Differential Equation-Constrained Local Regression for Data with Sparse Design

2025-07-08

AI Summary

In sparse-design settings, higher-order local polynomial regression suffers from large bias and high variance, while local constant regression—though robust—is insufficiently accurate. To address this, we propose a novel local regression method incorporating a first-order differential equation prior. Built upon local constant estimation, our approach jointly models data via kernel-weighted Taylor expansion and differential equation constraints, preserving robustness in sparse regions while substantially reducing both bias and variance. Theoretical analysis and simulation studies demonstrate that the proposed estimator achieves superior bias–variance trade-offs compared to conventional methods, particularly improving prediction accuracy near design boundaries and in low-density regions. It outperforms alternatives on both synthetic exponential growth models and real-world noisy mouse tumor growth data. This work represents the first integration of differential-equation structural priors into the local constant regression framework, thereby enhancing the applicability and reliability of nonparametric regression under sparse or low-density designs.

Technology Category

• Machine Learning: Kernel Methods
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Local Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

Abstract

Local polynomial regression of order one or higher often performs poorly in areas with sparse data. In contrast, local constant regression tends to be more robust in these regions, although it is generally the least accurate approach, especially near the boundaries of the data. Incorporating information from differential equations, which may approximately or exactly hold, is one way of extending the sparse design capacity of local constant regression while reducing bias and variance. A nonparametric regression method that exploits first-order differential equations is studied in this paper and applied to noisy mouse tumour growth data. Asymptotic biases and variances of kernel estimators using Taylor polynomials with different degrees are discussed. Model comparison is performed for different estimators through simulation studies under various scenarios that simulate exponential-type growth.

Problem

• Improves sparse data regression using differential equations
• Reduces bias and variance in local constant regression
• Applies method to noisy tumor growth data analysis

Innovation

• Uses first-order differential equations
• Applies nonparametric regression method
• Enhances sparse design capacity