Gaussian process regression of conditioned linear-drift diffusion process

Abstract

Stochastic differential equations (SDEs) are flexible models for describing system behaviors from various contexts. We propose a way to adopt the Gaussian process regression method to approximate the conditional probability distribution of the state vector whose evolution is described by SDE with linear drift. We mathematically derive the exact covariance structure of a multi-dimensional diffusion process with linear drift, and use it as a kernel function in multi-output Gaussian process regression. The usage of our approach differs according to the knowledge regarding the parameters in SDE. For the case when the coefficients of the SDEs are known, we firstly compare our approach with sampling-based methodologies for estimating the conditional probability distribution of diffusion processes by taking Cox-Ingersoll-Ross model as an example. The results show that our approach is consistent with sampling algorithms considering the estimated mean and variance. We then illustrate how to adopt our approach when the coefficients of the SDEs are unknown. We compare the performance of Gaussian process regression with the kernel function we derived to those with conventional kernel candidates. When applied to multi-dimensional Ornstein-Uhlenbeck process and inlet/outlet pressure data from gas regulators, our method considerably reduces the mean square error of prediction tasks, especially when there is information that comes across dimensions.