### The Monitoring of Dirichlet Compositional Data

#### Abstract

Compositional data are used in many applications such as Cement, Asphalt, and many other Chemical industries. Such data represent random variables whose values must sum up to a certain constant. Quality engineers and technicians require monitoring compositional data and detecting the source of the irregularity in the process as soon as it happens. Throughout the literature, complicated methods were introduced to monitor compositional data. Such methods are computationally complex and can lead to difficulties in interpreting the results. The Dirichlet distribution is commonly used in the literature to model compositional data. In this study, we propose three simple methods to monitor the mean vector of the Dirichlet distribution. The first method is based on a MEWMA control chart. The second method is based on transforming the Dirichlet random variables into beta random variables and then monitoring them using multiple EWMA control charts, while the third method uses multiple EWMA control charts for transformed independent random variables. Using a simulation technique, the performance of the three methods is investigated, and the three methods performed very well under different sample sizes, many random variables, and values of the distribution parameters. When the process is out-of-control, the source of the out-of-control signal can be detected using Method 2 and Method 3. Method 2 maintained its good performance with a probability 0.99 of correctly detecting the source of the signal. Method 3 performed well except for the case of Dirichlet parameter values less than one. However, it maintained almost a probability of correct detection of at least 90% in most cases. The three proposed methods are simple, do not need complicated calculations, and can easily be applied and used by practitioners.

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D. C. Montgomery, Introduction to Statistical Quality Control, Sixth Edition. 2009.

L. Foley, D. Dumuid, A. J. Atkin, T. Olds, and D. Ogilvie, â€œPatterns of health behaviour associated with active travel: A compositional data analysis,â€ Int. J. Behav. Nutr. Phys. Act., 2018, doi: 10.1186/s12966-018-0662-8.

T. P. Quinn, I. Erb, G. Gloor, C. Notredame, M. F. Richardson, and T. M. Crowley, â€œA field guide for the compositional analysis of any-omics data,â€ Gigascience, 2019, doi: 10.1093/gigascience/giz107.

K. Pearson, â€œMathematical contributions to the theory of evolution. â€”On a form of spurious correlation which may arise when indices are used in the measurement of organs,â€ Proc. R. Soc. London, vol. 60, no. 359â€“367, pp. 489â€“498, Dec. 1897, doi: 10.1098/rspl.1896.0076.

V. Pawlowsky-Glahn and A. Buccianti, Compositional Data Analysis: Theory and Applications. 2011.

J. Aitchison, The Statistical Analysis of Compositional Data. 1986.

P. Praus, â€œRobust multivariate analysis of compositional data of treated wastewaters,â€ Environ. Earth Sci., 2019, doi: 10.1007/s12665-019-8248-6.

J. J. Egozcue, V. Pawlowsky-Glahn, and G. B. Gloor, â€œLinear association in compositional data analysis,â€ Austrian J. Stat., 2018, doi: 10.17713/ajs.v47i1.689.

V. Pawlowsky-Glahn, â€œPeter Filzmoser, Karel Hron, Matthias Templ: Applied compositional data analysis, with worked examples in R,â€ Stat. Pap., vol. 61, no. 2, pp. 921â€“922, 2020, doi: 10.1007/s00362-020-01163-7.

R. A. Boyles, â€œUsing the chi-square statistic to monitor compositional process data,â€ J. Appl. Stat., 1997, doi: 10.1080/02664769723567.

G. Yang, D. B. H. Cline, R. L. Lytton, and D. N. Little, â€œTernary and Multivariate Quality Control Charts of Aggregate Gradation for Hot Mix Asphalt,â€ J. Mater. Civ. Eng., 2004, doi: 10.1061/(asce)0899-1561(2004)16:1(28).

M. Vives-Mestres, J. Daunis-I-Estadella, and J. A. MartÃn-FernÃ¡ndez, â€œIndividual T2 control chart for compositional data,â€ J. Qual. Technol., 2014, doi: 10.1080/00224065.2014.11917958.

J. J. Egozcue, V. Pawlowsky-Glahn, G. Mateu-Figueras, and C. BarcelÃ³-Vidal, â€œIsometric Logratio Transformations for Compositional Data Analysis,â€ Math. Geol., 2003, doi: 10.1023/A:1023818214614.

M. Vives-Mestres, J. Daunis-I-Estadella, and J. A. MartÃn-FernÃ¡ndez, â€œOut-of-control signals in three-part compositional T2 control chart,â€ 2014, doi: 10.1002/qre.1583.

M. Vives-Mestres, J. Daunis-i-Estadella, and J. A. MartÃn-FernÃ¡ndez, â€œSignal interpretation in Hotellingâ€™s T2 control chart for compositional data,â€ IIE Trans. (Institute Ind. Eng., 2016, doi: 10.1080/0740817X.2015.1125042.

K. P. Tran, P. Castagliola, G. Celano, and M. B. C. Khoo, â€œMonitoring compositional data using multivariate exponentially weighted moving average scheme,â€ Qual. Reliab. Eng. Int., 2018, doi: 10.1002/qre.2260.

F. Alt and K. Jain, â€œMultivariate quality controlMultivariate quality control,â€ in Encyclopedia of Operations Research and Management Science, S. I. Gass and C. M. Harris, Eds. New York, NY: Springer US, 2001, pp. 544â€“550.

A. Ongaro and S. Migliorati, â€œA generalization of the dirichlet distribution,â€ J. Multivar. Anal., 2013, doi: 10.1016/j.jmva.2012.07.007.

DOI: http://dx.doi.org/10.18517/ijaseit.11.5.13429

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