算子匹配的空间回归

中文摘要 AI 辅助

标准空间回归模型 $Y = X^\ op\eta + U$，其中协变量 $X$ 和具有 Matérn 协方差的高斯随机场 $U$，将协变量效应与空间依赖性解耦。在若干应用中，这在机制上是不合理的，因为协变量可能驱动产生残差协方差的动力学，并且在统计上存在问题，因为 $\eta$ 的广义最小二乘估计可能因空间自混杂而产生偏差。我们提出算子匹配的空间回归，其中协变量既以逐点方式进入，又作为生成随机效应的随机偏微分方程（SPDE）中的确定性强迫项，将表观效应分解为局部成分和算子介导成分。该公式作为由 $X$ 强迫的输运-松弛方程的稳态出现，赋予参数物理解释。我们表明，自混杂通过构造得到解决，并且当算子参数已知时，以及当它们在扩展域渐近下被联合估计时，通过标准协方差方法可实现的回归系数的最大似然推断是校准的。我们引入了一种计算高效的有限元实现，并通过离散化误差的界限加以证明。在温度-海拔回归和空气质量监测中的应用表明，该方法匹配或优于替代方法，并产生物理上有意义的结果。

英文摘要

A standard spatial regression model $Y = X^\topβ+ U$, with covariates $X$ and a Gaussian random field $U$ with Matérn covariance, decouples covariate effects from spatial dependence. In several applications this is mechanistically implausible, as covariates likely drive the dynamics generating the residual covariance, and it is statistically problematic, since the generalised least-squares estimator of $β$ may be biased by spatial self-confounding. We propose operator-matched spatial regression, where covariates enter both pointwise and as deterministic forcings in a stochastic partial differential equation (SPDE) that generates the random effect, partitioning the apparent effect into local and operator-mediated components. The formulation arises as the steady state of a transport--relaxation equation forced by $X$, giving the parameters physical interpretations. We show that self-confounding is resolved by construction and that maximum likelihood inference for the regression coefficients, implementable through standard covariance-based methods, is calibrated when the operator parameters are known and, under expanding-domain asymptotics, when they are jointly estimated. A computationally efficient finite element implementation is introduced and justified via bounds on the discretisation error. Applications to temperature--altitude regression and air-quality monitoring show that the method matches or outperforms alternatives and produces physically meaningful results.

发表机构

• King Abdullah University of Science and Technology（阿卜杜拉国王科技大学）

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