7. Conclusions
The RKHS approach we have introduced in this paper provides a natural framework for a formal unified theory of variable selection for functional data. The “sparse” models (those where the variable selection techniques are fully justified) appear as particular cases in this setup. As a consequence, it is possible to derive asymptotic consistency results as those obtained in the paper. Likewise, it is also possible to consider the problem of estimating the “true” number of relevant variables in a consistent way, as we do in Section 4. This is in contrast with other standard proposals for which the number of variables is previously fixed as an input, or it is determined using cross validation and other computationally expensive methods. Then, our proposal is more firmly founded in theory and, at the same time, provides a much faster method in practice, which is important when dealing with large data sets.
The empirical results we have obtained are encouraging. In short, according to our experiments, the RKHS-based method works better than other variable selection methods is those sparse models that fulfill the ideal theoretical conditions we need. In the non sparse model considered in the simulations, the RKHS method is slightly outperformed by other proposals (but still behaves reasonably). Finally, in the “neutral” field of real data examples the performance looks also satisfactory and competitive.
Last but not least, from a general, methodological point of view, this paper represents an additional example of the surprising usefulness of reproducing kernels in statistics. Additional examples can be found in [3, 5, 17, 21, 32].
