Sequential algorithm for detecting changes in the variance of time series

Abstract

We consider one of the sequential parametric methods for detection of the so-called “disorder” of a discrete random process, i.e. spontaneous change of its probabilistic characteristics. Among the variety of the algorithms, the most common are those based on modified sequential analysis, usually referred as cumulative sums algorithms (CUSUM-algorithms). The aim of the work is to study the CUSUM-algorithm designed to detect changes in the variance of the Gaussian time series. The initial statement of the problem is formulated. The probabilistic characteristics of the algorithm are studied by the method of simulation experiment. The dependences of the average interval between false alarms and the average delay time in the detection of the disorder on the value of the decisive threshold for different values of the indicator characterizing the value of the variance change in the disorder are obtained. It is shown that the algorithm under consideration is more effective for detecting an increase in the variance compared to the case of its possible decrease. A method for synthesizing the controlling algorithm with the specified probabilistic characteristics is proposed. Study of the stability of the method in relation to the inaccuracy of setting the variance for the initial state without a disorder revealed that even relatively small errors in the value of the variance lead to rather large deviations of the actual probabilistic characteristics of the algorithm from those specified in the synthesis procedure. This poses rather stringent requirements for the number of observations when the variance is estimated from the experimental data. A simplified relation for determination of the sample size required for estimation of the standard deviation with a given permissible relative error at the selected confidence probability is presented. The results of the study can be used in construction of the control cards designed to solve the problems of statistical management of various processes.