On Prior Distributions for Orthogonal Function Sequences

2025-08-21

AI Summary

This paper addresses the challenge of constructing orthogonal functional basis priors in functional principal component analysis (FPCA). We propose a novel hierarchical Bayesian prior that sequentially builds orthogonal function sequences via conditional normal distributions, incorporating learnable adaptive orthogonality constraints to jointly regulate orthogonality and smoothness. Unlike conventional approaches, our prior explicitly embeds orthogonality within the Bayesian hierarchy, with hyperparameters estimated in a data-driven manner—ensuring posterior functional bases are near-orthogonal and sufficiently smooth. Inference is performed using Markov Chain Monte Carlo (MCMC). Extensive simulation studies and empirical analysis on Tokyo human mobility trajectories demonstrate substantial improvements over baselines: enhanced orthogonality of estimated principal components, higher accuracy in functional reconstruction, and greater efficiency in low-rank representation. The method thus yields more interpretable, statistically principled FPCA models.

Technology Category

• Machine Learning: Bayesian Learning
• Reasoning under Uncertainty: Relational Probabilistic Models
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We propose a novel class of prior distributions for sequences of orthogonal functions, which are frequently required in various statistical models such as functional principal component analysis (FPCA). Our approach constructs priors sequentially by imposing adaptive orthogonality constraints through a hierarchical formulation of conditionally normal distributions. The orthogonality is controlled via hyperparameters, allowing for flexible trade-offs between exactness and smoothness, which can be learned from the observed data. We illustrate the properties of the proposed prior and show that it leads to nearly orthogonal posterior estimates. The proposed prior is employed in Bayesian FPCA, providing more interpretable principal functions and efficient low-rank representations. Through simulation studies and analysis of human mobility data in Tokyo, we demonstrate the superior performance of our approach in inducing orthogonality and improving functional component estimation.

Problem

• Developing priors for orthogonal function sequences in statistical models
• Constructing adaptive orthogonality constraints through hierarchical formulations
• Improving interpretability and efficiency in functional principal component analysis

Innovation

• Hierarchical conditional normal distributions with orthogonality constraints
• Hyperparameter-controlled trade-off between exactness and smoothness
• Bayesian FPCA with nearly orthogonal posterior estimates