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

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.

💡 Research Summary

The paper addresses a central problem in sequential change‑detection: estimating the Average Run Length (ARL), i.e., the expected number of observations before a false alarm occurs, for moving‑sum (also called moving‑average or filtered‑derivative) algorithms. Traditional ARL evaluation relies on either solving large Markov‑chain equations or performing extensive Monte‑Carlo simulations, both of which become computationally prohibitive as the window length L grows.
The authors propose a fundamentally different approach: they rewrite the ARL as a series of multivariate tail probabilities. Let (S_t) denote the weighted sum of the most recent L observations, and let (h) be the detection threshold. The first‑exceedance time is (T=\min{t: S_t\ge h}). By the identity
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