Multiscale Kernel Function for Multiscale Geographical Weighted Regression

Jinbiao Yan

doi:10.1080/24694452.2025.2608176

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Multiscale Kernel Function for Multiscale Geographical Weighted Regression

2026-02-06

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DOI：10.1080/24694452.2025.2608176

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Abstract

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Multiscale geographically weighted regression (MGWR) allows explanatory variables to exhibit spatially nonstationary relationships across different spatial scales. Traditional MGWR, however, assumes that the spatial nonstationarity of each variable is influenced by a single spatial scale. This study relaxes that assumption, proposing that spatial nonstationarity is not only variable-specific but that even a single variable can be shaped by multiple spatial scales simultaneously. The proposed extension, termed double multiscale geographically weighted regression (M2GWR), incorporates a linear multiscale kernel function into the MGWR. An efficient coefficient estimation technique for M2GWR is developed by leveraging advancements in algorithms and hardware. Additionally, two pseudo-bandwidth parameters are introduced as diagnostics for spatial nonstationarity, allowing for the detection and interpretation of variable-specific multiscale effects. Simulation experiments demonstrate that M2GWR consistently outperforms MGWR in coefficient estimation accuracy (mean absolute error [MAE]) and model fit (corrected Akaike’s information criterion [AICc]). The pseudo-bandwidth parameters effectively differentiate multiscale patterns among variables and, to some extent, capture the combined multiscale influences acting on individual variables. Computational results also reveal substantial improvements in efficiency. Real-world applications further validate M2GWR’s effectiveness, showing gains in R2, adjusted R2, AICc, local condition numbers, and reduced spatial autocorrelation in residuals. Overall, our findings position M2GWR as a robust and efficient framework for modeling complex spatially nonstationary relationships.

Keywords:

cost-effective

multiscale geographically weighted regression

multiscale kernel function

spatial nonstationarity relationship