Dynamic Linear Models (DLMs) are commonly employed for time series analysis due to their versatile structure, simple recursive updating, ability to handle missing data, and probabilistic forecasting. However, the options for count time series are limited: Gaussian DLMs require continuous data, while Poisson-based alternatives often lack sufficient modeling flexibility. We introduce a novel semiparametric methodology for count time series by warping a Gaussian DLM. The warping function has two components: a (nonparametric) transformation operator that provides distributional flexibility and a rounding operator that ensures the correct support for the discrete data-generating process. We develop conjugate inference for the warped DLM, which enables analytic and recursive updates for the state space filtering and smoothing distributions. We leverage these results to produce customized and efficient algorithms for inference and forecasting, including Monte Carlo simulation for offline analysis and an optimal particle filter for online inference. This framework unifies and extends a variety of discrete time series models and is valid for natural counts, rounded values, and multivariate observations. Simulation studies illustrate the excellent forecasting capabilities of the warped DLM. The proposed approach is applied to a multivariate time series of daily overdose counts and demonstrates both modeling and computational successes.

翻译：动态线性模型（DLM）因其灵活的结构、简单的递归更新、处理缺失数据的能力以及概率预测功能，被广泛应用于时间序列分析。然而，用于计数时间序列的模型选择较为有限：高斯DLM要求数据连续，而基于泊松分布的替代模型往往缺乏足够的建模灵活性。我们提出了一种新颖的半参数方法，通过扭曲高斯DLM来处理计数时间序列。扭曲函数包含两个组成部分：一个（非参数）变换算子，提供分布灵活性；一个取整算子，确保离散数据生成过程具有正确的支撑集。我们发展了扭曲DLM的共轭推断方法，实现了状态空间滤波和平滑分布的分析递归更新。利用这些结果，我们设计了定制化且高效的推断与预测算法，包括用于离线分析的蒙特卡洛模拟和用于在线推断的最优粒子滤波器。该框架统一并扩展了多种离散时间序列模型，适用于自然计数、四舍五入值以及多元观测数据。模拟研究展示了扭曲DLM卓越的预测能力。该提议方法被应用于每日过量计数多元时间序列，并验证了其建模与计算的成功性。