A Binning Approach to Quickest Change Detection with Unknown Post-Change Distribution

The problem of quickest detection of a change in distribution is considered under the assumption that the pre-change distribution is known, and the post-change distribution is only known to belong to a family of distributions distinguishable from a discretized version of the pre-change distribution. A sequential change detection procedure is proposed that partitions the sample space into a finite number of bins, and monitors the number of samples falling into each of these bins to detect the change. A test statistic that approximates the generalized likelihood ratio test is developed. It is shown that the proposed test statistic can be efficiently computed using a recursive update scheme, and a procedure for choosing the number of bins in the scheme is provided. Various asymptotic properties of the test statistic are derived to offer insights into its performance trade-off between average detection delay and average run length to a false alarm. Testing on synthetic and real data demonstrates that our approach is comparable or better in performance to existing non-parametric change detection methods.Comment: Double-column 13-page version sent to IEEE. Transaction on Signal Processing. Supplementary material include

Quickest Change Detection with Leave-one-out Density Estimation

The problem of quickest change detection in a sequence of independent observations is considered. The pre-change distribution is assumed to be known, while the post-change distribution is completely unknown. A window-limited leave-one-out (LOO) CuSum test is developed, which does not assume any knowledge of the post-change distribution, and does not require any post-change training samples. It is shown that, with certain convergence conditions on the density estimator, the LOO-CuSum test is first-order asymptotically optimal, as the false alarm rate goes to zero. The analysis is validated through numerical results, where the LOO-CuSum test is compared with baseline tests that have distributional knowledge

Quickest Change-Point Detection with Sampling Right Constraints

The quickest change-point detection problems with sampling right constraints are considered. Specially, an observer sequentially takes observations from a random sequence, whose distribution will change at an unknown time. Based on the observation sequence, the observer wants to identify the change-point as quickly as possible. Unlike the classic quickest detection problem in which the observer can take an observation at each time slot, we impose a causal sampling right constraint to the observer. In particular, sampling rights are consumed when the observer takes an observation and are replenished randomly by a stochastic process. The observer cannot take observations if there is no sampling right left. The causal sampling right constraint is motivated by several practical applications. For example, in the application of sensor network for monitoring the abrupt change of its ambient environment, the sensor can only take observations if it has energy left in its battery. With this additional constraint, we design and analyze the optimal detection and sampling right allocation strategies to minimize the detection delay under various problem setups. As one of our main contributions, a greedy sampling right allocation strategy, by which the observer spends sampling rights in taking observations as long as there are sampling rights left, is proposed. This strategy possesses a low complexity structure, and leads to simple but (asymptotically) optimal detection algorithms for the problems under consideration. Specially, our main results include: 1) Non-Bayesian quickest change-point detection: we consider non-Bayesian quickest detection problem with stochastic sampling right constraint. Two criteria, namely the algorithm level average run length (ARL) and the system level ARL, are proposed to control the false alarm rate. We show that the greedy sampling right allocation strategy combined with the cumulative sum (CUSUM) algorithm is optimal for Lorden\u27s setup with the algorithm level ARL constraint and is asymptotically optimal for both Lorden\u27s and Pollak\u27s setups with the system level ARL constraint. 2) Bayesian quickest change-point detection: both limited sampling right constraint and stochastic sampling right constraint are considered in the Bayesian quickest detection problem. The limited sampling right constraint can be viewed as a special case of the stochastic sampling right constraint with a zero sampling right replenishing rate. The optimal solutions are derived for both sampling right constraints. However, the structure of the optimal solutions are rather complex. For the problem with the limited sampling right constraint, we provide asymptotic upper and lower bounds for the detection delay. For the problem with the stochastic sampling right constraint, we show that the greedy sampling right allocation strategy combined with Shiryaev\u27s detection rule is asymptotically optimal. 3) Quickest change-point detection with unknown post-change parameters: we extend previous results to the quickest detection problem with unknown post-change parameters. Both non-Bayesian and Bayesian setups with stochastic sampling right constraints are considered. For the non-Bayesian problem, we show that the greedy sampling right allocation strategy combined with the M-CUSUM algorithm is asymptotically optimal. For the Bayesian setups, we show that the greedy sampling right allocation strategy combined with the proposed M-Shiryaev algorithm is asymptotically optimal

Data-efficient quickest change detection

In the classical problem of quickest change detection, a decision maker observes a sequence of random variables. At some point of time, the distribution of the random variables changes abruptly. The objective is to detect this change in distribution with minimum possible delay, subject to a constraint on the false alarm rate. In many applications of quickest change detection, changes are rare and there is a cost associated with taking observations or acquiring data. For such applications, the classical quickest change detection model is no longer applicable. In this dissertation we extend the classical formulations by adding an additional penalty on the cost of observations used before the change point. The objective is to find a causal on-off observation control policy and a stopping time, to minimize the detection delay, subject to constraints on the false alarm rate and the cost of observations used before the change point. We show that two-threshold generalizations of the classical single-threshold tests are asymptotically optimal for the proposed formulations. The nature of optimality is strong in the sense that the false alarm rates of the two-threshold tests are at least as good as the false alarm rates of their classical counterparts. Also, the delays of the two-threshold tests are within a constant of the delays of their classical counterparts. These results indicate that an arbitrary but fixed fraction of observations can be skipped before change without any loss in asymptotic performance. A detailed performance analysis of these algorithms is provided, and guidelines are given for the design of the proposed tests, on the basis of the performance analysis. An important result obtained through this analysis is that the two constraints, on the false alarm rate and the cost of observations used before the change, can be met independent of each other. Numerical studies of these two-threshold algorithms also reveal that they have good trade-off curves, and perform significantly better than the approach of fractional sampling, where classical single threshold tests are used and the constraint on the cost of observations is met by skipping observations randomly. We first study the problem in Bayesian and minimax settings and then extend the results to more general quickest change detection models, namely, model with unknown post-change distribution, a sensor network model, and a multi-channel model