The problem of system identification for the Kalman filter, relying on the expectation-maximization (EM) procedure to learn the underlying parameters of a dynamical system, has largely been studied assuming that observations are sampled at equally-spaced time points. However, in many applications this is a restrictive and unrealistic assumption. This paper addresses system identification for the continuous-discrete filter, with the aim of generalizing learning for the Kalman filter by relying on a solution to a continuous-time It\^o stochastic differential equation (SDE) for the latent state and covariance dynamics. We introduce a novel two-filter, analytical form for the posterior with a Bayesian derivation, which yields analytical updates which do not require the forward-pass to be pre-computed. Using this analytical and efficient computation of the posterior, we provide an EM procedure which estimates the parameters of the SDE, naturally incorporating irregularly sampled measurements. Generalizing the learning of latent linear dynamical systems (LDS) to continuous-time may extend the use of the hybrid Kalman filter to data which is not regularly sampled or has intermittent missing values, and can extend the power of non-linear system identification methods such as switching LDS (SLDS), which rely on EM for the linear discrete-time Kalman filter as a sub-unit for learning locally linearized behavior of a non-linear system. We apply the method by learning the parameters of a latent, multivariate Fokker-Planck SDE representing a toggle-switch genetic circuit using biologically realistic parameters, and compare the efficacy of learning relative to the discrete-time Kalman filter as the step-size irregularity and spectral-radius of the dynamics-matrix increases.

翻译：针对卡尔曼滤波器的系统辨识问题，通常依赖期望最大化（EM）过程学习动力系统的潜在参数，且现有研究大多假设观测数据在等间隔时间点采样。然而在许多应用中，这一假设具有局限性和不现实性。本文研究连续-离散滤波器的系统辨识问题，旨在通过依赖潜状态和协方差动力学的连续时间伊藤随机微分方程解来推广卡尔曼滤波器的学习过程。我们提出一种新颖的贝叶斯推导双滤波器解析形式，该形式产生无需预计算前向传递的解析更新。利用这种解析且高效的后验计算方法，我们提供了可自然纳入非均匀采样测量值的SDE参数估计EM过程。将潜在线性动力系统的学习推广到连续时间，可拓展混合卡尔曼滤波器在不规则采样或存在间歇缺失值数据中的应用，并可增强非线性系统辨识方法（如切换线性动力系统）的能力——后者依赖离散时间线性卡尔曼滤波器的EM过程作为子单元学习非线性系统的局部线性化行为。我们通过使用生物物理合理参数学习代表拨动开关基因回路的潜在多变量福克-普朗克SDE参数来应用该方法，并随着步长不规则性和动力学矩阵谱半径的增加，比较了与离散时间卡尔曼滤波器的学习效能差异。