A Route Map for Successful Applications of Geographically Weighted Regression

Geographically Weighted Regression (GWR) is increasingly used in spatial analyses of social and environmental data. It allows spatial heterogeneities in processes and relationships to be investigated through a series of local regression models rather than a single global one. Standard GWR assumes that relationships between the response and predictor variables operate at the same spatial scale, which is frequently not the case. To address this, several GWR variants have been proposed. This paper describes a route map to decide whether to use a GWR model or not, and if so which of three core variants to apply: a standard GWR, a mixed GWR or a multiscale GWR (MS-GWR). The route map comprises 3 primary steps that should always be undertaken: (1) a basic linear regression, (2) a MS-GWR, and (3) investigations of the results of these in order to decide whether to use a GWR approach, and if so for determining the appropriate GWR variant. The paper also highlights the importance of investigating a number of secondary issues at global and local scales including collinearity, the influence of outliers, and dependent error terms. Code and data for the case study used to illustrate the route map are provided.     

Single and Multiscale Models of Process Spatial Heterogeneity

Recent work in local spatial modeling has affirmed and broadened interest in multivariate local spatial analysis. Two broad approaches have emerged: Geographically Weighted Regression (GWR) which follows a frequentist perspective and Bayesian Spatially Varying Coefficients models. Although several comparisons between the two approaches exist, recent developments, particularly in GWR, mean that these are incomplete and missing some important axes of comparison. Consequently, there is a need for a more thorough comparison of the two families of local estimators, including recent developments in multiscale variants and their relative performance under controlled conditions. We find that while both types of local models generally perform similarly on a series of criteria, some interesting and important differences exist.     

The Multiple Testing Issue in Geographically Weighted Regression

This article describes the problem of multiple testing within a Geographically Weighted Regression framework and presents a possible solution to the problem which is based on a family‐wise error rate for dependent processes. We compare the solution presented here to other solutions such as the Bonferroni correction and the Byrne, Charlton, and Fotheringham proposal which is based on the Benjamini and Hochberg False Discovery Rate. We conclude that our proposed correction is superior to others and that generally some correction in the conventional t‐test is necessary to avoid false positives in GWR.     

Exploring the Relations Between Riverbank Erosion and Geomorphological Controls Using Geographically Weighted Logistic Regression

The relations between riverbank erosion and geomorphological variables that are thought to control or influence erosion are commonly modelled using regression. For a given river, a single regression model might befitted to data on erosion and its geomorphological controls obtained along the river's length. However, it is likely that the influence of some variables may vary with geographical location (i.e., distance upstream). For this reason, the spatially stationary regression model should be replaced with a non‐stationary equivalent. Geographically weighted regression (GWR) is a suitable choice. In this paper, GWR is extended to predict the binary presence or absence of erosion via the logistic model. This extended model was applied to data obtained from historical archives and a spatially intensive field survey of a length of 42 km of the Afon Dyfi in West Wales. The model parameters and the residual deviance of the model varied greatly with distance upstream. The practical implication of the result is that different management practices should be implemented at different locations along the river. Thus, the approach presented allowed inference of spatially varying management practice as a consequence of spatially varying geomorphological process.     

The Spatial Association of Demographic and Population Health Characteristics with COVID-19 Prevalence Across Districts in India

In less-developed countries, the lack of granular data limits the researcher's ability to study the spatial interaction of different factors on the COVID-19 pandemic. This study designs a novel database to examine the spatial effects of demographic and population health factors on COVID-19 prevalence across 640 districts in India. The goal is to provide a robust understanding of how spatial associations and the interconnections between places influence disease spread. In addition to the linear Ordinary Least Square regression model, three spatial regression models—Spatial Lag Model, Spatial Error Model, and Geographically Weighted Regression are employed to study and compare the variables explanatory power in shaping geographic variations in the COVID-19 prevalence. We found that the local GWR model is more robust and effective at predicting spatial relationships. The findings indicate that among the demographic factors, a high share of the population living in slums is positively associated with a higher incidence of COVID-19 across districts. The spatial variations in COVID-19 deaths were explained by obesity and high blood sugar, indicating a strong association between pre-existing health conditions and COVID-19 fatalities. The study brings forth the critical factors that expose the poor and vulnerable populations to severe public health risks and highlight the application of geographical analysis vis-a-vis spatial regression models to help explain those associations.     

Modélisation spatiale de la pauvretéà Montréal: apport méthodologique de la régression géographiquement pondérée

Spatial Modeling of Poverty in Montréal: Methodological Contribution of the Geographically Weighted Regression
The Island of Montréal is particularly concerned with the issue of poverty. In 2000, 29 percent of its inhabitants lived under the low income cut-offs as defined by Statistics Canada. However, poverty is not a homogeneous phenomenon at the intra-urban scale, and identifying and categorizing spaces of poverty has become a main concern for ongoing researches. According to this way of thinking, this paper proposes an analysis of the factors influencing the geographical distribution of poverty on the Island of Montréal. To be able to identify properly the various profiles of poverty, this analysis uses a specific methodology, the geographically weighted regression (GWR), and compares its results with the ones of a classical regression model. At the global level, the most important factors to explain poverty are in order: unemployment, lone-parent families, one person households, recent immigrants, part time or part year workers, school dropouts. At the local level,     

A Note on the Mixed Geographically Weighted Regression Model*

Abstract. A mixed, geographically weighted regression (GWR) model is useful in the situation where certain explanatory variables influencing the response are global while others are local. Undoubtedly, how to identify these two types of the explanatory variables is essential for building such a model. Nevertheless, It seems that there has not been a formal way to achieve this task. Based on some work on the GWR technique and the distribution theory of quadratic forms in normal variables, a statistical test approach is suggested here to identify a mixed GWR model. Then, this note mainly focuses on simulation studies to examine the performance of the test and to provide some guidelines for performing the test in practice. The simulation studies demonstrate that the test works quite well and provides a feasible way to choose an appropriate mixed GWR model for a given data set.     

协同学视角下的村域贫困风险耐受度诊断——以陕西省平利县为例

以陕西省平利县79个贫困村为研究对象，基于协同理论，建立贫困风险耐受度评价模型，利用主成分分析与地理加权回归方法，辨识平利县村域贫困风险因子，评测贫困村风险耐受度，并与贫困发生率进行对比验证。结果显示：自然本底、区位-生产资料、内外助力和生产生活保障等四个贫困风险因子作用方向有所差别；全县贫困风险耐受度处于中下水平；比对贫困风险耐受指数模型与贫困发生率现状，发现二者分布趋势相似性特征明显，未出现异常现象。后续应建立针对性的风险预测和防范机制，适量减少直接性福利供给，激发农户脱贫的内生动力，提升贫困户风险抵御力和恢复力，构建持续稳定的脱贫-振兴的扶贫新模式，持续监测县域贫困村贫困风险耐受度的时序变动特征。     

Geographically Weighted Quantile Regression (GWQR): An Application to U.S. Mortality Data

In recent years, techniques have been developed to explore spatial nonstationarity and to model the entire distribution of a regressand. The former is mainly addressed by geographically weighted regression (GWR), and the latter by quantile regression (QR). However, little attention has been paid to combining these analytical techniques. The goal of this article is to fill this gap by introducing geographically weighted quantile regression (GWQR). This study briefly reviews GWR and QR, respectively, and then outlines their synergy and a new approach, GWQR. The estimations of GWQR parameters and their standard errors, the cross‐validation bandwidth selection criterion, and the nonstationarity test are discussed. We apply GWQR to U.S. county data as an example, with mortality as the dependent variable and five social determinants as explanatory covariates. Maps summarize analytic results at the 5, 25, 50, 75, and 95 percentiles. We found that the associations between mortality and determinants vary not only spatially, but also simultaneously across the distribution of mortality. These new findings provide insights into the mortality literature, and are relevant to public policy and health promotion. Our GWQR approach bridges two important statistical approaches, and facilitates spatial quantile‐based statistical analyses. En los últimos años se han desarrollado diversas técnicas para explorar tanto la heterocedasticidad (o no estacionariedad) espacial, así como para modelar toda la distribución de una variable dependiente. El primer tema ha sido abordado principalmente por la regresión ponderada geográficamente (Geographically Weighted Regression ‐GWR), y el segundo por la regresión por cuantiles (Quantile Regression‐QR). La combinación de ambas técnicas analíticas, sin embargo, ha recibido mucho menos atención. El objetivo de este artículo es llenar dicho vacío mediante la propuesta de una regresión geográficamente ponderada por cuantiles (Geographically Weighted Quantile Regression‐ GWQR). Los autores resumen brevemente las técnicas GWR y QR respectivamente, y luego esbozan sus propiedades sinérgicas. Luego presentan la nueva técnica propuesta: GWQR. Los autores abordan los temas de las estimaciones de los parámetros GWQR y sus errores estándar, el criterio de selección del ancho de banda de la validación cruzada (cross‐validation bandwidth), y la prueba heterocedasticidad espacial. Como ejemplo se aplica GWQR a datos de la tasa de mortalidad como variable dependiente y cinco determinantes sociales como variables independientes para los condados de los Estados Unidos. Los patrones espaciales se presentan en mapas con los resultados del análisis para los percentiles 5, 25, 50, 75, y 95. Los resultados muestran que las asociaciones entre la mortalidad y sus factores determinantes no sólo varían espacialmente, sino también de forma simultánea a través de la distribución de la tasa de mortalidad. Estos nuevos hallazgos coinciden con la literatura de los estudios de mortalidad, y son relevantes para aplicaciones de política pública y promoción de la salud. El enfoque GWQR representa un puente conceptual y metodológico entre dos enfoques estadísticos importantes a la vez que hace más factible el análisis estadístico espacial por cuantiles. 近年来,可用于探讨空间非平稳性和模拟回归变数分布的技术得到发展。前者主要用地理加权回归方法（GWR）处理,后者采用分位数回归（QR）处理。然而对这些分析技术的结合使用却很少关注。本文试图通过提出地理加权分位数回归（GWQR）来填补这一空白。在分别简要回顾了GWR和QR方法的基础上,基于两个方法的协同应用提出了GWQR新方法,进而讨论了GWQR的参数估计、标准误差、带宽选择标准的交叉验证和非平稳性检验。本文将死亡率作为因变量及五个社会因子作为解释变量,进行了美国县域单元的案例研究,绘制了0.05、0.25、0.5、0.75和0.95不同百分位点的分析结果图。研究发现,死亡人数不仅与解释变量的空间分布相关,同时也与其地理分布相关。这些新发现不仅可促进对死亡率相关成果的深入分析,同时也与公共政策和健康促进有关。GWQR方法架构了QR和GWR两种重要统计方法之间的纽带,也促进了基于分位数的空间统计分析方法的发展。     

基于GWR模型的合肥居住地价影响因素研究

以合肥市主城区为例,基于2010-2014年居住用地的出让数据,运用地统计法、GWR模型等方法,对合肥市居住地价的空间异质性及其影响因素进行研究。研究表明：①合肥市居住地价的空间分布呈现出显著的多中心的空间结构,地价的峰值区分别以老城区、政务区天鹅湖及滨湖新区塘西河公园为中心呈现圈层式分布;②不同的地价影响因素表现出不同的空间分布特征,其中容积率对居住地价的贡献度空间差异最大,其次是宗地面积,主干路次之,交通站点对居住地价的贡献度最小;③厘清各影响因素对地价的作用机制,建立动态的数字地价模型,不仅能促进土地资源的集约利用,重塑城市的空间结构,而且能为城市整体价值的发挥提供重要的理论支撑。     

A Comparison of Spatially Varying Regression Coefficient Estimates Using Geographically Weighted and Spatial‐Filter‐Based Techniques

Geographically weighted regression (GWR) is a technique that explores spatial nonstationarity in data‐generating processes by allowing regression coefficients to vary spatially. It is a widely applied technique across domains because it is intuitive and conforms to the well‐understood framework of regression. An alternative method to GWR that has been suggested is spatial filtering, which it has been argued provides a superior alternative to GWR by producing spatially varying regression coefficients that are not correlated with each other and which display less spatial autocorrelation. It is, therefore, worthwhile to examine these claims by comparing the output from both methods. We do this by using simulated data that represent two sets of spatially varying processes and examining how well both techniques replicate the known local parameter values. The article finds no support that spatial filtering produces local parameter estimates with superior properties. The results indicate that the original spatial filtering specification is prone to overfitting and is generally inferior to GWR, while an alternative specification that minimizes the mean square error (MSE) of coefficient estimates produces results that are similar to GWR. However, since we generally do not know the true coefficients, the MSE minimizing specification is impractical for applied research.     

港口连通性及其影响因素的时空差异——以中国环渤海港口为例

为明确港口连通性及其影响因素的时空差异，基于港口供应链视角，从港口面向内陆、内贸和外贸连通能力构建港口连通性模型，并借助空间计量模型对2002-2017年间中国环渤海港口连通性及其影响因素的时空差异进行测度。结果表明：①除天津、青岛和大连港连通性最强外，内陆、内贸和外贸连通性最强分别为日照、唐山和烟台港，连通性最弱分别为威海、丹东和盘锦港，黄骅港增速最快；②连通性分布具有多核心-边缘特征和多门户港口并存格局；③连通性影响因素具有空间相关性和异质性，但均对连通性有正向促进作用。本文旨在使决策者明确港口运输的连通能力及其影响因素，为港口规划布局和提升在港口供应链中地位提供决策支持。     

Inference in Multiscale Geographically Weighted Regression

A recent paper expands the well-known geographically weighted regression (GWR) framework significantly by allowing the bandwidth or smoothing factor in GWR to be derived separately for each covariate in the model—a framework referred to as multiscale GWR (MGWR). However, one limitation of the MGWR framework is that, until now, no inference about the local parameter estimates was possible. Formally, the so-called “hat matrix,” which projects the observed response vector into the predicted response vector, was available in GWR but not in MGWR. This paper addresses this limitation by reframing GWR as a Generalized Additive Model, extending this framework to MGWR and then deriving standard errors for the local parameters in MGWR. In addition, we also demonstrate how the effective number of parameters can be obtained for the overall fit of an MGWR model and for each of the covariates within the model. This statistic is essential for comparing model fit between MGWR, GWR, and traditional global models, as well as for adjusting multiple hypothesis tests. We demonstrate these advances to the MGWR framework with both a simulated data set and a real-world data set and provide a link to new software for MGWR (MGWR1.0) which includes the novel inferential framework for MGWR described here.     

Spatial Patterns in the Association between the Prevalence of Asthma and Determinants of Health

The World Health Organization endorses the study of diseases from the perspective of the Determinants of Health (DH), that is, the circumstances in which people are born and raised, the environment in which they grow up and age and their lifestyle. The aim of this study is to analyze the spatial behavior of the prevalence of asthma in Aragon, a Mediterranean region in Spain, under the DH approach. The methodological process entailed building a spatial database collating asthma prevalence as dependent variable, and lifestyle, socioeconomic, and climate indicators as explanatory factors, and then evaluating the spatial variability of the relationships by combining the Principal Component Analysis (PCA), Multiscale Geographically Weighted Regression (MGWR) models and cartographic design techniques. MGWR evidenced spatially varying relationships operating at different scales. Lifestyles seem closely tied to the prevalence of asthma in most of the study area while urban functionality and local climate patterns seem to boost prevalence rates in some specific enclaves. Consequently, the social and environmental conditions that characterize the study area translate into several DH scenarios modulating the spatial distribution of asthma. This differential DH behavior detected by local regression models is relevant to guiding and refining public health decision-making.     

沈阳市住房价格空间分异格局及其影响因素研究

本文以我国典型的老工业城市沈阳为例来分析其住房价格空间分异特征与影响机理。通过大数据方法获取该市1450个住宅小区的房价及特征数据，利用Kriging空间插值法模拟其房价空间分布格局，并从社区、公共配套设施和交通出行等方面构建地理加权回归模型，探究各因子对房价空间分布的影响机理。结果表明：①沈阳市住房价格呈现出多中心的空间结构，且长白区域已成为新的价格峰值区。②特征因素对住房价格的影响具有显著的空间异质性，其中，公共配套设施和地铁站对房价表现出较高的影响力，并对住房价格的作用程度呈现明显空间差异性。③受“强政府、弱市场”等的长期影响，政府调控下的城市资源分配不均衡成为沈阳等老工业城市住房价格空间分异的根本原因。     

Exploring Heterogeneities with Geographically Weighted Quantile Regression: An Enhancement Based on the Bootstrap Approach

Geographically weighted quantile regression (GWQR) has been proposed as a spatial analytical technique to simultaneously explore two heterogeneities, one of spatial heterogeneity with respect to data relationships over space and one of response heterogeneity across different locations of the outcome distribution. However, one limitation of GWQR framework is that the existing inference procedures are established based on asymptotic approximation, which may suffer computation difficulties or yield incorrect estimates with finite samples. In this article, we suggest a bootstrap approach to address this limitation. Our bootstrap enhancement is first validated by a simulation experiment and then illustrated with an empirical U.S. mortality data. The results show that the bootstrap approach provides a practical alternative for inference in GWQR and enhances the utilization of GWQR.     

A SPATIAL HEDONIC APPROACH ON LAND USE CHANGE ANTICIPATIONS*

ABSTRACT This paper estimates land use conversion anticipation when zoning is the primary tool of land use regulation. Extending the Capozza and Helsey model (1989) to the case of uncertainty in future land use zoning, a spatialized hedonic model is proposed to estimate such anticipation phenomena at a fine level (cadastral unit). Estimations use Mixed Geographically Weighted Regression (MGWR) techniques with a two‐stage model that links agricultural and developable land markets. This allows for mapping varying spatial parameters that measure anticipation effects within the theoretical framework. Results confirm the influence of anticipation on agricultural land prices in the Provence region. Moreover, the level of data spatialization allows us to take into account intra‐municipalities' heterogeneity of land use conversion anticipation.     

Some Notes on Parametric Significance Tests for Geographically Weighted Regression

The technique of geographically weighted regression (GWR) is used to model spatial 'drift' in linear model coefficients. In this paper we extend the ideas of GWR in a number of ways. First, we introduce a set of analytically derived significance tests allowing a null hypothesis of no spatial parameter drift to be investigated. Second, we discuss 'mixed' GWR models where some parameters are fixed globally but others vary geographically. Again, models of this type may be assessed using significance tests. Finally, we consider a means of deciding the degree of parameter smoothing used in GWR based on the Mallows C_p statistic. To complete the paper, we analyze an example data set based on house prices in Kent in the U.K. using the techniques introduced.     

Spatial and Scale‐Dependent Controls on North American Pan‐Arctic Minimum River Discharge

Spatial patterns of minimum monthly river discharge in the North American Pan‐Arctic and its potential controls are explored with geographically weighted regression (GWR). Minimum discharge is indicative of soil water conditions; therefore, understanding spatial variability of its controls may provide insights into patterns of hydrologic change. Here, GWR models are applied to determine a suitable combination of independent variables selected from a set of eight variables. A model specification with annual mean river discharge, temperature at time of minimum discharge, and biome describes well the spatial patterns in minimum discharge. However, minimum discharge in larger watersheds is influenced more by temperature and biome distributions than it is in small basins, suggesting that scale is critical for understanding minimum river discharge. This study is the first to apply GWR to explore spatial variation in Pan‐Arctic hydrology. Factores de control espaciales y dependientes de escala en las descargas fluviales mínimas de ríos Pan‐Articos en Norteamérica. El artículo explora los patrones espaciales de caudales fluviales mínimos mensuales la región pan‐ártica de Norteamérica y sus posibles factores de control haciendo uso de una regresión ponderada geográficamente (geographically weigted regression‐GWR). Los caudales mínimos son indicadores de las condiciones del agua en el suelo, y por lo tanto el entendimiento de la variabilidad espacial de los factores que los controlan puede ayudar a comprender los patrones de cambio hidrológico. En el presente estudio, varios modelos de tipo GWR son aplicados para determinar una combinación adecuada de variables independientes seleccionadas a partir de un conjunto de ocho variables. El modelo que utiliza la media anual media de descarga fluvial, la temperatura en el momento de caudal mínimo, y el bioma, proporciona una buena descripción de los patrones espaciales en la descarga mínima. Sin embargo, en las cuencas hidrográficas grandes, la descarga mínima está más influenciada por la temperatura y la distribución de los biomas que en el caso de cuencas más pequeñas, lo que sugiere que la escala es fundamental para entender la descarga mínima fluvial. Este estudio es el primero en aplicar GWR para comprender la variación espacial en la hidrología de la región pan‐ártica. 基于GWR（地理加权回归模型）对北美泛北极地区月份最小河流流量的空间模式和潜在控制进行研究。最小流量暗示水土条件;因此,理解空间分异及控制可深刻理解水文变化的模式。GWR可从8个变量中提取一组独立变量的适当组合。通过年均河流流量、最小流量时的温度和生物群落,来描述最小下泄流量的空间格局。在大范围流域中,最小流量受到温度和生物群落分布的影响大于在小规模的流域,揭示出在河流最小流量分析中尺度是非常重要的。本文首次将GWR应用于泛北极水文空间异质性分析。     

Geographical and Temporal Weighted Regression (GTWR)

Both space and time are fundamental in human activities as well as in various physical processes. Spatiotemporal analysis and modeling has long been a major concern of geographical information science (GIScience), environmental science, hydrology, epidemiology, and other research areas. Although the importance of incorporating the temporal dimension into spatial analysis and modeling has been well recognized, challenges still exist given the complexity of spatiotemporal models. Of particular interest in this article is the spatiotemporal modeling of local nonstationary processes. Specifically, an extension of geographically weighted regression (GWR), geographical and temporal weighted regression (GTWR), is developed in order to account for local effects in both space and time. An efficient model calibration approach is proposed for this statistical technique. Using a 19‐year set of house price data in London from 1980 to 1998, empirical results from the application of GTWR to hedonic house price modeling demonstrate the effectiveness of the proposed method and its superiority to the traditional GWR approach, highlighting the importance of temporally explicit spatial modeling.