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

The classical method of studying a cumulative sum control scheme of the decision interval type has been to regard the scheme as a sequence of sequential tests, to determine the average sample number for these component tests and hence to study the average run length for the scheme. A different approach in which the operation of the scheme is regarded as forming a Markov chain is set out. The transition probability matrix for this chain is obtained and then the properties of this matrix used to determine not only the average run lengths for the scheme, but also moments and percentage points of the run-length distribution and exact probabilities of run length. The method may be used with any discrete distribution and also, as ani accurate approximation, with any continuous distribution for the random variable which is to be controlled. Examples are given for the cases of a Poisson random variable and a normal random variable.

An approach to the probability distribution of cusum run length

Detection of Abrupt Changes: Theory and Applications.

Cumulative Sum (CUSUM) control schemes are widely used in industry for process and measurement control. Most CUSUM applications have been for continuous variables. There have been fewer uses of CUSUM control schemes when the response is a count such as the number of defects per unit or the occurrence of an accident. This article describes design and implementation procedures for counted data CUSUM's (these are sometimes called CUSUM's for attributes). These CUSUM's are easy to design and implement; they can be used to detect both increases and decreases in the count level. Enhancements to the CUSUM scheme, including the fast initial response (FIR) feature and the robust CUSUM are discussed. These enhancements speed up the detection of changes in the count level and guard against the effects of atypical or outlier observations.

Counted Data CUSUM's

The fast initial response (FIR) feature for cumulative sum (CUSUM) quality-control schemes permits a more rapid response to an initial out-of-control situation than does a standard CUSUM quality-control scheme. This feature is especially valuable at start-up or after a CUSUM has given an out-of-control signal. This article presents the average run length and the distribution of run length for CUSUM schemes with the FIR feature and compares FIR CUSUM schemes to standard CUSUM schemes. The comparisons show that if the process starts out in control, the fast initial response feature has little effect; however, if the process mean is not at the desired level, an out-of-control signal will be given faster when the FIR feature is used.

A. F. Bissell

Papers

Cusum Techniques for Quality Control