3.06.2.1 Martingales

Some important and useful results regarding martingales are as follows:

Kolmogorov’s inequality can be considered to be a generalization of Markov’s inequality, which is given by

(6.2)

As we will see in the following sections, quickest change detection procedures often involve comparing a stochastic sequence to a threshold to make decisions. Martingale inequalities often play a crucial role in the design of the threshold so that the procedure meets a false alarm constraint. We now state one of the most useful results regarding martingales.

3.06.2.2 Stopping times

Sometimes the definition of a stopping time also requires that be finite almost surely, i.e., that .

Stopping times are essential to sequential decision making procedures such as quickest change detection procedures, since the times at which decisions are made are stopping times with respect to the observation sequence. There are two main results concerning stopping times that are of interest.

Like martingale inequalities, the optional stopping theorem is useful in the false alarm analysis of quickest change detection procedures. Both the optional stopping theorem and Wald’s identity also play a key role in the delay analysis of quickest change detection procedures.

3.06.2.3 Renewal and nonlinear renewal theory

As we will see in subsequent sections, quickest change detection procedures often involve comparing a stochastic sequence to a threshold to make decisions. Often the stochastic sequence used in decision-making can be expressed as a sum of a random walk and possibly a slowly changing perturbation. To obtain accurate estimates of the performance of the detection procedure, one needs to obtain an accurate estimate of the distribution of the overshoot of the stochastic sequence when it crosses the decision threshold. Under suitable assumptions, and when the decision threshold is large enough, the overshoot distribution of the stochastic sequence can be approximated by the overshoot distribution of the random walk. It is then of interest to have asymptotic estimates of the overshoot distribution, when a random walk crosses a large boundary.

Consider a sequence of i.i.d. random variables (with Y denoting a generic random variable in the sequence) and let

and

The quantity of interest is the distribution of the overshoot . If are i.i.d. positive random variables with cumulative distribution function (c.d.f.) , then can be viewed as inter-arrival times of buses at a stop. The overshoot is then the time to next bus when an observer is waiting for a bus at time b. The distribution of the overshoot, and hence also of the time to next bus, as is a well known result in renewal theory.

When the are i.i.d. but not necessarily non-negative, and , then the following concept of ladder variables can be used. Let

Note that if , then is a positive random variable. Also, if

then the distribution of is the same as the overshoot distribution for the sum of a sequence of i.i.d. positive random variables (each with distribution equal to that of ) crossing the boundary b. Therefore, by applying Theorem 5, we have the following result.

Techniques for computing the required quantities involving the distribution of the ladder height in Theorem 6 can be found in [17].

As mentioned earlier, often the stochastic sequence considered in quickest change detection problem can be written as a sum of a random walk and a sequence of small perturbations. Let

and

Then,

Therefore, assuming that , Wald’s Identity (see Theorem 4) implies that

(6.3)

(6.4)

Thus,

If and are finite then it is easy to see that

where is as defined in Table 6.1.

But if we can characterize the overshoot distribution of when it crosses a large threshold then we can obtain better approximations for . Nonlinear renewal theory allows us to obtain distribution of the overshoot when satisfies some properties.

If indeed is a slowly changing sequence, then the distribution of , as , is equal to the asymptotic distribution of the overshoot when the random walk crosses a large positive boundary, as stated in the following result.

3.06.3.1 The Bayesian i.i.d. setting

Here it is assumed that conditioned on the change point , the random variables are i.i.d. with probability density function (p.d.f.) before the change point, and i.i.d. with p.d.f. after the change point. The change point is modeled as geometric with parameter , i.e., for

(6.11)

where is the indicator function. The goal is to choose a stopping time on the observation sequence to solve (6.10).

A solution to (6.10) is provided in Theorem 8 below. Let denote the observations up to time n. Also let

(6.12)

be the a posteriori probability at time n that the change has taken place given the observation up to time n. Using Bayes’ rule, can be shown to satisfy the recursion

(6.13)

where

(6.14)

, is the likelihood ratio, and .

Note that with equality iff almost surely. We will assume that

See Figure 6.2a for a plot of and as a function of . Figure 6.2b shows a typical evolution of the optimal Shiryaev algorithm.

We now discuss some alternative descriptions of the Shiryaev algorithm. Let

and

We note that is the likelihood ratio of the hypotheses “” and “” averaged over the change point:

(6.19)

where as defined before . Also, is a scaled version of :

(6.20)

Like can also be computed using a recursion:

It is easy to see that and have one-to-one mappings with the Shiryaev statistic .

We will later see that defining the Shiryaev algorithm using the statistic (6.19) will be useful in Section 3.06.3.2, where we discuss the Bayesian quickest change detection problem in a non-i.i.d. setting. Also, defining the Shiryaev algorithm using the statistic (6.20) will be useful in Section 3.06.4 where we discuss quickest change detection in a minimax setting.

3.06.3.2 General asymptotic Bayesian theory

As mentioned earlier, when the observations are not i.i.d. conditioned on the change point , then finding an exact solution to the problem (6.10) is difficult. Fortunately, a Bayesian asymptotic theory can be developed for quite general pre- and post-change distributions [12]. In this section we discuss the results from [12] and provide a glimpse of the proofs.

We first state the observation model studied in [12]. When the process evolves in the pre-change regime, the conditional density of given is . After the change happens, the conditional density of given is given by .

As in the i.i.d. case, we can define the a posteriori probability of change having taken place before time n, given the observation up to time n, i.e.,

(6.24)

with the understanding that the recursion (6.13) is no longer valid, except for the i.i.d. model.

We note that in the non-i.i.d. case also is the likelihood ratio of the hypotheses “” and “.” If , then following (6.19), can be written for a general change point distribution as

where

If is geometrically distributed with parameter , the above expression reduces to

In fact, can even be computed recursively in this case:

(6.25)

with .

In [12], it is shown that if there exists q such that

(6.26)

( for the i.i.d. model), and some additional conditions on the rates of convergence are satisfied, then the Shiryaev algorithm (6.21) is asymptotically optimal for the Bayesian optimization problem of (6.10) as . In fact, minimizes all moments of the detection delay as well as the moments of the delay, conditioned on the change point. The asymptotic optimality proof is based on first finding a lower bound on the asymptotic moment of the delay of all the detection procedures in the class , as , and then showing that the Shiryaev stopping time (6.21) achieves that lower bound asymptotically.

To state the theorem, we need the following definitions. Let q be the limit as specified in (6.26), and let . Then define

Thus, is the last time that the log likelihood sum falls outside an interval of length around q. In general, existence of the limit q in (6.26) only guarantees , and not the finiteness of the moments of . Such conditions are needed for existence of moments of detection delay of . In particular, for some , we need:

(6.27)

and

(6.28)

Now, define

The parameter d captures the tail parameter of the distribution of . If is “heavy tailed” then , and if has an “exponential tail” then . For example, for the geometric prior with parameter .

From the above theorem, the following corollary easily follows.

(6.34)

A similar result can be concluded for the conditional moments as well.

3.06.3.3 Performance analysis for i.i.d. model with geometric prior

We now present the second order asymptotic analysis of the Shiryaev algorithm for the i.i.d. model, provided in [12] using the tools from nonlinear renewal theory introduced in Section 3.06.2.3.

When the observations are i.i.d. conditioned on the change point, condition (6.26) is satisfied and

where is the K-L divergence between the densities and (see Definition 4). From Thereom 9, it follows that for ,

Also, it is shown in [12] that if

then conditions (6.27) and (6.28) are satisfied, and hence from Corollary 1,

(6.35)

Note that the above statement provides the asymptotic delay performance of the Shiryaev algorithm. However, the bound for can be quite loose and the first order expression for the (6.35) may not provide good estimate for if the is not very small. In that case it is useful to have a second order estimate based on nonlinear renewal theory, as obtained in [12].

First note that (6.25) for the i.i.d. model will reduce to

(6.36)

with . Now, let , and recall that . Then, it can be shown that

(6.37)

Therefore the Shiryaev algorithm can be equivalently written as

We now show how the asymptotic overshoot distribution plays a key role in second order asymptotic analysis of and . Since, implies that , we have,

Thus,

and we see that is a function of the overshoot when crosses a from below.

Similarly,

Following the developments in Section 3.06.2.3, if the sequence satisfies some additional conditions, then we can write³:

It is shown in [12] that is a slowly changing sequence, and hence the distribution of , as , is equal to the asymptotic distribution of the overshoot when the random walk crosses a large positive boundary. We define the following quantities: the asymptotic overshoot distribution of a random walk

(6.38)

its mean

(6.39)

and its Laplace transform at 1

(6.40)

Also, the sequence satisfies some additional conditions and hence the following results are true.

In Table 6.2, we compare the asymptotic expressions for and given in Theorem 10 with simulations. As can be seen in the table, the asymptotic expressions get more accurate as approaches 0.

In Figure 6.3 we plot the as a function of for Gaussian observations. For a constraint of that is small, , and

giving a slope of to the trade-off curve.

When , the observations contain more information about the change than the prior, and the tradeoff slope is roughly . On the other hand, when , the prior contains more information about the change than the observations, and the tradeoff slope is roughly . The latter asymptotic slope is achieved by the stopping time that is based only on the prior:

This is also easy to see from (6.14). With small, , and the recursion for reduces to

Expanding we get . The desired expression for the mean delay is obtained from the equation .

3.06.4.1 Minimax algorithms based on the Shiryaev algorithm

Recall that the Shiryaev algorithm is given by (see (6.20) and (6.23)):

where

Also recall that has the recursion:

Setting in the expression for we get the Shiryaev-Roberts (SR) statistic [21]:

(6.56)

with the recursion:

(6.57)

It is shown in [9] that the SR algorithm is the limit of a sequence of Bayes tests, and in that limit it is asymptotically Bayes efficient. Also, it is shown in [20] that the SR algorithm is second order asymptotically optimal for (6.55), as , i.e., its delay is within a constant of the best possible delay over the class . Further, in [22], the SR algorithm is shown to be exactly optimal with respect to a number of other interesting criteria.

It is also shown in [9] that a modified version of the SR algorithm, called the Shiryaev-Roberts-Pollak (SRP) algorithm, is third order asymptotically optimal for (6.55), i.e., its delay is within a constant of the best possible delay over the class , and the constant goes to zero as . To introduce the SRP algorithm, let be the quasi-stationary distribution of the SR statistic above:

The new recursion, called the Shiryaev-Roberts-Pollak (SRP) recursion, is given by,

with distributed according to .

Although the SRP algorithm is strongly asymptotically optimal for Pollak’s formulation of (6.55), in practice, it is difficult to compute the quasi-stationary distribution . A numerical framework for computing efficiently is provided in [23]. Interestingly, the following modification of the SR algorithm with a specifically designed starting point is found to outperform the SRP procedure uniformly over all possible values of the change point [20]. This modification, referred to as the SR-r procedure, has the recursion:

It is shown in [20] that the SR-r algorithm is also third order asymptotically optimal for (6.55), i.e., its delay is within a constant of the best possible delay over the class , and the constant goes to zero as .

Note that for an arbitrary stopping time, computing the metric (6.54) involves taking supremum over all possible change times, and computing the metric (6.51) involves another supremum over all possible past realizations of observations. While we can analyze the performance of the proposed algorithms through bounds and asymptotic approximations, as we will see in Section 3.06.4.2, it is not obvious how one might evaluate and for a given algorithm in computer simulations. This is in contrast with the Bayesian setting, where (see (6.7)) can easily be evaluated in simulations by averaging over realizations of change point random variable .

Fortunately, for the CuSum algorithm (6.43) and for the Shiryaev-Roberts algorithm (6.58), both and are easy to evaluate in simulations due to the following lemma.

Note that the above proof crucially depended on the fact that both the CuSum algorithm and the Shiryaev-Roberts algorithm start with the initial value of 0. Thus it is not difficult to see that Lemma 2 does not hold for the SR-r algorithm, unless of course . Lemma 2 holds partially for the SRP algorithm since the initial distribution makes the statistic stationary in n. As a result is the same for every n. However, as mentioned previously, is difficult to compute in practice, and this makes the evaluation of and in simulations somewhat challenging.

3.06.4.2 Optimality properties of the minimax algorithms

In this section we first show that the algorithms based on the Shiryaev-Roberts statistics, SR, SRP, and SR-r are asymptotically optimal for Pollak’s formulation of (6.55). We need an important theorem that is proved in [22].

We now state the optimality proof for the procedures SR, SR-r and SRP. We only provide an outline of the proof to illustrate the fundamental ideas behind the result.

It is shown in [22] that is also equivalent to the average delay when the change happens at a “far horizon”: . Thus, the SR algorithm is also optimal with respect to that criterion.

The following corollary follows easily from the above two theorems. Recall that in the minimax setting, an algorithm is called third order asymptotically optimum if its delay is within an term of the best possible, as the goes to zero. An algorithm is called second order asymptotically optimum if its delay is within an term of the best possible, as the goes to zero. And an algorithm is called first order asymptotically optimal if the ratio of its delay with the best possible goes to 1, as the goes to zero.

Furthermore, by Lemma 2, we also have:

In [6] the asymptotic optimality of the CuSum algorithm (6.43) as is established for Lorden’s formulation of (6.53). First, ergodic theory is used to show that choosing the threshold ensures . For the above choice of threshold , it is shown that

Then the exact optimality of the SPRT [25] is used to find a lower bound on the of the class :

These arguments establish the first order asymptotic optimality of the CuSum algorithm for Lorden’s formulation. Furthermore, as we will see later in Theorem 13, Lai [10] showed that:

Thus by Lemma 2, the first order asymptotic optimality of the CuSum algorithm extends to Pollak’s formulation (6.55), and we have the following result.

In Figure 6.5, we plot the trade-off curve for the CuSum algorithm, i.e., plot as a function of . Note that the curve has a slope of .

3.06.4.3 General asymptotic minimax theory

In [10], the non-i.i.d. setting earlier discussed in Section 3.06.3.2 is considered, and asymptotic lower bounds on the and for stopping times in are obtained under quite general conditions. It is then shown that the an extension of the CuSum algorithm (6.43) to this non-i.i.d. setting achieves this lower bound asymptotically as .

Recall the non-i.i.d. model from Section 3.06.3.2. When the process evolves in the pre-change regime, the conditional density of given is . After the change happens, the conditional density of given is given by . Also

The CuSum algorithm can be generalized to the non-i.i.d. setting as follows.

Then the following result is proved in [10].

Further

and under the additional condition of

(6.71)

we have

Thus, if we set , then

and

which is asymptotically equal to the lower bound in (6.70) up to first order. Thus is first-order asymptotically optimum within the class of tests that meet the constraint of .

3.06.6.1 Quickest change detection with unknown pre- or post-change distributions

In the previous sections we discussed quickest change detection problem when both the pre- and post-change distributions are completely specified. Although this assumption is a bit restrictive, it helped in obtaining recursive algorithms with strong optimality properties in the i.i.d. setting, and allowed the development of asymptotic optimality theory in a very general non-i.i.d. setting. In this section, we provide a review of the existing results for the case where this assumption on the knowledge of the pre- and post-change distributions is relaxed.

Often it is reasonable to assume that the pre-change distribution is known. This is the case when changes occur rarely and/or the decision maker has opportunities to estimate the pre-change distribution parameters before the change occurs. When the post-change distribution is unknown but the pre-change distribution is completely specified, the post-change uncertainty may be modeled by assuming that the post-change distribution belongs to a parametric family . Approaches for designing algorithms in this setting include the generalized likelihood ratio (GLR) based approach or the mixture based approach. In the GLR based approach, at any time step, all the past observations are used to first obtain a maximum likelihood estimate of the post-change parameter , and then the post-change distribution corresponding to this estimate of is used to compute the CuSum, Shiryaev-Roberts or related statistics. In the mixture based approach, a prior is assumed on the space of parameters, and the statistics (e.g., CuSum or Shiryaev-Roberts), computed as a function of the post change parameter, are then integrated over the assumed prior.

In the i.i.d. setting this problem is studied in [6] for the case when the post-change p.d.f. belongs to a single parameter exponential family , and the following generalized likelihood ratio (GLR) based algorithm is proposed:

(6.73)

If , and is of the order of , then it is shown in [6] that as the FAR constraint ,

and for each post-change parameter ,

Here, the notation is used to denote the when the post-change observations have p.d.f . Note that is the best one can do asymptotically for a given FAR constraint of , even when the exact post-change distribution is known. Thus, this GLR scheme is uniformly asymptotically optimal with respect to the Lorden’s criterion (6.53).

In [27], the post-change distribution is assumed to belong to an exponential family of p.d.f.s as above, but the GLR based test is replaced by a mixture based test. Specifically, a prior is assumed on the range of and the following test is proposed:

(6.74)

For the above choice of threshold, it is shown that

and for each post-change parameter , if is in a set over which has a positive density, as the FAR constraint ,

Thus, even this mixture based test is uniformly asymptotically optimal with respect to the Lorden’s criterion.

Although the GLR and mixture based tests discussed above are efficient, they do not have an equivalent recursive implementation, as the CuSum test or the Shiryaev-Roberts tests have when the post-change distribution is known. As a result, to implement the GLR or mixture based tests, we need to use the entire past information , which grows with n. In [10], asymptotically optimal sliding-window based GLR and mixtures tests are proposed, that only used a finite number of past observations. The window size has to be chosen carefully and is a function of the FAR. Moreover, the results here are valid even for the non-i.i.d. setting discussed earlier in Section 3.06.4.3. Recall the non-i.i.d. model from Section 3.06.3.2, with the prechange conditional p.d.f. given by , and the post-change conditional p.d.f. given by . Then the generalized CuSum (6.68) GLR and mixture based algorithms are given, respectively, by:

(6.75)

and

(6.76)

It is shown that for a suitable choice of and , under some conditions on the likelihood ratios and on the distribution G, both of these tests satisfy the FAR constraint of . In particular,

and as ,

Moreover, under some conditions on the post-change parameter ,

Thus, under the conditions stated, the above window-limited tests are also asymptotically optimal. We remark that, although finite window based tests are useful for the i.i.d. setting here, we still need to store the entire history of observations in the non-i.i.d. setting to compute the conditional densities, unless the dependence is finite order Markov. See [10,28] for details.

To detect a change in mean of a sequence of Gaussian observations in an i.i.d. setting, i.e., when and , the GLR rule discussed above (6.75) (with and ) reduces to

(6.77)

This test is studied in [29] and performance of the test, i.e., expressions for and are obtained. In [28], it is shown that a window-limited modification of the above scheme (6.77) can also be designed.

When both pre- and post-change distributions are not known, again GLR based or mixture based tests have been studied and asymptotic properties characterized. In [30], this problem has been studied in the i.i.d setting (Bayesian and non-Bayesian), when both pre- and post-change distributions belong to an exponential family. For the Bayesian setting, the change point is assumed to be geometric and there are priors (again from an exponential family) on both the pre- and post-change parameters. GLR and mixture based tests are proposed that have asymptotic optimality properties. For a survey of other algorithms when both the pre- and post-change distributions are not known, see [28].

In [31], rather than developing procedures that are uniformly asymptotically optimal for each possible post-change distribution, robust tests are characterized when the pre- and post-change distributions belong to some uncertainty classes. The objective is to find a stopping time that satisfies a given false alarm constraint (probability of false alarm or the depending on the formulation) for each possible pre-change distribution, and minimizes the detection delay (Shiryaev, Lorden or the Pollak version) supremized over each possible pair of pre- and post-change distributions. It is shown that under some conditions on the uncertainty classes, the least favorable distribution from each class can be obtained, and the robust test is the classical test designed according to the lease favorable distribution pair. It is shown in [31] that although robust test is not uniformly asymptotically optimal, it can be significantly better than the GLR based test of [6] for some parameter ranges and for moderate values of . The robust solution also has the added advantage that it can be implemented in a simple recursive manner, while the GLR test does not admit a recursive solution in general, and may require the solution to a complex nonconvex optimization problem at every time instant.

3.06.6.2 Data-efficient quickest change detection

In the classical problem of quickest change detection that we discussed in Section 3.06.3, a change in the distribution of a sequence of random variables has to be detected as soon as possible, subject to a constraint on the probability of false alarm. Based on the past information, the decision maker has to decide whether to stop and declare change or to continue acquiring more information. In many engineering applications of quickest change detection there is a cost associated with acquiring information or taking observations, e.g., cost associated with taking measurements, or cost of batteries in sensor networks, etc. (see [32] for a detailed motivation and related references). In [32], Shiryaev’s formulation (Section 3.06.3) is extended by including an additional constraint on the cost of observations used in the detection process. The observation cost is captured through the average number of observations used before the change point, with the understanding that the cost of observations after the change point is included in the delay metric.

In order to minimize the average number of observations used before , at each time instant, a decision is made on whether to use the observation in the next time step, based on all the available information. Let , with if it is been decided to take the observation at time n, i.e., is available for decision making, and otherwise. Thus, is an on-off (binary) control input based on the information available up to time , i.e.,

with denoting the control law and I defined as:

Here, represents if , otherwise is absent from the information vector .

Let represent a policy for data-efficient quickest change detection, where is a stopping time on the information sequence . The objective in [32] is to solve the following optimization problem:

(6.78)

Here, and stand for average detection delay, probability of false alarm and average number of observations used, respectively, and and are given constraints.

Define,

Then, the two-threshold algorithm from [32] is:

With the DE-Shiryaev algorithm reduces to the Shiryaev algorithm. When the DE-Shiryaev algorithm is employed, the a posteriori probability typically evolves as depicted in Figure 6.6a. It is shown in [32] that for a fixed , as :

(6.80)

and

(6.81)

Here, is the asymptotic overshoot distribution of the random walk , when it crosses a large positive boundary. It is shown in [12] that these are also the performance expressions for the Shiryaev algorithm. Thus, the and of the DE-Shiryaev algorithm approach that of the Shiryaev algorithm as , i.e., the DE-Shiryaev algorithm is asymptotically optimal.

The DE-Shiryaev algorithm is also shown to have good delay-observation cost trade-off curves: for moderate values of probability of false alarm, for Gaussian observations, the delay of the algorithm is within 10% of the Shiryaev delay even when the observation cost is reduced by more than 50%. Furthermore, the DE-Shiryaev algorithm is substantially better than the standard approach of fractional sampling scheme, where the Shiryaev algorithm is used and where the observations to be skipped are determined a priori in order to meet the observation constraint (see Figure 6.6b).

In most practical applications, prior information about the distribution of the change point is not available. As a result, the Bayesian solution is not directly applicable. For the classical quickest change detection problem, an algorithm for the non-Bayesian setting was obtained by taking the geometric parameter of the prior on the change point to zero (see Section 3.06.4.1). Such a technique cannot be used in the data-efficient setting. This is because when an observation is skipped in the DE-Shiryaev algorithm in [32], the a posteriori probability is updated using the geometric prior. In the absence of prior information about the distribution of the change point, it is by no means obvious what the right substitute for the prior is. A useful way to capture the cost of observations in a minimax setting is also needed.

In [33], the minimax formulation of [9] is used to propose a minimax formulation for data-efficient quickest change detection. We observe that in the two-threshold algorithm , when the change occurs at a far horizon, it is the fraction of observations taken before change that is controlled. This insight is used to formulate a duty cycle based metric to capture the cost of taking observations before the change point. Also, we note that the duration for which observations are not taken in the algorithm in [32], is also a function of the undershoot of the a posteriori probability when it goes below the threshold B. Given the fact that for the DE-Shiryaev algorithm, has the interpretation of the likelihood ratio of the hypotheses “” and “,” the undershoots essentially carry the information on the likelihood ratio. It is shown in [33] that this insight can be used to design a good test in the non-Bayesian setting. This algorithm is called the DE-CuSum algorithm and it is shown that it inherits good properties of the DE-Shiryaev algorithm. The DE-CuSum algorithm is also asymptotically optimal in a sense similar to (6.80) and (6.81), has good trade-off curves, and performs significantly better than the standard approach of fractional sampling.

3.06.6.3 Distributed sensor systems

In the previous sections, we provided a summary of existing work on quickest change detection and classification in the single sensor (equivalently, centralized multisensor) setting. For the problem of detecting biological and chemical agents, the setting that is more relevant is one where there is a set of distributed sensors collecting the data relevant for detection, as shown in Figure 6.7. Based on the observations that the sensors receive, they send messages (which could be local decisions, but not necessarily) to a fusion center where a final decision about the hypothesis or change is made.

Since the information available for detection is distributed across the sensors in the network, these detection problems fall under the umbrella of distributed (or decentralized) detection, except in the impractical setting where all the information available at the sensors is immediately available without any errors at the fusion center. Such decentralized decision making problems are extremely difficult. Without certain conditional independence assumptions across sensors, the problem of finding the optimal solutions, even in the simplest case of static binary hypothesis testing, is computationally intractable [34–39]. Decentralized dynamic decision making problems, of which the quickest change detection problem is a special case, are even more challenging due to the fact that information pattern in these problems is non-classical, i.e., the different decision makers have access to different pieces of information [40].

The problem of decentralized quickest change detection in distributed sensor systems was introduced in [41], and is described as follows. Consider the distributed multisensor system with L sensors, communicating with a fusion center shown in Figure 6.7. At time n, an observation is made at sensor . The changes in the statistics of the sequences are governed by the event. Based on the information available at time n, a message is sent from sensor to the fusion center. The fusion center may possibly feedback some control signals to the sensors based on all the messages it has received so far. For example, the fusion center might inform the sensors how to update their local decision thresholds. The final decision about the change is made at the fusion center.

There are two main approaches to generating the messages at the sensors. In the first approach, the sensors can be thought of simply quantizing their observations, i.e., is simply a quantized version of . The goal then is to choose these quantizers over time and across the sensors, along with a decision rule at the fusion center, to provide the best tradeoff between detection delay and false alarms. In the second approach, the sensors perform local change detection, using all of their observations, and the fusion center combines these decisions to make the final decision about the change.

The simplest observation model for the decentralized setting is one where the sensors are affected by the change at the same time, and the observations are i.i.d. in time at each sensor and independent across sensors in both the pre-change and post-change modes. This model was introduced in [41], and studied in a Bayesian setting with a geometric prior on the change point for the case of quantization at the sensors. It was shown that, unlike the centralized problem for which the Shiryaev test is optimal (see Section 3.06.3), in the decentralized setting the optimization problem is intractable in even for this simple observation model. Some progress can be made if we allow for feedback from the fusion center [41]. Useful results can be obtained in the asymptotic setting where the probability of false (Bayesian formulation) or false alarm rate (minimax formulation) go to zero. These results can be summarized as follows (see, e.g., [12,42,43] for more details):

For the case where the sensors make local decisions about the change, it is reasonable to assume that the local detection procedures use (asymptotically) optimum centralized (single sensor) statistics. For example, in the Bayesian setting, the Shiryaev statistic described in Algorithm 1 can be used, and in the minimax setting one of the statistics described in Section 3.06.4.1 can be used depending on the specific minimax criterion used. The more interesting aspect of the decision-making here is the fusion rule used to combine the individual sensor decisions. There are three main basic options that can be considered [43]:

It was first shown by [44] that the procedure using CuSum statistics at the sensors is globally first order asymptotically optimal under Lorden’s criterion (6.53) if the sensor thresholds are chose appropriately. That is, the first order performance is the same as that of the centralized procedure that has access to all the sensor observations. A more detailed analysis of minimax setting was carried out in [45], in which procedures based on using CuSum and Shiryaev-Roberts statistics at the sensors were studied under Pollak’s criterion (6.55). It was again shown that is globally first order asymptotically optimal, and that and are not.

For the Bayesian case, if the sensors use Shiryaev statistics, then both and can be shown to be globally first order asymptotically optimal, with an appropriate choice of sensor thresholds [43,46]. The procedure does not share this asymptotic optimality property.

Interestingly, while tests based on local decision making at the sensors can be shown to have the same first order performance as that of the centralized test, simulations reveal that these asymptotics “kick in” at unreasonably low values of false alarm probability (rate). In particular, even schemes based on binary quantization at the sensors can perform better than the globally asymptotically optimum local decision based tests at moderate values of false alarm probability (rate) [43]. These results point to the need for further research on designing procedures that perform local detection at the sensors that provide good performance at moderate levels of false alarms.

3.06.6.4 Variants of quickest change detection problem for distributed sensor systems

In Section 3.06.6.3, it is assumed for the decentralized quickest change detection problem, that the change affects all the sensors in the system simultaneously. In many practical systems it is reasonable to assume that the change will be seen by only a subset of the sensors. This problem can be modeled as quickest change detection with unknown post-change distribution, with a finite number of possibilities. A GLRT based approached can of course be used, in which multiple CuSum tests are run in parallel, corresponding to each possible post-change hypotheses. But this can be quite expensive from an implementation view point. In [47], a CuSum based algorithm is proposed in which, at each sensor a CuSum test is employed, the CuSum statistic is transmitted from the sensors to the fusion center, and at the fusion center, the CuSum statistics from all the sensors are added and compared with a threshold. This test has much lower computational complexity as compared to the GLR based test, and is shown to be asymptotically as good as the centralized CuSum test, as the goes to zero. Although this test is asymptotically optimal, the noise from the sensors not affected by change can degrade the performance for moderate values of false alarm rate. In [48], this work of [47], is extended to the case where information is transmitted from the sensors only when the CuSum statistic at each sensor is above a certain threshold. It is shown that this has the surprising effect of suppressing the unwanted noise and improving the performance. In [49], it is proposed that a soft-thresholding function should be used to suppress these noise terms, and a GLRT based algorithm is proposed to detect presence of a stationary intruder (with unknown position) in a sensor network with Gaussian observations. A similar formulation is described in [50].

The Bayesian decentralized quickest change detection problem under an additional constraint on the cost of observations used is studied in [51]. The cost of observations is captured through the average number of observations used until the stopping time and it is shown that a threshold test similar to the Shiryaev algorithm is optimal. Recently, this problem has been studied in a minimax setting in [52] and asymptotically minimax algorithms have been proposed. Also, see [53,54] for other interesting energy-efficient algorithms for quickest change detection in sensor networks.

1. Statistical process control (SPC): As discussed in the introduction, algorithms are required that can detect a sudden fault arising in an industrial process or a production process. In recent years algorithms for SPC with sampling rate and sampling size control have also been developed to minimize the cost associated with sampling [56,57]. See [58,59] and the references therein for some recent contributions.

2. Sensor networks: As discussed in [32], quickest change detection algorithms can be employed in sensor networks for infrastructure monitoring [60], or for habitat monitoring [61]. Note that in these applications, the sensors are deployed for a long time, and the change may occur rarely. Therefore, data-efficient quickest change detection algorithms are needed (see Section 3.06.6.2).

3. Computer network security: Algorithms for quickest change detection have been applied in the detection of abnormal behavior in computer networks due to security breaches [62–64].

4. Cognitive radio: Algorithms based on the CuSum algorithm or other quickest change detection algorithms can be developed for cooperative spectrum sensing in cognitive radio networks to detect activity of a primary user. See [53,65–67].

5. Neuroscience: The evolution of the Shiryaev algorithm is found to be similar to the dynamics of the Leaky Integrate-and-Fire model for neurons [68].

6. Social networks: It is suggested in [69,70] that the algorithms from the change detection literature can be employed to detect the onset of the outbreak of a disease, or the effect of a bioterroist attack, by monitoring drug sales at retail stores.

1. Transient change detection: It is assumed throughput this article that the change is persistent, i.e., once the change occurs, the system stays in the post-change state forever. In many applications it might be more appropriate to model change as transient, i.e., the system only stays in the post-change state for a finite duration and then returns to the pre-change state; see e.g., [71–73]. In this setting, in addition to false alarm and delay metrics, it may be of interest to consider metrics related to the detection of the change while the system is still in the change state.

2. Change propagation: In applications with multiple sensors, unlike the model assumed in Section 3.06.6.3, it may happen that the change does not affect all the sensors simultaneously [74]. The change may propagate from one sensor to the next, with the statistics of the propagation process being known before hand [75].

3. Multiple change points in networks: In some monitoring application there may be multiple change points that affect different sensors in a network, and the goal is to exploit the relationship between the change points and the sensors affected to detect the changes [76].

4. Quickest change detection with social learning: In the classical quickest change detection problem, to compute the statistic at each time step, the decision maker has access to the entire past observations. An interesting variant of the problem is quickest change detection with social learning, when the time index is replaced by an agent index, i.e., when the statistic is updated over agents and not over time, and the agents do not have access to the entire past history of observations but only to some processed version (e.g., binary decisions) from the previous agent; see [77–79].

5. Change detection with simultaneous classification: In many applications, the post-change distribution is not uniquely specified and may come from one of multiple hypotheses , in which case along with detecting the change it of interest to identify which hypothesis is true. See, e.g., [80,81].

6. Synchronization issues: If a quickest change detection algorithm is implemented in a sensor network where sensors communicate with the fusion center using a MAC protocol, the fusion center might receive asynchronous information from the sensors due to networking delays. It is of interest to develop algorithms that can detect changes while handling MAC layer issues; see, e.g., [50].

See Vol. 1, Chapter 11 Parametric Estimation

See Vol. 1, Chapter 12 Adaptive Filters

See Vol. 1, Chapter 18 Introduction to Probabilistic Graphical Models

See this Volume, Chapter 5 Distributed Signal Detection