On Combining Wavelets Expansion and Sparse Linear Models for Regression on Metabolomic Data and Biomarker Selection

Wavelet thresholding of spectra has to be handled with care when the spectra are the predictors of a regression problem. Indeed, a blind thresholding of the signal followed by a regression method often leads to deteriorated predictions. The scope of this article is to show that sparse regression methods, applied in the wavelet domain, perform an automatic thresholding: the most relevant wavelet coefficients are selected to optimize the prediction of a given target of interest. This approach can be seen as a joint thresholding designed for a predictive purpose. The method is illustrated on a real world problem where metabolomic data are linked to poison ingestion. This example proves the usefulness of wavelet expansion and the good behavior of sparse and regularized methods. A comparison study is performed between the two-steps approach (wavelet thresholding and regression) and the one-step approach (selection of wavelet coefficients with a sparse regression). The comparison includes two types of wavelet bases, various thresholding methods, and various regression methods and is evaluated by calculating prediction performances. Information about the location of the most important features on the spectra was also obtained and used to identify the most relevant metabolites involved in the mice poisoning.     

Interpretable sparse SIR for functional data

We propose a semiparametric framework based on sliced inverse regression (SIR) to address the issue of variable selection in functional regression. SIR is an effective method for dimension reduction which computes a linear projection of the predictors in a low-dimensional space, without loss of information on the regression. In order to deal with the high dimensionality of the predictors, we consider penalized versions of SIR: ridge and sparse. We extend the approaches of variable selection developed for multidimensional SIR to select intervals that form a partition of the definition domain of the functional predictors. Selecting entire intervals rather than separated evaluation points improves the interpretability of the estimated coefficients in the functional framework. A fully automated iterative procedure is proposed to find the critical (interpretable) intervals. The approach is proved efficient on simulated and real data. The method is implemented in the R package SISIR available on CRAN at https://cran.r-project.org/package=SISIR.