CuSum for sequential change diagnosis

The problem of quickly detecting the occurrence of an unusual event that happens on one of multiple independent data streams is considered. In the considered problem, all data streams at the initial are under normal state and are generated by probability distribution P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> . At some unknown time, an unusual event happens and the distribution of one data stream is modified to P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> while the distributions of the rest remain unchange. The observer can only observe one data stream at one time. With his sequential observations, the observer wants to design an online stopping rule and a data stream switching rule to minimize the detection delay, namely the time difference between the occurrence of the unusual event and the time of raising an alarm, while keeping the false alarm rate under control. We model the problem under non-Bayesian quickest detection framework, and propose a detection procedure based on the CUSUM statistic. We show that this proposed detection procedure is asymptotically optimal.

Sequential change detection is a fundamental problem in statistics and signal processing, with the CUSUM procedure widely used to achieve minimax detection delay under a prescribed false alarm rate when pre- and post-change distributions are fully known. However, in many practical settings, raw observations cannot be shared with a trusted central curator, and privacy must be enforced at the data source, which prevents the computation of exact CUSUM statistics. We therefore introduce a local differentially private (DP) variant called LDP-CUSUM, which first applies a local DP mechanism to transform the raw data into privatized observations and then applies a CUSUM procedure to detect the change. We derive closed-form bounds on the average run length to false alarm and on the worst-case average detection delay, explicitly characterizing the tradeoff among privacy level, false alarm rate, and detection efficiency. Numerical simulations and a real-data case study were conducted to demonstrate the detection efficiency of our proposed LDP-CUSUM across various scenarios.

Suppose F and G are unknow continuous distributions and one can observe sequential a series of independent random vectors (X1,Y1),(X2,Y2),...such that (Xi,Yi)'s initially have distribution F×F and at some unknow time their distribution may become F×G. Namely,a change in the distribution of the Y observations may occur for some reason, while the X observation maintain their distribution. We coinsider the case where the X observation maintain their distribution We consider the case where the post-change distribution is a Lehmann alternative of F, i.e.., G = Fδ for some δ > 0. The problem is to detect the change as soon as possible after its occurrence, subject to constriant on the rate of false alarms Let An k denote the likehood ratio of the ranks of the combined data(X1,...,Xn,Y1,...,Yn) for the test of no change versus na change to a Lehmann alternative at k+1 in the Y sequence. We consider the nonparmetric Shiryaev-Roberts stopping rule based on An k and compute its average run length to dectection by decoupling method

The problem of quickest detection of a change in the mean of a sequence of independent observations is studied. The pre-change observations are assumed to be stationary, while the post-change observations are allowed to be non-stationary. The case where the pre-change distribution is known is studied first, and then the extension where only the mean and variance of the pre-change distribution are known. No knowledge of the post-change distributions is assumed other than that the means of the observations are above some pre-specified threshold larger than the pre-change mean. For the case where the pre-change distribution is known, a test is derived that asymptotically minimizes the worst-case detection delay over all possible post-change distributions, as the false alarm rate goes to zero. Towards deriving this asymptotically optimal test, some new results are provided for the general problem of asymptotic minimax robust quickest change detection in non-stationary settings. Then, the limiting form of the optimal test is studied as the gap between the pre- and post-change means goes to zero, called the Mean-Change Test (MCT). It is shown that the MCT can be designed with only knowledge of the mean and variance of the pre-change distribution. The performance of the MCT is also characterized when the mean gap is moderate, under the additional assumption that the distributions of the observations have bounded support. The analysis is validated through numerical results for detecting a change in the mean of a beta distribution. The use of the MCT in monitoring pandemics is also demonstrated.

The problem of sequentially detecting a moving anomaly is studied, in which the anomaly affects different parts of a sensor network over time. Each network sensor is characterized by a pre- and post-change distribution. Initially, the observations of each sensor are generated according to the corresponding pre-change distribution. After some unknown but deterministic time instant, a moving anomaly emerges, affecting different sets of sensors as time progresses. Our goal is to design a stopping procedure to detect the emergence of the anomaly as quickly as possible, subject to false alarms constraints. The problem is studied in a quickest change detection framework where it is assumed that the evolution of the anomaly is unknown but deterministic. A modification of Lorden's detection delay is proposed to account for the trajectory of the anomaly that maximizes the detection delay of a detection procedure. It is established that a Cumulative Sum-type test solves the resulting sequential detection problem exactly when the sensors are homogeneous. For the case of heterogeneous sensors, the proposed detection scheme can be modified to provide a first-order asymptotically optimal algorithm.

We sequentially observe independent observations $X_1, X_2,\ldots$ such that initially they have distribution $G_0$; at some unknown time $\nu$ they become stochastically larger, having distribution $G_1$. Neither $G_0$ nor $G_1$ is fully specified. We wish to detect that a change has taken place as soon as possible after its occurrence, subject to a constraint on the rate of false alarms. We derive a family of nonparametric sequential procedures based on ranks, with noncontiguous alternatives in mind. Large-sample approximations to the operating characteristics are obtained analytically. The proposed procedures all possess robustness of validity, because they are based on ranks. Near-optimal sensitivity can be obtained for specific alternatives by choosing an appropriate procedure. For example, when observations are normally distributed with one standard deviation shift in mean postchange, an appropriate nonparametric surveillance scheme yields 97% asymptotic relative efficiency, as compared to the optimal procedure when all distributional parameters are known. Our procedures are computationally feasible. Monte Carlo experiments confirm the applicability of the asymptotic calculations, including high levels of efficiency, for sample sizes met in practice.

The problem of joint sequential change detection and isolation in a multichannel system is considered. It is assumed that a disruption occurs at some unknown time, and changes the distributions of the observations in an unknown subset of channels. The problem is to quickly detect the change, and at the same time to reliably isolate the affected channels. A novel scheme is proposed for this task, which admits a recursive structure, is scalable with respect to the number of channels, and does not require any prior information about the change-point. Its performance is analyzed in the special case that the number of affected channels is known. Specifically, explicit critical values are obtained for the control of the false alarm rate and the conditional probability of wrong isolation below arbitrary levels to be prescribed by the practitioner. Finally, the asymptotic optimality of the average detection delay of the proposed scheme is established as the error probabilities go to 0 and the effect of the prior distribution for the change point vanishes in the limit.

An adaptive threshold estimation scheme for abrupt changes detection algorithm in a cement rotary kiln

ABSTRACTThe quickest multidecision change detection/isolation problem is the generalization of the quickest changepoint detection problem to the case of M post-change hypotheses. It is necessary to detect the change in distribution as soon as possible and to indicate which hypothesis is true after a change occurs. Both the rate of false alarms and the misidentification (false isolation) rate should be controlled by given levels. Several detection/isolation procedures that asymptotically minimize the tradeoff between the expected detection delay and the false alarm/false isolation rates in the worst-case scenario are discussed.

A sensor network is considered where at each sensor a sequence of random variables is observed. At each time step, a processed version of the observations is transmitted from the sensors to a common node called the fusion center. At some unknown point in time the distribution of observations at an unknown subset of the sensor nodes changes. The objective is to detect the change in distribution as quickly as possible, subject to constraints on the false alarm rate, the cost of observations taken at each sensor, and the cost of communication between the sensors and the fusion center. Minimax formulations are proposed for the above problem and distributed algorithms are proposed in which on-off observation control and censoring is used at each sensor to meet the constraints on data. It is shown that the proposed algorithms are asymptotically optimal for the proposed formulations, as the false alarm rate goes to zero. The asymptotic optimality of the proposed algorithms implies that an arbitrary but fixed fraction of data can be skipped without any loss in asymptotic performance as compared to the scheme where all the observations are used for decision making. It is also shown, via numerical studies, that the proposed algorithms perform significantly better than those based on fractional sampling, in which the classical algorithms from the literature are used and the constraint on the cost of observations is met by skipping a fixed fraction of observations either deterministically or randomly, independent of the observation process.

We consider a class of stochastic linear systems that are subject to jumps of unknown magnitudes in the state variables occurring at unknown times. This model can be used when considering such problems as the estimation of systems subject to possible component failures and the tracking of vehicles capable of abrupt maneuvers. Using Kalman-Bucy filtering and generalized likelihood ratio techniques, we devise an adaptive filtering system for state estimation and the detection of the jumps. An example that illustrates the dynamical properties of our filtering scheme is discussed in detail.

Warranty databases managed by most world-leading manufacturers are constantly expanding in the big data era. An important application of warranty databases is to detect unobservable reliability problems that emerge at design and/or manufacturing stages, through modeling and analysis of warranty claims data. Usually, serious reliability problems will result in certain abnormal patterns in warranty claims, which can be captured by appropriate statistical methods. In this article, a dynamic control charting scheme is developed for early detection of reliability problems by monitoring warranty claims one period after another, over the product life cycle. Instead of specifying a constant control limit, we determine the control limits progressively by considering stochastic product sales and non-homogeneous failure processes, simultaneously. The false alarm rate at each time period is controlled at a desired level, based on which abrupt changes in field reliability, if any, will be detected in a timely manner. Furthermore, a maximum-likelihood-based post-signal diagnosis scheme is presented to aid in identifying the most probable time of problem occurrence (i.e., change point). It is shown, through in-depth simulation studies and a real case study, that the proposed scheme is able to detect an underlying reliability problem promptly and meanwhile estimate the change point with an acceptable accuracy. Finally, a moving window approach concerning only recent production periods is introduced to extend the original model so as to mitigate the “inertia” problem.

This article proposes a likelihood-ratio-based method for recovering sparse reflectivity series from noisy seismic signals. Unlike the existing methods, the proposed method is specifically developed to be adaptive to the variance of the observational noise. This enhances the robustness of the reflectivity series recovering, as shown experimentally. The proposed method is briefly summarized as follows: First, the convolutional model of seismic signals is reformulated as a discrete-time linear system. In this linear system, the nonzero reflectivity impulses are treated as abrupt state changes with unknown magnitudes occurring at unknown times. Second, a Kalman filter is applied to the linear system without considering the changes. Third, the likelihood ratio as a function of both the magnitudes and occurring times is evaluated by using the Kalman filtering innovations. Fourth, the magnitudes and occurring times of the changes are estimated by alternately fixing one parameter and maximizing the ratio with the other parameter. Finally, the reflectivity series are reconstructed from the estimated magnitudes and occurring times of the changes. Experimental results verified the reliability of the reflectivity series recovery from noisy seismic traces.

An adaptive array algorithm is proposed to mit- igate against the presence of co-channel interference or hos- tile jammers for a GPS receiver. The unknown time and frequency offset estimation for the receiver, employs Markov Chain Monte Carlo (MCMC) integration, utilizing a novel, multi- dimensional, Bayesian, global optimization strategy for initializ- ing a Metropolis-Hastings proposal distribution. The technique enables the design of a high performance multi-user receiver, capable of overcoming the near-far problem, mitigates multipath and interference and provides significantly improved time and frequency estimation performance over standard matched filter algorithm approaches.

An anomaly detection system based on variable N-gram features and one-class SVM