Bayesian Information Criterion for Fitting the Optimum Order of Markov Chain Models: Methodology and Application to Air Pollution Data

Yousif Alyousifi, Kamarulzaman Ibrahim, Mahmod Othamn, Wan Zawiah Wan Zin, Nicolas Vergne and Abdullah Al-Yaari
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Yousif Alyousifi: Department of Mathematics and Statistics, Faculty of Science, Universiti Putra Malaysia, Serdang 43400, Selangor, Malaysia
Kamarulzaman Ibrahim: Department of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Bangi 43600, Selangor, Malaysia
Mahmod Othamn: Fundamental and Applied Sciences Department, Universiti Teknologi PETRONAS, Seri Iskandar 32160, Perak, Malaysia
Wan Zawiah Wan Zin: Department of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Bangi 43600, Selangor, Malaysia
Nicolas Vergne: CNRS, Laboratoire de Mathématiques Raphaël Salem, Normandie University, UNIROUEN, 76000 Rouen, France
Abdullah Al-Yaari: Department of Mathematics, Faculty of Applied Science, Thamar University, Dhamar 00967, Yemen

Mathematics, 2022, vol. 10, issue 13, 1-16

Abstract: The analysis of air pollution behavior is becoming crucial, where information on air pollution behavior is vital for managing air quality events. Many studies have described the stochastic behavior of air pollution based on the Markov chain (MC) models. Fitting the optimum order of MC models is essential for describing the stochastic process. However, uncertainty remains concerning the optimum order of such models for representing and characterizing air pollution index (API) data. In this study, the optimum order of the MC models for hourly and daily API sequences from seven stations in the central region of Peninsular Malaysia is identified, based on the Bayesian information criteria (BIC), contributing to exploring an adequate explanation of the probabilistic dependence of air pollution. A summary of the statistics for the API was calculated prior to the analysis. The Markov property and the divergence for the empirically estimated transition matrix of an MC sequence are also investigated. It is found from the analysis that the optimum order varies from one station to another. At most stations, for both observed and simulated API data, the second and third orders of the MC models are found to be optimum for hourly API occurrences, while the first-order MC is found to be most fitting for describing the dynamics of the daily API. Overall, fitting the optimum order of the MC model for the API data sequence captured the delay effect of air pollution. Accordingly, we concluded that the air quality standard lies within controllable limits, except for some infrequent occurrences of API values exceeding the unhealthy level.

Keywords: chi-squared test; high-order Markov chain; log-likelihood function; Markov property; maximum likelihood estimation; R software (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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