Hierarchical Total Variations and Doubly Penalized ANOVA Modeling for Multivariate Nonparametric Regression
提出一种基于全变差和经验范数双重惩罚的新方法，用于多元非参数回归中的函数方差分析建模，能稀疏选择成分函数及其基函数，在预测精度和模型简洁性上优于MARS、树提升和随机森林。
For multivariate nonparametric regression, functional analysis of variance (ANOVA) modeling aims to capture the relationship between a response and covariates by decomposing the unknown function into various components, representing main effects, two-way interactions, etc. Such an approach has been pursued explicitly in smoothing spline ANOVA modeling and implicitly in various greedy methods such as MARS. We develop a new method for functional ANOVA modeling, based on doubly penalized estimation using total-variation and empirical-norm penalties, to achieve sparse selection of component functions and their basis functions. For this purpose, we formulate a new class of hierarchical total variations, which measures total variations at different levels including main effects and multi-way interactions, possibly after some order of differentiation. Furthermore, we derive suitable basis functions for multivariate splines such that the hierarchical total variation can be represented as a regular Lasso penalty, and hence we extend a previous backfitting algorithm to handle doubly penalized estimation for ANOVA modeling. We present extensive numerical experiments on simulations and real data to compare our method with existing methods including MARS, tree boosting, and random forest. The results are very encouraging and demonstrate notable gains from our method in prediction or classification accuracy and simplicity of the fitted functions. Supplementary materials for this article are available online.