UC Santa Barbara

Basis Selection from Multiple Libraries (BSML) procedure was proposed by Sklar et al. (2013) to estimate spatially inhomogeneous functions with linear combinations of basis functions that are adaptively selected from multiple libraries. This methodology proves to be successful but suffers from certain drawbacks. First, the BSML procedures utilize the generalized degrees of freedom (GDF) to measure the complexity of a modeling procedure. This approach can be computationally demanding, since the GDF needs to be estimated at every step of the forward selection process. Another drawback of the BSML procedures is its greedy nature when searching for basis functions. At each forward selection step, the BSML procedures search for only one basis function to add to the current model.

We first propose two procedures called Adaptive LASSO Basis Selection (ALBS) and its modified version ALBS-2 to address the first drawback. The idea is to penalize the bases differently based on their induced reductions of RSS and the libraries they are from. This regularization approach shows no clear advantages over the BSML procedures in terms of performance according to our simulations.

We then propose the look-ahead procedure (LAP) to address both drawbacks of the BSML procedures. LAP avoids the estimation of GDF by treating the inflated degrees of freedom (IDF) of each library as tuning parameters, and then estimate them using cross validation. Moreover, LAP adopts less greedy search rules in the forward selection so that either one or a pair of basis functions can be added to the current model after each iteration. Extensive simulations show that the LAP can outperform the BSML procedures and other widely-used modeling procedures both in terms of computation speed and mean squared error. Interesting real data application in geology and meteorology using the look-ahead procedure are also provided.