Fast variable selection makes Karhunen-Loève decomposed Gaussian process BSS-ANOVA a speedy and accurate choice for dynamic systems identification

Many approaches for scalable GPs have focused on using a subset of data as inducing points. Another promising approach is the Karhunen-Lo\`eve (KL) decomposition, in which the GP kernel is represented by a set of basis functions which are the eigenfunctions of the kernel operator. Such kernels have the potential to be very fast, and do not depend on the selection of a reduced set of inducing points. However KL decompositions lead to high dimensionality, and variable selection thus becomes paramount. This paper reports a new method of forward variable selection, enabled by the ordered nature of the basis functions in the KL expansion of the Bayesian Smoothing Spline ANOVA kernel (BSS-ANOVA), coupled with fast Gibbs sampling in a fully Bayesian approach. The new algorithm determines how high the orders of included terms should reach, balancing model fidelity with model complexity using $L^0$ penalties inherent in Bayesian and Akaike information criteria. The inference speed and accuracy makes the method especially useful for modeling dynamic systems, by modeling the derivative in a dynamic system as a static problem, then integrating the learned dynamics using a high-order scheme. The methods are demonstrated on two dynamic datasets: a `Susceptible, Infected, Recovered' (SIR) toy problem, with the transmissibility used as forcing function, along with the experimental `Cascaded Tanks' benchmark dataset. Comparisons on the static prediction of derivatives are made with a random forest (RF), a residual neural network (ResNet), and the Orthogonal Additive Kernel (OAK) inducing points scalable GP, while for the timeseries prediction comparisons are made with LSTM and GRU recurrent neural networks (RNNs).