Многомерный алгоритм кумулятивных сумм для обнаружения изменений характеристик временных рядов

Abstract

The article presents a cumulative sum algorithm intended to detect a sudden step-like change in the probabilistic characteristics of a monitored time series when such a change (“disorder”) is associated with a simultaneous change in both the location characteristics and the dispersion characteristics of the corresponding distribution functions. In the general case of a multidimensional time series, the disorder is associated with a jump in the values of the mathematical expectation vector (the vector of means) and covariance matrix entries. To solve this problem, it is proposed to use a preliminary linear transformation of the time series values, as a result of which the covariance matrix is transformed to the unity form before disordering and to the diagonal form after disordering. The change in the vector of means is analyzed, and the main relations describing the considered detection algorithm are derived. It is noted that by using the above-mentioned linear transformation it is possible to simplify the obtaining of the reference data necessary for synthesizing the monitoring algorithm with the predetermined properties. As an example, a particular case of a one-dimensional time series and a disorder in the form of a simultaneous change in the mean and variance is considered. For this case, reference data obtained by applying the simulation method are given, using which it is possible to find the monitoring algorithm triggering threshold and estimate the average delay time of detecting the specified disorder from the given interval between false alarms. This study is a logical continuation and further development of the approach to construction of multidimensional algorithms for detecting disorders [1].