AI 中文总结
该研究提出局部拟线性模型，扩展了带微分方程约束的局部多项式回归范式，经模拟与火种燃烧实验验证，其在数据稀缺场景下的估计性能优于无约束局部线性回归。
AI 中文摘要
我们提出局部拟线性（LQL）模型，这是一种针对一般一阶线性常微分方程（ODE）$g'(x)=a(x)g(x)+b(x)$的带微分方程约束的局部多项式回归框架，扩展了此前针对指数增长模型的带微分方程约束的局部多项式回归（DE-constrained LPR）研究。我们推导了任意泰勒阶数$k$的闭式DE-constrained局部多项式（DE1-$k$）估计量，确定了其渐近条件偏差与方差，并提出了当$a(x)$和$b(x)$未知时的两种估计方法。针对两种结构不同的ODE开展的模拟研究显示，采用自动阶数选择的DE1-$k$估计相比无约束局部线性回归，可同时降低估计误差与ODE一致性误差。随后，我们将该框架应用于Albini（1979）的火种燃烧速率实验，用受物理驱动的强制对流ODE模拟风驱火种的密度损失曲线；在最稀疏的物种直径组中，DE-constrained估计量的表现优于局部线性回归，此时物理结构对弥补稀缺数据最为有用。我们还检验了LQL模型对错误设定的鲁棒性，针对局部拟指数替代模型开展了验证。总体而言，这些结果将DE-constrained回归范式扩展至广泛的受物理驱动的线性模型类别，并为火科学及其他具备机制知识但数据稀缺的应用领域提供了实用估计工具。
英文摘要
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.
