Multivariate cumulative sum control chart for compositional data with known and estimated process parameters. (15th March 2022)

• Record Type: Journal Article
• Authors: Imran, Muhammad; Sun, Jinsheng; Zaidi, Fatima Sehar; Abbas, Zameer; Nazir, Hafiz Zafar
• Is Part Of: Quality and reliability engineering international. Volume 38:Number 5(2022)
• Page Start: 2691
• Page End: 2714
• Publication Date: 2022-03-15
• Subjects: CoDa -- Markov chain -- MCUSUM -- parameter estimation -- performance; Reliability (Engineering) -- Periodicals; Quality control -- Periodicals; High technology -- Periodicals; 620.00452
• DOI: 10.1002/qre.3099
• Languages: English
• ISSNs: 0748-8017

Abstract

Abstract: This article uses the classic multivariate cumulative sum ( MCUSUM $\mathrm{MCUSUM}$ ) chart scheme proposed by Crossier (1988) to present a new modified MCUSUM $\mathrm{MCUSUM}$ chart for compositional data ( CoDa $\mathrm{CoDa}$ ). For this purpose, the data are first transformed using isometric log‐ratio ( ilr $\operatorname{ilr}$ ) coordinates representation to eliminate the constant sum constraint of CoDa $\mathrm{CoDa}$ . The MCUSUM $\mathrm{MCUSUM}$ ‐ CoDa $\mathrm{CoDa}$ control chart has been defined along with the performance measures of the proposed chart using the average run length ( ARL $\mathrm{ARL}$ ). Besides, the Markov chain method has been used to study the ARL $\mathrm{ARL}$ performance of the proposed chart. Assuming that the ilr $\operatorname{ilr}$ transformed data are normally distributed, the proposed MCUSUM $\mathrm{MCUSUM}$ ‐ CoDa $\mathrm{CoDa}$ charts have been compared with existing competitors such as T 2 $T^2$ ‐ CoDa $\mathrm{CoDa}$ and MEWMA $\mathrm{MEWMA}$ ‐ CoDa $\mathrm{CoDa}$ charts. The comparison shows that the proposed chart has better performance than the T 2 $T^2$ ‐ CoDa $\mathrm{CoDa}$ control charts, while the performance of the proposed chart is comparable with the MEWMA $\mathrm{MEWMA}$ ‐ CoDa $\mathrm{CoDa}$ chart. The effect of the estimated mean vector and variance‐co‐variance matrix on run‐length characteristics of the proposed MCUSUM $\mathrm{MCUSUM}$ ‐ CoDa $\mathrm{CoDa}$ control chart has also been studied in this paper. For the ARL $\mathrm{ARL}$ performance of MCUSUM $\mathrm{MCUSUM}$ ‐ CoDa $\mathrm{CoDa}$ with estimated parameters Monte Carlo simulation has been adopted. The effect of the number of variables p $p$, sample size n $n$, and subgroup size m $m$ has also been studied on the data's upper control limit ( UCL $\mathrm{UCL}$ ) and ARL $\mathrm{ARL}$ . In the end, two illustrative examples of the particle size distribution of plants and production of muesli are provided to represent the practical implementation of the MCUSUM $\mathrm{MCUSUM}$ ‐ CoDa $\mathrm{CoDa}$ chart.