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.
KeywordsFunctional regression SIR Lasso Ridge regression Interval selection
The authors thank the two anonymous referees for relevant remarks and constructive comments on a previous version of the paper.
- Allen, R.G., Pereira, L.S., Raes, D., Smith, M.: Crop evapotranspiration-guidelines for computing crop water requirements-fao irrigation and drainage paper 56. FAO, Rome 300(9), D05109 (1998)Google Scholar
- Coudret, R., Liquet, B., Saracco, J.: Comparison of sliced inverse regression aproaches for undetermined cases. J. Soc. Fr. Stat. 155(2), 72–96 (2014). http://journal-sfds.fr/index.php/J-SFdS/article/view/278
- Fromont, M., Tuleau, C.: Functional classification with margin conditions. In: Lugosi, G., Simon, H. (eds.) Proceedings of the 19th Annual Conference on Learning Theory (COLT 2006), Springer (Berlin/Heidelberg), Pittsburgh, PA, USA, Lecture Notes in Computer Science, vol. 4005, pp. 94–108 (2006). https://doi.org/10.1007/11776420_10
- Grollemund, P., Abraham, C., Baragatti, M., Pudlo, P.: Bayesian functional linear regression with sparse step functions. Preprint (2018). arXiv:1604.08403
- Li, K.: Sliced inverse regression for dimension reduction. J. Am. Stat. Assoc. 86(414), 316–342 (1991). http://www.jstor.org/stable/2290563
- Lin, Q., Zhao, Z., Liu, J.: On consistency and sparsity for sliced inverse regression in high dimensions. Preprint (2018). arXiv:1507.03895
- Meyer, D., Dimitriadou, E., Hornik, K., Weingessel, A., Leisch, F.: e1071: Misc Functions of the Department of Statistics, Probability Theory Group (Formerly: E1071), TU Wien. R package version 1.6-7 (2015)Google Scholar
- Park, A., Aston, J., Ferraty, F.: Stable and predictive functional domain selection with application to brain images. Preprint (2016). arXiv:1606.02186