Local Quasi-Linear Models: Kernel Differential Equation Regression and Fire Data Analysis

2026-08-04

AI Summary

This study addresses the challenge of nonparametric regression under data sparsity when observations are governed by a first-order linear ordinary differential equation (ODE), where conventional methods struggle to balance flexibility with physical consistency. The authors propose a locally quasi-linear (LQL) model that embeds the ODE constraint into a local polynomial regression framework, leveraging kernel methods to jointly estimate the unknown function and its derivative. This work extends differential equation-constrained regression to general first-order linear ODEs, deriving closed-form estimators based on arbitrary-order Taylor expansions and establishing their asymptotic properties. An automated order-selection strategy is introduced to enhance both estimation accuracy and ODE consistency. In simulations and Albini firebrand combustion experiments, LQL substantially reduces estimation error and ODE inconsistency, particularly excelling in the sparsest species–diameter groups, thereby demonstrating its practical utility in fire science and related domains.

Abstract

We introduce the local quasi-linear (LQL) model, a differential equation-constrained local polynomial regression framework for the general first-order linear ordinary differential equation (ODE) $g'(x)=a(x)g(x)+b(x)$, extending prior work on differential equation-constrained local polynomial regression (DE-constrained LPR) for the exponential growth model. We derive closed-form DE-constrained local polynomial (DE1-$k$) estimators for arbitrary Taylor degree $k$, establish their asymptotic conditional bias and variance, and propose two approaches for estimating $a(x)$ and $b(x)$ when they are unknown. A simulation study across two structurally different ODEs shows that DE1-$k$ estimation with automatic degree selection reduces both estimation error and ODE-consistency error relative to unconstrained local linear regression. We then apply the framework to the firebrand burning-rate experiment of Albini (1979), modelling the density-loss curve of wind-driven firebrands with a physically motivated forced-convection ODE; the DE-constrained estimator outperforms local linear regression in the sparsest species-diameter groups, where physical structure is most valuable in compensating for scarce data. We further examine the robustness of the LQL model to misspecification against a local quasi-exponential alternative. Together, these results extend the DE-constrained regression paradigm to a broad class of physically motivated linear models and provide a practical estimation tool for fire science and other application areas where mechanistic knowledge is available but data are sparse.

Problem

• differential equation-constrained regression
• local polynomial regression
• sparse data
• first-order linear ODE
• firebrand burning-rate

Innovation

• local quasi-linear model
• differential equation-constrained regression
• local polynomial regression
• ODE-consistency
• firebrand burning-rate