Distributed Target Detection in Sensor Networks Using Scan Statistics

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Distributed Target Detection in Sensor Networks Using Scan Statistics Marco Guerriero, Student Member, IEEE, Peter Willett, Fellow, IEEE and Joseph Glaz

Abstract We introduce a sequential procedure to detect a target with distributed sensors in a two dimensional region. The detection is carried out in a mobile fusion center which successively counts the number of binary decisions reported by local sensors lying inside its moving field of view. This is a two-dimensional scan statistic — an emerging tool from the statistics field that has been applied to a variety of anomaly detection problems such as of epidemics or computer intrusion, but that seems to be unfamiliar to the signal processing community. We show that an optimal size of the field of view exists. We compare the sequential two-dimensional scan statistic test and two other tests. Results for system level detection are presented. Index Terms—Sensor Network, Scan Statistics, Two-Dimensional Anomaly Detection.

I. I NTRODUCTION Distributed detection using multiple sensors and optimal fusion rules have been extensively investigated (e.g., [2] [3] [4]). The case where the communication links are not ideal but instead are subject to error has been addressed in [5], and that with dependent sensors is studied in [6]. For the fusion scheme, Chair and Varshney [7] showed that the optimal fusion rule is the likelihood ratio test on the decisions from sensors and becomes a threshold detector on the weighted sum of binary sensor decisions and the weight is obtained using the local detection and false alarm probability at each sensor under each hypothesis. M. Guerriero and P. Willett are with the ECE Department, University of Connecticut, Storrs, U-2157, CT 06269 USA, Phone: +1 8604862195, Fax: +1 8604862447, Emails: {marco.guerriero, willett}@engr.uconn.edu. J. Glaz is with the Department of Statistics, University of Connecticut, Storrs, U-4120, CT 06269-4120 USA, Fax: +1 8604864113, E-mail: [email protected] This research was supported by the Office of Naval Research under contract N00014-07-1-0055. An abbreviated version of this manuscript was presented in [1].

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In [8] [9] the authors propose an appealing architecture for sensor networks (SN) in which information is accumulated in a traveling Mobile Agent (MA) (think of a wheeled rover collecting data from a haphazardly deployed dynamic population of functional ground sensors, or of a surface vessel collecting data from a drifting underwater sensor network), which sequentially queries the sensors that fall in its current (and changing) field of view (FOV). The MA, playing the role of the fusion center, makes the final decision about the presence of a target. There are sequential versions of this (e.g., [10]) that seek to use an MA to decide as quickly as possible which among a pair of hypotheses is true. Most distributed detection references begin with the assumption that the local sensors’ detection performances – their probabilities of detection and false alarm rate – are known to the fusion center. For a dynamic target and passive sensors, it is difficult to estimate local sensors’ performances via experiment: these performances are time varying as the target moves through the sensor field, and, in any case, it is expensive to transmit them to the fusion center. In [11] a fusion rule that uses the total number of detections (“1”s) transmitted from the local sensors as the test statistic was proposed for the case where the total number of sensors in the SN was known. In [12] the authors studied the case of random deployment of sensors within the SN: even the number of sensors was unknown and was modeled as a random variable, and via this prior information the system level decision at the fusion center compared number of detections made by the local sensors to a threshold. Most distributed detection references also begin with the assumption that the hypotheses are homogeneous: either H0 (target absent) is true at all sensors or H1 (target present) is uniformly true. For example, while the counting rule is a very reasonable approach, it neither localizes the target nor makes any attempt to use information about the contiguity of sensors that report a target. We believe we can do better, since, more realistically, a target is a local disturbance. In this paper we combine the three previous ideas: we introduce a two-dimensional MA-based sequential procedure for detecting and localizing a target, counting only binary detection-level sensor data, and testing for any spatial inhomogeneity of such reports. The tool, of emerging importance in fields such as internet safety (detections denote network intrusion attempts, plotted versus time) or epidemiology (detections represent reported illness, plotted versus location), is the scan statistic [13], [14], [15]. Scan statistics look for events that are clustered amongst a background of those that are sporadic. In a nutshell, a scan-statistic test slides a window across its observation domain, sums the number of events (detections) within this window, and continually tests the count that it finds against a threshold (see Figure 1). Now, assuming that the window matches the shape of the “cluster” that is sought, the scan statistic is clearly a generalized likelihood ratio test (GLRT) [16]; so what is new? The answer is that DRAFT

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quite precise expressions have been developed for the exceedance probability for a scan statistic, and it is exactly the lack of ability to set a reliable threshold that often makes a GLRT unappetizing. One could, of course, circumvent this by testing the maximum amongst non-overlapping windows (as opposed to a sliding one), since in that case their counts would be independent, and the maximal order-statistic trivial to calculate. But there would a high price to be paid in detectability if the cluster did not precisely match one of these non-intersecting scans. The scan statistic is appealing both ways: it has excellent detection performance without regard to cluster location, and by accounting for the count dependence between overlapping scans it has reliable probability of false alarm. In addition, we will show that an optimal window size for the MA’s field of view exists, and we compute it. In light of the above, it also seems reasonable to compare the two-dimensional scan statistic test to: •

a two-dimensional non-overlapping test, and

•

a counting rule test [12],

both of which use the total number of detections reported by local sensors. The informing situation is in Figure 2. We test for a “target,” whose presence is suggested by randomlydeployed sensors; for example, these elements may sense a magnetic disturbance, acoustic signature, or even bistatic reflections from a remote radio source. Each sensor can generate a spurious false alarm with probability Pf a . More interesting, we assume that the sensor’s probability of detection decreases with increasing distance between it and the target, due to an inverse-square power law. A MA “drags” a sliding window around the area, and if the number of reporting sensors that are encompassed exceeds a threshold level that is determined by scan-statistic theory, a detection is declared. The remainder of this paper is organized as follows. A general overview of scan statistics together with a detailed mathematical formulation of the two-dimensional scan statistic is given in Section II. Section III discusses the target and signal models adopted and analytical/simulation results for the Bernoulli case where the number of sensors in the SN is known. The optimal field of view of the MA scenario is dealt with in Section III-D, while Section IV is devoted to consider the case of Poisson sensor field where the number of sensors in the SN is unknown. Sect. IV-B provides a comparison with the non-overlapping test. Finally, Section IV-C provides comparative performance of the fixed counting rule test [11] and the proposed test. Concluding remarks are presented in Section V.

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1

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Fig. 1. Scanning the unit time interval with a window of length w = 0.2. The ∗ represent times of occurrence of N = 19 events, S.2 = 8 The bar represents the window that contains the maximal number of events; that is, the scan statistic.

II. T WO -D IMENSIONAL S CAN S TATISTICS A. Definitions The detection and analysis of clustering of events is of great importance in many areas of science and technology including: epidemiology [17], bionformatics [18], biosurveillance [19], ecology [20], medicine [21], quality control and reliability [22]. Theory and applications of scan statistics, as well as recent advances, have been presented in [13], [14] and [23]. The following example serves as a good introduction to scan statistics. Let us consider N independent observations from a uniform distribution on the interval (0, T ) (see Figure 1). Let Sw be the largest number of observations in any window of length w in the interval (0, T ). Sw is called a scan statistic and it has been used in testing the null hypothesis – that the N points in (0, T ) are uniformly distributed – against a clustering alternative [24]. Scan statistics have been also formulated for a sequence of independent and identically distributed (i.i.d.) discrete random variables X1 , . . . , XT . A discrete scan statistic has been defined as follows: Sm = max {1 6 i 6 T − m + 1; Xi + . . . + Xi+m−1 } ,

(1)

where m is the length of the scanning window. For the special case of iid 0 − 1 Bernoulli observations, the event Sm = m implies that the longest run of 10 s is of length m. One can view the discrete scan statistic Sm as an extension of the longest run statistic. Since in this article we employ a scan statistic for observed counts of signals in a two dimensional DRAFT

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region, we introduce below a two dimensional scan statistic. This scan statistic deals with the following situation. A region R of Euclidean space is tessellated or subdivided into cells (which will be denoted by the symbol a, and are presumably much smaller than the window to be used when scanning). Data are available in the form of nonnegative integral counts Ya on cells a. In addition, a “size” value Aa is associated with each cell a. The cell sizes Aa are known and fixed, while the cell counts Ya are independent random variables. Two distributional settings are commonly studied: •

Bernoulli - Ya ∼ Bernoulli(pa ), where pa is a parameter attached to cell a with 0 < pa < 1.

•

Poisson - Aa is a positive real number and Ya ∼ P oisson(λa , Aa ), where λa > 0 is an unknown parameter attached to cell a.

The scan statistic seeks to identify hot spots or clusters of cells that have an elevated response compared with the rest of the region. The following is a statement of the null and alternative hypotheses in the Bernoulli setting: •

H0 : pa is the same and known for all cells in region R – that is, there is no hot spot.

•

H1 : There is a nonempty zone Z (connected union of cells) and parameter values 0 < p0 , p1 < 1

such that

  p1 , for all cells a in Z pa =  p , for all cells a in R − Z 0

(2)

with p1 > p0 . The zone Z specified in H1 is an unknown parameter of the model. The null model, H0 , is the limit of H1 as p1 → p0 ; however, the parameter Z is not identifiable in the limit. We now assume that the region R previously introduced is rectangular, which is defined by [0, T1 ] × [0, T2 ]. Let hi =

Ti Ni

> 0, where Ni are positive integers, i = 1, 2. In many applications the exact locations

of the observed events in the rectangular region are unknown. What is usually available are the counts in small rectangular subregions. For 1 6 i 6 N1 and 1 6 j 6 N2 , let Xij be the number of events that have been observed in the rectangular subregion (made up of cells) [(i − 1)h1 , ih1 ] × [(j − 1)h2 , jh2 ]. We are interested in detecting unusual clustering of these events under the null hypothesis that Xij are independent and identically distributed (i.i.d.) nonnegative integer valued random variables from a specified distribution. For 1 6 i1 6 N1 − m1 + 1 and 1 6 i2 6 N2 − m2 + 1, define: Yi1 ,i2 =

i2 +m X2 −1 i1 +m X1 −1 j=i2

Xij

(3)

i=i1

to be the number of events in a rectangular region comprising m1 by m2 adjacent rectangular subregions with area h1 h2 and the south-west corner located at the point ((i1 − 1)h1 , (i2 − 1)h2 ). If Yi1 ,i2 exceeds a DRAFT

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preassigned value of k , we will say that k events are clustered within the inspected region. We define a twodimensional discrete scan statistic as the largest number of events in any m1 by m2 adjacent rectangular subregions with area h1 h2 and the south-west corner located at the point ((i1 − 1)h1 , (i2 − 1)h2 ): Sm1 ×m2 ;N ×N = max {Yi1 ,i2 ; 1 6 i1 6 N1 − m1 + 1, 1 6 i2 6 N2 − m2 + 1}

(4)

For simplicity we assume that N1 = N2 = N and m1 = m2 = m. We also abbreviate Sm×m;N ×N to Sm×m . The scan statistic Sm×m , defined in equation (4), has been used in [25] to test the null hypothesis of randomness that assumes the Xij s are i.i.d. Bernoulli random variables with parameter P (Xij = 1) = p0 , and similarly to test a constant-intensity Poisson null hypothesis. For the alternative

hypothesis of clustering one can specify the region Z in (2) to be a rectangular region given by: R(i1 , i2 ) = [(i1 − 1)h1 , (i1 + m − 1)h1 ] × [(i2 − 1)h2 , (i2 + m − 1)h2 ]

(5)

such that for any i1 6 i 6 i1 + m − 1 and i2 6 j 6 i2 + m − 1, Xij has a Bernoulli distribution with parameter P (Xij = 1) = p1 , where p1 > p0 or Poisson distribution with mean θ1 > θ0 , respectively. For (i, j) ∈ / [i1 , i1 + m − 1] × [i2 , i2 + m − 1], Xij is distributed according to the distribution specified by the null hypothesis. We assume that i1 and i2 are unknown. If m is known Sm×m is a generalized likelihood ratio test (GLRT) for testing the above hypotheses [26]. In this case, the null hypothesis is rejected whenever Sm×m > k , where k is determined from P (Sm×m > k) = α, the probability of type I error. For this two-dimensional scan statistic to be useful, accurate approximations for P (Sm×m > k) are needed. There are no exact results available for P (Sm×m > k). B. Bernoulli Model Let Xij be i.i.d. Bernoulli random variables with parameter P (Xij = 1) = p0 , where 0 < p0 < 1. For small and moderate values of P (Sm×m > k) a quite accurate approximation has been derived in [27]: · ¸ 2 [P {Sm×m (m,m)6k−1}](N −m−1) P (Sm×m > k) ≈ 1 − [P {S 2(N −m−1)(N −m) m×m (m,m+1)6k−1}] (6) 2 × [P {Sm×m (m + 1, m + 1) 6 k − 1}](N −m) where: P {Sm×m (m, m) 6 k − 1} = Fb (k − 1; m2 , p0 )

P {Sm×m (m, m + 1) 6 k − 1} =

k−1 X

Fb2 (k − 1 − s; m, p)b(s; m(m − 1), p0 )

(7)

(8)

s=0 DRAFT

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P {Sm×m (m + 1, m + 1) 6 k − 1} =

k−1 X

k−1 X

1 X

b1 (s1 )b1 (s2 )b2 (t1 )b2 (t2 ) s1 ,s2 =0 t1 ,t2 =0 ij =0,16j64 P4 P4 ij × p0 j=1 (1 − p0 )4− j=1 ij Fb (x; (m −

1)2 , p0 ) (9)

and where for i = 1, 2, b1 (si ) = b(si ; m − 1, p0 ) b2 (ti ) = b(ti ; m − 1, p0 ) µ ¶ N b(j; N, w) = wj (1 − w)N −j j Fb (i; N, w) =

i X

(10) (11)

b(j; N, w)

j=0

x = min(k − 1 − s1 − t1 − i1 , k − 1 − s2 − t1 − i2 , k − 1 − s2 − t1 − i2 , k − 1 − s1 − t2 − i3 , k − 1 − s2 − t2 − i4 )

(12)

Bounds and approximations for P (Sm×m > k) and are discussed in great detail in [23], [28]. C. Poisson Model Let Xij be i.i.d. Poisson random variables with mean θ0 > 0. For 1 6 i1 6 N1 − m1 + 1 and 1 6 i2 6 N2 − m2 + 1 define the events Ai1 ,i2 =

i2 +m X2 −1 i1 +m X1 −1 j=i2

Then,

ÃN

1

P (Sm1 ×m2 6 k − 1) = P

(13)

Xij > k

i=i1

−m \1 +1 N2 −m \2 +1 i1 =1

! Aci1 ,i2

.

(14)

i2 =1

where we have denoted as Ac the complement event of A. To simplify the presentation of the results, we will discuss here only the case of N1 = N2 = N and m1 = m2 = m. For fixed values of 1 6 i1 6 ´ ³T N −m+1 c Ai1 ,i2 using a product-type approximation N1 − m1 + 1, one can accurately approximate P i2 =1

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discussed in [26]: P

ÃN −m+1 \

! Aci1 ,i2

i2 =1

´ ! N −m+1  ³Ti2 c P A Y j=1 i1 ,j  ³T ´ = P Aci1 ,i2 i2 −1 c i2 =1 i2 =m+2 P j=1 Ai1 ,j ³ ´  Ti2 ! N −m+1 Ãm+1 c P A Y \ j=i −m i1 ,j  ³T 2 ´ ≈ P Aci1 ,i2 i2 −1 c i2 =m+2 P i2 =1 j=i2 −m Ai1 ,j µ ¶N −2m qm,2m ≈ qm,2m qm,2m−1 Ãm+1 \

(15)

(16)

(17)

where qm,m+l−1 = P (Ac1,1 ∩ Ac1,2 ∩ · · · ∩ Ac1,l )

(18)

for 1 6 l 6 N − m + 1. The quantities qm,m+l−1 are evaluated recursively in [29] for one-dimensional scan statistics. For self-consistency we report in Appendix A how to compute those quantities for a twodimensional scan statistics. Two very accurate approximations for P (Sm×m > k) for the Poisson model are presented in the literature. Following the product-type approximation, the first of them is given by [25]:

µ P (Sm×m > k) = 1 − qm,2m−1

qm,2m

¶(N −2m+1)(N −m+1)

(19)

qm,2m−1

The second belongs to a class of approximations in [27] and it is given by: 2

P (Sm×m > k) ≈ 1 − ×

[P {Sm×m (m + 1, m + 1) 6 k − 1}](N −m)

[P {Sm×m (m + 1, m) 6 k − 1}](N −m−1)(N −m)

(qm,2m−1 )(N −2m)(N −m−1) . (qm,2m )(N −2m+1)(N −m−1)

(20)

Based on extensive simulation we have concluded that approximation (20) is more accurate than approximation (19), and therefore it will be used in this article. Upper and lower bounds for these approximations have been developed in [30]. III. A T WO -D IMENSIONAL B ERNOULLI G RID OF S ENSORS A. Observations Model Let us consider a scenario as depicted in Figure 3, where identical sensors are deployed in a grid pattern in the sensor field, which we consider to be a square of area b2 . Let us denote with (xs , ys ) for s = 1, . . . , M the coordinates of sensor s. Noises at local sensors are i.i.d. and follow the standard

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‘

Fov=fovh

b TARGET SENSOR WHICH DETECTS

TARGET POWER

Fig. 2. The MA is depicted with its field of view which we consider to be a square of size Fov . Only the sensors which detect the target are depicted. The different level curves correspond to a different target power. The MA traveling across the sensor network simply counts how many sensors are inside its field of view. We assume that the target (the black bold square) is a-priori uniformly distributed within the sensor network.

2: Gaussian distribution with zero mean and variance σw 2 ws ∼ N (0, σw )

s = 1, . . . , M

(21)

We assume that sensors make their local decisions independently without collaborating with other sensors. We design each sensor s to decide between the following (composite) hypotheses H0 : rs = ws H1 : rs = ys + ws

where rs is the received signal at sensor s, ys =

as ds ,

(22)

as are i.i.d. GRV (Gaussian random variables) with

zero-mean and variance σ 2 (σ 2 represents the power of the signal that is emitted by the target at distance ds = 1m) and ds is the distance between the target and local sensor s: ds =

p (xs − xt )2 + (ys − yt )2

(23)

and (xt , yt ) are the unknown coordinates of the target. We further assume that the location of the target also follows a uniform distribution within the sensor field. In (22) we used a model as in [31], [32], [33]. Assuming all the local sensors use the same threshold τ to make a decision and according to the DRAFT

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MA’s Field of View

cell c(i,j) with size h

Target

5

4.5

4

3.5

y

3

2.5

2

1.5

1

0.5

0

0

0.5

1

1.5

2

2.5 x

3

3.5

4

4.5

5

Fig. 3. A two-dimensional Bernoulli grid of sensors is depicted. M = 625 sensors are deployed in a grid pattern in the sensor field, which we consider to be a square of area b2 . The sensors which detect the target (the red bold circle) tend to form a 2 cluster around the target (the black bold square). Here we used b = 5,σ 2 = 1, σw = 4, pf a = 0.05.

Neyman-Pearson lemma [16], we have the local sensor-level false alarm rate and probability of detection given by: pf a = 2Q pd s

= 2Q

µq à r

¶ τ 2 σw

!

τ 2 + σ2 σw d2

(24)

s

where Q(x) =

R∞ x

2

ξ √1 e− 2 2π

dξ is the unit Gaussian exceedance function. We denote the binary data from

local sensor s as Is = {0, 1} (s = 1, . . . , M ). Is takes the value 1 when there is a detection; otherwise, it takes the value 0. Remark 1: Note that a different signal power attenuation model can also be used, yielding different expressions for pf a and pds . For example, following the same model as in [12] we have: ys2 =

P0 , 1 + γdns

(25)

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where P0 is the signal power emitted by the target at distance ds = 0, n is the signal decay exponent and γ is an adjustable constant, where a larger γ implies faster signal power decay. We will assume the model as in (22) but note that our results hold more generally. B. The Scan Statistic Assuming we know the total number of sensors M , we divide the square into M small sub-squares. The location of the sensor inside each small sub-square (cell) is known1 . Letting h =

b N,

where N is

such that N 2 = M , we divide the square of area b2 into M cells such that each cell of area h2 contains only one sensor2 . Let us denote by c(i, j) the cell [(i − 1)h, ih] × [(j − 1)h, jh] (see Figure 3). We define Xij as the binary data from the local sensor s inside c(i, j) with 1 6 i 6 N and 1 6 j 6 N . It is easy

to verify that

M X

Is =

s=1

N X N X

(26)

Xij

j=1 i=1

For 1 6 i 6 N and 1 6 j 6 N , let us denote with Xij = Is be the number of events that have been observed in the cell c(i, j) by the sensor s. Xij represents the binary data (“1” or “0”) of the sensor s inside the cell c(i, j). It is straightforward that under H0 (target absent), the Xij are i.i.d. Bernoulli random variables with P (Xij = 1) = pf a , whereas under H1 (target present), the Xij are discrete conditionally independent random variables with P (Xij = 1) = pds (i,j) , meaning that they are not identically-distributed3 . Now we assume that under H1 , pds (i,j) can be approximated with pf a for a sensor s whose distance to the target is such that d2s >> pds (i,j)

 s  = 2Q

σ2 2 . σw

 τ 2 σw +

 ≈ 2Q 2

σ d2s

In fact we have that µr

τ 2 σw

¶

(27)

= pf a

2 . Since the target power attenuates as function of the distance from the target, we expect for d2s À σ 2 /σw

that there is a cluster of sensors which are stronger (closer to the target), under the hypothesis H1 . 1 Our results hold more generally: we do not need to know the exact location of the sensor inside the cell. We can also assume that the location of the sensor follows a uniform distribution in the cell. 2

This assumption is related to the theory of two-dimensional scan statistic [15]; further, we will remove this assumption in the case of Poisson model. 3 pds (i,j) , which is the local detection probability for the sensor s which is located in the cell c(i, j); pds (i,j) is greater than pf a , but is otherwise unknown.

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C. False Alarm Probability at the MA The MA travels across the sensor network (see Figure 2) and sequentially collects the local binary decisions from sensors that lie inside its overlapping FOV’s, which we consider to be squares of size Fov = fov h (fov integer). The sequential fusion rule at the MA for 1 6 i1 6 N − fov + 1 and 1 6 i2 6 N − fov + 1 is given by:

where Yi1 ,i2 =

  Yi ,i > k ⇒ decide H1 , 1 2  otherwise ⇒ MA continues to scan.

Pi2 +fov −1 Pi1 +fov −1 j=i2

i=i1

(28)

Xij . At the MA the probability of false alarm Pf a is: Pf a = P (Sfov ×fov > k| H0 )

(29)

We note that equation (29) can be evaluated using the approximation as in equation (6) after substituting fov with m and pf a with p0 .

Remark 2: The key point is that while the exceedance probability for one location of the FOV (rectangular scan region or window) of (4) is trivial – and similarly for many non-overlapping windows it is easy – to obtain a powerful GLRT we need to “drag” the window as with the MA. The exceedance probability for (4) is not at all obvious for overlapping windows, but that is exactly what the scan statistics literature provides. The contribution and importance of scan statistics is their ability to do that, to provide an answer to the question: “How many counts constitute an abnormality?”. In Figure 4 we have plotted the global probability of false alarm Pf a for the MA versus the local probability of false alarm pf a for the sensor. In Figure 4, the curves obtained by using the scan statistic approximations in equation (6) and those by simulations (based on 10000 Monte Carlo runs) are plotted. Figure 4 shows that the approximation in equation (6) is very accurate for small values of Pf a , as expected from the approximations of scan statistic theory [15]. The question of the optimum field-ofview Fov = fov h (with fov integer) to use arises and it will be addressed in the next section. D. Optimized MA Window Size In this section we are interested in finding an answer to the following problem: Does an optimum Fov , which maximizes the probability of detection Pd for the MA, while maintaining fixed the probability of false alarm Pf a , exist? Does the following optimization problem have a solution? max

Fov : Pf a =α

Pd

(30)

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0

10

Simulation Theory

−1

fa

p (MA)

10

−2

10

−3

10

0.02

0.025

0.03

0.035 p sensor

0.04

0.045

0.05

fa

Probability of false alarm for the MA Pf a versus probability of false alarm for the local sensor pf a . Here 2 we have b = 5, σ 2 = 1, σw = 4, M = 625, k = 7, fov = 6 and Fov = 1.2.

Fig. 4.

where Pd is defined as Pd = P (Sfov ×fov > k| H1 )

(31)

If the “signature” of an anomaly was known and matched a possible MA window the scan statistic (in this case Sf ov×f ov is the GLRT for testing the two hypotheses H0 and H1 [26]), then the question would scarcely be worth asking since the optimum Fov coincides with the “signature” of the anomaly. However, according to the model of Section III-A the density of sensors that report “1”is unknown and decreases smoothly as the distance from the source increases and it and while it would be appealing to make the window large enough that all or most of them be encompassed, it is clear that doing so increases the likelihood of detection of a spurious cluster. It will be recalled that for hypothesis testing problems involving discrete distributions, it is usually not possible to choose a critical region consisting of realizable values of the statistic of size exactly α, where α is some prescribed value. The use of a randomized test [34] in order to solve the constrained

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optimization problem in (30) is needed. By defining α1 and α2 as follows: P (Sfov ×fov > k1 ) ≈ α1 < α P (Sfov ×fov > k1 − 1) ≈ α2 > α

(32)

then we use a “coin-flip” decision with probability p=

α − α1 α2 − α1

(33)

whenever Sfov ×fov = k1 . As we can see in Figure 5, there does exist an optimal field-of-view Fov that maximizes the probability opt of detection Pd for the MA. By employing this optimum Fov , a significant improvement in Pd can be opt achieved. In figure 6 the optimal Fov versus σ – σ 2 represents the power of the signal that is emitted

by the target at distance ds = 1m – is plotted. Not surprisingly, we observe an approximately linear opt relationship between the optimal scan-box dimension Fov and the target amplitude, and also can see

that there is a saturation point beyond that the optimal size of the MA’s field of view stays constant (the power of the signal is so high that all the sensors report “1”s.) An interesting alternative approach uses Multiple Window Discrete Scan Statistics [35]. Since the shape and the size of the cluster of the alarmed sensors (sensors that report “1”) is unknown, it might prove to be ineffective to arbitrarily choose a window size for performing the test. The testing procedure is based on several fixed window size scan statistics: Sf ov1 ×f ov1 , . . . , Sf ovn ×f ovn , where f ov1 6 . . . 6 f ovn . This testing procedure will reject H0 if for some 1 6 j 6 n, Sf ovj ×f ovj > kj , where:   n [ ¢ ¡ Sf ovj ×f ovj > kj = αm P  

(34)

j=1

is the desired probability of false alarm for the MA. The MA can be thought of as dragging a family of windows – some large and some small – simultaneously. Approximations for determining the thresholds kj are given in [35] where numerical results for the power of the multiple window scan statistic test are

presented along with a comparison with a multiple tests approach [36]. IV. A T WO -D IMENSIONAL P OISSON F IELD OF S ENSORS A. The Scan Statistic In Section III-B we studied the case of a grid of sensors where the number of the sensors was known and was equal to M . It is probably more realistic to assume that: DRAFT

15

0.86

0.84 Fopt=1.2 ov

Pd (MA)

0.82

0.8

0.78

0.76

0.74

1

1.5

2

2.5

3

3.5

4

Fov

Fig. 5. The probability of detection Pd for the MA versus the field-of-view Fov is plotted for a fixed value of α. Here we 2 have b = 5, σ 2 = 1, σw = 4, pf a = 0.04, M = 625, and α = 0.1 Simulations are based on 10000 runs.

•

the sensors are deployed randomly in the SN; and

•

some of them are out of communication range of the MA, malfunctioning, or out of battery power.

Therefore, at any given time, the total number of sensors that work properly in the SN is a random variable. We assume that the initial distribution of sensors over the space is a homogeneous Poisson process with intensity λ, therefore the number of sensors M within the sensor field is a random variable that follows a Poisson distribution: p(M ) =

−λb λM b e , M

M = 0, . . . , +∞

(35)

where λb = λb2 . Given M , the locations of the sensors are i.i.d., and follow an uniform distribution in the sensor field

  f (xs , ys ) =

1 b2

 0

0 6 xs , ys 6 b,

otherwise

for s = 1, . . . , M , where (xs , ys ) are the coordinates of sensor s. Since each sensor decision is independent and based on the signal strength at its location, the local decision-making at each sensor can be viewed as a DRAFT

16

3

2.5

Fopt ov

2

1.5

1 Fopt = 1.2 ov 0.5

1

2

3

4

σ

5

6

7

8

opt Fig. 6. The optimal Fov versus σ (σ 2 is the power of the signal that is emitted by the target at distance ds = 1m) is plotted 2 for a fixed value of α. Here we have b = 5, σw = 4, pf a = 0.04, M = 625, and α = 0.1 Simulations are based on 10000 runs.

location-dependent thinning procedure [37] of the original sensor distribution with probability pf a and pds , under H0 and H1 respectively, and the distribution of the alarmed sensors (sensors with Is = 1) forms a nonhomogeneous spatial Poisson process. Hence the distribution of alarmed sensors is a nonhomogeneous Poisson process whose local intensity is given by λpf a and λpds under H0 and H1 respectively. We measure the sensor field directly via a scan statistic; that is, the sensors are Bernoulli with random location. Unfortunately there are no extant results for scan statistics in this case. However, there are results for fixed numbers of Poisson variates. To exploit these, we model that the sensors still be located on a grid, but now the “gridded” sensor reports being of the number of “1”s observed within each sensor’s hinterland, a rectangle (w.l.o.g. but with nicer notation, a square) of dimension h1 × h2 = h2 , as in Section II-C (see Figure 7). Then it is straightforward that under H0 (target absent), Xij are i.i.d. Poisson random variables with mean λ0 = λh2 pf a , whereas under H1 (target present) Xij are discrete conditionally independent Poisson random variables but not identically distributed with mean λs(i,j) = λh2 pds (i,j) . Now we assume that under H1 , pds (i,j) can be approximated with pf a as in (27)

and the same analysis as described for the Bernoulli case can be done. In Figure 8 we have plotted

DRAFT

17

Poisson field of sensors

square of dimension h2 5

4.5

4

3.5

y

3

2.5

2

1.5

1

0.5

0

0

0.5

1

1.5

2

2.5 x

3

3.5

4

4.5

5

A two-dimensional Poisson field of sensors is depicted. The sensors which detect the target (the red bold circle) tend to form a cluster around the target (the black bold square). Here we used λ = 20, b = 5,σ 2 = 1, 2 = 4, pf a = 0.05. σw Fig. 7.

the global probability of false alarm for the MA Pf a versus the local probability of false alarm pf a . In Figure 8, the curves obtained by using the scan statistic approximations in (20) and those by simulations (based on 10000 Monte Carlo runs) are plotted. Figure 8 shows that the approximation in (20) is more accurate. In fact, a detailed simulation analysis confirmed that the approximation in (20) works better than the approximation in (6) for the Bernoulli case, especially for small values of pf a . B. The Disjoint-Window Test Here we study the performance of a non-overlapping (or disjoint-window) test for the Poisson model described in the section IV. The size (probability of false alarm) of such a test is trivial to calculate, due to the independence of the windows’ counts, versus the scan statistic expressions. But how much do we lose by constraining the window to “hop” rather than slide? We now suppose that the MA travels across the sensor sensor network and scans the network using

DRAFT

18

0

10

pfa (MA)

Simulation Theory

−1

10

−2

10

0.05

0.055

0.06

0.065

0.07

0.075 pfa sensor

0.08

0.085

0.09

0.095

0.1

Fig. 8. Probability of false alarm for the MA Pf a o versus probability of false alarm for the local sensor pf a . Here 2 we have b = 5, σ 2 = 1, σw = 4, λ = 20, N = 10, k = 6, fov = 2 and Fov = 1.

Non-overlapping test

Scan statistic

Fov=fovh

b TARGET SENSOR WHICH DETECTS

Fig. 9. The MA travels across the Poisson sensor field. The overlapping windows (the two-dimensional scan statistic test) and the non-overlapping windows (non-overlapping test) are depicted. The non-overlapping test is sensitive to the relative target position with respect to the scanning window, which turns out to affect the power of the test.

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19

0.9 Scan Statistic Disjoint−Window Test 0.85

Pd

0.8

0.75

0.7

Center of the window : xt =2.5, yt=2.5

0.65 yt = 2.5

2

2.1

2.2

2.3

2.4

2.5 xt

2.6

2.7

2.8

2.9

3

Fig. 10. Probability of detection for the MA Pd versus the target location in the x direction (xt ) is plotted for the scan statistic 2 test and the disjoint-window test respectively. Here we have b = 5, σ 2 = 1, σw = 4, λ = 20, N = 25, fov = 5, Fov = 1, pf a = 0.05, and α = 0.1 for both tests (in order to get the prescribed value α = 0.1 the use of the randomized test as previous described is needed). Simulation are based on standard Monte Carlo counting process, involving 10000 runs.

non-overlapping windows4 . It collects all the detections reported by local sensors which lie in its field of view. Here we assume that the field of view of the MA is a square of size Fov . We divide the square of size b in W windows of size Fov , with W =

b2 2 Fov

being an integer. We need to evaluate the probability

distribution of the highest number of counts detected over the whole set of windows. This calculation can be done exactly. In each window the number of alarmed sensor S under H0 is dictated by the following Poisson distribution p(S, λFov |H0 ) =

e−λFov λSFov S!

(36)

2 p . Let us define with Z = max (S , i = 1, . . . , W ) the highest number of counts where λFov = λFov i fa

detected over the all windows. It can be easily shown that the probability density function of Z is given 4

We use interchangeably the terms window and field of view.

DRAFT

20

1

0.9 Scan Statistics Disjoint−Window Test 0.8

0.7

Pd MA

0.6

0.5

0.4

0.3

0.2

0.1

0

0

0.1

0.2

0.3

0.4

0.5 Pfa MA

0.6

0.7

0.8

0.9

1

Fig. 11. ROC curves obtained by simulations for the scan statistic test and the disjoint-window test. Here we have b = 5, 2 σ 2 = 1, σw = 4, λ = 20, N = 25, fov = 5, Fov = 1, pf a = 0.02. Simulation are based on standard Monte Carlo counting process, involving 10000 runs.

1

Scan Statistics Disjoint−Window Test

0.9

Pd MA

0.8

0.7

0.6

0.5

0.4

−2

10

−1

10 Pfa (MA)

0

10

Fig. 12. ROC curves in semilogarithmic scale obtained by simulations for the scan statistic test and the disjoint-window test. 2 Here we have b = 5, σ 2 = 1, σw = 4, λ = 20, N = 25, fov = 5, Fov = 1, pf a = 0.02. Simulation are based on standard Monte Carlo counting process, involving 10000 runs.

DRAFT

21

by:

!W −k à !k à l−1 W µ ¶ X X e−λFov λnFov e−λFov λlFov W P (Z = l) = n! l! k k=1

(37)

n=0

At the MA the null hypothesis (H0 ) is rejected whenever Z > k , and Pf a = P (Z > k) = 1 −

k−1 X

(38)

P (Z = l) .

l=0

is the probability of false alarm. In Figure 10 the probability of detection for the MA Pd versus xt (the x target location) for a fixed value α of the probability of false alarm for the MA Pf a is plotted. The power of the scan statistic test is more robust than the disjoint-window test with respect to the target position (see Figure 10). The performances of the disjoint-window test are strictly related to the relative position of the target with respect to the center of mass of the disjoint-window. The two-dimensional scan statistic test outperforms the non-overlapping test as we can see from the Receiver Operating Characteristic (ROC) curves in Figures 11 and 12, especially at low Pf a (M A). C. The Counting Rule Test In [12] the counting decision fusion rule, which is based on the total number of local detections from local sensors, was proposed: according to [11] at the fusion center the system-level decision is made by first counting the number of detections made by local sensors and then comparing it with a threshold T :  N M N X  >T X X decide H1 , Xij (39) Λ= Is =  6T decide H , 0 s=1 j=1 i=1 where Xij = {0, 1} is the local decision made by sensor s which is located in the cell c(i, j). In Figure 13, the ROC curves obtained by simulations (based on 1000 Monte Carlo runs) for the counting rule and the scan statistic test are shown. We note that the ROC corresponding to the optimum opt field-of-view Fov of the MA clearly demonstrates the the substantial improvement over the ROC obtained

for the counting rule. Figure 14 provides the histograms of the number of sensors Ns which report their decision to the MA (scan statistic) and to the fusion center (counting rule). For the simulations carried out, the average scan

number of sensors which report their decision to the MA (we denote it with N s

) is about 17 when

no target is present in the sensor network and 19 when it is. In the counting rule case we have instead count

that N s

is respectively 31 and 42.4 – the scan-statistic test is more efficient. We mention in passing DRAFT

22

that a multiple scan statistic [23] can statistically characterize the number of those windows. 1 Scan Statistics with Fopt =1.2 ov Counting rule 0.9

Pd MA

0.8

0.7

0.6

0.5

0.4

−2

−1

10

10

0

10

Pfa MA

Fig. 13. ROC curves in semilogarithmic scale obtained by simulations for the scan statistic test and the counting rule test. Here 2 we have b = 5, M = 625, σ 2 = 1, σw = 4, N = 25, pf a = 0.05 . Simulation are based on standard Monte Carlo counting process, involving 10000 runs.

V. C ONCLUSIONS AND F UTURE D IRECTIONS We have proposed and studied a new sequential test based on two-dimensional scan statistics, based on the number of counts reported by a sliding window “dragged” across an area of interest by a mobile agent. This test is particularly appropriate when the phenomenon of interest is a localized disturbance, such as the magnetic or acoustic signature of a low-observable target that can only be registered at sensors that are nearby. The form of the test is obvious, and matches with the GLRT under some reasonable assumptions; but the key feature of a scan statistic is its precise approximation for the falsealarm rate (exceedance probability). We have shown how the system parameters can affect the detection performance, and numerical results corroborate the theory. An important result is that the proposed fusion rule outperforms both the counting fusion rule and the “hopping window” test. We note that the scan statistic approach might be thought a (two-dimensional) “quickest” detection scheme, in that it stops as soon as a departure from the null hypothesis is declared. Ongoing research includes using a Multiple Window Discrete Scan Statistic [35] to improve on our model, and to offer analysis for the probability of detection. Further, in [38], [39] anomalies are sought DRAFT

23

Counting test − H Target Present

Scan Statistic − H Target Present 1

2500

1

800

2000

600

1500 400 1000 200

500 0

0

20

40

0 20

60

30

N

s

0

1000

40

800

30

600

20

400

10

200

20

70

s

0

10

60

Counting test − H Target Absent

50

0

50

N

Scan Statistic − H Target Absent

0

40

30

40

0 10

50

N

20

30

40

50

N

s

s

Fig. 14. Number of sensors Ns which report their decision to the MA (scan statistic) and to the fusion center (counting rule). 2 Here we have b = 5, M = 625, σ 2 = 1, σw = 4, N = 25, pf a = 0.05. Simulation are based on standard Monte Carlo counting process, involving 10000 runs.

by comparing two successive scans: if in the second there is a region that contains significantly more detections than in the first, a detection is declared. That work is very interesting, and these references include performance measures for the asymptotic case; but for an exact means to compute the threshold a scan statistic seems very appropriate. A PPENDIX A In order to obtain qm,2m we use the following recursive formulas as in [29] but with a slight modification in order to account for the two-dimensional scan statistics: qm,2m =

k X

bk2(m) (y)

(40)

y=0

DRAFT

24

bk2(m) (y) =

 P Pk−y+η k y   b2m−1 (ν, y − η)f (η)  η=0 ν=0      0

(41)

for y 6 k, for y > k

For m < i < 2m, bki (x, y) are calculated recursively, using bkm+1 (x, y) =

 P k−x  f (ν)f (y)f (ν)fm−1 (x) ν=0

 0

for x + y 6 k,

(42)

for x + y > k

and bki (x, y) =

 P Pk−(x+y)+η y   bi−1 (x + ν, y − η)×  η=0 ν=0  f (ν)f (η)f2m−i (x)

f2m−i+1 (x+y)     0 for x + y > k

(43)

for x + y 6 k,

In the above equations we used the functions fm (y) and f (y) which are defined as:   m X m X  fm (y) = P Xij = y  

(44)

i=1 j=1

f (y) = f1 (y)

R EFERENCES [1] M. Guerriero, P. Willett, and J. Glaz, “Target detection in sensor network using a zamboni and scan statistics,” in Proceedings of ICASSP 2008, Las Vegas, NV, 2008. [2] J. N. Tsitsiklis, “Decentralized detection,” in Advances in Signal Processing, H. V. Poor and J. B. Thomas, Eds.

JAI

Press, 1993, pp. 297–344. [3] R. Viswanathan and P. K. Varshney, “Distributed detection with multiple sensors: Part I – fundamentals,” Proc. IEEE, vol. 85, no. 1, pp. 54–63, Jan. 1997. [4] R. S. Blum, A. Kassam, and H. V. Poor, “Distributed detection with multiple sensors: Part II – advanced topics,” Proc. IEEE, vol. 85, no. 1, pp. 64–79, Jan. 1997. [5] B. Chen, R. Jiang, T. Kasetkasem, and P. Varshney, “Channel aware decision fusion in wireless sensor networks,” IEEE Trans. Signal Process., vol. 52, no. 12, pp. 3454–3458, 2004. [6] R. Blum and S. Kassam, “Optimum distributed detection of weak signals in dependent sensors,” IEEE Trans. Inf. Theory, vol. IT-38, no. 3, pp. 1066–1079, May 1992. [7] Z. Chair and P. Varshney, “Optimal data fusion in multiple sensor detection systems,” IEEE Trans. Aerosp. Electron. Syst., vol. 22, no. 1, pp. 98–101, 1986.

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[8] L. Tong, Q. Zhao, and S. Adireddy, “Sensor networks with mobile agents,” in Proceedings of MILCOM 2003, Boston, MA, October 2003. [9] P. Willett and L. Tong, “One aspect to cross-layer design in sensor networks,” in Proceedings of MILCOM 2004, Monterey, CA, USA, Oct. 2004, pp. 688–693. [10] S. Marano, V. Matta, P. Willett, and L. Tong, “Cross-layer design of sequential detectors in sensor networks,” IEEE Trans. on Signal Processing, vol. 54, no. 11, pp. 4105–4117, November 2006. [11] R. Niu, P. K. Varshney, M. H. Moore, and D. Klamer, “Decision fusion in a wireless sensor network with a large number of sensors,” in Proc. 7th IEEE International Conference on Information Fusion (ICIF’04), Stockolm, Sweden, 2004. [12] R. Niu and P. Varshney, “Performance analysis of distributed detection in a random sensor field,” IEEE Trans. Signal Process., vol. 56, no. 1, pp. 339–349, Jan. 2008. [13] N. Balakrishnan and M. Koutras, Runs and Scans with Applications. New York: Wiley, 2002. [14] J. Fu and W. Lou, Distribution Theory of Runs and Patterns and its Applications. Singapore: World Scientific Publishing, 2003. [15] J. Glaz and N. Balakrishnan, Scan Statistics and Applications. Boston: Birkhauser, 1999. [16] H. V. Poor, An Introduction to Signal Detection and Estimation. New York: Springer-Verlag, 1988. [17] K. P. Kleinman, A. M. Abrams, M. Kulldorff, and R. Platt, “A model adjusted space-time scan statistic with an application to syndromic surveillance,” Epidemiol. Infect., vol. 133, pp. 409–419, 2005. [18] J. Hoh and J. Ott, “Scan statistics to scan markers for susceptibility of genes,” Journal of Computational and Graphical Statistics, vol. 10, pp. 296–328, 2001. [19] H. S. Burkom, “Biosurveillance applying scan statistic with multiple, disparate data sources,” J. Urban Health, vol. 80, pp. 57–65, 2003. [20] G. P. Patil and C. Taillie, “Upper level set scan statistic for detecting arbitrarily shaped hotspots,” Environmental and Ecological Statistics, vol. 11, pp. 183–197, 2004. [21] D. Q. Naiman and C. Priebe, “Computing scan statistic p-values using importance sampling, with applications to genetics and medical image analysis,” Proceedings National Academy Sciences USA, vol. 97, pp. 9615–9617, 2000. [22] J. Glaz, Scan statistics. Encyclopedia of Statistics in Quality and Reliability.

New York, Wiley (in press), 2007.

[23] J. Glaz, J. Naus, and S. Wallenstein, Scan Statistics. New York: Springer-Verlag, 2001. [24] J. Naus, “The distribution of the size of the maximum cluster of points on a line,” J. Amer. Statist. Assoc., vol. 60, pp. 532–538, 1965. [25] J. Chen and J. Glaz, “Two-dimensional discrete scan statistics,” Statistics and Probability Letters, vol. 31, pp. 59–68, 1996. [26] J. Glaz and J. Naus, “Tight bounds and approximations for scan statistic probabilities for discrete data,” Ann. Appl. Probab., vol. 1, pp. 306–318, 1991. [27] M. V. Boutsikas and M. Koutras, “Reliability approximations for markov chain imbeddable systems,” Methodology and Computing in Applied Probability, vol. 2, pp. 393–412, 2000. [28] ——, “Bounds for the distribution of the two-dimensional binary scan statistics,” Probability in the Engineering and Informational Sciences, vol. 17, pp. 509–525, 2003. [29] V. V. Karwe and J. I. Naus, “New recursive methods for scan statistic probabilities,” Computational Statistics and Data Analysis, vol. 23, pp. 389–402, 1997. [30] T. Akiba and H. Yamamoto, “Evaluation for the reliability of a large 2-dimensional rectangular k-within-consecutive -(r,s)-out-of-(m,n): F system,” Journal of Japan Industrial Management Association, vol. 53, pp. 208–219, 2005.

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[31] Z. Wang and P. Willett, “A performance study of some transient detectors,” IEEE Trans. Signal Process., vol. 48, no. 9, pp. 2682–2685, September 2000. [32] ——, “A variable threshold page procedure for detection of transient signals,” IEEE Trans. Signal Process., vol. 53, no. 11, pp. 4397–4402, November 2005. [33] A. Nuttall, “Detection capability of linear-and-power processor for random burst signals of unknown location,” in NUWCNPT Tech. Rep. 10822, 1997. [34] S. Kotz, D. L. Banks, and C. B. Read, Encyclopedia of Statistical Sciences.

New York: Wiley-Interscience, 1997.

[35] J. Glaz and Z. Zhang, “Multiple window discrete scan statistics,” Journal of Applied Statistics, vol. 31, no. 8, pp. 967–980, 2004. [36] R. J. Simes, “An improved Bonferroni procedure for multiple tests of significance,” Biometrika, vol. 73, pp. 751–754, 1986. [37] J. F. C. Kingman, Poisson processes. Oxford: Clarendon Press, 1993. [38] T. He, S. Ben-David, and L. Tong, “Nonparametric change detection and estimation in large scale sensor networks,” IEEE Trans. Signal Process., vol. 54, no. 4, pp. 1204–1217, 2006. [39] ——, “Change detection and estimation in large sensor networks: Linear complexity algorithms,” in Proceedings of the 24th Army Science Conference, Orlando, FL, USA, 2004.

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