Approximation of Average Run Length of Moving Sum Algorithms Using Multivariate Probabilities

TL;DR

This paper analyzes the approximation of the average run length in moving sum algorithms using multivariate probabilities, providing convergence conditions, error bounds, and validation through simulations.

Contribution

It introduces a series-based approximation method for ARL in moving sum algorithms, with convergence analysis and error bounds, validated by simulations.

Findings

Series approximation is applicable to moving average and filtered derivative algorithms.

The paper provides convergence conditions and error bounds for the series approach.

Simulation results confirm the effectiveness of the approximation method.

Abstract

Among the various procedures used to detect potential changes in a stochastic process the moving sum algorithms are very popular due to their intuitive appeal and good statistical performance. One of the important design parameters of a change detection algorithm is the expected interval between false positives, also known as the average run length (ARL). Computation of the ARL usually involves numerical procedures but in some cases it can be approximated using a series involving multivariate probabilities. In this paper, we present an analysis of this series approach by providing sufficient conditions for convergence and derive an error bound. Using simulation studies, we show that the series approach is applicable to moving average and filtered derivative algorithms. For moving average algorithms, we compare our results with previously known bounds. We use two special cases to illustrate our observations.