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Smog in China has been a major issue in recent years. As the main component of smog, PM 2.5 has received a lot of attention from the public, as the pollution caused by PM 2.5 is believed to have negative impacts in many aspects. Accordingly, modeling PM 2.5 concentrations has become research of interest for the sake of future prediction and control. Based on the data collected from monitoring stations nationwide, we find that PM 2.5 concentrations show both spatial and temporal dependency. In this paper, we adopt a grouped spatial-temporal model to depict the distribution of PM 2.5 , capturing the spatial heterogeneity in the patterns of PM 2.5 concentrations. An estimation method based on the EM algorithm is used since the group label assignment is unknown, and the results are further explained. Finally, an outlier detection procedure based on the grouped spatial-temporal model is presented. [ABSTRACT FROM AUTHOR]

Quantile regression is a method of fundamental importance. How to efficiently conduct quantile regression for a large dataset on a distributed system is of great importance. We show that the popularly used one-shot estimation is statistically inefficient if data are not randomly distributed across different workers. To fix the problem, a novel one-step estimation method is developed with the following nice properties. First, the algorithm is communication efficient. That is the communication cost demanded is practically acceptable. Second, the resulting estimator is statistically efficient. That is its asymptotic covariance is the same as that of the global estimator. Third, the estimator is robust against data distribution. That is its consistency is guaranteed even if data are not randomly distributed across different workers. Numerical experiments are provided to corroborate our findings. A real example is also presented for illustration. [ABSTRACT FROM AUTHOR]