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.     

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.     

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.     

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.     

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。     

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.     

Comparative Spatial Filtering in Regression Analysis

One approach to dealing with spatial autocorrelation in regression analysis involves the filtering of variables in order to separate spatial effects from the variables’ total effects. In this paper we compare two filtering approaches, both of which allow spatial statistical analysts to use conventional linear regression models. Getis’ filtering approach is based on the autocorrelation observed with the use of the G_i local statistic. Griffith's approach uses an eigenfunction decomposition based on the geographic connectivity matrix used to compute a Moran's I statistic. Economic data are used to compare the workings of the two approaches. A final comparison with an autoregressive model strengthens the conclusion that both techniques are effective filtering devices, and that they yield similar regression models. We do note, however, that each technique should be used in its appropriate context.     

Identifying the Impacts of Land-Use Spatial Patterns on Street-Network Accessibility Using Geospatial Methods

While the land use-street network nexus is well acknowledged, evidence for the one-way impacts of land-use patterns on street accessibility is still inadequate. The measurements of land-use patterns and street accessibility lack systematic knowledge. Their empirical correlations also lack geographical variability, constraining site-specific land-use practices. Therefore, this study overcame the aforementioned limitations by examining the two-level spatial models to formulate accessibility-oriented land plans, using a well-developed Chinese city as an example. Firstly, two landscape metrics—Euclidean Nearest-Neighbor Distance (ENN) and Similarity Index (SIMI)—were used to quantify the intra- and inter-land-use configurations, respectively. Both city-level and local accessibility were measured using spatial design network analysis. Performing both ordinary least squares (OLS) and geographically weighted regression (GWR) models, results identified the statistically significant effects of inter-land-use patterns on two-level street accessibility. An exception was that land-use configurations within residential and industrial regions were irrelevant to street accessibility. We also found GWR was a better-fitting model than OLS when estimating locally-varied accessibility, suggesting hierarchical multiscale land-use planning. Overall, locally heterogeneous evidence in this study can substantialize land use-street network interactions and support the decision-making and implementation of place-specific accessibility-oriented land use.     

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.     

Extremal Approaches to Estimating Spatial Interaction

Two recent theoretical approaches to the gravity model of spatial interaction are examined with emphasis on the dissimilar physical and statistical theories from which they were respectively derived. Formal relationships between the two methods are demonstrated. The two approaches jointly indicate a general method of generating new hypotheses of gravity flows. This method is discussed and evaluated, and some questions of theory comparability are explored. Comments on some of the computational features of each of the two models conclude the paper.     

Spatial Characteristics and Comparisons of Interaction and Median Clustering Models

Cluster analysis has been pursued from a number of directions for identifying interesting relationships and patterns in spatial information. A major emphasis is currently on the development and refinement of optimization‐based clustering models for the purpose of exploring spatially referenced data. Within this context, two basic methods exist for identifying clusters that are most similar. An interesting feature of these two approaches is that one method approximates the relationships inherent in the other method. This is significant given that the approximation approach is invariably utilized for cluster detection in spatial and aspatial analysis. A number of spatial applications are investigated which highlight the differences in clusters produced by each model. This is an important contribution because the differences are in fact quite significant, yet these contrasts are not widely known or acknowledged.     

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.     

MODELING SPATIAL AUTOCORRELATION IN SPATIAL INTERACTION DATA: AN APPLICATION TO PATENT CITATION DATA IN THE EUROPEAN UNION*

ABSTRACT Spatial interaction models of the gravity type are widely used to model origin–destination flows. They draw attention to three types of variables to explain variation in spatial interactions across geographic space: variables that characterize an origin region of a flow, variables that characterize a destination region of a flow, and finally variables that measure the separation between origin and destination regions. This paper outlines and compares two approaches, the spatial econometric and the eigenfunction‐based spatial filtering approach, to deal with the issue of spatial autocorrelation among flow residuals. An example using patent citation data that capture knowledge flows across 112 European regions serves to illustrate the application and the comparison of the two approaches.     

Local Forms of Spatial Analysis

Local forms of spatial analysis focus on exceptions to the general trends represented by more traditional global forms of spatial analysis. There is currently a rapid expansion in the development of such techniques but their history almost exactly parallels that of Geographical Analysis, with the first examples of local analysis appearing in the late 1960s. Indeed, Geographical Analysis has published many of the significant contributions in this field. This paper reviews the development of local forms of spatial analysis and assesses the current situation. Following a discussion on the nature and importance of local analysis, examples are given of local forms of point pattern analysis; local graphical approaches; local measures of spatial dependency; the spatial expansion method; adaptive filtering; multilevel modeling; geographically weighted regression; random coefficients models; autoregressive models; and local forms of spatial interaction models.     

Territorial Keys for Bringing Closer the Territorial Agenda of the EU and Europe 2020

The recent developments of the economic theory suggest that due attention to territorial context increases efficiency and improves delivery of the policies. This in turn calls for better linkages between spatial and socio-economic efforts. The paper analyses the concept of policy territorialization and proposes policy tools for that purpose. The relevant theoretical models are used, mainly evolutionary economics and new economic geography. The key outcome is a set of territorial keys supposed to enhance territorial approach in developmental policies. Also some plausible ways of making use of those keys are proposed and then tested using Polish territory as a case study.     

Exploring the Use of Computer Vision Metrics for Spatial Pattern Comparison

Detection of changes in spatial processes has long been of interest to quantitative geographers seeking to test models, validate theories, and anticipate change. Given the current “data-rich” environment of today, it may be time to reconsider the methodological approaches used for quantifying change in spatial processes. New tools emerging from computer vision research may hold particular potential to make significant advances in quantifying changes in spatial processes. In this article, two comparative indices from computer vision, the structural similarity (SSIM) index, and the complex wavelet structural similarity (CWSSIM) index were examined for their utility in the comparison of real and simulated spatial data sets. Gaussian Markov random fields were simulated and compared with both metrics. A case study into comparison of snow water equivalent spatial patterns over northern Canada was used to explore the properties of these indices on real-world data. CWSSIM was found to be less sensitive than SSIM to changing window dimension. The CWSSIM appears to have significant potential in characterizing change and/or similarity; distinguishing between map pairs that possess subtle structural differences. Further research is required to explore the utility of these approaches for empirical comparison cases of different forms of landscape change and in comparison to human judgments of spatial pattern differences.     

The Right to Rule by Thumb: A Comment on Epistemology in “A Route Map for Successful Applications of Geographically-Weighted Regression”

Comber et al. provide an important contribution to the future of quantitative geography and Geographical Analysis. The contribution is chiefly in their development of a “GWR Route Map,” a diagram showing the sequence of analytical steps that “successful” specification searches in local modeling tend to follow. Geographically weighted techniques have been rapidly expanding, both in terms of complexity, users, and disciplinary reach. With geographically weighted methods now in so many more analysts' hands, any new rule of thumb will have a major imprint. But, by what right does the thumb rule the analysts? That is, what “counts” as valid knowledge about local models in general? In the following comment, I argue that we probably should use theory, not route maps to decide specifications. But, if we are pressed to build route maps, we sorely need better epistemological foundations for them. I discuss a few previous examples of strongly grounded route maps and offer a few paths to these better grounds as well as two ways to the exit.     

我国乡村转型发展时空分异格局与影响机制分析

乡村转型发展是乡村振兴的重要内容,揭示我国乡村转型发展时空格局及驱动力对推进乡村振兴战略实施具有现实意义.以我国省域单元作为研究对象,基于对乡村转型发展内涵特征解读,从人口—土地—产业—社会维度上构建乡村转型发展评价体系,综合运用投影寻踪模型、ESDA模型、时空跃迁分析及GWR模型等方法研究我国乡村转型发展时空格局及其...     

ECONOMIC GEOGRAPHY WITHIN AND BETWEEN EUROPEAN NATIONS: THE ROLE OF MARKET POTENTIAL AND DENSITY ACROSS SPACE AND TIME*

ABSTRACT In explaining the uneven spatial distribution of economic activity, urban economics, and new economic geography (NEG) dominate recent research in economics. A main difference between these two approaches is that NEG stresses the role of spatial linkages whereas urban economics does not do so. We estimate simple versions of these two views on economic geography and also establish if the relevance of spatial linkages varies across aggregation levels or time. For our sample of 14 European countries and 213 corresponding regions, we find that spatial linkages are more important at the country level and that its relevance varies across time.     

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。