Distributed Detection/Isolation Procedures for Quickest Event Detection in Large Extent Wireless Sensor Networks
aa r X i v : . [ s t a t . A P ] J u l Distributed Detection/Isolation Procedures forQuickest Event Detection inLarge Extent Wireless Sensor Networks
K. Premkumar § , Anurag Kumar † , and Joy Kuri ‡ Abstract
We study a problem of distributed detection of a stationary point event in a large extent wirelesssensor network (
WSN ), where the event influences the observations of the sensors only in the vicinity ofwhere it occurs. An event occurs at an unknown time and at a random location in the coverage region(or region of interest (
ROI )) of the
WSN . We consider a general sensing model in which the effect of theevent at a sensor node depends on the distance between the event and the sensor node; in particular, inthe Boolean sensing model, all sensors in a disk of a given radius around the event are equally affected.Following the prior work reported in [1], [2], [3], the problem is formulated as that of detecting the eventand locating it to a subregion of the
ROI as early as possible under the constraints that the averagerun length to false alarm (
ARL2FA ) is bounded below by γ , and the probability of false isolation ( PFI )is bounded above by α , where γ and α are target performance requirements. In this setting, we proposedistributed procedures for event detection and isolation (namely MAX , ALL , and
HALL ), based on thelocal fusion of
CUSUM s at the sensors. For these procedures, we obtain bounds on the maximum mean § K. Premkumar’s work in this paper was done during his doctoral work at the Indian Institute of Science, Bangalore, India.He is currently with the Hamilton Institute, National University of Ireland, Maynooth, Ireland(e–mail: [email protected]). † Anurag Kumar is with the Department of Electrical Communication Engineering, Indian Institute of Science, Bangalore –560 012, India (e–mail: [email protected]). ‡ Joy Kuri is with the Centre for Electronics Design and Technology, Indian Institute of Science, Bangalore – 560 012, India(e–mail: [email protected]).This is a revised and expanded version of a paper that was presented in the 47th Annual Allerton Conference onCommunication, Control, and Computing, 2009. This work was supported by a Project on Wireless Sensor Networks, fundedby DRDO, Government of India. The work of the second author was also supported, in part, by the Department of Science andTechnology, through a J.C. Bose Fellowship.
November 10, 2018 DRAFT detection/isolation delay (
SADD ), and on
ARL2FA and
PFI , and thus provide an upper bound on
SADD as min { γ, /α } → ∞ . For the Boolean sensing model, we show that an asymptotic upper bound on themaximum mean detection/isolation delay of our distributed procedure scales with γ and α in the sameway as the asymptotically optimal centralised procedure [2]. Index Terms
Disorder problem, distributed quickest change detection, detection with distance dependent sensing,fusion of
CUSUM s, multi–decision change–point detection, multi–hypothesis change detection
I. I
NTRODUCTION
Event detection is an important application for which a wireless sensor network (
WSN ) is deployed. Anumber of sensor nodes (or “motes”) that can sense, compute, and communicate are deployed in a regionof interest (
ROI ) in which the occurrence of an event (e.g., crack in a structure) has to be detected. In ourwork, we view an event as being associated with a change in the distribution (or cumulative distributionfunction) of a physical quantity that is sensed by the sensor nodes . Thus, the work we present in thispaper is in the framework of quickest detection of change in a random process. In the case of smallextent networks, where the coverage of every sensor spans the whole
ROI , and where we assume thatan event affects all the sensor nodes in a statistically equivalent manner, we obtain the classical changedetection problem whose solution is well known (see, for example, [4], [5], [6]). In [7] and [8], we havestudied variations of the classical problem in the
WSN context, where there is a wireless communicationnetwork between the sensors and the fusion centre [7], and where there is a cost for taking sensormeasurements [8].However, in the case of large extent networks, where the
ROI is large compared to the coverage regionof a sensor, an event (e.g., a crack in a huge structure, gas leakage from a joint in a storage tank) affectssensors that are in its proximity; further the effect depends on the distances of the sensor nodes fromthe event. Since the location of the event is unknown, the post–change distribution of the observationsof the sensor nodes are not known . In this paper, we are interested in obtaining procedures for detectingand locating an event in a large extent network. This problem is also referred to as change detection andisolation (see [1], [2], [3], [9], [10]). Since the
ROI is large, a large number of sensors are deployed tocover the
ROI , making a centralised solution infeasible. In our work, we seek distributed algorithms fordetecting and locating an event, with small detection delay, subject to constraints on false alarm andfalse isolation . The distributed algorithms require only local information from the neighborhood of eachnode.
November 10, 2018 DRAFT
A. Discussion of Related Literature
The problem of sequential change detection/isolation with a finite set of post–change hypotheses wasintroduced by Nikiforov [1], where he studied the change detection/isolation problem with the observationsbeing conditionally independent, and proposed a non–Bayesian procedure which is shown to be maximummean detection/isolation delay optimal, as the average run lengths to false alarm and false isolation goto ∞ . Lai [10] considered the multi–hypothesis change detection/isolation problem with stationary pre–change and post–change observations, and obtained asymptotic lower bounds for the maximum meandetection/isolation delay.Nikiforov also studied a change detection/isolation problem under the average run length to false alarm( ARL2FA ) and the probability of false isolation (
PFI ) constraints [2], in which he showed that a
CUSUM –like recursive procedure is asymptotically maximum mean detection/isolation delay optimal among theprocedures that satisfy
ARL2FA > γ and PFI α asymptotically, as min { γ, /α } → ∞ . Tartakovskyin [3] also studied the change detection/isolation problem where he proposed recursive matrix CUSUM and recursive matrix Shiryayev–Roberts tests, and showed that they are asymptotically maximum meandelay optimal over the constraints
ARL2FA > γ and PFI α asymptotically, as min { γ, /α } → ∞ .Malladi and Speyer [11] studied a Bayesian change detection/isolation problem and obtained a meandelay optimal centralised procedure which is a threshold based rule on the a posteriori probability ofchange corresponding to each post–change hypothesis.Centralised procedures incur high communication costs and distributed procedures would be desirable.In this paper, we study distributed procedures based on CUSUM detectors at the sensor nodes where the
CUSUM detector at sensor node s is driven only by the observations made at node s . Also, in the caseof large extent networks, the post–change distribution of the observations of a sensor node, in general,depends on the distance between the event and the sensor node which is unknown. B. Summary of Contributions
1) As the
WSN considered is of large extent, the post–change distribution is unknown, and couldbelong to a set of alternate hypotheses. In Section III, we formulate the event detection/isolationproblem in a large extent network in the framework of [2], [3] as a maximum mean detec-tion/isolation delay minimisation problem subject to an average run length to false alarm (
ARL2FA )and probability of false isolation (
PFI ) constraints.2) We propose distributed detection/isolation procedures
MAX , ALL , and
HALL ( H ysteresis modified ALL ) for large extent networks in Section IV. The procedures
MAX and
ALL are extensions of
November 10, 2018 DRAFT the decentralised procedures
MAX [6] and
ALL [9], [12], which were developed for small extentnetworks. The distributed procedures are energy–efficient compared to the centralised procedures.Also, the known centralised procedures are applicable only for the Boolean sensing model.3) In Section IV, we first obtain bounds on
ARL2FA , PFI , and maximum mean detection/isolationdelay (
SADD ) for the distributed procedures
MAX , ALL , and
HALL . These bounds are then appliedto get an upper bound on the
SADD for the procedures when
ARL2FA > γ , and PFI α ,where γ and α are some performance requirements. For the case of the Boolean sensing model,we compare the SADD of the distributed procedures with that of Nikiforov’s procedure [2] (acentralised asymptotically optimal procedure) and show that the an asymptotic upper bound on themaximum mean detection/isolation delay of our distributed procedure scales with γ and α in thesame way as that of [2]. II. S YSTEM M ODEL
Let
A ⊂ R be the region of interest ( ROI ) in which n sensor nodes are deployed. All nodes areequipped with the same type of sensor (e.g., acoustic). Let ℓ ( s ) ∈ A be the location of sensor node s , anddefine ℓ := [ ℓ (1) , ℓ (2) , · · · , ℓ ( n ) ] . We consider a discrete–time system, with the basic unit of time beingone slot, indexed by k = 0 , , , · · · , the slot k being the time interval [ k, k + 1) . The sensor nodes areassumed to be time–synchronised (see, for example, [13]), and at the beginning of every slot k > , eachsensor node s samples its environment and obtains the observation X ( s ) k ∈ R . A. Change/Event Model
An event (or change) occurs at an unknown time T ∈ { , , · · · } and at an unknown location ℓ e ∈ A .We consider only stationary (and permanent or persistent) point events, i.e., an event occurs at a pointin the region of interest, and having occurred, stays there forever . Examples that would motivate sucha model are 1) gas leakage in the wall of a large storage tank, 2) excessive strain at a point in a large2–dimensional structure. In [14] and [15], the authors study change detection problems in which theevent stays only for a finite random amount of time.An event is viewed as a source of some physical signal that can be sensed by the sensor nodes. Let h e be the signal strength of the event . A sensor at a distance d from the event senses a signal h e ρ ( d ) + W , In case, the signal strength of the event is not known, but is known to lie in an interval [ h, h ] , we work with h e = h asthis corresponds to the least Kullback–Leibler divergence between the “ event not occurred ” hypothesis and the “ event occurred ”hypothesis. See [16] for change detection with unknown parameters for a collocated network. November 10, 2018 DRAFT where W is a random zero mean noise, and ρ ( d ) is the distance dependent loss in signal strength whichis a decreasing function of the distance d , with ρ (0) = 1 . We assume an isotropic distance dependentloss model, whereby the signal received by all sensors at a distance d (from the event) is the same. Example 1 The Boolean model (see [17]): In this model, the signal strength that a sensor receives isthe same (which is given by h e ) when the event occurs within a distance of r d from the sensor and is 0otherwise. Thus, for a Boolean sensing model, ρ ( d ) = , if d r d , otherwise . Example 2 The power law path–loss model (see [17]) is given by ρ ( d ) = d − η , for some path loss exponent η > . For free space, η = 2 . B. Detection Region and Detection Partition
In Example 2, we see that the signal from an event varies continuously over the region. Hence, unlikethe Boolean model, there is no clear demarcation between the sensors that observe the event and those thatdo not. Thus, in order to facilitate the design of a distributed detection scheme with some performanceguarantees, in the remainder of this section, we will define certain regions around each sensor.
Definition 1
Given < µ h e , the Detection Range r d of a sensor is defined as the distance fromthe sensor within which the occurrence of an event induces a signal level of at least µ , i.e., r d := sup { d : h e ρ ( d ) ≥ µ } . In the above definition, µ is a design parameter that defines the acceptable detection delay. For a givensignal strength h e , a large value of µ results in a small detection range r d (as ρ ( d ) is non–increasing in d ). We will see in Section IV-F (Eqn. (17)) that the SADD of the distributed change detection/isolationprocedures we propose, depends on the detection range r d , and that a small r d (i.e., a large µ ) resultsin a small SADD , while requiring more sensors to be deployed in order to achieve coverage of the
ROI .We say that a location x ∈ ROI is detection–covered by sensor node s , if k ℓ ( s ) − x k r d . For anysensor node s , D ( s ) := { x ∈ A : k ℓ ( s ) − x k r d } is called its detection–coverage region (see Fig. 1). Weassume that the sensor deployment is such that every x ∈ A is detection–covered by at least one sensor November 10, 2018 DRAFT
24 31 A A A AA A A Fig. 1.
Partitioning of A in a large WSN by detection regions : (a simple example) The coloured solid circles around eachsensor node denote their detection regions. The four sensor nodes divide the
ROI , indicated by the square region, into regions A , · · · , A such that region A i is detection–covered by a unique set of sensors N i . For example, A is detection covered bythe set of sensors N = { , , } , etc. (Fig. 1). For each x ∈ A , define N ( x ) to be the largest set of sensors by which x is detection–covered,i.e., N ( x ) := { s : x ∈ D ( s ) } . Let C ( N ) = {N ( x ) : x ∈ A} . C ( N ) is a finite set and can have at most n − elements. Let N = |C ( N ) | . For each N i ∈ C ( N ) , we denote the corresponding detection–coveredregion by A i = A ( N i ) := { x ∈ ROI : N ( x ) = N i } . Evidently, the A i , i N , partition the ROI .We say that the
ROI is detection–partitioned into a minimum number of subregions , A , A , · · · , A N ,such that the subregion A i is detection–covered by a unique set of sensors N i , and A i is the maximaldetection–covered region of N i , i.e., ∀ i = i ′ , N i = N i ′ and A i ∩ A i ′ = ∅ . See Fig. 1 for an example. C. Sensor Measurement Model
Before change, i.e., for k < T , the observation X ( s ) k at the sensor s is just the zero mean sensornoise W ( s ) k , the probability density function (pdf) of which is denoted by f ( · ) ( pre–change pdf ). Afterchange, i.e., for k > T with the location of the event being ℓ e , the observation of sensor s is given by X ( s ) k = h e ρ ( d e,s )+ W ( s ) k where d e,s := k ℓ ( s ) − ℓ e k , the pdf of which is denoted by f ( · ; d e,s ) ( post–changepdf ). The noise processes { W ( s ) k } are independent and identically distributed (iid) across time and acrosssensor nodes. In the rest of the paper, we consider f ( · ) to be Gaussian with mean 0 and variance σ . We denote the probability measure when the change happens at time T and at location ℓ e by P ( d ( ℓ e )) T {·} ,where d ( ℓ e ) = [ d e, , d e, , · · · , d e,n ] , and the corresponding expectation operator by E ( d ( ℓ e )) T [ · ] . In the caseof Boolean sensing model, the post–change pdfs depend only on the detection subregion where the eventoccurs, and hence, we denote the probability measure when the event occurs at ℓ e ∈ A i and at time T November 10, 2018 DRAFT by P ( i ) T {·} , and the corresponding expectation operator by E ( i ) T [ · ] . D. Local Change Detectors
We compute a
CUSUM statistic C ( s ) k , k > at each sensor s based only on its own observations. The CUSUM procedure was proposed by Page [5] as a solution to the classical change detection problem(
CDP , in which there is one pre–change hypothesis and only one post–change hypothesis). The optimalityof
CUSUM was shown for conditionally iid observations by Moustakides in [18] for a maximum meandelay metric introduced by Pollak [19] which is
SADD ( τ ) := sup T > E T [ τ − T | τ > T ] .The driving term of CUSUM should be the log likelihood–ratio (LLR) of X ( s ) k defined as Z ( s ) k ( d e,s ) :=ln (cid:16) f ( X ( s ) k ; d e,s ) f ( X ( s ) k ) (cid:17) . As the location of the event ℓ e is unknown, the distance d e,s is also unknown. Hence,one cannot work with the pdfs f ( · ; d e,s ) . We propose to drive the CUSUM at each node s with Z ( s ) k ( r d ) ,where we recall that r d is the detection range of a sensor. Based on the CUSUM statistic C ( s ) k , k > ,sensor s computes a sequence of local decisions D ( s ) k ∈ { , } , k > , where 0 represents no–changeand 1 represents change. For each set of sensor nodes N i that detection partitions the ROI , we define τ ( N i ) , the stopping time (based on the sequence of local decisions D ( s ) k s for all s ∈ N i ) at which the setof sensors N i detects the event. The way we obtain the local decisions D ( s ) k from the CUSUM statistic C ( s ) k , k > , and the way these local decisions determine the stopping times τ ( N i ) , varies from rule torule. Specific rules for local decision and the fusion of local decisions will be described in Section IV(also see [20]).An implementation strategy for our distributed event detection/isolation procedure can be the following.We assume that the sensors know to which detection sensor sets N i s they belong. This could be doneby initial configuration or by self–organisation. When the local decision of sensor s is 1, it broadcaststhis fact to all sensors in its detection neighbourhood. In practise, the broadcast range of these radios issubstantially larger than the detection range. Hence, the local decision of s is learnt by all sensors s ′ thatbelong to N i to which s belongs. When any node learns that all the sensors in N i have reached the localdecision 1, it transmits an alarm message to the base station [21]. A distributed leader election algorithmcan be implemented so that only one, or a controlled number of alarms is sent. This alarm message iscarried by geographical forwarding [22]. A system that utilises such local fusion (but with a differentsensing and detection model) was developed by us and is reported in [23]. November 10, 2018 DRAFT
E. Influence Region
After a set of nodes N i declares an event, the event is isolated to a region associated with N i called theinfluence region. In the Boolean sensing model, if an event occurs in A i , then only the sensors s ∈ N i observe the event, while the other sensors s ′ / ∈ N i only observe noise. On the other hand, in the powerlaw path–loss model, sensors s ′ / ∈ N i can also observe the event, and the driving term of the CUSUM sof sensors s ′ may be affected by the event. The mean of the driving term of CUSUM of any sensor s isgiven by E f ( · ; d e,s ) [ Z ( s ) k ( r d )] = ( h e ρ ( r d )) σ (cid:18) ρ ( d e,s ) ρ ( r d ) − (cid:19) . (1)Thus, the mean of the increment that drives CUSUM of node s decreases with d e,s and becomes negativewhen ρ ( d e,s ) < ρ ( r d ) . In this region, we are interested in finding T E , the expected time for the CUSUM statistic C ( s ) k to cross the threshold c . Define τ ( s ) := inf n k : C ( s ) k > c o , and hence, T E = E ( d ( ℓ e ))1 (cid:2) τ ( s ) (cid:3) . Lemma 1
If the distance between sensor node s and the event, d e,s is such that ρ ( d e,s ) < ρ ( r d ) , then T E > exp( ω c ) where ω = 1 − ρ ( d ) ρ ( r d ) .Proof: From (Eqn. 5.2.79 pg. 177 of) [24], we can show that E ( d ( ℓ e ))1 (cid:2) τ ( s ) (cid:3) > exp( ω c ) where ω is the solution to the equation E ( d ( ℓ e ))1 h e ω Z ( i ) k ( r d ) i = 0 , which is given by ω = 1 − ρ ( d ) ρ ( r d ) (see Eqn. (1)).We would be interested in T E > exp( ω · c ) for some < ω < . We now define the influence range of a sensor as follows. Definition 2 Influence Range of a sensor, r i , is defined as the distance from the sensor within whichthe occurrence of an event can be detected within a mean delay of exp ( ω c ) where ω is a parameter ofinterest and c is the threshold of the local CUSUM detector. Using Lemma 1, we see that r i = min { d ′ :2 ρ ( d ′ ) (1 − ω ) ρ ( r d ) } .A location x ∈ A is influence covered by a sensor s if k ℓ ( s ) − x k r i , and a set of sensors N j is saidto influence cover x if each sensor s ∈ N j influence covers x .From Lemma 1, we see that by having a large value of ω , i.e., ω close to 1, the sensors that arebeyond a distance of r i from the event take a long time to cross the threshold. However, we see from November 10, 2018 DRAFT
24 31 A A A AA A A (a) Detection and influence regionsof the Boolean model
24 31 (b) Detection and influence regionsof the power law path loss modelFig. 2.
Influence and detection regions : A simple example of partitioning of A in a large WSN . The coloured solid circlesaround each sensor node denote their detection regions. The four sensor nodes, in the figure, divide the
ROI , indicated by thesquare region, into regions A , · · · , A such that region A i is detection–covered by a unique set of sensors N i . The dashedcircles represent the influence regions. In the Boolean model, the influence region of a sensor coincides with its detection region. the definition of influence range that a large value of ω gives a large influence range r i . We will seefrom the discussion in Section II-F that a large influence range results in the isolation of the event to alarge subregion of A . On the other hand, from Section IV-E, we will see that a large ω decreases theprobability of false isolation, a performance metric of change detection/isolation procedure, which wedefine in Section III.We define the influence–region of sensor s as T ( s ) := { x ∈ A : k ℓ ( s ) − x k r i } . For the Booleansensing model, r i = r d , and hence, D ( s ) = T ( s ) for all s n , and for the power law path–losssensing model, r i > r d , and hence, D ( s ) ⊂ T ( s ) for all s n (see Fig. 2).Recalling the sets of sensors N i , i N , defined in Section II-B, we define the influence regionof the set of sensors N i as the region B i such that each x ∈ B i is within the influence range of all thesensors in N i , i.e., B i := B ( N i ) := T s ∈N i T ( s ) . Note that A ( N i ) = T s ∈N i D ( s ) ! T T s ′ / ∈N i D ( s ′ ) ! ,where D is the complement of the set D , and D ( s ) ⊆ T ( s ) . Hence, A ( N i ) ⊆ B ( N i ) . For the power lawpath–loss sensing model, D ( s ) ⊂ T ( s ) for all s n , and hence, A ( N i ) ⊂ B ( N i ) for all i N .For the Boolean sensing model, A ( N i ) = B ( N i ) T T s ′ / ∈N i D ( s ′ ) ! , and hence A ( N i ) = B ( N i ) only when N i = { , , · · · , n } . Thus, for a general sensing model, A ( N i ) ⊆ B ( N i ) . We note here that in theBoolean and the power law path loss models, an event which does not lie in the detection subregion of N i , but lies in its influence subregion (i.e., ℓ e ∈ B ( N i ) \ A ( N i ) ) can be detected due to N i because ofthe stochastic nature of the observations; in the power law path loss sensing model, this is also because November 10, 2018 DRAFT0 of the difference in losses ρ ( d e,s ) between different sensors. Remark:
The definition of the detection and influence ranges have involved two design parameters µ and ω which can be used to “tune” the performance of the distributed detection schemes that we develop. F. Isolating the Event
In Section II D, we provided an outline of a class of distributed detection procedures that will yielda stopping rule. On stopping, a decision for the location of the event is made, which is called isolation.
In Section IV, we will provide specific distributed detection/isolation procedures in which stopping willbe due to one of the sensor sets N i .An event occurring at location ℓ e ∈ A i can influence sensors s ′ which influence cover ℓ e , and hence,the detection can be due to sensors N i = N j which influence cover ℓ e . Thus, we isolate the event to theinfluence region of the sensors that detect the event. Because of noise, detection can be due to a sensorset N h which does not influence cover the event. Such an error event is called false isolation.An event occurring at ℓ e ∈ A i is influence covered by sensors s ′ ∈ N ( ℓ e ) := { s : k ℓ ( s ) − ℓ e k r i } .Hence, the detection due to any N j ⊆ N ( ℓ e ) corresponds to the isolation of the event, and that due to N j
6⊆ N ( ℓ e ) corresponds to false isolation. Note that in the case of Boolean sensing model N ( ℓ e ) = N i .In Section III, we formulate the problem of quickest detection of an event and isolating the event toone of the influence subregions B , B , · · · , B N under a false alarm and false isolation constraint.III. P ROBLEM F ORMULATION
We are interested in studying the problem of distributed event detection/isolation in the setting devel-oped in Section II. Given a sample node deployment (i.e., given ℓ ), and having chosen a value of thedetection range, r d , we partition the ROI , A into the detection–subregions, A , A , · · · , A N . Let N i bethe set of sensors that detection–cover the region A i . Having chosen the influence range r i , the influenceregion B i of the set of sensor nodes N i can be obtained. We define the following set of hypotheses H : event not occurred , H T,i : event occurred at time T in subregion A i , T = 1 , , · · · , i = 1 , , · · · , N. The event occurs in one of the detection subregions A i , but we will only be able to isolate it toone of the influence subregions B i that is consistent with the A i (see Section II-F). We study distributedprocedures described by a stopping time τ , and an isolation decision L ( τ ) ∈ { , , · · · , N } (i.e., the tuple ( τ, L ) ) that detect an event at time τ and locate it to L ( τ ) (i.e., to the influence region B L ( τ ) ) subject November 10, 2018 DRAFT1 to a false alarm and false isolation constraint. The false alarm constraint considered is the average runlength to false alarm
ARL2FA , and the false isolation constraint considered is the probability of falseisolation
PFI , each of which we define as follows.
Definition 3
The
Average Run Length to False Alarm
ARL2FA of a change detection/isolation proce-dure τ is defined as the expected number of samples taken under null hypothesis H to raise an alarm,i.e., ARL2FA ( τ ) := E ∞ [ τ ] , where E ∞ [ · ] is the expectation operator (with the corresponding probability measure being P ∞ {} ) whenthe change occurs at infinity. Definition 4
The
Probability of False Isolation
PFI of a change detection/isolation procedure τ isdefined as the supremum of the probabilities of making an incorrect isolation decision, i.e., PFI ( τ ) := max i N sup ℓ e ∈A i max j N, N j ( ℓ e ) P ( d ( ℓ e ))1 { L ( τ ) = j } where we recall that N ( ℓ e ) = { s : k ℓ ( s ) − ℓ e k r i } is the set of sensors that influence covers ℓ e ∈ A i .In the case of Boolean sensing model, the post–change pdfs depend only on the index i of the detectionsubregion where the event occurs, and hence, the PFI is given by
PFI ( τ ) := max i N max j N, N j i P ( i )1 { L ( τ ) = j } . In [2], Nikiforov defined the probability of false isolation, also, over the set of all possible change times,as
SPFI ( τ ) := sup i N sup j = i N sup T > P ( i ) T { L ( τ ) = j | τ > T } . Define the following classesof change detection/isolation procedures, ∆( γ, α ) := { ( τ, L ) : ARL2FA ( τ ) > γ, SPFI ( τ ) α } , e ∆( γ, α ) := { ( τ, L ) : ARL2FA ( τ ) > γ, PFI ( τ ) α } . We define the supremum average detection delay
SADD performance for the procedure τ , in the samesense as Pollak [19] (also see [2]), as the maximum mean number of samples taken under any hypothesis H T,i , i = 1 , , · · · , N , to raise an alarm, i.e., SADD ( τ ) := sup ℓ e ∈A sup T > E ( d ( ℓ e )) T [ τ − T | τ > T ] . November 10, 2018 DRAFT2
We are interested in obtaining an optimal procedure τ that minimises the SADD subject to the averagerun length to false alarm and the probability of false isolation constraints, inf sup ℓ e ∈A sup T > E ( d ( ℓ e )) T [ τ − T | τ > T ] subject to ARL2FA ( τ ) > γ PFI ( τ ) α. The change detection/isolation problem that we pose here is motivated by the framework of [1], [2],[3], which we discuss in the next subsection.
A. Centralised Recursive Solution for the Boolean Sensing Model
In [2], Nikiforov and in [3], Tartakovsky studied a change detection/isolation problem that involves
N > post–change hypotheses (and one pre–change hypothesis). Thus, their formulation can be appliedto our problem. But, in their model, the pdf of X k for k > T , under hypothesis H T,i , g i is completelyknown. It should be noted that in our problem, in the case of power law path–loss sensing model, thepdf of the observations under any post–change hypothesis is unknown as the location of the event isunknown. The problem posed by Nikiforov [2] is inf ( τ,L ) ∈ ∆( γ,α ) sup i N sup E ( i ) T [ τ − T | τ > T ] , (2)and that by Tartakovsky [3] is inf ( τ,L ) ∈ e ∆( γ,α ) sup i N sup E ( i ) T [ τ − T | τ > T ] . (3)Nikiforov [2] and Tartakovsky [3] obtained asymptotically optimal centralised change detection/isolation procedures as min { γ, α } → ∞ , the SADD of which is given by the following theorem.
Theorem 1 (Nikiforov 03)
For the N –hypotheses change detection/isolation problem (for the Booleansensing model) defined in Eqn. (2) , the asymptotically maximum mean delay optimal detection/isolationprocedure τ ∗ has the property, SADD ( τ ∗ ) ∼ max ln γ min i N KL ( g i , g ) , − ln( α )min i N, j = i N KL ( g i , g j ) , as min (cid:26) γ, α (cid:27) → ∞ , where KL ( · , · ) is the Kullback–Leibler divergence function, and g i is the pdf of the observation X k for k > T under hypothesis H T,i . Remark:
Since, ∆( γ, α ) ⊆ e ∆( γ, α ) , the asymptotic upper bound on SADD for τ ∗ is also an upper boundfor the SADD over the set of procedures in e ∆( γ, α ) . November 10, 2018 DRAFT3
In the case of Boolean sensing model, for any post–change hypothesis H T,i , only the set of sensornodes that detection cover (which is the same as influence cover) the subregion A i switch to a post–change pdf f (and the distribution of other sensor nodes continues to be f ). Since the pdf of thesensor observations are conditionally i.i.d., the pdf of the observation vector, in the Boolean sensingmodel, corresponds to the post–change pdf g i of the centralised problem studied by Nikiforov [2] and byTartakovsky [3]. Thus, their problem directly applies to our setting with the Boolean sensing model. Inour work, however, we propose algorithms for the change detection/isolation problem for the power lawsensing model as well. Also, the procedures proposed by Nikiforov and by Tartakovsky are (while beingrecursive) centralised, whereas we propose distributed procedures which are computationally simple.In Section IV, we propose distributed detection/isolation procedures MAX , HALL and
ALL and analysetheir false alarm (
ARL2FA ), false isolation (
PFI ) and the detection delay (
SADD ) properties.IV. D
ISTRIBUTED C HANGE D ETECTION /I SOLATION P ROCEDURES
In this section, we study the procedures
MAX and
ALL for change detection/isolation in a distributedsetting. Also, we propose a distributed detection procedure “
HALL ,” and analyse the
SADD , the
ARL2FA ,and the
PFI performance.
A. The
MAX
Procedure
Tartakovsky and Veeravalli proposed a decentralised procedure
MAX for a collocated scenario in[6]. We extend the
MAX procedure to a large
WSN under the
ARL2FA and
PFI constraints. RecallingSection II, each sensor node i employs CUSUM for local change detection between pdfs f and f ( · ; r d ) .Let τ ( i ) be the random time at which the CUSUM statistic of sensor node i crosses the threshold c forthe first time. At each time k , the local decision of sensor node i , D ( i ) k is defined as D ( i ) k := , for k < τ ( i ) , for k > τ ( i ) . The global decision rule τ MAX declares an alarm at the earliest time slot k at which all sensor nodes j ∈ N i for some i = 1 , , · · · , N have crossed the threshold c . Thus, τ MAX , ( N i ) := inf n k : D ( j ) k = 1 , ∀ j ∈ N i o = min n τ ( j ) : j ∈ N i o τ MAX := min n τ MAX , ( N i ) : 1 i N o . i.e., the MAX procedure declares an alarm at the earliest time instant when the
CUSUM statistic of allthe sensor nodes N i corresponding to hypothesis H T,i of some i have crossed the threshold at least once. November 10, 2018 DRAFT4 kV ( i ) U ( i ) V ( i ) U ( i ) = : U ( i ) c C ( i ) k R ( i ) R ( i ) Fig. 3.
ALL and
HALL : Evolution of
CUSUM statistic C ( i ) k of node i plotted vs. k . Note that at time k = V ( i ) j , R ( i ) j is theexcess above the threshold. The isolation rule is L ( τ ) = arg min i N { τ MAX , ( N i ) } , i.e., to declare that the event has occurred in theinfluence region B L ( τ ) = B ( N L ( τ ) ) corresponding to the set of sensors N L ( τ ) that raised the alarm. B. ALL
Procedure
Mei, [9], and Tartakovsky and Kim, [25], proposed a decentralised procedure
ALL , again for a collocatednetwork. We extend the
ALL procedure to a large extent network under the
ARL2FA and the
PFI constraints. Here, each sensor node i employs CUSUM for local change detection between pdfs f and f ( · ; r d ) . Let C ( i ) k be the CUSUM statistic of sensor node i at time k . The
CUSUM in the sensornodes is allowed to run freely even after crossing the threshold c . Here, the local decision of sensor node i is D ( i ) k := , if C ( i ) k < c , if C ( i ) k > c. The global decision rule τ ALL declares an alarm at the earliest time slot k at which the local decision ofall the sensor nodes corresponding to a set N i , for some i = 1 , , · · · , N , are 1, i.e., τ ALL , ( N i ) := inf n k : D ( j ) k = 1 , ∀ j ∈ N i o = inf n k : C ( j ) k > c, ∀ j ∈ N i o τ ALL := min n τ ALL , ( N i ) : 1 i N o . The isolation rule is L ( τ ) = arg min i N { τ ALL , ( N i ) } , i.e., to declare that the event has occurred inthe influence region B L ( τ ) = B ( N L ( τ ) ) corresponding to the set of sensors N L ( τ ) that raised the alarm. C. HALL
Procedure
Motivated by
ALL , and the fact that sensor noise can make the
CUSUM statistic fluctuate around thethreshold, we propose a local decision rule which is 0 when the
CUSUM statistic has visited zero andhas not crossed the threshold yet and is 1 otherwise. We explain the
HALL procedure below.
November 10, 2018 DRAFT5
The following discussion is illustrated in Fig. 3. Each sensor node i computes a CUSUM statistic C ( i ) k based on the LLR of its own observations between the pdfs f ( · ; r d ) and f . Define U ( i )0 := 0 . Define V ( i )1 as the time at which C ( i ) k crosses the threshold c (for the first time) as: V ( i )1 := inf n k : C ( i ) k > c o (see Fig. 3 where the “overshoots” R ( i ) k , at V ( i ) k , are also shown). Note that inf ∅ := ∞ . Next define U ( i )1 := inf n k > V ( i )1 : C ( i ) k = 0 o . Now starting with U ( i )1 , we can recursively define V ( i )2 , U ( i )2 etc. in the obvious manner (see Fig. 3). Eachnode i computes the local decision D ( i ) k based on the CUSUM statistic C ( i ) k as follows: D ( i ) k = , if V ( i ) j k < U ( i ) j for some j , otherwise. (4)The global decision rule is a stopping time τ HALL defined as the earliest time slot k at which all thesensor nodes in a region have a local decision , i.e., τ HALL , ( N i ) := inf n k : D ( j ) k = 1 , ∀ j ∈ N i o ,τ HALL := min n τ HALL , ( N i ) : 1 i N o . The isolation rule is L ( τ ) = arg min i N { τ HALL , ( N i ) } , i.e., to declare that the event has occurred inthe influence region B L ( τ ) = B ( N L ( τ ) ) corresponding to the set of sensors N L ( τ ) that raised the alarm. Remark:
The procedures
HALL , MAX and
ALL differ only in their local decision rule; the global decisionrule as a function of { D ( i ) k } is the same for HALL , MAX and
ALL . For the distributed procedures
MAX , ALL , and
HALL , we analyse the
ARL2FA in Section IV-D, the
PFI in Section IV-E, and the
SADD performance in Section IV-F.
D. Average Run Length to False Alarm (
ARL2FA ) From the previous sections, we see that the stopping time of any procedure ( MAX , ALL , or
HALL ) isthe minimum of the stopping times corresponding to each N i , i.e., τ procedure := min n τ procedure , ( N i ) : 1 i N o . Under the null hypothesis H , the CUSUM statistics C ( s ) k s of sensors s ∈ N i are driven by independentnoise processes, and hence, C ( s ) k s are independent. But, there can be a sensor that is common to twodifferent N i s, and hence, τ procedure , ( N i ) s, in general, are not independent. We provide asymptotic lowerbounds for the ARL2FA for
MAX , HALL , and
ALL , in the following theorem.
November 10, 2018 DRAFT6
Theorem 2
For local
CUSUM threshold c , ARL2FA ( τ MAX ) > exp ( a MAX c ) · (1 + o (1)) (5) ARL2FA ( τ HALL ) > exp ( a HALL c ) · (1 + o (1)) (6) ARL2FA ( τ ALL ) > exp ( a ALL c ) · (1 + o (1)) (7) ( o (1) → as c → ∞ ), where for any arbitrarily small δ > , a MAX = a HALL = 1 − δ , a ALL = m − δ , where m = min {N i \ S j = i, j ∈I N j : i ∈ I} , I is the set of indices of the detection sets that are minimalin the partially order of set inclusion among the detection sets. Proof:
See Appendix A.Thus, for procedure , for a given
ARL2FA requirement of γ , it is sufficient to choose the threshold c as c = ln γa procedure (1 + o (1)) , as γ → ∞ . (8) E. Probability of False Isolation (
PFI ) A false isolation occurs when the hypothesis H T,i is true for some i and the hypothesis H T,j = H T,i is declared to be true at the time of alarm, and the event does not lie in the region B ( N j ) . The followingtheorem provide asymptotic upper bounds for the PFI for each of the procedures
MAX , ALL , and
HALL . Theorem 3
For local
CUSUM threshold c , PFI ( τ MAX ) exp ( − b MAX c ) B MAX · (1 + o (1)) (9) PFI ( τ HALL ) exp ( − b HALL c ) B HALL · (1 + o (1)) (10) PFI ( τ ALL ) exp ( − b ALL c ) B ALL · (1 + o (1)) . (11) where o (1) → as c → ∞ , and b MAX = b HALL = mξω −
1+ ¯ mn , b ALL = mξω − n , ω = 1 forBoolean sensing model, ξ is 2 for Boolean sensing model and is 1 for path–loss sensing model, m =min {|N j \ N ( ℓ e ) | : 1 i N, ℓ e ∈ A i , j N, N j
6⊆ N ( ℓ e ) } and ¯ m = max {|N j \ N ( ℓ e ) | : 1 i N, ℓ e ∈ A i , j N, N j
6⊆ N ( ℓ e ) } , and B MAX , B HALL , and B ALL are positive constants.Proof:
See Appendix B.Thus, for a given
PFI requirement of α , the threshold c for should satisfy c = − ln B procedure − ln αb procedure (1 + o (1)) = − ln αb procedure (1 + o (1)) , as α → . (12) November 10, 2018 DRAFT7
F. Supremum Average Detection Delay (
SADD ) In this section, we analyse the
SADD performance of the distributed detection/isolation procedures.We observe that for any sample path of the observation process, for the same threshold c , the MAX ruleraises an alarm first, followed by the
HALL rule, and then by the
ALL rule. This ordering is due to thefollowing reason. For each sensor node s , let τ ( s ) be the first time instant at which the CUSUM statistic C ( s ) k crosses the threshold c (denoted by V ( i )1 in Figure 3). Before time τ ( s ) , the local decision is 0 forall the procedures, MAX , ALL , and
HALL . For
MAX , for all k > τ ( s ) , the local decision D ( s ) k = 1 .Thus, the stopping time of MAX is at least as early as that of
HALL and
ALL . The local decision of
ALL is 1 ( D ( s ) k = 1 ) only at those times k for which C ( s ) k > c . However, even when C ( s ) k < c , the localdecision of HALL is 1 if V ( s ) j k < U ( s ) j (see Figure 3) for some j . Thus, the local decisions of MAX , HALL , and
ALL are ordered as, for all k > , D ( s ) k ( MAX ) > D ( s ) k ( HALL ) > D ( s ) k ( ALL ) , and hence, τ MAX , ( N i ) τ HALL , ( N i ) τ ALL , ( N i ) . Each of the stopping times MAX , HALL , or
ALL is the minimumof stopping times corresponding to the sets of sensors {N i : i = 1 , , · · · , N } , i.e., τ procedure = min { τ procedure , ( N i ) : i = 1 , , · · · , N } where “ procedure ” can be MAX or HALL or ALL . Hence, we have τ MAX τ HALL τ ALL . (13)From [9], we see that sup T > E ( i ) T h τ ALL , ( N i ) − T | τ ALL , ( N i ) > T i = cI (1 + o (1)) (14)where I is the Kullback–Leibler divergence between the post–change and the pre–change pdfs. For ℓ e ∈ A i , we have ∀ s ∈ N i , d e,s r d . Also, since τ ALL τ ALL , ( N i ) , we have sup ℓ e ∈A i sup T > E ( d ( ℓ e )) T h τ ALL − T | τ ALL > T i sup ℓ e ∈A i sup T > E ( d ( ℓ e )) T h τ ALL , ( N i ) − T | τ ALL > T i (15)From Appendix C, Eqn. (15) becomes, sup ℓ e ∈A i sup T > E ( d ( ℓ e )) T h τ ALL − T | τ ALL > T i sup ℓ e ∈A i sup T > E ( d ( ℓ e )) T h τ ALL , ( N i ) − T | τ ALL , ( N i ) > T i = c KL ( f ( · ; r d ) , f ) (1 + o (1)) (16)From the above equation, and from Eqn. (13), we have SADD ( τ MAX ) SADD ( τ HALL ) SADD ( τ ALL ) c KL ( f ( · ; r d ) , f ) (1 + o (1)) , as c → ∞ , (17) November 10, 2018 DRAFT8
Remark:
Recall from Section II-B that µ = h e ρ ( r d ) . We now see that µ governs the detectiondelay performance, and µ can be chosen such that a requirement on SADD is met. Thus, to achieve arequirement on
SADD , we need to choose r d appropriately. A small value of r d (gives a large µ andhence,) gives less detection delay compared to a large value of r d . But, a small r d requires more sensorsto detection–cover the ROI .In the next subsection, we discuss the asymptotic minimax delay optimality of the distributed proceduresin relation to Theorem 1.
G. Asymptotic Upper Bound on
SADD
For any change detection/isolation procedure to achieve a
ARL2FA requirement of γ and PFI require-ment of α , a threshold c is chosen such that it satisfies Eqns. 8 and 12, i.e., c = max (cid:26) ln γa procedure , − ln αb procedure (cid:27) (1 + o (1)) . (18)Therefore, from Eqn.(17), the SADD is given by
SADD ( τ procedure ) KL ( f ( · ; r d ) , f ) · max (cid:26) ln γa procedure , − ln αb procedure (cid:27) (1 + o (1)) . (19)where o (1) → as min { γ, α } → ∞ . Note that as r d decreases, KL ( f ( · ; r d ) , f ) = h e ρ ( r d ) σ increases.Thus, to achieve a smaller detection delay, the detection range r d can be decreased, and the number ofsensors n can be increased to cover the ROI .We can compare the asymptotic
SADD performance of the distributed procedures
HALL , MAX and
ALL against Theorem 1 for the Boolean sensing model. For Gaussian pdfs f and f , the KL divergencebetween the hypotheses H T,i and H T,j is given by KL ( g i , g j ) = Z ln Q s ∈N i f ( x ( s ) ) Q s ′ / ∈N i f ( x ( s ′ ) ) Q s ∈N j f ( x ( s ) ) Q s ′ / ∈N j f ( x ( s ′ ) ) ! Y s ∈N i f ( x ( s ) ) Y s ′ / ∈N i f ( x ( s ′ ) ) d x = Z ln Y s ∈N i f ( x ( s ) ) f ( x ( s ) ) ! − ln Y s ∈N j f ( x ( s ) ) f ( x ( s ) ) Y s ∈N i f ( x ( s ) ) Y s ′ / ∈N i f ( x ( s ′ ) ) d x = X s ∈N i KL ( f , f ) − X s ∈N j ∩N i KL ( f , f ) + X s ∈N j \N i KL ( f , f )= |N i ∆ N j | KL ( f , f ) where the operator ∆ represents the symmetric difference between the sets. Thus, from Theorem 1 for November 10, 2018 DRAFT9
Gaussian f and f , we have SADD ( τ ∗ ) KL ( f , f ) · max (cid:26) ln γa ∗ , − ln αb ∗ (cid:27) (1 + o (1)) , where a ∗ = min i N |N i | , and b ∗ = min i N j N, N j i |N i ∆ N j | . The
SADD performance of the distributed procedure with the Boolean sensing model is
SADD ( τ procedure ) KL ( f , f ) · max (cid:26) ln γa procedure , − ln αb procedure (cid:27) (1 + o (1)) . (20)where o (1) → as min { γ, α } → ∞ . Thus, the asymptotically optimal upper bound on SADD (whichcorresponds to the optimum centralised procedure τ ∗ ) and that of the distributed procedures ALL , HALL ,and
MAX scale in the same way as ln γ/ KL ( f , f ) and − ln α/ KL ( f , f ) .V. N UMERICAL R ESULTS
We consider a deployment of 7 nodes with the detection range r d = 1 , in a hexagonal ROI (see Fig. 4)such that we get N = 12 detection subregions, and N = { , , , } , N = { , , } , N = { , , , } , N = { , , } , N = { , , , } , N = { , , } , N = { , , , } , N = { , , } , N = { , , , } , N = { , , } , N = { , , , } , and N = { , , } . The pre–change pdf considered is f ∼ N (0 , ,and the detection range and the influence range considered are r d = 1 . and r i = 1 . respectively.We compute the SADD , the
ARL2FA and the
PFI performance of
MAX , HALL , ALL , and Nikiforov’sprocedure ([2]) for the Boolean sensing model with f ∼ N (1 , , and plot the SADD vs log( ARL2FA ) performance in Fig. 5(a), of the change detection/isolation procedures for PFI × − . The local CUSUM threshold c that yields the target ARL2FA and other simulation parameters and results aretabulated in Table I. To obtain the
SADD the event is assumed to occur at time 1, which correspondsto the maximum mean delay (see [19], [26]). We observe from Fig. 5(a) that the
SADD performanceof
MAX is the worst and that of
Nikiforov ′ s is the best. Also, we note that the performance of thedistributed procedures, ALL and
HALL , are very close to that of the optimal centralised procedure.For eg., for a requirement of
ARL2FA = 10 (and PFI × − ), we observe from Fig. 5(a) that SADD ( τ MAX ) = 26 . , SADD ( τ HALL ) = 13 . , SADD ( τ ALL ) = 12 . , and SADD ( τ ∗ ) = 11 . . Since MAX does not make use of the the dynamics of C ( s ) k beyond τ s , it’s SADD vs ARL2FA performance ispoor. On the other hand,
ALL and
HALL make use of C ( s ) k for all k and hence, give a better performance.For the same sensor deployment in Fig. 4, we compute the SADD and the
ARL2FA for the square lawpath loss ( η = 2 ) sensing model given in Section II. Also, the signal strength h e is taken to be unity. November 10, 2018 DRAFT0 Fig. 4.
Sensor nodes placement : 7 sensor nodes (which are numbered 1,2, · · · ,7) represented by small filled circles are placedin the hexagonal
ROI A . The sensor nodes partition the ROI into the detection subregions A , A , · · · , A (for both theBoolean and the power law path loss sensing models). TABLE IS IMULATION PARAMETERS AND RESULTS FOR THE B OOLEAN SENSING MODEL FOR
PFI × − Detection/ No. of Threshold 99% Confidence interval 99% Confidence intervalIsolation MC c ARL2FA ARL2FA lower
ARL2FA upper
SADD SADD lower
SADD upper procedure runs
MAX HALL ALL Nikiforov November 10, 2018 DRAFT1 S ADD ( sa m p l es ) ARL2FA (samples)
ALLHALLMAXNikiforov (a)
SADD vs ARL2FA for the Boolean model S ADD ( sa m p l es ) ARL2FA (samples)
ALLHALLMAX (b)
SADD vs ARL2FA for the square law path loss modelFig. 5.
SADD versus
ARL2FA (for
PFI × − ) for MAX , HALL , ALL and Nikiforov’s procedure for the Boolean andthe square law path loss sensing models. In the Boolean sensing model, the system parameters are f ∼ N (0 , , f ∼ N (0 , ,and in the case of path loss sensing model, the parameters are f ∼ N (0 , , h e = 1 , r d = 1 . , r i = 1 . .TABLE IIS IMULATION PARAMETERS AND RESULTS FOR THE SQUARE LAW PATH LOSS SENSING MODEL FOR
PFI × − Detection/ No. of Threshold 99% Confidence interval 99% Confidence intervalIsolation MC c ARL2FA ARL2FA lower
ARL2FA upper
SADD SADD lower
SADD upper procedure runs
MAX HALL ALL Thus, the sensor sets ( N i s) and the detection subregions ( A i s) are the same as in the Boolean model, wedescribed above. Since r d is taken as 1, f ( · ; r d ) ∼ N (1 , . Thus, the LLR of observation X ( s ) k is givenby ln (cid:16) f ( X ( s ) k ; r d ) f ( X ( s ) k ) (cid:17) = X ( s ) k − , which is the same as that in the Boolean sensing model. Hence, under theevent not occurred hypothesis , the ARL2FA under the path loss sensing model is the same as that of the
November 10, 2018 DRAFT2
Boolean sensing model. The
CUSUM threshold c that yields the target ARL2FA s and other parametersand results are tabulated in Table II. To obtain the
SADD the event is assumed to occur at time 1, andat a distance of r i from all the nodes of N i that influence covers the event (which corresponds to themaximum detection delay). We plot the SADD vs log( ARL2FA ) in Fig. 5(b). The ordering on SADD forany
ARL2FA across the procedures is the same as that in the Boolean model, and can be explained inthe same manner. The ambiguity in ℓ e affects f ( · ; d e,s ) and shows up as large SADD values.VI. C
ONCLUSION
We consider the quickest distributed event detection/isolation problem in a large extent
WSN with apractical sensing model which incorporates the reduction in signal strength with distance. We formulatethe change detection/isolation problem in the optimality framework of [2] and [3]. We propose distributeddetection/isolation procedures,
MAX , ALL and
HALL and show that as min { ARL2FA , / PFI } → ∞ , the
SADD performance of the distributed procedures grows in the same scale as that of the optimal centralisedprocedure of Tartakovsky [3] and Nikiforov [2].A
PPENDIX AP ROOF OF T HEOREM N i , i = 1 , , · · · , N , we choose the collection of indices I ⊆ { , , · · · , N } such that any two sensor sets N i , N j , i, j ∈ I , are not partially ordered by set inclusion. For each i ∈ I ,define the set of sensors that are unique to the sensor set N i , M i := N i \ S j = i,j ∈I N j ⊆ N i . The sets M , M , · · · , M |I| are disjoint. Under the null hypothesis, H , the observations of sensors in the sensorsets M , M , · · · , M |I| are iid, with the pdf f ∼ N (0 , σ ) . For every N i , there exists M j such that M j ⊆ N i , so that τ rule , ( N i ) > τ rule , ( M j ) . Hence, τ procedure = min { τ procedure , ( N i ) : i = 1 , , · · · , N } > min { τ procedure , ( M i ) : i ∈ I} =: b τ rule . Hence, E ∞ h τ rule i > E ∞ hb τ rule i > e mc · P nb τ rule > e mc o (by the Markov inequality)or, E ∞ (cid:2) τ rule (cid:3) e mc > P nb τ rule > e mc o = Y i ∈I P ∞ n τ rule , ( M i ) > e mc o . (21)We analyse P ∞ (cid:8) τ rule , ( M i ) > e mc (cid:9) as c → ∞ , for ALL , MAX , and
HALL . For
ALL , P ∞ n τ ALL , ( M i ) = k o P ∞ n C ( s ) k > c, ∀ s ∈ M i o = Y s ∈M i P ∞ n C ( s ) k > c o e − cm i (using Wald’s inequality)Therefore , P ∞ n τ ALL , ( M i ) k o k · e − cm i P ∞ n τ ALL , ( M i ) > e mc o > − e − c ( m i − m ) . November 10, 2018 DRAFT3
Hence, for any m < m i , we have lim inf c →∞ P ∞ (cid:8) τ ALL , ( M i ) > e mc (cid:9) = 1 . A large m (which is smallerthan all m i s) is desirable. Thus, a good choice for m is a ALL = min { m i : i ∈ I} − δ . for some arbitrarilysmall δ > . Hence, from Eqn. (21), E ∞ h τ ALL i > exp ( a ALL c ) (1 + o (1)) (22)For MAX , at the stopping time of
MAX , at least one of the
CUSUM statistics is above the threshold c , P ∞ n τ MAX , ( M i ) = k o P ∞ n C ( s ) k > c, for some s ∈ M i o X s ∈M i P ∞ n C ( s ) k > c o m i e − c (using Wald’s inequality) . (23)Therefore, for any arbitrarily small δ > , P ∞ n τ MAX , ( M i ) > e (1 − δ ) c o > − m i e − δc lim inf c →∞ P ∞ n τ MAX , ( M i ) > e (1 − δ ) c o = 1 . (24)Let a MAX = 1 − δ . For any arbitrarily small δ > , we see from Eqn. (21), E ∞ h τ MAX i > exp ((1 − δ ) c ) (1 + o (1)) =: exp ( a MAX c ) (1 + o (1)) , (25)For HALL , for the same threshold c , the stopping time of HALL is after that of
MAX . Hence, τ HALL > τ MAX . Hence, E ∞ (cid:2) τ HALL (cid:3) > E ∞ (cid:2) τ MAX (cid:3) > exp ((1 − δ ) c ) (1 + o (1)) (from Eqn. (25)). Thus, for a ALL := 1 − δ , for any arbitrarily small δ > , E ∞ h τ HALL i > exp ( a ALL c ) (1 + o (1)) (26)A PPENDIX BP ROOF OF T HEOREM ℓ e ∈ A i . The probability of false isolation when the detection is due to N j
6⊆ N ( ℓ e ) is P ( d ( ℓ e ))1 n τ rule = τ rule , ( N j ) o = P ( d ( ℓ e ))1 n τ rule , ( N j ) τ rule , ( N h ) , ∀ h = 1 , , · · · , N o P ( d ( ℓ e ))1 n τ rule , ( N j ) τ rule , ( N i ) o = ∞ X k =1 P ( d ( ℓ e ))1 n τ rule , ( N i ) = k o P ( d ( ℓ e ))1 n τ rule , ( N j ) k | τ rule , ( N i ) = k o = ∞ X k =1 P ( d ( ℓ e ))1 n τ rule , ( N i ) = k o " k X t =1 P ( d ( ℓ e ))1 n τ rule , ( N j ) = t | τ rule , ( N i ) = k o November 10, 2018 DRAFT4 A. PFI ( τ ALL ) – Boolean Sensing Model P ( i )1 n τ ALL , ( N j ) = t | τ ALL , ( N i ) = k o P ( i )1 n C ( s ) t > c, ∀ s ∈ N j | τ ALL , ( N i ) = k o P ∞ n C ( s ) t > c, ∀ s ∈ N j \ N i o exp ( −|N j \ N i | c ) (using Wald’s inequality) . Therefore, P ( i )1 n τ ALL , ( N j ) τ ALL , ( N i ) o exp ( −|N j \ N i | c ) · E ( i )1 h τ ALL , ( N i ) i exp ( − ( |N j \ N i | c − ln( c ))) · α |N i | (1 + o (1)) . Hence , PFI ( τ ALL ) max i N max j N, N j i P ( i )1 n τ ALL , ( N j ) τ ALL , ( N i ) o exp ( − ( mc − ln( c ))) nα (1 + o (1)) (27)where n = min {|N i : i = 1 , , · · · , N |} , m = min i N, j N, N j i {|N j \ N i |} . For any n , there exists c ( n ) such that for all c > c ( n ) , c < e c/n . Using this inequality, for sufficiently large c PFI ( τ ALL ) exp (cid:0) − (cid:0)(cid:0) m − n (cid:1) c (cid:1)(cid:1) nα (1 + o (1)) = exp( − b ALL · c ) B ALL (1 + o (1)) , where b ALL = m − /n and B ALL = nα . B. PFI ( τ MAX ) – Boolean Sensing Model P ( i )1 n τ MAX , ( N j ) = t | τ MAX , ( N i ) = k o P ( i )1 n τ ( s ) t, ∀ s ∈ N j | τ MAX , ( N i ) = k o P ∞ n τ ( s ) t, ∀ s ∈ N j \ N i | τ MAX , ( N i ) = k o = Y s ∈N j \N i t X n =1 P ∞ n τ ( s ) = n | τ MAX , ( N i ) = k o = Y s ∈N j \N i t X n =1 P ∞ n C ( s ) n > c o exp ( − m ji c ) t m ji Hence, P ( i )1 n τ MAX , ( N j ) τ MAX , ( N i ) o exp ( − m ji c ) E ( i )1 h ( τ MAX , N i ) m ji i exp ( − m ji c ) c m ji α m ji (1 + o (1))= exp ( − ( m ji c − (1 + m ji ) ln( c ))) α m ji (1 + o (1)) November 10, 2018 DRAFT5
Let m = min i N, j N, N j i m ji , ¯ m = max i N, j N, N j i m ji , and α ∗ = min i N, j N, N j i α m ji .Therefore, PFI ( τ MAX ) max i N max j N, N j i P ( i )1 n τ MAX , ( N j ) τ MAX , ( N i ) o exp ( − ( mc − (1 + ¯ m ) ln( c ))) α ∗ (1 + o (1)) . For any n , there exists c ( n ) such that for all c > c ( n ) , c < e c/n . Hence, for sufficiently large c PFI ( τ MAX ) max i N max j N, N j i P ( i ) T n τ MAX , ( N j ) τ MAX , ( N i ) o exp (cid:0) − (cid:0) m −
1+ ¯ mn (cid:1) c (cid:1) α ∗ (1 + o (1)) = exp( − b MAX · c ) B MAX (1 + o (1)) , where b MAX = m − ((1 + ¯ m ) /n ) and B MAX = α ∗ . C. PFI ( τ HALL ) – Boolean Sensing Model P ( i )1 n τ HALL , ( N j ) = t | τ HALL , ( N i ) = k o P ( i )1 n τ ( s ) t, ∀ s ∈ N j | τ HALL , ( N i ) = k o which has the same form as that of MAX . Hence, from the analysis of
MAX , it follows that P ( i )1 n τ HALL , ( N j ) τ HALL , ( N i ) o exp ( − m ji c ) E ( i )1 h ( τ HALL , ( N i ) ) m ji i exp ( − m ji c ) c m ji |N i | m ji α m ji (1 + o (1))= exp ( − ( m ji c − (1 + m ji ) ln( c ))) (cid:20) α |N i | (cid:21) m ji (1 + o (1)) PFI ( τ HALL ) max i N max j N, N j i P ( i )1 n τ HALL , ( N j ) τ HALL , ( N i ) o exp ( − ( mc − (1 + ¯ m ) ln( c ))) α ∗ (1 + o (1)) . where α ∗ = min i N, j N, N j i ( α · |N i | ) m ji . For any n there exists c ( n ) such that for all c >c ( n ) , c < e c/n . Hence, for sufficiently large c PFI ( τ HALL ) max i N max j N, N j i P ( i )1 n τ HALL , ( N j ) τ HALL , ( N i ) o exp (cid:0) − (cid:0) m −
1+ ¯ mn (cid:1) c (cid:1) α ∗ (1 + o (1)) = exp( − b HALL · c ) B HALL (1 + o (1)) , where b HALL = m − ((1 + ¯ m ) /n ) and B HALL = α ∗ . November 10, 2018 DRAFT6
PFI – P
ATH –L OSS S ENSING M ODEL
Lemma 2
For s ∈ N j \M e and for t > T , (with the pre–change pdf f ∼ N (0 , σ ) and the post–changepdf f ∼ N ( h e ρ ( r s ) , σ ) ) P ( d ( ℓ e ))1 n C ( s ) t > c o exp (cid:16) − ω c (cid:17) · exp (cid:16) − αω (cid:17) − exp (cid:16) − αω (cid:17) , where we recall that the parameter ω defines the influence range, and α = KL ( f , f ) .Proof: For s ∈ N j \ M e and for t > T , P ( d ( ℓ e ))1 n C ( s ) t > c o = P ( d ( ℓ e ))1 ( max n t n X k =1 ln f ( X ( s ) k ; r s ) f ( X ( s ) k ) ! > c ) ∞ X n =1 P ( d ( ℓ e ))1 ( n X k =1 ln f ( X ( s ) k ; r s ) f ( X ( s ) k ) ! > c ) = T − X n =1 P ∞ ( n X k =1 ln f ( X ( s ) k ; r s ) f ( X ( s ) k ) ! > c ) + ∞ X n = T P ( d ( ℓ e ))1 ( n X k =1 ln f ( X ( s ) k ; r s ) f ( X ( s ) k ) ! > c ) = T − X n =1 P ∞ ( n X k =1 ln f ( X ( s ) k ; r s ) f ( X ( s ) k ) ! > c ) + ∞ X n = T P ( d ( ℓ e ))1 ( T − X k =1 ln f ( X ( s ) k ; r s ) f ( X ( s ) k ) ! + n X k = T ln f ( X ( s ) k ; r s ) f ( X ( s ) k ) ! > c ) = T − X n =1 P ∞ ( n X k =1 X ( s ) k > σ h e ρ ( r s ) ( c + nα ) ) + ∞ X n = T P ∞ ( n X k =1 X ( s ) k > σ h e ρ ( r s ) c + nh e (cid:18) ρ ( r s )2 − ρ ( d e,s ) (cid:19) + ( T − h e ρ ( d e,s ) ) T − X n =1 P ∞ ( n X k =1 X ( s ) k > σ h e ρ ( r s ) ( c + nα ) ) + ∞ X n = T P ∞ ( n X k =1 X ( s ) k > n · h e ρ ( r s )2 ω + c · σ h e ρ ( r s ) ) ∞ X n =1 P ∞ ( exp θ n X k =1 X ( s ) k ! > exp (cid:18) θσ h e ρ ( r s ) ( c + nαω ) (cid:19)) for any θ > . Hence , P ( d ( ℓ e ))1 n C ( s ) t > c o ∞ X n =1 exp (cid:18) − θσ h e ρ ( r s ) ( c + nαω ) (cid:19) (cid:18) E ∞ (cid:20) e θX ( s )1 (cid:21)(cid:19) n = ∞ X n =1 exp (cid:18) − θσ h e ρ ( r s ) ( c + nαω ) + nσ θ (cid:19) Since the above inequality holds for any θ > , we have P ( d ( ℓ e ))1 n C ( s ) t > c o ∞ X n =1 min θ> exp (cid:18) − θσ h e ρ ( r s ) ( c + nαω ) + nσ θ (cid:19) The minimising θ is c + nαω nh e ρ ( r s ) . Therefore, for θ = c + nαω nh e ρ ( r s ) , P ( d ( ℓ e ))1 n C ( s ) t > c o ∞ X n =1 exp (cid:18) − ( c + nαω ) αn (cid:19) . Note that − ( c + αω n ) αn + ( c + αω ( n − α ( n −
1) = − αω c α ( n − n November 10, 2018 DRAFT7
Therefore, by iteratively computing the exponent, we have exp (cid:18) − ( c + αω n ) αn (cid:19) = exp (cid:18) − ( c + αω ) α (cid:19) · exp (cid:18) − αω n − (cid:19) exp (cid:18) c α (cid:18) − n (cid:19)(cid:19) exp (cid:18) − ( c + αω ) α (cid:19) · exp (cid:18) − αω n − (cid:19) exp (cid:18) c α (cid:19) or ∞ X n =1 exp (cid:18) − ( c + αω n ) αn (cid:19) exp (cid:16) − ω c (cid:17) · exp (cid:16) − αω (cid:17) − exp (cid:16) − αω (cid:17) =: β D. PFI ( τ ALL ) – Path Loss Sensing Model P ( d ( ℓ e ))1 n τ ALL , ( N j ) = t | τ ALL , ( N i ) = k o P ( d ( ℓ e ))1 n C ( s ) t > c, ∀ s ∈ N j | τ ALL , ( N i ) = k o P ( d ( ℓ e ))1 n C ( s ) t > c, ∀ s ∈ N j \ N ( ℓ e ) | τ ALL , ( N i ) = k o = Y s ∈N j \N ( ℓ e ) P ( d ( ℓ e ))1 n C ( s ) t > c o β |N j \N ( ℓ e ) | ( from Lemma 2 ) Therefore , P ( d ( ℓ e ))1 n τ ALL , ( N j ) τ ALL , ( N i ) o β |N j \N ( ℓ e ) | E ( d ( ℓ e ))1 h τ ALL , ( N i ) i β |N j \N ( ℓ e ) | cα |N i | (1 + o (1)) Let m = min i N,ℓ e ∈A i , j N, N j ( ℓ e ) |N j \ N ( ℓ e ) | and n = min {|N i | : i = 1 , , · · · , N } . Define K =max i N,ℓ e ∈A i , j N, N j ( ℓ e ) exp (cid:18) − αω (cid:19) − exp (cid:18) − αω (cid:19) |N j \N ( ℓ e ) | . Therefore, PFI (cid:16) τ ALL (cid:17) max i N sup ℓ e ∈A i max j N, N j ( ℓ i ) P ( d ( ℓ e ))1 n τ ALL , ( N j ) τ ALL , ( N i ) o K exp (cid:0) − (cid:0) mω c − ln( c ) (cid:1)(cid:1) αn (1 + o (1)) . For any n there exists c ( n ) such that for all c > c ( n ) , c < e c/n . Hence, for sufficiently large c PFI (cid:16) τ ALL (cid:17) K exp (cid:0) − (cid:0) mω − n (cid:1) c (cid:1) αn (1 + o (1)) = exp( − b ALL ,d · c ) B ALL , d (1 + o (1)) where b ALL ,d = ( mω / − (1 /n ) and B ALL ,d = αn/K . November 10, 2018 DRAFT8 E. PFI ( τ MAX ) – Path Loss Sensing Model P ( d ( ℓ e ))1 n τ MAX , ( N j ) = t | τ MAX , ( N i ) = k o P ( d ( ℓ e ))1 n τ ( s ) t, ∀ s ∈ N j \ N ( ℓ e ) | τ MAX , ( N i ) = k o = Y s ∈N j \N ( ℓ e ) P ( d ( ℓ e ))1 n τ ( s ) t | τ MAX , ( N i ) = k o Y s ∈N j \N ( ℓ e ) t X n =1 P ( d ( ℓ e ))1 n C ( s ) n > c o β |N j \N ( ℓ e ) | · t |N j \N ( ℓ e ) | ( from Lemma 2 ) P ( d ( ℓ e ))1 n τ MAX , ( N j ) τ MAX , ( N i ) o β |N j \N ( ℓ e ) | · E ( d ( ℓ e ))1 h ( τ MAX , ( N i ) ) |N j \N ( ℓ e ) | i β |N j \N ( ℓ e ) | · c |N j \N ( ℓ e ) | α |N j \N ( ℓ e ) | (1 + o (1)) Let m = min i N,ℓ e ∈A i , j N, N j ( ℓ e ) |N j \ N ( ℓ e ) | , ¯ m = max i N,ℓ e ∈A i , j N, N j e |N j \ N ( ℓ e ) | , anddefine K = max i N,ℓ e ∈A i , j N, N j ( ℓ e ) exp (cid:18) − αω (cid:19) − exp (cid:18) − αω (cid:19) |N j \N ( ℓ e ) | . Therefore, PFI ( τ MAX ) max i N sup ℓ e ∈A i max j N, N j ( ℓ e ) P ( d ( ℓ e ))1 n τ MAX , ( N j ) τ MAX , ( N i ) o Kα ∗ exp (cid:16) − (cid:16) mω c − (1 + ¯ m ) ln( c ) (cid:17)(cid:17) (1 + o (1)) . where α ∗ = min i N,ℓ e ∈A i , j N, N j ( ℓ e ) α |N j \N ( ℓ e ) | . For any n there exists c ( n ) such that for all c > c ( n ) , c < e c/n . Hence, for sufficiently large c PFI ( τ MAX ) Kα ∗ exp (cid:18) − (cid:18) mω − mn (cid:19) c (cid:19) (1 + o (1)) = exp( − b MAX ,d · c ) B MAX , d (1 + o (1)) , where b MAX , d = ( mω ) − (
1+ ¯ mn ) and B MAX ,d = α ∗ K . F. PFI ( τ HALL ) – Path Loss Sensing Model P ( d ( ℓ e ))1 n τ HALL , ( N j ) = t | τ HALL , ( N i ) = k o P ( d ( ℓ e ))1 n τ ( s ) t, ∀ s ∈ N j \ N ( ℓ e ) | τ HALL , ( N i ) = k o which has the same form as that of MAX . Hence, from the analysis of
MAX , it follows that P ( d ( ℓ e ))1 n τ HALL , ( N j ) τ HALL , ( N i ) o β |N j \N ( ℓ e ) | E ( d ( ℓ e ))1 h ( τ HALL , ( N i ) ) |N j \N ( ℓ e ) | i β |N j \N ( ℓ e ) | c |N j \N ( ℓ e ) | ( α |N i | ) |N j \N ( ℓ e ) | (1 + o (1)) Therefore,
PFI ( τ HALL ) max i N sup ℓ e ∈A i max j N, N j ( ℓ e ) P ( d ( ℓ e ))1 n τ HALL , ( N j ) τ HALL , ( N i ) o Kα ∗ exp (cid:16) − (cid:16) mω c − (1 + ¯ m ) ln( c ) (cid:17)(cid:17) (1 + o (1)) . November 10, 2018 DRAFT9
Therefore for large c, PFI Kα ∗ exp (cid:18) − (cid:18) mω − mn (cid:19) c (cid:19) (1 + o (1)) = exp( − b HALL ,d · c ) B HALL , d (1 + o (1)) , where α ∗ = min i N,ℓ e ∈A i , j N, N j ( ℓ e ) ( α · |N i | ) |N j \N ( ℓ e ) | , b HALL , d = ( mω / − (1 + ¯ m ) /n , and B HALL ,d = α ∗ /K . A PPENDIX C SADD
FOR THE B OOLEAN AND THE P ATH LOSS M ODELS
Fix i, i N . For each change time T > , define F T = σ ( X ( s ) k , s ∈ N , k T ) , and for ℓ e ∈ A i , F ( i ) T = σ ( X ( s ) k , s ∈ N i , k T ) . From [9] (Theorem 3, Eqn. (24)), ess sup E ( d ( ℓ e )) T (cid:16) ( τ rule , ( N i ) − T ) + |F ( i )( T − (cid:17) cI (1 + o (1)) , as c → ∞ , (28)Define F { τ rule , ( N i ) > T } as the σ -field generated by the event { τ rule , ( N i ) > T } , and similarly define the σ -field F { τ rule > T } . Evidently F { τ rule , ( i ) > T } ⊂ F ( i )( T − and F { τ rule > T } ⊂ F ( T − . By iterated conditionalexpectation, E ( d ( ℓ e )) T (cid:16) ( τ rule , ( N i ) − T ) + |F { τ rule > T } (cid:17) ess sup E ( d ( ℓ e )) T (cid:16) ( τ rule , ( N i ) − T ) + |F ( T − (cid:17) (29)We can further assert that E ( d ( ℓ e )) T (cid:16) ( τ rule , ( N i ) − T ) + |F ( T − (cid:17) a . s . = E ( d ( ℓ e )) T (cid:16) ( τ rule , ( N i ) − T ) + |F ( i )( T − (cid:17) Using this observation with Eqn. 29 and Eqn. 28, we can write, as c → ∞ , E ( d ( ℓ e )) T (cid:16) ( τ rule , ( N i ) − T ) + |F { τ rule > T } (cid:17) cI (1 + o (1)) (30)Finally, E ( d ( ℓ e )) T (cid:0) ( τ rule , ( N i ) − T ) + | τ rule > T (cid:1) I { τ rule > T } a . s . = E ( d ( ℓ e )) T (cid:0) ( τ rule , ( N i ) − T ) + |F { τ rule > T } (cid:1) I { τ rule > T } .We conclude, from 30, that, as c → ∞ , E ( d ( ℓ e )) T (cid:0) ( τ rule , ( N i ) − T ) + | τ rule > T (cid:1) cI (1 + o (1)) .R EFERENCES [1] I. V. Nikiforov, “A generalized change detection problem,”
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