A Kernel Learning Method for Backward SDE Filter

Abstract

In this paper, we develop a kernel learning backward SDE filter method to estimate the state of a stochastic dynamical system based on its partial noisy observations. A system of forward backward stochastic differential equations is used to propagate the state of the target dynamical model, and Bayesian inference is applied to incorporate the observational information. To characterize the dynamical model in the entire state space, we introduce a kernel learning method to learn a continuous global approximation for the conditional probability density function of the target state by using discrete approximated density values as training data. Numerical experiments demonstrate that the kernel learning backward SDE is highly effective and highly efficient.

A PDE-based Adaptive Kernel Method for Solving Optimal Filtering Problems

“…we track 300 time steps. The diffusion coefficient is chosen as σ 1 = σ 2 = 0.5, σ 3 = σ 4 = 0.3, and we locate two platforms at (2,6) and (10,12) , respectively. To demonstrate the stability of our method compared with other state-of-the-art methods, we assume that there's an unexpected turn in the target moving direction at the time instant t = 1.2, which would challenge the robustness of optimal filtering methods.…”

“…In many scenarios in the optimal filtering problem, the filtering density appears to be a bell-shaped function. This makes kernel method (especially with Gaussian type kernels) an effective way to construct approximations for the target filtering density [2,3]. Since optimal filtering is often used to solve practical application problems in real time, efficiency of an optimal filtering method is essential.…”