Abstract

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.

 抽象的

高维、大规模和长序列动态系统的推理是现代数据科学中的一项具有挑战性的任务。现有的算法，例如粒子滤波器或顺序重要性采样器，不能很好地适应系统的维度和数据集的样本大小，并且经常遇到长序列数据的样本简并问题。最近提出的朗之万集成卡尔曼滤波器（LEnKF）以一致的方式解决了这些困难。但它不能应用于动态系统包含未知参数的情况。本文提出了所谓的随机逼近-LEnKF来联合估计动态系统的状态和未知参数，其中参数是在随机逼近马尔可夫链蒙特框架下根据LEnKF模拟的状态变量进行动态估计的卡罗（MCMC）。在温和的条件下，我们证明了其参数估计的一致性和状态变量模拟的遍历性。该算法可用于长序列、大规模、高维动态系统的不确定性量化。数值结果表明其相对于现有算法的优越性。我们采用所提出的算法通过长短期记忆（LSTM）网络对海面温度进行状态空间建模，这表明它在现代数据科学中遇到的复杂动态系统的统计分析中具有巨大潜力。本文的补充材料可在线获取。