A series expansion formula of the scale matrix with applications in change-point detection

TL;DR

This paper develops a new series expansion formula for the scale matrix of Markov additive processes and applies it to improve analysis of the CUSUM change-point detection method for phase-type distributed data.

Contribution

It introduces a novel series expansion formula for the scale matrix of Markov additive processes and applies it to derive exact performance metrics for CUSUM detection.

Findings

Derived exact formulas for average run length and detection delay.

Provided a new analytical approach for phase-type distributions.

Enhanced understanding of CUSUM performance in complex distributions.

Abstract

We introduce a new Levy fluctuation theoretic method to analyze the cumulative sum (CUSUM) procedure in sequential change-point detection. When observations are phase-type distributed and the post-change distribution is given by exponential tilting of its pre-change distribution, the first passage analysis of the CUSUM statistic is reduced to that of a certain Markov additive process. We develop a novel series expansion formula of the scale matrix for Markov additive processes of finite activity, and apply it to derive exact expressions of the average run length, average detection delay, and false alarm probability under the CUSUM procedure.