Scientific Model Building Vs Mathematic Approaches to Statistics: With Applications from Process Monitoring (Often Using Examples with a Headstart, Lucas and Crosier, (1982))

• Activity Number: 559 - Spectral Analysis, Process Monitoring, and Sampling
• Type: Contributed
• Date/Time: Thursday, August 11, 2022 : 10:30 AM to 12:20 PM
• Sponsor: Section on Physical and Engineering Sciences
• Author(s): James M. Lucas*
• Companies: J. M. Lucas and Associates
• Keywords: process monitoring; headstart; Bernoulli CUSUM; geometric CUSUM; steady-state distribution; model building

Abstract

The properties of a steady-state distribution were developed for CUSUM procedures by Crosier (1986) and for EWMA procedures by Lucas and Saccucci (1989). Crosier named two different types of steady-state distributions, cyclic and conditional, that were first described by Taylor (1968). Knoth (2021) discussed the mathematical properties of steady-state distributions published up to that time. Saccucci et al. (2022), while taking “A closer Look at the equivalence of Bernoulli and geometric CUSUMS” performed a sensitivity analysis of cyclic steady-state distributions by “returning” to different states when they we making the Markov-chain ergodic in their approach calculating cyclic steady-state vectors. When writing their papers, Crosier, Lucas and Saccucci were all unaware that finding the steady-state vector s by solving Ps=s was solving the eigen vector problem with ?=1 the largest eigenvalue of a stochastic (Markov) matrix; this was clearly explained by Knoth (2021). Crosier, Lucas and Saccucci modelled a how a practitioner would monitor an in-control process to obtain the steady-state distribution. We discuss whether more understanding of the mathematics would have helped.