State space representation for smoothing spline ANOVA models

Smoothing Spline ANOVA (SS-ANOVA) is an important tool for multivariate spline regression. Using the standard SS-ANOVA algorithms RKPACK, a high-dimensional smooth function can be estimated in the order of 0 (N-3) (where N is the sample size). This imposes a heavy computational burden and prevents the practical applicability of SS-ANOVA. In this article, we address this challenge as a signal extraction problem and develop a state space representation for SS-ANOVA. We then reformulate the proposed state space model into an equivalent model for a univariate time series. This model can be fitted more efficiently by adopting the modified Kalman filter smoother algorithm. Representing an SS-ANOVA model in the state space form not only leads to high computational efficiency, but also allows the algorithm to be implemented in an online setting, which is of particular importance to our real data application. Although we focus development on a two-dimensional setting, straightforward extension can yield efficient estimation of higher dimensional functions. The performances of the state space and the RKPACK algorithms are compared to demonstrate the computational savings. An application to electroencephalogram (EEG) data is used for illustration.