Multivariate cumulative sum control chart for compositional data with known and estimated process parameters

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 T2$T^2$‐CoDa$\mathrm{CoDa}$ and MEWMA$\mathrm{MEWMA}$‐CoDa$\mathrm{CoDa}$ charts. The comparison shows that the proposed chart has better performance than the T2$T^2$‐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.

Comprehensive Scientometric Analysis and Longitudinal SDG Mapping of Quality and Reliability Engineering International Journal

“…Based on APY, the most recent topic from the cluster is "compositional data", with an APY score of 2019.6. Hu et al [60] and Williams et al [47] feature as the top-cited articles on measurement error and process quality in our list, cited 49 and 230 times, respectively, while Imran et al [61] (2022) article is the most recent on compositional data.…”

Effect of Measurement Error on the Multivariate CUSUM Control Chart for燙ompositional Data

“…To assess the MCUSUM-CoDa CC's run-length efficiency, the authors implement a Markov chain approximation suggested by Crosier [42]. The Markov Chain model needed to measure the MCUSUM-CoDa chart's ARL is given in [43].…”

Evaluating the performance of variable sampling interval HotellingT2charting scheme for compositional data in the presence of measurement error

“…The multivariate exponentially weighted moving average (MEWMA) CoDa CC has been studied by Tran et al, 23 , while Zaidi et al 24 evaluated the impact of ME on the MEWMA CCs for CoDa. The estimation of the parameters for the multivariate cumulative sum (MCUSUM) CC for CoDa has been studied 25 and also the effects of ME on the MCUSUM CoDa CC has been evaluated. 26 The MEWMA CC for CoDa has been studied using VSS.…”

“….00 370.00 370.00 370.00 370.00 370.00 370.00 370.00 370.00 370.00 370.00 370.00 370.00 370.00 370.00 370.00 370.00 370.00 370.00 370.00 370.00 0 25. 202.04 369.15 369.15 357.35 357.35 228.90 228.90 199.29 368.45 368.45 356.65 356.65 228.20 228.20 196.97 363.15 363.26 350.87 350.94 223.03 223.13 0.50 129.68 368.30 368.30 345.40 345.40 157.97 157.97 122.53 367.52 367.52 344.62 344.62 157.19 157.19 118.93 362.72 362.82 339.39 339.44 152.51 152.59 0.75 90.82 367.46 367.46 334.10 334.10 116.30 116.30 84.55 .00 370.00 370.00 370.00 370.00 370.00 370.00 370.00 370.00 370.00 370.00 370.00 370.00 370.00 370.00 370.00 370.00 370.00 370.00 370.00 370.00 0.25 221.24 361.35 365.56 357.35 357.35 328.68 346.46 199.29 360.65 364.86 356.65 356.65 327.98 345.76 196.97 355.35 359.67 350.87 350.94 322.81 340.69 0.50 146.48 353.03 361.21 345.40 345.40 294.46 325.30 122.53 352.25 360.43 344.62 344.62 293.68 324.52 118.93 347.45 355.72 339.39 339.44 289.00 319.93 0.75 105.22 345.03 356.95 334.10 334.10 265.73 306.20 84.55 .66 313.25 313.25 220.27 273.10 47.50 328.90 347.67 312.26 312.26 219.27 272.10 46.40 326.00 344.83 309.10 309.13 216.44 269.32 1.50 48.35 322.73 344.64 303.62 303.62 202.04 258.67 38.01 321.67 343.58 302.56 302.56 200.98 257.61 37.12 319.27 341.23 299.94 299.97 198.64 255.31 1.75 38.24 315.82 340.69 294.46 294.46 186.11 245.45 31.04 314.69 339.56 293.33 293.33 184.97 244.31 30.93 313.09 337.99 291.58 291.60 183.41 242.78 2.00 30.11 309.15 336.82 285.76 285.76 172.07 233.28 26.59 307.88 335.55 284.48 284.48 170.80 232.01 26.46 306.85 334.54 283.36 283.37 169.79 231.02 ARL and ATS curves corresponding to the without measurement error scenario.…”