Effect-Wise Inference for Smoothing Spline ANOVA on Tensor-Product Sobolev Space

2026-02-02

AI Summary

Existing functional ANOVA methods struggle with interaction effects in effect-wise inference, often lacking theoretical guarantees or being restricted to pointwise analysis. This work proposes a unified framework for effect-wise inference by leveraging an orthogonal decomposition of tensor-product Sobolev spaces, enabling systematic analysis of main and interaction effects in smoothing spline ANOVA. For the first time, it achieves rigorous theoretical guarantees for effect-wise inference in models involving interactions: main effects attain the optimal one-dimensional convergence rate, while interaction effects achieve nearly optimal rates. The framework also establishes Wald-type tests and functional Bahadur representations. Theoretically, both inference rates and test power reach the minimax separation boundary up to logarithmic factors. Numerical simulations and an empirical analysis of Colorado temperature data demonstrate superior performance over existing methods.

Technology Category

• Machine Learning: Kernel Methods
• Reasoning under Uncertainty: Graphical Models
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

Abstract

Functional ANOVA provides a nonparametric modeling framework for multivariate covariates, enabling flexible estimation and interpretation of effect functions such as main effects and interaction effects. However, effect-wise inference in such models remains challenging. Existing methods focus primarily on inference for entire functions rather than individual effects. Methods addressing effect-wise inference face substantial limitations: the inability to accommodate interactions, a lack of rigorous theoretical foundations, or restriction to pointwise inference. To address these limitations, we develop a unified framework for effect-wise inference in smoothing spline ANOVA on a subspace of tensor product Sobolev space. For each effect function, we establish rates of convergence, pointwise confidence intervals, and a Wald-type test for whether the effect is zero, with power achieving the minimax distinguishable rate up to a logarithmic factor. Main effects achieve the optimal univariate rates, and interactions achieve optimal rates up to logarithmic factors. The theoretical foundation relies on an orthogonality decomposition of effect subspaces, which enables the extension of the functional Bahadur representation framework to effect-wise inference in smoothing spline ANOVA with interactions. Simulation studies and real-data application to the Colorado temperature dataset demonstrate superior performance compared to existing methods.

Problem

• effect-wise inference
• smoothing spline ANOVA
• tensor-product Sobolev space
• interaction effects
• functional ANOVA

Innovation

• effect-wise inference
• smoothing spline ANOVA
• tensor-product Sobolev space
• orthogonality decomposition
• Bahadur representation