A Stochastic Approximation-Langevinized Ensemble Kalman Filter Algorithm for State Space Models with Unknown Parameters

Inference for high-dimensional, large scale and long series dynamic systems is a challenging task in modern data science. The existing algorithms, such as particle filter or sequential importance sampler, do not scale well to the dimension of the system and the sample size of the dataset, and often suffers from the sample degeneracy issue for long series data. The recently proposed Langevinized ensemble Kalman filter (LEnKF) addresses these difficulties in a coherent way. However, it cannot be applied to the case that the dynamic system contains unknown parameters. This article proposes the so-called stochastic approximation-LEnKF for jointly estimating the states and unknown parameters of the dynamic system, where the parameters are estimated on the fly based on the state variables simulated by the LEnKF under the framework of stochastic approximation Markov chain Monte Carlo (MCMC). Under mild conditions, we prove its consistency in parameter estimation and ergodicity in state variable simulations. The proposed algorithm can be used in uncertainty quantification for long series, large scale, and high-dimensional dynamic systems. Numerical results indicate its superiority over the existing algorithms. We employ the proposed algorithm in state-space modeling of the sea surface temperature with a long short term memory (LSTM) network, which indicates its great potential in statistical analysis of complex dynamic systems encountered in modern data science. Supplementary materials for this article are available online.

对高维、大规模及长序列动态系统的推理是现代数据科学领域的一项极具挑战性的任务。现有的算法，例如粒子滤波或顺序重要性采样器，在处理系统维度和数据集样本量方面扩展性不佳，且对于长序列数据往往遭受样本退化问题的困扰。近期提出的Langevin化集合卡尔曼滤波器（LEnKF）以连贯的方式解决了这些难题。然而，该算法无法应用于动态系统包含未知参数的情形。本文提出了所谓的随机逼近-LEnKF，以联合估计动态系统的状态和未知参数，其中参数估计是在基于随机逼近马尔可夫链蒙特卡洛（MCMC）框架下，由LEnKF模拟的状态变量实时进行的。在温和的条件下，我们证明了其在参数估计方面的一致性和状态变量模拟方面的遍历性。所提出的算法可用于对长序列、大规模和高维动态系统的不确定性量化。数值结果表明，该算法优于现有算法。我们将所提出的算法应用于海面温度的状态空间建模，并使用长短期记忆（LSTM）网络，这表明其在现代数据科学中遇到复杂动态系统的统计分析中具有巨大的潜力。本文的补充材料可在网上获取。