  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.
