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 knots. 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 considerable gains from our method in both prediction or classification accuracy and simplicity of the fitted functions.
Total variations. Use of total-variation penalties has often been restricted to univariate and bivariate functions in statistics and computer science, although mathematical theory of multivariate functions of bounded variations seems to be well studied . For univariate smoothing, @cite_9 studied regression splines using total-variation penalties with an arbitrary differentiation order @math . A closely related method is trend filtering . Additive modeling has also been pursued using total variations on univariate functions in low- and high-dimensional settings [e.g.,][] petersen2016convex,sadhanala2017additive,yang2018backfitting .
@cite_9
2072081687
In this paper least squares penalized regression estimates withtotal variation penalities are considered. It is shown that theseestimators are least squares splines with locally data adaptive placed knotpoints. Algorithms and asymptotic properties are discussed.