Beta–negative binomial auto‐regressions for modelling integer‐valued time series with extreme observations

A Systematic Review of INGARCH Models for Integer-Valued Time Series

Count time series are widely available in fields such as epidemiology, finance, meteorology, and sports, and thus there is a growing demand for both methodological and application-oriented research on such data. This paper reviews recent developments in integer-valued generalized autoregressive conditional heteroscedasticity (INGARCH) models over the past five years, focusing on data types including unbounded non-negative counts, bounded non-negative counts, Z-valued time series and multivariate counts. For each type of data, our review follows the three main lines of model innovation, methodological development, and expansion of application areas. We attempt to summarize the recent methodological developments of INGARCH models for each data type for the integration of the whole INGARCH modeling field and suggest some potential research topics.

Beta-negative binomial nonlinear spatio-temporal random effects modeling of COVID-19 case counts in Japan

Coronavirus disease 2019 (COVID-19) caused by the SARS-CoV-2 virus has spread seriously throughout the world. Predicting the spread, or the number of cases, in the future can facilitate preparation for, and prevention of, a worst-case scenario. To achieve these purposes, statistical modeling using past data is one feasible approach. This paper describes spatio-temporal modeling of COVID-19 case counts in 47 prefectures of Japan using a nonlinear random effects model, where random effects are introduced to capture the heterogeneity of a number of model parameters associated with the prefectures. The negative binomial distribution is frequently used with the Paul-Held random effects model to account for overdispersion in count data; however, the negative binomial distribution is known to be incapable of accommodating extreme observations such as those found in the COVID-19 case count data. We therefore propose use of the beta-negative binomial distribution with the Paul-Held model. This distribution is a generalization of the negative binomial distribution that has attracted much attention in recent years because it can model extreme observations with analytical tractability. The proposed beta-negative binomial model was applied to multivariate count time series data of COVID-19 cases in the 47 prefectures of Japan. Evaluation by one-step-ahead prediction showed that the proposed model can accommodate extreme observations without sacrificing predictive performance.

Modeling air quality level with a flexible categorical autoregression

To study urban air quality, this paper proposes a novel categorical time series model, which is based on a linear combination of bounded Poisson distribution and discrete distribution to describe the dynamic and systemic features of air quality, respectively. Daily air quality level data of three major cities in China, including Beijing, Shanghai and Guangzhou, are analyzed. It is concluded that the air quality in Beijing is the worst among the three cities but is gradually improving, and its dynamics is also the most pronounced. Theoretically, the design of our model increases the flexibility of the probabilistic structure while ensuring a dynamic feedback mechanism without high computational stress. We estimate the parameters through an adaptive Bayesian Markov chain Monte Carlo sampling scheme and show the satisfactory finite sample performance of the model through simulation studies.