Tensor product P-splines using a sparse mixed model formulation

A new approach to represent P-splines as a mixed model is presented. The corresponding matrices are sparse allowing the new approach can find the optimal values of the penalty parameters in a computationally efficient manner. Whereas the new mixed model P-splines formulation is similar to the original P-splines, a key difference is that the fixed effects are modelled explicitly, and extra constraints are added to the random part of the model. An important feature ensuring that the entire computation is fast is a sparse implementation of the Automated Differentiation of the Cholesky algorithm. It is shown by means of two examples that the new approach is fast compared to existing methods. The methodology has been implemented in the R-package LMMsolver available on CRAN ( https://CRAN.R-project.org/package=LMMsolver ).