This book is downloadable from http://www.irisa.fr/sisthem/kniga/. Many monitoring problems can be stated as the problem of detecting a change in the parameters of a static or dynamic stochastic system. The main goal of this book is to describe a unified framework for the design and the performance analysis of the algorithms for solving these change detection problems. Also the book contains the key mathematical background necessary for this purpose. Finally links with the analytical redundancy approach to fault detection in linear systems are established. We call abrupt change any change in the parameters of the system that occurs either instantaneously or at least very fast with respect to the sampling period of the measurements. Abrupt changes by no means refer to changes with large magnitude; on the contrary, in most applications the main problem is to detect small changes. Moreover, in some applications, the early warning of small - and not necessarily fast - changes is of crucial interest in order to avoid the economic or even catastrophic consequences that can result from an accumulation of such small changes. For example, small faults arising in the sensors of a navigation system can result, through the underlying integration, in serious errors in the estimated position of the plane. Another example is the early warning of small deviations from the normal operating conditions of an industrial process. The early detection of slight changes in the state of the process allows to plan in a more adequate manner the periods during which the process should be inspected and possibly repaired, and thus to reduce the exploitation costs.

Detection of abrupt changes: theory and application

Economic Control of Quality of Manufactured Product.

In this article, the authors consider optimal decision making in two-alternative forced-choice (TAFC) tasks. They begin by analyzing 6 models of TAFC decision making and show that all but one can be reduced to the drift diffusion model, implementing the statistically optimal algorithm (most accurate for a given speed or fastest for a given accuracy). They prove further that there is always an optimal trade-off between speed and accuracy that maximizes various reward functions, including reward rate (percentage of correct responses per unit time), as well as several other objective functions, including ones weighted for accuracy. They use these findings to address empirical data and make novel predictions about performance under optimality.

/pdf/the-physics-of-optimal-decision-making-a-formal-analysis-of-2ghxc7vpjh.pdf

The physics of optimal decision making: a formal analysis of models of performance in two-alternative forced-choice tasks.

https://www.robots.ox.ac.uk/~davidc/pubs/NDreview2014.pdf

Review: A review of novelty detection

Fault detection, isolation, and reconfiguration (FDIR) is an important and challenging problem in many engineering applications and continues to be an active area of research in the control community. This paper presents a survey of the various model-based FDIR methods developed in the last decade. In the paper, the FDIR problem is divided into the fault detection and isolation (FDI) step, and the controller reconfiguration step. For FDI, we discuss various model-based techniques to generate residuals that are robust to noise, unknown disturbance, and model uncertainties, as well as various statistical techniques of testing the residuals for abrupt changes (or faults). We then discuss various techniques of implementing reconfigurable control strategy in response to faults.

A Survey of Fault Detection, Isolation, and Reconfiguration Methods

A problem of optimal stopping is formulated and simple rules are proposed which are asymptotically optimal in an appropriate sense. The problem is of central importance in quality control and also has applications in reliability theory and other areas.

/pdf/procedures-for-reacting-to-a-change-in-distribution-26xm7vpnhv.pdf

Procedures for reacting to a change in distribution

A random walk, {Sn}∞n = 0 , having positive drift and starting at the origin, is stopped the first time Sn > t ≧ 0. The present paper studies the "excess," Sn - t, when the walk is stopped. The main result is an upper bound on the mean of the excess, uniform in t. Through Wald's equation, this gives an upper bound on the mean stopping time, as well as upper bounds on the average sample numbers of sequential probability ratio tests. The same elementary approach yields simple upper bounds on the moments and tail probabilities of residual and spent waiting times of renewal processes.

/pdf/on-excess-over-the-boundary-1brgtkyc6u.pdf

On Excess Over the Boundary

A simple combination of one-sided sequential probability ratio tests, called a 2-SPRT, is shown to approximately minimize the expected sample size at a given point θ0 among all tests with error probabilities controlled at two other points, θ1 and θ2. In the symmetric normal and binomial testing problems, this result applies directly to the Kiefer-Weiss problem of minimizing the maximum over θ of the expected sample size. Extensive computer calculations for the normal case indicate that 2-SPRT's have efficiencies greater than 99% regardless of the size of the error probabilities. Accurate approximations to the error probabilities and expected sample sizes of these tests are given.

/pdf/2-sprt-s-and-the-modified-kiefer-weiss-problem-of-minimizing-wg3nlr07ki.pdf

2-SPRT'S and The Modified Kiefer-Weiss Problem of Minimizing an Expected Sample Size

Combinations of one-sided sequential probability ratio tests (SPRT's) are shown to be "nearly optimal" for problems involving a finite number of possible underlying distributions. Subject to error probability constraints, expected sample sizes (or weighted averages of them) are minimized to within o(1) asymptotically. For sequential decision problems, simple explicit procedures are proposed which "do exactly what a Bayes solution would do" with probability approaching one as the cost per observation, c, goes to zero. Exact computations for a binomial testing problem show that efficiencies of about 97% are obtained in some "small-sample" cases.

https://projecteuclid.org/download/pdf_1/euclid.aos/1176343737

Nearly-optimal sequential tests for finitely many parameter values

Suppose a process yields independent observations whose distributions belong to a family parameterized by theta E Theta. When the process is in control, the observations are i.i.d. with a known parameter value theta(0). When the process is out of control, the parameter changes. We apply an idea of Robbins and Siegmund [Proc. Sixth Berkeley Symp. Math. Statist. Probab. 4 (1972) 37-41] to construct a class of sequential tests and detection schemes whereby the unknown post-change parameters are estimated. This approach is especially useful in situations where the parametric space is intricate and mixture-type rules are operationally or conceptually difficult to formulate. We exemplify our approach by applying it to the problem of detecting a change in the shape parameter of a Gamma distribution, in both a univariate and a multivariate setting.

/pdf/nonanticipating-estimation-applied-to-sequential-analysis-5bwpi95aci.pdf

Gary Lorden

Papers

Procedures for reacting to a change in distribution

On Excess Over the Boundary

2-SPRT'S and The Modified Kiefer-Weiss Problem of Minimizing an Expected Sample Size

Nearly-optimal sequential tests for finitely many parameter values

Nonanticipating estimation applied to sequential analysis and changepoint detection