Bayesian Quickest Change Detection Under Energy Constraints

Transcription

1 Bayesian Quickest Change Detection Under Energy Constraints Taposh Banerjee and Venugopal V. Veeravalli ECE Department and Coordinated Science Laboratory University of Illinois at Urbana-Champaign, Urbana, IL Abstract In the classical version of the Bayesian quickest change detection problem proposed by Shiryaev in the nineteen sixties, there is a sequence of observations whose distribution changes at a random time, and the goal is to minimize the average detection delay, subject to a constraint on the probability of false alarm. We consider this quickest change detection problem with an additional constraint on the average energy consumed in sensing the observations. The optimal algorithm for this problem has a three threshold structure, in contrast to the single threshold Shiryaev test that is optimal for the classical Bayesian quickest change detection problem. We provide an asymptotic analysis of the three threshold policy for the case where the probability of false alarm is small, the average energy consumption is large, and the change event is rare. The analysis yields approximations for the average detection delay, probability of false alarm and average energy consumption, which can be used to optimize the thresholds to achieve desired operating points. The asymptotic analysis also reveals that the three threshold policy can be approximated by a simpler two threshold policy. The advantage of the two threshold policy is that the thresholds can be set directly using constraints on the probability of false alarm and average energy consumption. We provide extensive simulation results that corroborate our analytical findings. I. INTRODUCTION In the Bayesian quickest change detection problem proposed by Shiryaev, there is a sequence of random variables, {X n }, whose distribution changes at a random time Γ. It is assumed that before Γ, {X n } are independent and identically distributed i.i.d.) with density f, and after Γ they are i.i.d. with density f. The distribution of Γ is assumed to be known and modeled as a geometric random variable. The objective is to find a stopping time τ, at which time the change is declared, such that the average detection delay is minimized subject to a constraint on the probability of false alarm. In this paper we extend Shiryaev s formulation by explicitly accounting for the energy consumed in the detection process. We model the energy consumption through the number of observations taken before the change is declared. The motivation for this model comes from the consideration of modern wireless sensor networks, where the sensors are battery operated and the most effective way to save energy at the sensors is to switch the sensor between on and off states. The objective is to choose the control policy that turns the sensors on and off along with the stopping time τ so that the average detection This research partially supported by the National Science Foundation under grant CCF 8-369, through the University of Illinois at Urbana-Champaign. delay is minimized subject to constraints on the probability of false alarm and the average energy consumption. There have been other formulations of the Bayesian quickest change detection problem that are relevant to sensor networks. For example in the decentralized quickest change detection problem, a set of geographically distributed sensors observe the change and send quantized versions of their observations to a fusion center where a final decision about the change is made 2, 3, 4. The new design problem is then to jointly optimize the sensor and fusion center decision rules to minimize the average detection delay subject to a constraint on the probability of false alarm. In 5, the energy consumed for communication between sensors and fusion center is taken into account more explicitly by considering the ana channel connecting the sensors and the fusion center. To the best of our knowledge, the paper by Premkumar and Kumar 6 is the only other work in the context of quickest change detection in sensor networks that takes the approach of switching the sensors between on and off states to save energy. In 6, there is a fusion center that decides how many sensors need to be turned on each time step. The optimization problem studied in 6 is to minimize a linear combination of the detection delay and energy consumption average number of sensors turned on in each time step), subject to a constraint on the probability of false alarm. A Lagrangian version of the problem is posed within the framework of dynamic programming, similar to the way in which the Shiryaev test is derived for the classical quickest change detection problem. It is shown that, just as in the Shiryaev test, the a posteriori probability p k of the change having happened before time k, given all the information up to time k, serves as a sufficient statistic for both the stopping rule as well as the control law that determines how many sensors need to be turned on at each time k. Some structural properties of the optimal stopping rule and control law are also found. In particular, as is to be expected, it is optimal to stop and declare the change when p k exceeds a certain threshold. It is also conjectured, based on numerical results, that the number of sensors to be turned on increases with p k initially, peaks at a certain value, and then decreases with p k ; the intuition behind this behavior being that it is not necessary to waste sensing resources if the change is either very unlikely or very likely, given the observations. The goal of this paper is do develop a deeper understanding

2 of the on-off control policy for energy efficient sensing. In particular, while it is important to understand the structural properties of the optimal control policy and the stopping rule, it is even more useful from an application viewpoint to be able to characterize the performance of the optimal policy as a function of the parameters that characterize the optimal policy. By performance analysis, we mean obtaining analytical expressions and approximations for the average detection delay, probability of false alarm, and average energy consumption as a function of the parameters. Performance analysis of sequential and quickest change detection procedures is notoriously difficult, and requires sophisticated tools from renewal and nonlinear renewal theory 7, 8. For example, for the classical Bayesian change detection problem, while the optimal test structure was derived by Shiryaev more than forty years ago, it was only recently that a complete performance analysis of this test using nonlinear renewal theory was provided 9. In order to make progress with the analysis of energy aware quickest change detection, we therefore consider the simplest setting where there is only one sensor in the system and the goal of the control policy is to determine whether or not to turn on the sensor at each time step. The solution to the single sensor problem is also relevant to the distributed sensor case where the sensors act autonomously to detect changes within their sensing range, and the fusion center if one exists) simply fuses the decisions made by the sensors 4. In the following section, we set up the single sensor energy aware quickest change detection problem, and summarize the dynamic programming solution. In particular, by specializing the optimization problem considered in 6 to the single sensor case, we are able to analytically characterize the structure of an optimal policy. II. PROBLEM FORMULATION AND DYNAMIC PROGRAMMING SOLUTION As in the model for the classical Bayesian quickest change detection problem described in Section I, we have a sequence of random variables {X n }, which are i.i.d. with density f before the random change point Γ, and i.i.d. with density f after Γ. The change point Γ is modeled as geometric with parameter ρ, i.e., for < ρ <, π <, π k P{Γ k} π I {k} + π )ρ ρ) k I {k } where I is the indicator function, and π represents the probability of the change having happened before the observations are taken. Typically π is set to. In order to save energy, the sensor is switched between the on and off states. Let S k {, }, with S k if the sensor is on and X k is available for decision making, and S k otherwise. Note that S k is a control input based on the information available up to time k, i.e., S k µ k I k ), k, 2,... with µ denoting the control law and I defined as: I k S,...,S k, X S),...,X S k) k The choice of S is based on the prior π. As in the classical change detection problem, the end goal is to choose a stopping time on the observation sequence at which time the change is declared. Denoting the stopping time by τ, we can define the average detection delay ADD) as ADD E τ Γ) + Further, we can define the probability of false alarm PFA) as PFA Pτ < Γ) and the average energy consumption AEC) as τ AEC E S k. Let γ {τ, µ,..., µ τ } represent a policy for energy aware quickest change detection. We wish to solve the following optimization problem: minimize γ ADDγ), subject to PFAγ) α, and AECγ) β, ) where α and β are given constraints. The solution to this optimization problem can be obtained using dynamic programming and following steps similar to those given in 6. In the following, for completeness, we summarize these steps. We begin by considering the Lagrangian version of the optimization problem in ), i.e., Rγ) min γ ADDγ) + λ f PFAγ) + λ e AECγ) 2) where λ f and λ e are Lagrange multipliers. Let Θ k denote the state of the system at time k. After the stopping time τ is assumed that the state in a terminal state T and stay there. For k < τ, we have Θ k for k < Γ, and Θ k otherwise. Then we can write τ ADD E k I {Θk } and PFA EI {Θτ}. Furthermore, let D k denote the stopping decision variable at time k, i.e., D k if k < τ and D k otherwise. Then the optimization problem in 2) can be written as a minimization of an additive cost over time: with Rγ) min γ τ Eg k Θ k, D k, S k ) g k θ, d, s) I {θ T } I{θ} I {d} + λ f I {θ} I {d} + λ e I {s} I {d} Using standard arguments it can be seen that this optimization problem can be solved using infinite horizon dynamic programming with sufficient statistic belief state) given by: p k P{Θ k I k } P{Γ k I k }

3 Using Bayes rule, p k can be shown to satisfy the recursion { Φ ) p k ) if S k+ p k+ Φ ) X k+, p k ) if S k+ where and Φ ) X k+, p k ) Φ ) p k ) p k + p k )ρ 3) Φ ) p k )LX k+ ) Φ ) p k )LX k+ ) + Φ ) p k )) with LX k+ ) f X k+ )/f X k+ ) being the likelihood ratio, and p π. Note that the structure of recursion for p k is independent of time k. The optimal policy for the problem given in 2) can be obtained from the solution to the Bellman equation: p k ) min d k,s k+ λ f p k )I {dk } + I {dk } p k + with and B ) p k)i {sk+ } + λ e + B ) p k))i {sk+ } B ) p k) Φ ) p k )) B ) p k) EΦ ) X k+, p k )). 4) 5) It is easy to show by an induction argument see, e.g., 6) that, B ) and B ) are all non-negative concave functions on the interval,, and that ) B ) ) B) ). Also, by ensen s inequality p) EΦ) X, p)) B ) p), p, B ) From these properties of, B ) and B ), it is easy to show that the optimal policy γ τ, µ, µ,..., µ τ ) for the problem given in 2) has the following three threshold structure: { Sk+ µ if p k, B ) C, A kp k ) if p k B, C ) 6) τ inf {k : p k > A } with B C A. The optimal policy described in 6) has the following interpretation. Recall that p k is the a posteriori probability of the change having happened before time k, given all the information up to time k. If p k is either close to or close to, the test has high confidence about the change, and therefore does not take a new observation and relies on the prior. If p k is large enough then we stop and declare a change. Note that the dynamic programming argument only tells us that the optimal policy, i.e., the one that minimizes ADD subject to constraints on PFA and AEC, can be found within the class of three-threshold policies of the form given in 6). Not all three-threshold policies are optimal solutions to the constrained optimization problem described in ). We will therefore denote an arbitrary three-threshold policy by γa, B, C), with the understanding that the policy may not be optimal. In order to obtain the optimal thresholds A, B, C ) for given constraints PFA α and AEC β, it is still necessary to obtain analytical expressions for ADD, PFA and AEC, and to solve the optimization problem 6) on the space of thresholds, A, B, C). To this end, in the following section, we derive analytical approximations for ADD, PFA and AEC in terms of A, B, C) for any three-threshold policy γa, B, C). We will show in Section IV-A that the solution to the optimization problem to obtain A, B, C ), using our approximations, can be found within a class of two-threshold policies. This fact also greatly simplifies the problem of finding the thresholds to meet constraints on PFA and AEC, since the thresholds for the two-threshold policy can be expressed directly in terms of the constraints and no further optimization is required. III. ASYMPTOTIC ANALYSIS OF γa, B, C) In this section we derive asymptotic approximations for ADD, PFA and AEC for a three-threshold policy γa, B, C). To that end, we first convert the recursion for p k see 3) and 4)) to a form that is amenable to asymptotic analysis. Define, Z k p k p k for k. This new variable Z k has a one-to-one mapping with p k. By defining a A A, b B B, c C C, we can write the recursions 3) and 4) in terms of Z k. For k, Z k+ Z k + LX k+ ) + ρ) + + ρ e Z ) k, if Zk b, c) 7) and Z k+ Z k + ρ) + + ρ e Z k), if Zk / b, c) 8) with Z e Z + ρ ) + ρ) + LX )) I {Z / b,c)}. Here we have used the fact that S k+ if p k B, C), and S k+ otherwise see 6)). The crossing of thresholds A, B, C by p k is equivalent to the crossing of thresholds a, b, c by Z k. Thus the stopping time for γa, B, C) equivalently γa, b, c) with some abuse of notation) is τ inf {k : Z k > a}, In this section we study the asymptotic behavior of γa, b, c) in terms of Z k, under various limits of a, b, c and ρ. Specifically, we provide an asymptotic expression for ADD, for fixed b and ρ, as c, a. We also provide asymptotic expressions for PFA and AEC as c, a and ρ, for fixed b. Note that the limit of c, a corresponds to PFA going to zero and ADD, AEC ), and the limit of ρ corresponds to a rare change event. Figure shows a typical evolution of γa, b, c), i.e., of Z k using 7) and 8), starting at time. Note that for Z k b, c), recursion 7) is used, while outside that interval, recursion 8),

4 5 4 3 Z k Γ 6 τ 9 c 2 τ * a c b PFA and AEC to argue that the optimal three-threshold policy γ a, b, c ) for given constraints PFA α and AEC β, can be well approximated by a two threshold policy. We also provide numerical and simulation results to support the claim. We begin our analysis by first obtaining the asymptotic overshoot distribution for Z c) using non-linear renewal theory 7, 8. As mentioned above, this will be useful for the ADD analysis and will be critical to the PFA analysis. In what follows, we use E i and P i to denote, respectively, the expectation and probability measure under f i, i,. Also, gx) o) as x x is used to denote that gx) in the specified limit. Fig.. Evolution of Z k for f N, ), f N.5, ), ρ. and Z b, with A.99, C.98, and B.2. which only uses the prior ρ, is used. As a result Z k increases monotonically outside b, c). Define, τ c inf {k : Z k > c}. From Figure again, each time Z k crosses b from below, it can either increase to c point τ c ), and then monotonically increase to stop at a point τ), or it can go below b and approach b monotonically again from below, at which time it faces a similar set of alternatives. Thus the passage to threshold c possibly involves multiple cycles of the evolution of Z k below b. We will show in Section III-B that after the change point Γ, following a finite number of cycles below b, Z k grows up to cross c. We also show that the time spent, after Γ, on the cycles below b is insignificant as c, a. In fact we show that, asymptotically, the time to reach c is equal to the time taken by the classical Shiryaev algorithm to move from b to c. Note that for the classical Shiryaev algorithm Z k would evolve using only 7).) When Z k crosses c from below, it does so with an overshoot. Overshoots play significant role in the performance of many sequential algorithms see 7, 9) and is central to the performance of γa, b, c) as well. In Section III-C, we show that, asymptotically, PFA depends only on the overshoot Z c) as c, and on thresholds c and a, but is not a function of the threshold b. The overshoot distribution is also used to approximate the time for Z k to move from c to a. The energy consumed during the detection process is the total time spent by Z k between b and c. As c, a, Z k crosses c only after change point Γ, with high probability. The total energy consumed can thus be divided in to two parts: the energy consumption before Γ, which is the fraction of time Z k is above b and hence depends only on b), and the part consumed after Γ. In Section III-D we show that, the energy consumed after Γ is approximately equal to the delay itself. In Section III-E we provide numerical results to demonstrate that under various scenarios, for limiting as well as moderate values of a, b, c and ρ, our asymptotic expressions for ADD, PFA and AEC provide good approximations. In Section IV we use the asymptotic expressions for ADD, A. Asymptotic overshoot In this section we characterize the overshoot distribution of Z k as it crosses c as c. For this analysis, we can therefore assume that Z k < c. Also, in analyzing the trajectory of Z k, it useful to allow for arbitrary starting point Z shifting the time axis). We first combine the recursions in 7) and 8) to get: Z k+ Z k + I {Zk b} LX k+ ) + ρ) + + e Z k ρ ). By defining Y k LX k )+ ρ) and expanding the above recursion, we can write an expression for Z n : Z n n Y k + e Z + ρ ) n + + e Z k ρ ) n I {Zk <b} LX k ) n Y k + η n. 9) Here η n is used to represent all terms other than the first in the equation above. When b, Z k evolves as in the classical Shiryaev test statistic, and it is easy to see that in this case: η n e Z + ρ ) n + + e Z k ρ ) k n k e Z + ρ ρ) k f X i ) f X i ) It was shown in 9 that this {η n } sequence for b ) is a slowly changing sequence in the sense defined in 7. Using this result, we have been able to show that the sequence {η n } of 9) is also a slowly changing sequence. As a result, if Rx) is the asymptotic distribution of the overshoot when the random walk n Y k crosses a large positive boundary under P, then due to the slowly changing nature of {η n }, the asymptotic distribution of Z c, as c, is also equal Rx) see 7). i

5 Theorem. For fixed ρ and b, lim P Z τ c c c x τ c Γ Rx). ) B. Delay Analysis The PFA can be shown to have the following expression and bound 9: PFA E p τ A + e a e a. ) We will later show that this upper bound is tight if the gap between c and a is significant enough. Using this upper bound we can show: ADD E τ Γ) + Eτ Γ τ Γ + o)) as c, a 2) Based on the arguments leading to ), we can also show that Pτ c < Γ E p e c. As c, a, the conditional delay Eτ Γ τ Γ will be due to the sample paths in which τ c crosses c after the change point Γ. The overall delay over these paths is the sum of τ c Γ and τ τ c. This is proved in Lemma. We need the following definition. Let tx, y) be the constant time taken by Z k to move from Z x to y using the recursion 8), i.e. tx, y) inf{k : Z k > y, Z x, x, y / b, c)}. 3) Lemma. For fixed ρ and b, if tc, a) is bounded as c, a, then as c, a, Eτ Γ τ Γ Eτ τ c τ c Γ + Eτ c Γ τ c Γ) + o)) 4) In the following, we provide asymptotic expressions for Eτ τ c τ c Γ and Eτ c Γ τ c Γ. ) Asymptotic expression for Eτ c Γ τ c Γ: It was discussed in reference to Figure that each time Z k crosses b from below, it faces two alternatives, to cross c without ever coming back to b or to go below b and cross it again from below. It was mentioned that the passage to the threshold c is through multiple such cycles. Motivated by this we define the following stopping times λ and Λ. λ inf{k : Z k / b, c), Z b}. 5) Let {Z k b} denote the event that Z k crosses b from below. Then Λ inf{k : Z k > c or Z k b, Z b} 6) We can write Λ as a function of λ using 3): Λ λ+tz λ, b))i {Zλ <b}+λ I {Zλ >c} λ+tz λ, b)i {Zλ <b}. The significance of these stopping times is as follows. If we start the process at Z b and reset Z k to b each time it crosses b from below, then the time taken by Z k to move from b to c is the sum of a finite but random number of random variables with distribution of Λ, say Λ, Λ 2,...,Λ N. For i,...,n, Z Λi < b, and Z ΛN > c. N We will show in Lemma 3 that E Λ k is the dominant term in an upper bound to Eτ c Γ τ c Γ as c, a. The stopping time λ also appears in the expression for AEC. Specifically, since the time spent above b is the energy consumption, λ is precisely the energy consumed when Z k starts with Z b and stops at Λ. Define ADD s N E Λ k. Then N ADD s E Λ k E NE Λ E Λ P Z λ > c) E λ + E tz λ, b) {Z λ < b}p Z λ < b). P Z λ > c) The second equality follows from Wald s lemma 7, and the third follows because N GeomPZ λ > c)). Let Ψ represent the Shiryaev recursion, i.e., updating Z k using only 7). Define the stopping time: ν inf {k : ΨZ k ) > c, Z b} 7) which is the time for the Shiryaev test to reach c starting at b. We now show that ADD s is asymptotically equivalent to E ν. Lemma 2. For a fixed b, ADD s, the average time for Z k to cross c starting at b, under P, with Z k reset to b each time it crosses b from below, is asymptotically equal to the corresponding time taken by the Shiryaev recursion, i.e., ADD s E ν + o)) as c, a Using Lemma 2, we obtain an approximation for an upper bound to Eτ c Γ τ c Γ. Lemma 3. For a fixed b and ρ, we have as c, a, c < a Eτ c Γ τ c Γ E ν + o)). 8) 2) Second order approximation for E ν: A second order approximation for E ν is developed in 9: E ν c Eηb) + r + o) as c, 9) Df, f ) + ρ) where, Df, f ) is the K-L divergence between f and f, ηb) is the a.s. limit of the corresponding slowly changing sequence with Z o b see 9), and r xdrx). 2) with Rx) as in Theorem. Using the above second order approximation we show that Eτ c Γ τ c Γ is asymptotically greater than E ν.

6 Lemma 4. For fixed b and ρ, Eτ c Γ τ c Γ E ν + o)) as c. Lemmas 3 and 4 immediately imply the following theorem. Theorem 2. For a fixed b and ρ, we have as c, a, Eτ c Γ τ c Γ E ν + o)). 2) 3) Asymptotic expression for Eτ τ c τ c Γ: The time for Z k to reach a after it has crossed c is non-zero only if the overshoot Z c < a c. If the overshoot is x < a c, then the time taken is tc + x, a). Since, with c, the distribution of Z c is Rx), one can approximate Eτ τ c τ c Γ for large c, by averaging tc + x, a) over Rx). We first obtain upper and lower bounds on tx, y). These bounds will also be useful for computation of AEC. We use t to represent tx, y) for simplicity. First note that by definition see 3)), Z t > y Z t. Also, from 8) Thus equivalently Z t Z t + ρ + + ρ) e Zt ρ + + ρ) ezt ρ + ey + ρ). y + ρ + + e y ρ). y < Z t y + ρ + + e y ρ), e y < e Zt e y ρ + e y ρ). Further, the recursion 8) can be written in terms of e Z k for k : e Z k+ ρ + ez k ρ. Using this we can write an expression for e Zt : e Zt e x t ρ) t + ρ ρ) k ex + ρ). ρ) t Using the bounds for Z t obtained above, we get e y < ex + ρ) t ρ) ey ρ + e y ρ). This gives us bounds for t tx, y): +e y ρ +e x ) ρ ) tx, y) ) +e y +e y ρ) ρ ρ) +e x ρ ). 22) Remark: Note that as y, tx, y) ρ), which is the expression in 9) for large c, but without the contribution of the observations, Df, f ). y The bounds in 22) are used to prove the following theorem. Theorem 3. Let d a c. Then as c, a, d d x ρ) drx) + o) Eτ τ c τ c Γ d d x drx) + Rd) + o). ρ) 23) The upper and lower bounds differ by Rd), which being a distribution satisfies Rd). Further if d or ρ, then we have Eτ τ c τ c Γ ) d d x ρ) drx) + o)). C. PFA Analysis 24) We know from equation ) that PFA can be written as E p τ. We first obtain an expressions for PFA as a function of the overshoot when Z k crosses a. Lemma 5. For fixed ρ and b, as c, a PFA E p τ e a Ee Zτ a) + o)). From Lemma 5, it is evident that PFA depends on the overshoot when Z k crosses a as a. This overshoot in turn depends on the overshoot of Z k when it crosses c. Since the latter has an asymptotic distribution Theorem ) that depends only on densities f, f and prior ρ, and is independent of b, it is natural to expect that as c, PFA is completely characterized by the asymptotic distribution Rx) and is not a function of the threshold b. This is indeed true and is proved in the following. When Z k crosses c it can either directly jump above a with an overshoot greater than a c, i.e., {Z > a}, or cross c with an overshoot less than a c, i.e., {Z a}. In the former case, the false alarm is then a function of the asymptotic distribution Rx) as c. In the latter case, because Z k crosses a with the help of only the prior, the overshoot is small and goes to zero as ρ 8). As a, Z τ. Hence, + e Z τ ) ) ρ + ρ Z τ Z τ as a. ρ ρ Thus on the set {Z τ a}, Z τ a for ρ small enough, and in this case PFA e a. Based on this idea we have obtained an asymptotic expression for PFA. Theorem 4. For b fixed, as c, a, and ρ, ) PFA e a Ra c) + e c e x drx) + o)) a c 25)

7 Note that for a c, or A C, the expression for PFA reduces to the one obtained in 9. Also, for a large gap between a and c, PFA e a, hence for these cases the upper bound obtained in ) is tight. Note that we do not need the asymptote c to deduce this. Based on the discussion above, the tightness can be directly deduced from ). An important observation here is that PFA is independent of b. D. Computation of AEC As c, τ c Γ with high probability. As a result, the total energy consumed can be separated in two parts, one consumed before Γ and another after Γ. The average energy consumed before Γ is the fraction of time the process Z k is above b. The average energy consumed after Γ, for large c, is approximately the time taken by Z k to reach c. We obtain an asymptotic expression for AEC based on the above ideas. First note that, τ AEC E S k E S k E E S k τ c Γ Pτ c Γ) + E S k τ c < Γ Pτ c < Γ) S k τ c Γ + o)) as c. The last equality follows because τ c S k Γ on {τ c < Γ}. Also Pτ c < Γ) < e c as c. Define AEC as the average energy consumed before Γ, and AEC as the average energy consumed after Γ, conditioned on the event {τ c Γ}. The subscripts and are chosen to indicate that the computations are done with respect to the distributions P and P respectively. We can then write AEC as ) AEC E S k τ c Γ + o)) E Γ S k τ c Γ + E ) S k τ c Γ + o)) kγ AEC + AEC ) + o)) as c. We now obtain asymptotic expressions for AEC and AEC. ) Asymptotic expression for AEC : The energy consumption after Γ can be written as the difference between the time for Z k to reach c and the time spend by it below b. For this we define the variable T b E. Thus kγ {Zk <b} AEC E τ c Γ τ c Γ T b +. Lemma 6. For a given ρ and b, AEC E ν + o)) as c, a. 2) Asymptotic expression for AEC : For computation of AEC, we allow for the possibility that the process {Z k } started with Z z. Let tb) be the first time Z k crossed b from below, i.e., tb) tz, b). Using the fact that observations are used only after tb), we can write the following: Γ AEC E S k τ c Γ E E E Γ S k ; Γ > tb) τ c Γ S k Γ > tb), τ c Γ PΓ > tb) τ c Γ) Γ Γ ktb) S k Γ > tb), τ c Γ PΓ > tb) τ c Γ). We now show that PΓ > tb) τ c Γ) is independent of z as ρ. Lemma 7. PΓ > tb) τ c Γ) +o) as c, a, ρ + eb Proof: The proof uses the fact that the bounds in 22) +e converges to b ) rho), for a fixed x and y, when we take ρ. The details are omitted due to lack of space. Thus we can write, as c, a, ρ, Γ AEC E S k τ c Γ 26) E Γ ktb) S k Γ > tb), τ c Γ + e b + o). We now compute the right hand side of 26) as c, a, ρ. First note that lim E c,a Γ ktb) S k Γ > tb), τ c Γ E Γ ktb) S k Γ > tb), c. With λ defined as in 5), we compute the right hand side in the lemma below. Lemma 8. For a fixed b, as ρ, E Γ ktb) S k Γ > tb), c ρ E λ; c + o)). E λ; c + EtZ λ, b); c

8 The results of this section are summarized in the following theorem. Theorem 5. For fixed threshold b, we have following asymptotic expression for AEC: As c, a, ρ, AECAEC + AEC ) + o)) AEC ρ E λ; c + o)) E λ; c + EtZ λ, b); c + eb AEC E ν + o)). 3) Simpler Approximation for AEC : Setting c, and invoking Wald s lemma, we approximate E λ by, E λ EZ λ E η λ Df, f ) + ρ). In general its difficult to compute this expression and also EtZ λ, b). But AEC depends on E λ and EtZ λ, b) only E through the ratio λ E λ+etz λ,b). We have developed a good approximation for this ratio by using 24) and E λ r + + ρe b ) Df, f ) ρ). 27) Here, +ρe b ) is an approximation to E η λ by ignoring all the random terms after b is factored out of it. This extra b will cancel with the b in EZ λ b + EZ λ b. We approximate EZ λ b by r, the mean overshoot of the random walk k i Y k, with mean Df, f ) ρ), when it crosses a large boundary see 9)). The term EtZ λ, b) can be approximated using 22) and steps from the proof of Theorem 3. Specifically we approximate +e b EtZ λ, b) E. Numerical Results +e b x ) ρ ) drx). 28) In earlier sections we have obtained asymptotic expressions for PFA, ADD and AEC as a function of the system parameters: the thresholds a, b, c, the densities f and f, and the prior ρ. We summarize the results below for convenience of reference. Let νx, y) represent the time for Shiryaev algorithm to reach y, starting at x; so ν defined in 7) equals νb, c) in the new notation. Also, we write λ as λb) to indicate its dependence on b. The asymptotic approximations are: ADD Eτ c Γ τ c Γ + EtZ, a) τ c Γ a c a c x E νb, c) + drx) 29) ρ) PFA e a Ra c) + e c e x drx) 3) a c AEC AEC + AEC ρ E λb); c + e b E λb); c + EtZ λ, b); c + E νb, c) 3) Recall that E νb, c) was obtained in 9), based on the result from 9, and a simpler approximation for AEC was developed using 27) and 28). In Section IV we show that the optimal solution to the problem in ), with ADD, PFA, and AEC given by the expressions in 29), 3) and 3), has a two threshold structure. In this section we show that these expressions quite accurately reflect the actual behavior of the system in simulations. Together these two facts suggest that the twothreshold policy of Section IV is approximately optimal. In the following we assume that the observations are Gaussian with f N, ), and f N.75, ) for the simulations and analysis. In the simulations, the PFA values are computed using the expression E p τ given in ). This guarantees a faster convergence for small values of PFA. In Table I we show that E νb, c) is a good approximation for AEC and also for Eτ c Γ τ c Γ. We simulate the algorithm with ρ. and fix b 2.2, and c a. The Eτ c Γ τ c Γ and AEC thus obtained is compared with the E νb, c) computed from 9). We also provide the corresponding PFA values in the table. TABLE I ρ., f N, ), f N.75, ), b 2.2, AND c a. a Eτ c Γ τ c Γ AEC E ν PFA e e e e-5 Next we show the accuracy of approximation for PFA and AEC in Table II. For various values of ρ, a, b and c, we compare the approximations with simulations, and see that the approximations are indeed good. In Table III, we show that PFA is not a function of b for large values of c and a. We fix a 4.6 and c 3.89, and increase b from -2.2 to.85. We notice that PFA is unchanged in simulations when b is changed this way. This is also captured by the analysis and its quite accurate. TABLE II f N, ), f N.75, ) Simulations Analysis ρ a b c AEC PFA AEC PFA e e e e e e e e e e-4 Table IV shows the comparison of simulations and analysis for EtZ, a) τ c Γ, as provided in equation 29). We tabulate the result for various values of ρ and thresholds a and c. The values indicate that the approximation is accurate. IV. TWO-THRESHOLD STRUCTURE In this section, using the analytical results developed so far, i.e., using 29), 3) and 3), we argue that the threethreshold policy γa, b, c) can be well-approximated by a

9 TABLE III ρ., f N, ), f N.75, ) Simulations Analysis a b c PFA PFA e e e e e e e e e e-3 TABLE IV f N, ), f N.75, ) Simulations Analysis ρ a c EtZ, a) τ c Γ EtZ, a) τ c Γ two-threshold policy by showing the sense in which a twothreshold policy is optimal Section IV-A). By two-threshold policy we mean γa, b, c) with c a. This two threshold policy offers uniqueness of operating point and simplicity of design Section IV-B). A. Optimality of the two-threshold structure If the asymptotic expressions 29)-3) are taken to be the actual system performance, we can solve the optimization problem in ) by finding the thresholds a, b and c which minimize ADD for given constraints on PFA and AEC. We will now prove that the solution to this constrained optimization problem can be found within the class of two-threshold policies obtained by setting c a. If c a then the performance of γa, b, c) can be obtained by simply substituting c a in the various asymptotic expressions. We identify this as a separate policy and use b and a to name the two thresholds. Then the performance of this two-threshold policy would be: ADD E νb, a ) 32) PFA e a e x drx). 33) AEC AEC + AEC E λb ) ρ + e b E λb ) + EtZ λ, b ) + E νb, a )34) Theorem 6. The optimal solution to the problem in ) with ADD, PFA and AEC given by 29), 3) and 3) is a twothreshold policy. Proof: We claim that given any a, b and c, we can select some a and b and get at least as good of an operating point. The key to this argument is the independence of PFA from b and the fact that AEC and Eτ c Γ τ c Γ, being equal, can be controlled simultaneously. First select a such that the false alarm probabilities are same for the two policies: e a Ra c) + e c e x drx) e a e x drx). a c It is easy to see that a c. Now select b such that the following Shiryaev delays are equal: E νb, c) E νb, a ). Since, a c, we have b b. Since the Shiryaev delays are exactly the respective AEC s for the two algorithm, we see that: the two systems have the same PFA, and have the same post change energy consumption, i.e., AEC AEC. Also, AEC AEC since b b. Using this a and b we also get a smaller delay because E νb, a ) E νb, c)+ a c a c x drx). 35) ρ) Thus we have found a and b which gives the same PFA performance but at least as good ADD and AEC. Moreover, note the optimal a and b can be obtained directly on the constraints on PFA and AEC using 33) and 34), and no further optimization is required. Remark. We note that the constraint α on PFA can always be met with equality using 33) to set a. However, if the constraint β on AEC is too large, then it is possible that the solution for b obtained using 34) is such that b. In this case, observations are always taken and the resulting average stopping time is such that the resulting AEC is strictly less than β, i.e., the AEC constraint is not met with equality. Based on Theorem 6 and the accuracy of the asymptotic expressions demonstrated in the last section, we see that as long as the asymptotic expressions reflect the true system behavior, a three-threshold policy can always be well-approximated by a two-threshold policy. For cases in which the analytical expressions do not capture the actual system behavior, e.g., for moderate values of the system parameters, we show below using numerical and simulation results, that the arguments used in Theorem 6 can still be used to get almost as good an operating point by using two thresholds. For the data in Table V, we first choose various values of a, b and c. We then find the corresponding values of a and b using the technique mentioned above, i.e., we use the analysis results corresponding to the given values of a, b and c, to choose a and b. We then simulate both the systems to see if we get a better operating point using only two thresholds. The Table V tabulates the result for ρ.. The results clearly show that, for the parameters chosen, we get much better ADD and AEC, and almost the same PFA. In Table VI we show similar results for a wide range of thresholds and ρ. The table shows that, even for such a large range of ρ values and thresholds, we obtain as good or better operating points using just two thresholds.

10 TABLE V ρ., f N, ), f N.75, ) Three Threshold Two Threshold a b c ADDAECPFA a b ADDAECPFA e e e e e e e e e e-2 TABLE VI f N, ), f N.75, ) Three Threshold Two Threshold ρ a b c ADDAECPFA a b ADDAECPFA e e e e e e e e e e e e-3 B. Design of two-threshold policy In this section we start with the constraints α and β and use the analytical expressions 33) and 34) to choose a and b that meet these constraints. We then simulate the algorithm using these thresholds to check if the performance meets the desired constraints we started with. The findings are tabulated in Table VII. For comparison we also provide the ADD obtained from the analytical expression 32) using thresholds a and b. We see that the analytical expressions provide us with the means to design the two-threshold algorithm. As mentioned earlier, the ADD value obtained is unique for given α and β over the class of two-threshold policies, and is the minimum possible value. problem of finding the thresholds to meet constraints on PFA and AEC, since the thresholds for the two-threshold policy can be expressed directly in terms of the constraints and no further optimization is required. We provided extensive numerical and simulation results that validate our analysis. REFERENCES A. N. Shiryaev, On optimal methods in quickest detection problems, Theory Probab. Appl., Vol. 8, pp , V.V. Veeravalli, Decentralized quickest change detection, Information Theory, IEEE Transactions on, vol. 47, 2, pp Y. Mei. Information bounds and quickest change detection in decentralized decision systems. Information Theory, IEEE Transactions on, vol. 5, 25, pp A.G. Tartakovsky and V.V. Veeravalli, Quickest change detection in distributed sensor systems, Information Fusion, 23. Proceedings of the Sixth International Conference of, 23, pp L. Zacharias and R. Sundaresan, Decentralized sequential change detection using physical layer fusion, Proc. ISIT, France, un K. Premkumar and Anurag Kumar, Optimal Sleep/Wake Scheduling for Quickest Intrusion Detection Using Sensor Networks, IEEE INFOCOM, Arizona, USA, D Siegmund, Sequential analysis: tests and confidence intervals, Springer Series in Statistics, M. Woodroofe. Nonlinear Renewal Theory in Sequential Analysis. Philadelphia: SIAM, A.G. Tartakovsky and V.V. Veeravalli. General Asymptotic Bayesian Theory of Quickest Change Detection. SIAM Theory of Probability and its Applications, Vol 49, No. 3, pp , 25. D. P. Bertsekas. Dynamic Programming and Optimal Control, Vol. I and II 3rd ed.). Athena Scientific. 27. TABLE VII f N, ), f N.75, ) Constraints Two Threshold Analysis Simulations ρ α β a b ADD ADD AEC PFA. e e e e-5.5 5e e-4.5 3e e-6 V. CONCLUSIONS We posed an energy aware version of the classical Bayesian quickest change detection problem, where energy consumption is measured in terms of the number of observations taken before the change is declared. We showed via a dynamic programming argument that an optimal solution to the problem of minimizing ADD, under constraints on PFA and AEC, can be found within a class of three-threshold policies. We obtained asymptotic approximations for ADD, PFA and AEC for a generic three-threshold policy. The expressions showed the role of the three thresholds a, b, and c in determining the performance of the policy. A closer look at the analytical expressions showed that the policy that minimizes ADD, under given constraints on PFA and AEC, is well approximated by a two-threshold policy. This fact also greatly simplified the