# Comparative Performance Analysis of the Cumulative Sum Chart and the Shiryaev-Roberts Procedure for Detecting Changes in Autocorrelated Data

## TL;DR

This paper compares the effectiveness of CUSUM and Shiryaev-Roberts procedures in detecting changes in autocorrelated Gaussian AR(1) processes, revealing both are asymptotically optimal with CUSUM slightly outperforming SR.

## Contribution

First comprehensive performance comparison of CUSUM and SR procedures for autocorrelated data, including new insights into their asymptotic optimality and sensitivity to correlation changes.

## Key findings

- Both procedures are asymptotically second-order optimal.
- CUSUM outperforms SR slightly across parameters.
- Identifies worst-case post-change correlation for detection.

## Abstract

We consider the problem of quickest change-point detection where the observations form a first-order autoregressive (AR) process driven by temporally independent standard Gaussian noise. Subject to possible change are both the drift of the AR(1) process ($\mu$) as well as its correlation coefficient ($\lambda$), both known. The change is abrupt and persistent, and is of known magnitude, with $\vert\lambda\vert<1$ throughout. For this scenario, we carry out a comparative performance analysis of the popular Cumulative Sum (CUSUM) chart and its less well-known but worthy competitor -- the Shiryaev-Roberts (SR) procedure. Specifically, the performance is measured through Pollak's Supremum (conditional) Average Delay to Detection (SADD) constrained to a pre-specified level of the Average Run Length (ARL) to false alarm. Particular attention is drawn to the sensitivity of each procedure's SADD and ARL with respect to the value of $\lambda$ before and after the change. The performance is studied through the solution of the respective integral renewal equations obtained via Monte Carlo simulations. The simulations are designed to estimate the sought performance metrics in an unbiased and asymptotically strongly consistent manner, and to within a prescribed proportional closeness (also asymptotically). Our extensive numerical studies suggest that both the CUSUM chart and the SR procedure are asymptotically second-order optimal, even though the CUSUM chart is found to be slightly better than the SR procedure, irrespective of the model parameters. Moreover, the existence of a worst-case post-change correlation parameter corresponding to the poorest detectability of the change for a given ARL to false alarm is established as well. To the best of our knowledge, this is the first time the performance of the SR procedure is studied for autocorrelated data.