Алгоритм кумулятивных сумм для обнаружения изменений ковариационной матрицы многомерных временных рядов

Abstract

The problem of detecting spontaneous changes in the probabilistic characteristics (mismatching) of vector time series using the well-known cumulative sum algorithm (CUSUM) is considered. The case in which it is in the form of a stepwise change in the entries of the time series covariance matrix is studied. To solve this problem, it is proposed to use a preliminary linear transformation of the values of the observed time series, which ensures simultaneous transformation of the covariance matrix to the unity form before the change, and to the diagonal form after the change. Basic provisions that describe the corresponding detection algorithm and the relationships using which the above- mentioned linear transformation is implemented are given. It is shown that its application can significantly simplify the solution of matters concerned with synthesizing a control algorithm with the specified properties. This is due to the fact that the time series resulting from such transformation has uncorrelated components, and the changes boil down to deviation of the variances of these components from the original unity value. For the two-dimensional case, the procedure for synthesizing the control algorithm is considered in detail. For its practical implementation, the calculation relations obtained using the simulation method are presented that make it possible to find the threshold of the control algorithm from the given value of the interval between false alarms and to estimate the average delay time in detecting a mismatch. An example of implementing such synthesis is presented, which illustrates the entire process of constructing a controlling procedure.