Alyousifi Yousif, Ibrahim Kamarulzaman, Kang Wei, Zin Wan Zawiah Wan
Department of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor Malaysia.
Center for Geospatial Sciences, University of California, Riverside, CA USA.
J Environ Health Sci Eng. 2021 Jan 26;19(1):343-356. doi: 10.1007/s40201-020-00607-4. eCollection 2021 Jun.
Air pollution is a matter of concern among the public, especially for those living in urban and industrial areas. Markov chain modeling is often used to model the underlying dynamics of air pollution, which involves describing the transition probability of going from one air pollution state to another. Thus, estimating the transition probability matrix for the data of the air pollution index (API) is an essential process in the modeling. However, one may observe many zero probabilities in the transition probability matrix, especially when faced with a small sample, interpreting the results with respect to the climate condition less realistic. This study proposes 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. The robustness of the empirical Bayesian estimation is investigated based on Bayes risk. The transition probability matrices estimated based on the robust empirical Bayes method for the hourly API data collected from seven monitoring stations in Malaysia for the period 2012 to 2014 are used for determining the air pollution characteristics such as the mean residence time, the steady-state probability and the mean recurrence time. Furthermore, the proposed method has been evaluated by Monte Carlo simulations. Results suggest that it is quite effective in producing non-zero transition probability estimates, and superior to the maximum likelihood method in terms of minimizing the mean squared error for individual and entire transition probabilities. Therefore, the robust empirical Bayes method proves to be an improved approach to the estimation of the Markov chain. When applied to API data, it could provide important information on air pollution dynamics that may help guiding the development of proper strategies for managing the impact of air quality.
The online version contains supplementary material available at 10.1007/s40201-020-00607-4.
空气污染是公众关注的问题，尤其是对于生活在城市和工业区的人们。马尔可夫链建模常用于对空气污染的潜在动态进行建模，这涉及描述从一种空气污染状态转变为另一种状态的转移概率。因此，估计空气污染指数（API）数据的转移概率矩阵是建模中的一个关键过程。然而，人们可能会在转移概率矩阵中观察到许多零概率，特别是当面对小样本时，结合气候条件来解释结果就不太现实。本研究提出了一种稳健的经验贝叶斯方法，该方法结合了一种平滑计数矩阵中零频率的方法，有助于改进转移概率矩阵的估计。基于贝叶斯风险研究了经验贝叶斯估计的稳健性。使用基于稳健经验贝叶斯方法对2012年至2014年期间从马来西亚七个监测站收集的每小时API数据估计的转移概率矩阵来确定空气污染特征，如平均停留时间、稳态概率和平均重现时间。此外，通过蒙特卡罗模拟对所提出的方法进行了评估。结果表明，该方法在产生非零转移概率估计方面非常有效，并且在最小化单个和整个转移概率的均方误差方面优于最大似然法。因此，稳健经验贝叶斯方法被证明是一种改进的马尔可夫链估计方法。当应用于API数据时，它可以提供有关空气污染动态的重要信息，这可能有助于指导制定适当的策略来管理空气质量的影响。
在线版本包含可在10.1007/s40201-020-00607-4获取的补充材料。
