Latent Gaussian Count Time Series

This article develops the theory and methods for modeling a stationary count time series via Gaussian transformations. The techniques use a latent Gaussian process and a distributional transformation to construct stationary series with very flexible correlation features that can have any prespecified marginal distribution, including the classical Poisson, generalized Poisson, negative binomial, and binomial structures. Gaussian pseudo-likelihood and implied Yule–Walker estimation paradigms, based on the autocovariance function of the count series, are developed via a new Hermite expansion. Particle filtering and sequential Monte Carlo methods are used to conduct likelihood estimation. Connections to state space models are made. Our estimation approaches are evaluated in a simulation study and the methods are used to analyze a count series of weekly retail sales. Supplementary materials for this article are available online.

本文提出了基于高斯变换的平稳计数时间序列建模理论与方法。该方法借助隐高斯过程（latent Gaussian process）与分布变换，构建具备高度灵活相关特性的平稳序列，其边缘分布可任意预设，涵盖经典泊松（Poisson）、广义泊松（generalized Poisson）、负二项（negative binomial）及二项（binomial）分布结构。本文基于计数序列的自协方差函数，通过全新的埃尔米特（Hermite）展开方法，提出了高斯伪似然与隐含尤尔-沃克（Yule–Walker）估计范式。同时采用粒子滤波与序列蒙特卡洛（sequential Monte Carlo）方法开展似然估计工作。本文还建立了该建模方法与状态空间模型的关联。通过仿真实验对所提估计方法进行了性能验证，并将所提方法应用于每周零售销售额的计数序列分析。本文补充材料可在线获取。