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 $\operatorname{ilr}$ ) 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 $\operatorname{ilr}$ 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.