Modelling time series via automatic learning of basis functions

We present a model for time series consisting of an infinite mixture of basis functions, whereby the bases and the mixing process are modelled as posterior means of latent Gaussian processes (GPs). Conditional to observed data, the bases and the mixing process are learnt using a parametric approximation based on pseudo-observations, where the complexity and accuracy of the method are controlled by the number of pseudo-observations (N
_x
and N
_h
). The resulting model is linear the pseudo-observations, and its likelihood function has a complexity O(NN
_h
N
_x
), N
_x
<; N, N
_h
≪ N, which is lower than that of the standard GP O(N
³
) - where N is the number of observations. We validate the proposed approach using synthetic data, where we recovered latent GPs with five different kernels from noisy observations; and using a real-world heart-rate signal to assess the proposed model's computational complexity and performance.