The Stochastic Occupation Kernel (SOCK) Method for Learning Stochastic Differential Equations

TL;DR

This paper introduces SOCK, a kernel-based method for learning multivariate stochastic differential equations by estimating drift and diffusion functions through occupation kernels, avoiding likelihood intractability, and validated on benchmarks and real data.

Contribution

The paper develops a novel occupation kernel framework for SDE learning, extending to operator-valued kernels for diffusion estimation, and offers an efficient, likelihood-free learning procedure.

Findings

Accurate drift and diffusion estimation on simulated data.

Effective application to real-world Amyloid imaging data.

Outperforms existing methods in predictive accuracy.

Abstract

We present a novel kernel-based method for learning multivariate stochastic differential equations (SDEs). The method follows a two-step procedure: we first estimate the drift term function, then the (matrix-valued) diffusion function given the drift. Occupation kernels are integral functionals on a reproducing kernel Hilbert space (RKHS) that aggregate information over a trajectory. Our approach leverages vector-valued occupation kernels for estimating the drift component of the stochastic process. For diffusion estimation, we extend this framework by introducing operator-valued occupation kernels, enabling the estimation of an auxiliary matrix-valued function as a positive semi-definite operator, from which we readily derive the diffusion estimate. This enables us to avoid common challenges in SDE learning, such as intractable likelihoods, by optimizing a reconstruction-error-based objective. We propose a simple learning procedure that retains strong predictive accuracy while using Fenchel duality to promote efficiency. We validate the method on simulated benchmarks and a real-world dataset of Amyloid imaging in healthy and Alzheimer's disease subjects.