This article presents a filter for state-space models based on Bellman's dynamic programming principle applied to the mode estimator. The proposed Bellman filter generalises the Kalman filter including its extended and iterated versions, while remaining equally inexpensive computationally. The Bellman filter is also (unlike the Kalman filter) robust under heavy-tailed observation noise and applicable to a wider range of (nonlinear and non-Gaussian) models, involving e.g. count, intensity, duration, volatility and dependence. The Bellman-filtered states are shown to be convergent, in quadratic mean, towards a small region around the true state. (Hyper)parameters are estimated by numerically maximising a filter-implied log-likelihood decomposition, which is an alternative to the classic prediction-error decomposition for linear Gaussian models. Simulation studies reveal that the Bellman filter performs on par with (or even outperforms) state-of-the-art simulation-based techniques, e.g. particle filters and importance samplers, while requiring a fraction (e.g. 1%) of the computational cost, being straightforward to implement and offering full scalability to higher dimensional state spaces.
翻译:本文章根据Bellman的动态编程原则,为适用于模式估测器的州空间模型提供了一个过滤器。 Bellman 过滤器将Kalman过滤器(包括其扩展版和迭代版版)概括为Kalman过滤器,同时以同样廉价的方式计算。Bellman 过滤器(与Kalman过滤器不同)在重尾观测噪音下也很强大,并且适用于范围更广的(非线性和非Gauussian)模型,例如计数、强度、持续期、波动性和依赖性。 Bellman 过滤器状态显示,以二次等平均值,接近真实状态周围的小区域。 (Hyper) 参数通过从数字上最大化过滤器简化的日志相似性分解定位来估算。 这是对典型的预测器分解(非线性和非Gausian)模型的一种替代。模拟研究表明,Bellman 过滤器与(甚至超常规的) 状态模拟技术(例如粒子过滤器和重要采样器), 需要完全的精确度计算。