Non-parametric Estimation of Stochastic Differential Equations with Sparse Gaussian Processes
This provides a computationally efficient method for analyzing temporal data in fields like economy and paleoclimatology, but it is incremental as it builds on existing Gaussian process approximations.
The paper tackles the problem of estimating drift and diffusion terms in Stochastic Differential Equations from densely observed time series by introducing a non-parametric method using sparse Gaussian processes, validated on simulated and real data from economy and paleoclimatology to capture complex system behavior.
The application of Stochastic Differential Equations (SDEs) to the analysis of temporal data has attracted increasing attention, due to their ability to describe complex dynamics with physically interpretable equations. In this paper, we introduce a non-parametric method for estimating the drift and diffusion terms of SDEs from a densely observed discrete time series. The use of Gaussian processes as priors permits working directly in a function-space view and thus the inference takes place directly in this space. To cope with the computational complexity that requires the use of Gaussian processes, a sparse Gaussian process approximation is provided. This approximation permits the efficient computation of predictions for the drift and diffusion terms by using a distribution over a small subset of pseudo-samples. The proposed method has been validated using both simulated data and real data from economy and paleoclimatology. The application of the method to real data demonstrates its ability to capture the behaviour of complex systems.