Spatio-temporal markov chain modeling for air pollution index based on classical and Bayesian approach

Abstract

Research on air pollution has garnered much interest among researchers since the results found are useful for the management of air quality. Since data analysis is highly involved in the research, a prudent use of probability and statistics is demanded. Accordingly, in this study, spatio-temporal Markov chain modeling under both classical and Bayesian approaches are proposed in order to study the behavior of air pollution data in Malaysia. Data on air pollution index (API) available at 37 air monitoring stations in Peninsular Malaysia for the period of 2012 to 2014 are considered in the analysis. Since the underlying behavior of API can be described by stochastic dependence and spatial dependence, two different methods which are the Markov chain model based on classical and Bayesian approaches and spatial Markov chain model are proposed to deal with the behaviors respectively. In the first approach, the Markov chain based on the maximum likelihood method is applied to model the underlying dynamics of air pollution that involves describing the transition probability of going from one air pollution state to another. Although the Markov model appears to be quite flexible in representing the transitions between the different air pollution states, the resultant Markov matrix is found to include excessive zero probabilities, indicating no possibility of going from one state to another. Thus, the problem of zero probability in the transition probability matrix is addressed based on two techniques. Firstly, by applying the maximum a posteriori (MAP) method for estimating the transition probability matrix of the Markov chain model under three different priors, and secondly, by introducing a robust empirical Bayes method, which incorporates a method of smoothing the zero frequencies in the count matrix, contributing to an improved estimation of the transition probability matrix obtained. For investigating the spatio-temporal dynamics of API in Peninsular Malaysia, the second approach, which is the spatial Markov chain model, is proposed in order to explore the regional effect of air pollution in Peninsular Malaysia. The results show that the stochastic and spatial dependence of API can be adequately described using the Markov chain models. In particular, it is found that the robust empirical Bayes method is superior for estimating the transition probabilities of API data, producing more accurate estimates of the transition probability, with lower variance than those found by the classical method. Based on the results found by the robust empirical Bayes method, the transition between the same state of air pollution is highest for the moderate state as compared to the other states. On average, the modal length of duration of unhealthy states of air pollution is found to be about three days. The findings of the fitted spatial Markov chain indicate that the states of air pollution for a particular station are significantly dependent on the state of air quality of the neighboring stations. In general, this study provides comprehensive information on the trend, dynamics and characteristics of air pollution, which could help environmental experts for predicting the air pollution events that can impact the public health, and also, to develop proper strategies for controlling the air quality.