The Steady-State Behavior of Multivariate Exponentially Weighted Moving Average Control Charts

TL;DR

This paper provides accurate approximations of the steady-state distribution of the multivariate EWMA control chart statistic, revealing dimension-dependent behaviors and enabling precise calculation of performance measures.

Contribution

It introduces new methods to accurately approximate the steady-state densities of the MEWMA statistic and derives optimal smoothing constants for multivariate process monitoring.

Findings

Steady-state densities can be expressed as products of single-variable functions.

Large dimensions exhibit different steady-state behaviors compared to univariate cases.

Higher accuracy in calculating steady-state and worst-case average run lengths.

Abstract

Multivariate Exponentially Weighted Moving Average, MEWMA, charts are popular, handy and effective procedures to detect distributional changes in a stream of multivariate data. For doing appropriate performance analysis, dealing with the steady-state behavior of the MEWMA statistic is essential. Going beyond early papers, we derive quite accurate approximations of the respective steady-state densities of the MEWMA statistic. It turns out that these densities could be rewritten as the product of two functions depending on one argument only which allows feasible calculation. For proving the related statements, the presentation of the non-central chisquare density deploying the confluent hypergeometric limit function is applied. Using the new methods it was found that for large dimensions, the steady-state behavior becomes different to what one might expect from the univariate monitoring field. Based on the integral equation driven methods, steady-state and worst-case average run lengths are calculated with higher accuracy than before. Eventually, optimal MEWMA smoothing constants are derived for all considered measures.