Modeling Spatial Anisotropic Relationships Using Gradient-Based Geographically Weighted Regression

Abstract

Distance and direction play crucial roles in modeling the spatial nonstationarity relationship. Because Euclidean distance ignores the effect of direction, several modified geographically weighted regression (GWR) attempts have been made to model anisotropic relationships using various non-Euclidean distance metrics. These methods, however, adopt uniform parameters to define the non-Euclidean metrics over the whole study area, neglecting the varying numerical features existing in different regions. As a result, they fail to accurately depict spatial anisotropic relationships between variables. To address this issue, we propose a novel method called gradient-based geographically weighted regression (GGWR) that integrates the gradient of spatial relationships into GWR. Additionally, we introduce an l₀-norm regularization technique to achieve the parameter estimation of GGWR. Both simulated and actual data sets were used to validate the proposed method, and the experimental results demonstrate that the gradient field of the spatial relationship obtained by GGWR can effectively characterize the direction and intensity of variable relationships at various locations. Moreover, GGWR outperforms other models, including GWR, directional geographically weighted regression, and Minkowski distance-based geographically weighted regression, in terms of fitting accuracy, coefficient estimation accuracy, and interpretation of coefficient symbols. These findings indicate that the GGWR can be a valuable tool for modeling spatial anisotropic relationships by leveraging the spatial relationship gradient field.

距离和方向在空间非平稳关系模型中具有重要的作用。欧几里得距离忽略方向的影响, 因此, 现有研究对地理加权回归(GWR)进行了修订, 使用各种非欧几里得距离指数对各向异性关系进行建模。然而, 这些方法采用统一的参数来定义整个研究区域的非欧几里得指数, 忽略了不同区域的不同数值特征。因此, 这些方法无法准确描述变量之间的空间各向异性关系。我们提出基于梯度的地理加权回归(GGWR)新方法, 结合了空间关系的梯度和GWR。为了实现GGWR参数估计, 引入了L₀范数正则化技术。利用模拟和实际数据集, 我们对该方法进行验证。实验结果表明, GGWR获得的空间关系梯度场能有效地表征不同位置的变量关系的方向和强度。在拟合精度、系数估计精度和系数符号解释等方面, GGWR优于GWR、定向地理加权回归和Minkowski距离地理加权回归等模型。这表明, 利用空间关系梯度场, GGWR是模拟空间各向异性关系的重要方法。

La distancia y la dirección juegan roles cruciales en la modelización de la relación espacial de la no estacionalidad. Debido a que la distancia euclidiana ignora el efecto de la dirección, se han emprendido varios intentos de regresión geográficamente ponderada (GWR) modificada para modelizar las relaciones anisotrópicas usando varias métricas no euclidianas de distancia. Sin embargo, estos métodos adoptan parámetros uniformes para definir las métricas no euclidianas en toda el área de estudio, dejando de lado las características numéricas variables existentes en diferentes regiones. Como resultado, no logran representar con precisión las relaciones anisotrópicas espaciales entre las variables. Para dar solución a esta cuestión, proponemos un nuevo método denominado regresión geográficamente ponderada basada en gradiente (GGWR), que integra el gradiente de las relaciones espaciales en la GWR. Adicionalmente, presentamos una técnica de regularización 10-norm para lograr la estimación del parámetro de GGWR. Se usaron conjuntos de datos tanto simulados como reales para validar el método propuesto, y los resultados experimentales demuestran que el campo de gradiente de la relación espacial obtenida mediante GGWR puede caracterizar efectivamente la dirección e intensidad de las relaciones variables en varias locaciones. Aún más, la GGWR supera el desempeño de otros modelos, incluida la GWR, la regresión geográficamente ponderada direccional y la regresión geográficamente ponderada basada en distancia de Minkowski, en lo que concierne a la precisión del ajuste, precisión en el cálculo del coeficiente e interpretación de los símbolos de los coeficientes. Estos hallazgos indican que la GGWR puede ser una herramienta valiosa para modelizar las relaciones anisotrópicas espaciales aprovechando el campo de gradiente de la relación espacial.

Key Words:

• gradient-based geographically weighted regression
• GWR
• spatial nonstationarity
• spatial relationship gradient field (SRGF)
• velocity anisotropy

关键词:

• GWR
• 空间非平稳性
• 空间关系梯度场
• 速度各向异性

Palabras clave:

• anisotropía de velocidad
• campo de gradiente de la relación espacial (SRGF)
• GWR
• no estacionalidad espacial

Disclosure Statement

No potential conflict of interest was reported by the authors.

Acknowledgments

We would like to thank the anonymous reviewers for their insightful comments that have been very helpful in improving this article. The GGWR software is hosted at https://github.com/jbyanhynu/GGWR.

Additional information

Funding

Funding was provided by the Natural Science Foundation of China (Grant No. 42371419, 41961055, 42371254), Natural Science Foundation of Hunan Province, China (2023JJ40100), and Scientific Research Fund of Hunan Provincial Education Department (Grant No.22A0498) and the open fund project of the National-Local Joint Engineering Laboratory on Digital Preservation and Innovative Technologies (Grant No. 2021HSKFJJ015, CTCZ20K01).

Notes on contributors

Jinbiao Yan

JINBIAO YAN is a Lecturer in the Department of College of Geography and Tourism at Hengyang Normal University, Hengyang, Hunan Province 421010, China. E-mail: [email protected]. His research interests include spatial statistical analysis and GeoAI.

Bo Wu

BO WU is a Professor in the Department of Geography and Environment at Jiangxi Normal University, Nanchang, Jiangxi Province 330022, China. E-mail: [email protected]. His research interests include spatiotemporal data analysis, GeoAI, and remote sensing image processing.

Xiaoqi Duan

XIAOQI DUAN is a Lecturer in the Department of College of computer science and technology at Guizhou University, Guiyang, Guizhou Province 550025, China. E-mail: [email protected]. His research interests include spatiotemporal data mining, GeoAI, and urban study.