Matrix-free Penalized Spline Smoothing with Multiple Covariates

TL;DR

This paper introduces an efficient matrix-free method for high-dimensional penalized spline smoothing with multiple covariates, linking it to Bayesian mixed models and extending it to generalized regression, demonstrated on satellite data.

Contribution

It extends matrix-free penalized spline smoothing to multiple covariates, incorporates Bayesian mixed model formulation, and adapts the approach for generalized regression models.

Findings

Efficient matrix-free computation for high-dimensional smoothing.

Bayesian formulation enables automatic smoothing parameter selection.

Successful application to remote sensing satellite data.

Abstract

The paper motivates high dimensional smoothing with penalized splines and its numerical calculation in an efficient way. If smoothing is carried out over three or more covariates the classical tensor product spline bases explode in their dimension bringing the estimation to its numerical limits. A recent approach by Siebenborn and Wagner(2019) circumvents storage expensive implementations by proposing matrix-free calculations which allows to smooth over several covariates. We extend their approach here by linking penalized smoothing and its Bayesian formulation as mixed model which provides a matrix-free calculation of the smoothing parameter to avoid the use of high-computational cross validation. Further, we show how to extend the ideas towards generalized regression models. The extended approach is applied to remote sensing satellite data in combination with spatial smoothing.