with this observation, source detection reduces to the problem of deciding whether the counts observed by a spatially distributed network of radiation sensors correspond solely to background radiation, or whether they also include emission from a radioactive source with known parameters. In this setting, [25] explores the Signal-to-Noise Ratio (SNR) resulting from the combination of data from a network of radiation sensors, allowing for spatially varying background rates. The analysis is restricted, however, to uniform linear source motion and does not provide a decision test. The costs and benefits of using networked sensors for moving sources, together with a threshold test (based on the total number of recorded counts) are addressed in [45], assuming uniform background and constant geometry between source and sensor. 2 For the case of a stationary source and correlated sensor measurements, a distributed detection scheme is developed in [35] using the theory of copulas. The work in [41] studies detection (via Bayesian estimation) for a moving source, but the motion is required to be linear with constant velocity. Detection and parameter estimation for an unknown number of static radioactive point sources are treated in [13], [24]. Evidently, the networked detection problem for general source motion with spatially varying background intensity has yet to be studied. Motivated by the above, we pose the following problem: a spatially distributed sensor network observes-over a fixed-time interval-a time-inhomogeneous point process which is known a priori to be governed by one of two intensities. How should the local information be processed and communicated through the network in order to reach a reliable decision regarding which intensity governs the observed process? With respect to the spectrum of approaches from centralized to decentralized, we take in this paper an intermediate approach that combines the significantly lower communication cost of decentralized processing (not decision) with the enhanced accuracy of centralized decision making. For the case of a vector of Poisson processes whose intensities explicitly depend on time, 3 we develop an optimal-in the Neyman-Pearson sense-decision-making scheme that combines decentralized processing (local processing at each individual sensor) with centralized decision making via a fusion center. In particular, assuming that the relative motion between the suspected source and the sensors is deterministic and known, and that the sensor observations are conditionally independent, our method relies on the sensors communicating processed information in the form of locally-computed likelihood ratios to the fusion center. The fusion center then combines these messages to arrive at a decision, without the need for any additional information such as the location or the raw data of individual sensors.
As applied to radiation detection, our framework allows us to consider arbitrary continuous source motion in any number of dimensions allowing for sensor mobility and spatially varying background rates. In relation to sensor networks, the time-inhomogeneity in our problem leads to non-identically distributed sensor observations; this marks a departure from the frequently used independent identically distributed (i.i.d.) assumption.
The paper is organized as follows. In Section II, we state the problem and our technical assumptions. The main result (Theorem 1), which indicates how a global likelihood ratio test can be formulated based on local computation of sensor-specific likelihood ratios, is presented in Section III. The proof of this result and the supporting technical material are found in Section IV. Based on this analysis, we offer conservative lower and upper bounds on the probabilities of detection and false alarm, respectively, in Section V. Finally, a numerical example of a one-dimensional case of networked nuclear detection is developed in Section VI, highlighting the benefits of using multiple sensors. The results provided in this paper can be viewed as a building block toward a general decision-making framework that leverages networks of mobile sensor platforms to enhance detection capability in problems that involve time-inhomogeneous point processes.

II. PROBLEM STATEMENT AND ASSUMPTIONS
Consider a collection of sensors observing a point process generated by some physical phenomenon. The goal is to decide between two hypotheses regarding the state of the environment.
To this end, each sensor communicates a processed version of its observations to a fusion center, which combines all received messages to a binary decision (Fig. 1). With an eye towards applications such as detection of mobile radioactive sources where the point process can be observed only for a limited time, we require here that the decision be made within a fixed time interval. The problem is formulated as a binary hypothesis test based on measurements from an array of k sensors connected in a parallel network architecture. Figure 1 shows a realization of  Sensor observations are modeled as inhomogeneous Poisson processes, whose intensities are deterministic functions of time. In particular, for 1 ≤ i ≤ k, sensor i observes a Poisson process, whose time-dependent intensity under The explicit dependence of the intensities on time arises from the known motion of the target, which is a potential "source" of the point process with intensity ν i . Thus sensor observations are independent but not identically distributed. This is in agreement with the physics of the motivating application, since a gamma ray emitted from the source cannot pass through more than one sensor simultaneously (see Fig. 1), and the time varying nature of the distance between May 3, 2014 DRAFT source and sensor changes the arrival statistics on the sensor side [25].
The following assumptions will be imposed on β i and ν i .
The physical significance of these assumptions will become clear in Section VI.
The detection problem can now be summarized as follows: Suppose that T > 0 is the decision time; that is, the time by which a decision must be made. Given a single realization of a k-dimensional vector of Poisson processes over the time horizon [0, T ] (the k components corresponding to the k sensors), decide whether the intensities are given by the collection β i (·) or by the collection

III. MAIN RESULT
In this section, we state our main result, Theorem 1. For a collection of sensors with a fusion center, configured in a parallel network architecture, Theorem 1 gives a procedure for locally processing sensor information and transmitting compressed summaries (at a single time) to the fusion center to enable networked decision making that recovers the performance of a centralized scheme. Our starting point is a measurable space (Ω, F ), 4 on which a k-dimensional vector of with intensities β i (t), respectively. Hypothesis H 1 corresponds to a probability measure P 1 , with respectively. The decision problem is thus one of identifying the correct probability measure (P 0 versus P 1 ) on (Ω, F ) based on a realization of the k-dimensional process N t = (N t (1), . . . , N t (k)). 4 Here, Ω is the sample space and F is a σ-field on Ω. Equipping (Ω, F ) with a probability measure P gives the probability space (Ω, F , P).

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We will keep track of the flow of information using the filtration ( The interpretation is: for any event A ∈ F N t , an observer of the sample path s → N s , 0 ≤ s ≤ t, knows at time t whether or not the event A has occurred. The σ-field F N T thus represents the information generated by the totality of sensor observations up to t = T ; to wit, the information on which the decision must be based.
A test for deciding between hypotheses H 0 and H 1 on the basis of F N T observations can be thought of as a set T , two types of errors might occur. A "false alarm" occurs when the outcome ω ∈ A 1 (i.e. decide H 1 ) while H 0 is the correct hypothesis.
A "miss" occurs when ω ∈ Ω \ A 1 (i.e. decide H 0 ) while H 1 is the correct hypothesis. Clearly, the probability of false alarm is given by P 0 (A 1 ), while the probability of a miss is given by Then, the probability of detection is given by In the Neyman-Pearson framework, one is given an acceptable upper bound on the probability of false alarm α ∈ (0, 1), and the problem is to find an optimal test: 5 a set A * 1 ∈ F N T which maximizes the probability of detection over all tests whose probability of false alarm is less than or equal to α. The following result provides an optimal test that employs local information processing at the sensor level, to enable decisions at the fusion center that recover the optimal performance of a centralized Neyman-Pearson test. Assume that at decision time T , sensor i transmits to the fusion center the statistic performed at the fusion center, with and γ > 0 satisfying P 0 (L T ≥ γ) = α, 6 is optimal for F N T -observations in the sense that for any . Before continuing with the proof of Theorem 1, we highlight why optimal decision making through decentralized processing of information at the sensor level is possible in the case considered here.
Equivalently, if one were to consider a centralized framework (where information is continuously streamed from each of the sensors to the fusion center), {L T ≥ γ} would be the optimal test. Note, however, that each L T (i) in the product L T = k i=1 L T (i) can be computed locally at sensor i without any knowledge of the measurements at other sensors (see also Remark 2). Indeed, computation of L T (i) requires knowledge solely of the times (τ n (i) : n ≥ 1) at which counts have been recorded at sensor i, and the quantities β i (t), ν i (t), which are deterministic.
Consequently, each sensor simply needs to transmit its locally computed L T (i) to the fusion center at (the single) time t = T (in lieu of the element t → N t (i), t ∈ [0, T ] of function space).
The fusion center then forms the product L T = k i=1 L T (i) which is compared against γ to arrive at a decision. We thus retain the accuracy of centralized decision making while decentralizing most of the data processing, thereby accruing significant savings in communication costs.

IV. PROOF OF MAIN RESULT
The contents of this section are organized as follows. In Section IV-A, we provide precise definitions for various quantities of interest. In Section IV-B, we state and prove some results needed for the proof of Theorem 1. In Section IV-C, the proof of Theorem 1 is completed. 6 While there may not exist such γ for every α ∈ (0, 1), one can always find a sequence αn → 0 for which there exist γn > 0 with P0(LT ≥ γn) = αn. In other words, arbitrarily small upper bounds on probability of false alarm can be accommodated.

A. Definitions
Let (Ω, F , P) be a probability space. The sample space Ω is the set of all possible outcomes ω of a random experiment, F is the σ-field (or σ-algebra) of events, and P is a probability measure. A point process on [0, ∞) can be described by a sequence (τ n : n ≥ 0) of random variables defined on (Ω, F , P) taking values in [0, ∞] such that Here, τ n denotes the (random) time of the n-th occurrence of an event (such as a radiation counter registering a count). Associated to the sequence (τ n : n ≥ 0) is the stochastic process where 1 A denotes the indicator function of A, i.e.
Thus, N t counts the number of occurrences of the phenomenon prior to or at time t. N t is a counting process, i.e. N t is a Z + -valued process with N 0 = 0 such that the sample paths t → N t are non-decreasing, piecewise constant, right-continuous functions of t which increase in steps of size 1. We will also refer to N t as a point process [28].
A filtration (F t : t ≥ 0) is an increasing family of sub-σ-fields of F , i.e. F t ⊂ F for all t, and s ≤ t implies F s ⊂ F t . The σ-field F t represents the information available at time t.
A stochastic process (N t : t ≥ 0) taking values in R k is said to be adapted to the filtration F t if for all t ≥ 0, N t is F t -measurable; i.e. for any Borel measurable subset B of R k , the event be the smallest σ-field on Ω with respect to which all the random variables N s , 0 ≤ s ≤ t are measurable. Then, (F N t : t ≥ 0) is the filtration generated by the process N t and corresponds to the information available to an observer of the process N t . Clearly, if N t is F t -adapted, then F N t ⊂ F t for all t. Next, let us make precise what we mean by an inhomogeneous Poisson process.
Definition 1 (Inhomogeneous Poisson process): Suppose λ(t) is a nonnegative, measurable function such that t 0 λ(s)ds < ∞ for all t > 0. A point process N t on a probability space (Ω, F , P) adapted to the filtration (F t : t ≥ 0) is said to be a (P, F t )-Poisson process with 2) N t − N s is a Poisson random variable with parameter t s λ(u)du, i.e. for all n ∈ Z + ,

B. Useful Results
The primary result in this section is Proposition 2, which provides the probabilistic setup for the statement and proof (given in the next section) of Theorem 1. Our development proceeds through the following steps. We start with Proposition 1-a result of Brémaud [28]  In allowing stochastic intensities, as in Proposition 1 below, one has to have the technical requirement of predictability [28, Section I.3]. In our problem, however, the intensities are deterministic and automatically predictable. In terms of notation, we follow the convention of using λ t for a stochastic intensity versus λ(t) for a deterministic one.
Proposition 1 (Theorem VI.2.T3, [28]): Let (N t (1), . . . , N t (k)) be a k-variate point process adapted to the filtration F t on the given probability space (Ω, F , P 0 ), and let λ t (i), 1 ≤ i ≤ k, be the predictable (P 0 , F t )-intensities of N t (i), 1 ≤ i ≤ k, respectively. Let µ t (i), 1 ≤ i ≤ k, Let (τ n (i) : n ≥ 1) denote the jump times of N t (i), 1 ≤ i ≤ k, and define the process L t by where with the convention that 0 n=1 (. . . ) = 1. Suppose moreover that Define the probability measure P 1 on (Ω, F ) by Then, for each To underscore the role that deterministic intensities play in allowing decentralized processing in our problem, we briefly discuss some of the subtleties that arise in networked detection of point processes with stochastic intensities.

Remark 2:
In Proposition 1, the assumption of F t -predictability of λ t (i), µ t (i) (which are in general stochastic) implies that the latter potentially depend on all the information in F t , which includes the information generated by all the sample paths s → N s (i), s ∈ [0, t], 1 ≤ i ≤ k. 7 Consequently, for a sensor network which observes a point process with stochastic intensities, computation of the L t (i)'s cannot be decentralized as described in Remark 1 without incorporating a filtering component [28, Section VI.4]-finding the best estimatesL t (i) of L t (i) based on the locally available information F In such a decentralized processing scheme, the likelihood ratio to be compared to a threshold (at the fusion center) would be the processL t k i=1L t (i) evaluated at the decision time. This would entail, in general, some loss of performance in comparison to a fully centralized scheme. These extra considerations do not arise in our detection problem since our analysis focuses on a problem with deterministic intensities.
Next, we state Lemmas 1 and 2, which together verify (8) for our problem. The proofs of these Lemmas are given in the Appendix. Lemma 1 shows that the process (L t : t ∈ [0, T ]) defined by (10)-(11) is a martingale with mean one. Lemma 2 describes how the intensity of a Poisson process transforms under a time rescaling. Taken together, these lemmas enable us in Proposition 2 to obtain the probability measure P 1 on (Ω, F ) (corresponding to hypothesis H 1 ).
In the sequel, we denote the expectations corresponding to probability measures P 0 and P 1 by E 0 and E 1 , respectively.

Lemma 1: Under Assumptions 3, 4, the process (L
where is a nonnegative (P 0 , F N t )-martingale. By (P 0 , F N t )-martingale, we mean that 1) L t is adapted to the filtration Suppose (Ω, F , P) is a probability space equipped with the filtration (F t : t ≥ 0). Let λ(t) be a nonnegative, measurable function defined on [0, ∞) with t 0 λ(s)ds < ∞ for all t > 0. Let (X t : t ≥ 0) be a (P, F t )-Poisson process with intensity λ(t) (see Definition 1).
Proof: See Appendix.
We now state Proposition 2, which constructs a probability measure P 1 on (Ω, F ) corresponding to hypothesis H 1 . This completes the construction of the probabilistic model of our networked decision problem. As will be seen in the next section, the explicit construction of P 1 via the process L t and the probability measure P 0 , as described in the proof of Proposition 2, facilitates the application of Lemma 3 in proving Theorem 1.
, is a vector of independent F N t -Poisson processes whose components N t (i) admit intensities β i (t), 1 ≤ i ≤ k, with β i (t) satisfying Assumption 3. Then, there exists a probability Proof: Recall the process (L t : t ∈ [0, T ]) defined by (10), (11). Lemma 1 assures us that L T is a nonnegative random variable with E 0 [L T ] = 1. Hence, P 1 defined through is indeed a probability measure on (Ω, F ) which is absolutely continuous with respect to P 0 . We would now like to show that with respect to P 1 , N t (i) for 1 ≤ i ≤ k, are independent Poisson for u ∈ [0, 1]. By Lemma 2, eachÑ u (i), 1 ≤ i ≤ k is a (P 0 , G u )-Poisson process with intensitỹ . Denote by (τ n (i) : n ≥ 1) the sequence of jump times ofÑ u (i). Note thatτ n (i) = τ n (i)/T . Lettingν i (u) T ν i (T · u) for It is now easily checked that L t =L t/T for all t ∈ [0, T ], which implies in particular that We now apply Proposition 1 [28, Theorem VI.2.T3], using the rescaled time variable u ∈ [0, 1], filtration G u , withβ i (u) and 1 +ν i (u)/β i (u) in place of λ t (i) and µ t (i) respectively, to infer that N u (i) has intensityβ i (u) +ν i (u) with respect to P 1 . Using Lemma 2 in the "reverse" direction (i.e. interchanging t and u, replacing T by 1/T ), it now follows that the N t (i)'s, 1 ≤ i ≤ k, respectively. To complete the proof, it remains to show that the N t (i), 1 ≤ i ≤ k, are independent and Poisson under P 1 . Using Assertion (β) of [28,Theorem II.3.T8] (with X t ≡ 1), we get that for 1 ≤ i ≤ k, is a (P 1 , F N t )-martingale. By the Multichannel Watanabe Theorem [28, Theorem II.2.T6], it now follows that with respect to P 1 , the N t (i)'s are independent F N t -Poisson processes over t ∈ [0, T ] with intensities β i (t) + ν i (t), respectively.
Before concluding this section, we state the Neyman-Pearson Lemma which will be used in the next section to prove Theorem 1. The Neyman-Pearson Lemma describes an optimal rule for deciding between probability measures P 0 and P 1 on a measurable space (Ω, F ) on the basis of observations in the sub-σ-field G ⊂ F . Thus, for any event A ∈ G , it is known whether or not A has occurred. Recall that P 1 is absolutely continuous with respect to P 0 , both restricted to (Ω, G ), denoted P 1 P 0 if, whenever A ∈ G with P 0 (A) = 0, we have P 1 (A) = 0.
Then the decision strategy A * 1 = {L ≥ γ} is optimal for G -observations in the sense that for any A 1 ∈ G with P 0 (A 1 ) ≤ α, we have P 1 (A * 1 ) ≥ P 1 (A 1 ). See [28] for a proof. We will use this Lemma in the next section to prove Theorem 1, with F N T playing the role of G .

C. Proof of Theorem 1
LetP 0 ,P 1 be the restrictions of P 0 , P 1 respectively to F N T . Since P 1 P 0 on F , the restrictions of P 1 and P 0 to the smaller σ-field F N T inherit the absolute continuity, i.e.P 1 P 0 .
Hence, the Radon-Nikodym derivative exists; i.e. there exists a nonnegative, F N T -measurable random variable ξ, denoted dP 1 dP 0 , such that for any A ∈ F N T , Moreover, this Radon-Nikodym derivative is unique in the sense that ifξ is any nonnegative, F N T -measurable random variable satisfying (18) withξ replacing ξ, then ξ =ξ, P 0 -a.s. Since L T is a nonnegative, F N T -measurable random variable (by Lemma 1) which satisfies (18), it follows that P 0 -a.s. A direct application of Lemma 3 completes the proof.

V. PERFORMANCE ANALYSIS
Here we provide a lower bound on the probability of detection and an upper bound on the probability of false alarm when the proposed detection scheme is used. It turns out that bounds on both these probabilities involve the tails of (different) Poisson distributions. To compactly describe our results, we follow the notation of [48]. Tails): For λ > 0, j ∈ Z + , let p(λ, j) denote the Poisson distribution p(λ, j) = e −λ λ j j! .
The following quantities will also be of interest: It now follows from (10)-(11) that In the sequel we will also use the integer ceiling function · which assigns to a real number x the smallest integer greater than or equal to x.

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For γ > 0, consider the test A * 1 = {L T ≥ γ}. A lower bound on the probability of detection P 1 (L T ≥ γ), and an upper bound on the probability of false alarm P 0 (L T ≥ γ) can now be obtained as follows. Recalling Assumptions 3 and 4, define Note that if one can find − , + such that − ≤ L T ≤ + , then for j ∈ {0, 1}, it can be verified that − ≤ L T ≤ + . Next, note that Since N T (i) for 1 ≤ i ≤ k are independent Poisson random variables with parameters T 0 β i (s)ds with respect to P 0 , it follows that k i=1 N T (i) is a Poisson random variable with parameter B = k i=1 T 0 β i (s)ds with respect to P 0 . Under the probability measure P 1 , N T (i) for 1 ≤ i ≤ k are independent Poisson random variables with parameters T 0 [β i (s) + ν i (s)]ds. It follows that under

VI. APPLICATION TO NUCLEAR DETECTION
In this section, the framework developed above is applied to the problem of detecting radioactive materials in transit, using a network of spatially distributed sensors. The setting here May 3, 2014 DRAFT is simple but representative of a frequently encountered class of scenarios. Our method is not restricted, however, to this setting. Indeed, the results in the previous sections apply whenever the intensity of the suspected source and the motion of the source relative to the sensors are deterministic and known.
Radiation sensors always record background radiation (due to cosmic radiation and due to naturally occurring radioactive isotopes in the environment). In the absence of illicit nuclear material (hypothesis H 0 is true), the sensors simply measure background. If radioactive material is present (hypothesis H 1 is true), the sensors record the sum of the photons coming from background and the photons coming from the material. These two sources of radiation act independently, and one can treat each sensor as observing a single Poisson process whose intensity is the sum of intensities due to background and material (the source). The problem we face is to determine, in a fixed amount of time, whether a target passing in front of the sensors is a source of radiation.
The specific assumptions for this problem are as follows: The workspace is the horizontal plane, R 2 . We have k sensors uniformly spaced along the positive x-axis at locations x = 0, where β i is assumed to be varying between locations, from a minimum of β min = 2 counts per second (cps) to a maximum of β max = 8 cps, with the maximum appearing at the first and last sensor and the minimum occurring at the sensor in the middle (Fig. 2(a)).
We assume that a target is passing at a distance h = 0.362 m (the equivalent of 14 1 4 inches) from the x-axis, namely with a constant coordinate y = h, appearing first at some initial location To illustrate the derivation process, let us for the sake of argument assume that the acceptable probability of false alarm in this scenario is α = 10 −6 (see (23b)). With r i (t), 1 ≤ i ≤ k, May 3, 2014 DRAFT denoting the distance between sensor i and the potential source, the intensity ν i (t) at sensor i due to the source is modeled in [25] by 8 where a > 0 is the activity of the potential source (in cps) and χ > 0 (in m 2 ) is the sensors' cross-section coefficient. 9 We assume a numerical value for χa equal to what has been used in [25], but shielded in 3 cm of lead, dropping the source's perceived intensity by one order of magnitude to χa = 506.8 cps·m 2 . We also assume that no sensor is ever closer than distance h to the target, ensuring that ν i (t) is always bounded.
Since the location of the potential source at time t ∈ [0, T ] is (x 0 +vt, h), the distance r i (t) between the potential source and sensor i, Recalling (24) with for the decision time, we get from (21) we obtain h and for the ten-sensor array we have J = 2559.74 counts.
It is not hard to see that  Thus, for the case of the ten-sensor array, (22) evaluates to With reference to (21) and Fig. 2(a), we have B = 4387 17 counts, and with this we can attempt to numerically compute a threshold γ for the likelihood ratio test using (23b). It can be verified that the Poisson tail on the right hand side of (23b) falls below 10 −6 when the second argument of P (·, ·) increases to 338 (see Fig. 3(a)). We thus compute the value of γ for which log γ+J log D ≥ 338 counts, and obtain that with γ = 0.1718, the bound on the probability of false alarm P 0 falls at 8.5 × 10 −7 , which is below the acceptable error rate. The decision rule therefore is based on the test: which if true, suggests that the target is indeed a radioactive source. It should be mentioned that there is conservatism built in the bounds (22), which renders the probability of detection using (25) rather impractically small for the given false alarm rate. In addition, it is acknowledged that the illustrated method for obtaining a threshold makes the solution for γ very sensitive to changes in the underlying parameters h, v, χa and β min . Improving the bounds in (22)  parameters on the probability of detection. To illustrate that point, let us consider the possibility of using more sensors with the same spacing as before. Without changing the decision rule (25) (keeping the same threshold), the analysis shows ( Fig. 3(b)) how the upper bound on the probability of false alarm (PFA) estimated in (23b) not only falls monotonically with the addition of new sensors, but that there is a clear transition between the state where the sensor network decision is unreliable, and that where an alarm should be taken into account seriously. Such information can be useful for determining the maximum number of detectors that can get out of commission before significantly compromising the effectiveness of the system.

VII. CONCLUSIONS
A network of sensors can be deployed to optimally decide between two hypotheses regarding the statistics of a time-inhomogeneous point process in a way that preserves the accuracy of centralized decision making without incurring the increased communication cost. The sensors collect their measurements over a fixed-time interval, at the end of which a processed summary is communicated to a fusion center. In particular, each sensor transmits a locally computed likelihood ratio to the fusion center, which then compares the product of the sensor-specific likelihood ratios against a threshold to arrive at a decision. The analysis is based on the Neyman-May 3, 2014 DRAFT Pearson formulation. A set of conservative performance bounds on the error probabilities is provided and the framework is applied to the problem of detecting a moving radioactive source using an array of sensors. The work here supports the development of a general decision-making framework that leverages networks of mobile sensor platforms to enhance detection capability in problems that involve time-inhomogeneous point processes.

VIII. APPENDIX
Proof of Lemma 1: The process L t given by (10) and (11) is a particularization to our problem of the general process L t in Proposition 1. The latter admits the representation [28, where M t (i) = N t (i) − t 0 λ s (i)ds, and for t > 0, L t− = lim s t L s is the left limit of L t . The application of (26) to our problem, with L t given by (10), (11), yields where M t (i) = N t (i) − t 0 β i (s)ds. The non-negativity of L t is evident from (10), (11). To complete the proof, it thus suffices to show that each of the integrals on the right in (27) is a martingale. By [28,Theorem II.3.T8], for 1 ≤ i ≤ k, t 0 L s− (ν i (s)/β i (s))dM s (i) is a (P 0 , F N t )martingale whenever for t ≥ 0. 10 By Assumptions 3, 4, we get that where K = 1 + ν max /β min . Since K > 1 and the N t (i), 1 ≤ i ≤ k are independent with each N t (i) non-decreasing in t, we get where the last line follows from the fact that under P 0 , for t ≥ 0, each N t (i) is a Poisson random variable with parameter t 0 β i (s)ds. Proof of Lemma 2: Note thatλ(u) is nonnegative and measurable with u 0λ (v)dv < ∞ for all u > 0. Next, since X T ·u is F T ·u -measurable for all u ≥ 0, it follows that Y u is G u -adapted.
To complete the proof, we need to show that for 0 ≤ v ≤ u, Y u − Y v is independent of G v and is a Poisson random variable with parameter u vλ (τ )dτ . Since X T ·u − X T ·v is independent of F T ·v , it follows that Y u − Y v is independent of G v . Finally, for n ∈ Z + , we have where the last equality follows by making the change of variables τ = s/T .