A Parametric МА-Algorithm for Gaussian Time Series Change Point Detection from Mathematical Expectation

Abstract

The problem of online detection of a Gaussian time series change connected with a spontaneous abrupt change in its mathematical expectation is solved. For solving the problem, it is proposed to use a parametric controlling algorithm based on the Moving Average method or the MA- algorithm. It is noted that, despite the fact that this algorithm has been known almost since the first works in the field of statistical control, its properties, capabilities, and efficiency in comparison with other change detection algorithms are still (for a number of reasons) have been studied to a very poor extent. The purpose of this work is to thoroughly study the MA-algorithm’s characteristics with the aim to synthesize an optimal time series change detection procedure. The study was carried out using a simulation modeling method. The article presents the structure and element-wise description of the simulation experiment program, which fully replicates the MA-algorithm operation in the online mode. By using the developed program, it has been found that the conventional way of setting the control procedure deciding threshold h, the value of which should provide the preset value of the average time between false alarms, when an alarm signal about the appearance of a change is generated, although in reality the object remains in the "normal" state, is invalid. For a fixed series of the above-mentioned quantities, the correct values of the threshold h are obtained depending on the width of the moving averaging window N of the controlling MA-algorithm. Similarly, the values of the average delay time of producing an alarm signal when a change with the given fixed level  appears. Based on the data obtained, dependences of the control procedure efficiency indicator E on N for a set of different values of average time between false alarms and  have been found. It has been shown that for each such set there is a value of N at which the control procedure is most effective. This optimal procedure is compared with a similar procedure of the cumulative sum algorithm (a CUSUM algorithm), and it has been shown from the comparison results that the MA-algorithm is in general only slightly inferior to the CUSUM algorithm in efficiency, and in some cases even outperforms it.