Multivariate cumulative sum control chart for compositional data with known and estimated process parameters. (15th March 2022) - Imran, Muhammad; Sun, Jinsheng; Zaidi, Fatima Sehar; Abbas, Zameer; Nazir, Hafiz Zafar
Abstract: This article uses the classic multivariate cumulative sum ( MCUSUM MCUSUM ) chart scheme proposed by Crossier (1988) to present a new modified MCUSUM MCUSUM chart for compositional data ( CoDa CoDa ). For this purpose, the data are first transformed using isometric log‐ratio ( ilr ilr ) coordinates representation to eliminate the constant sum constraint of CoDa CoDa . The MCUSUM MCUSUM ‐ CoDa CoDa control chart has been defined along with the performance measures of the proposed chart using the average run length ( ARL ARL ). Besides, the Markov chain method has been used to study the ARL ARL performance of the proposed chart. Assuming that the ilr ilr transformed data are normally distributed, the proposed MCUSUM MCUSUM ‐ CoDa CoDa charts have been compared with existing competitors such as T 2 T^2 ‐ CoDa CoDa and MEWMA MEWMA ‐ CoDa CoDa charts. The comparison shows that the proposed chart has better performance than the T 2 T^2 ‐ CoDa CoDa control charts, while the performance of the proposed chart is comparable with the MEWMA MEWMA ‐ CoDa CoDa chart. The effect of the estimated mean vector and variance‐co‐variance matrix on run‐length characteristics of the proposed MCUSUM MCUSUM ‐ CoDa CoDa control chart has also been studied in this paper. For the ARL ARL performance of MCUSUM MCUSUM ‐ CoDa 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 UCL ) and ARL 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 MCUSUM ‐ CoDa CoDa chart.