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.     

Anisotropic Variance Functions in Geographically Weighted Regression Models

Most standard methods of statistical analysis used in the social and environmental sciences are built upon the basic assumptions of independence, homogeneity, and isotropy. A notable exception to this rule is the collection of methods used in geographical analysis, which have been designed to take into account serial dependence often observed in spatial data. In addition, recent developments, in particular the method of geographically weighted regression, have provided the tools to model non‐stationary processes, and thus evidence that challenges the assumption of homogeneity. The assumption of isotropy, however, although suspect, has received considerably less attention, and there is thus a need for tools to study anisotropy in a more systematic fashion. In this paper we expand the method of geographically weighted regression in a simple yet effective way to explore the topic of anisotropy in spatial processes. We discuss two different estimation situations and exemplify the proposed technical development by means of a case study. The results suggest that anisotropy issues might be a fairly common occurrence in spatial processes and/or in the statistical modeling of spatial processes.     

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.     

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.     

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.     

Geographically Weighted Cox Regression for Prostate Cancer Survival Data in Louisiana

The Cox proportional hazard model is one of the most popular tools in analyzing time-to-event data in public health studies. When outcomes observed in clinical data from different regions yield a varying pattern correlated with location, it is often of great interest to investigate spatially varying effects of covariates. In this paper, we propose a geographically weighted Cox regression model for sparse spatial survival data. In addition, a stochastic neighborhood weighting scheme is introduced at the county level. Theoretical properties of the proposed geographically weighted estimators are examined in detail. A model selection scheme based on the Takeuchi’s model robust information criteria is discussed. Extensive simulation studies are carried out to examine the empirical performance of the proposed methods. We further apply the proposed methodology to analyze real data on prostate cancer from the Surveillance, Epidemiology, and End Results cancer registry for the state of Louisiana.     

Geographically Weighted Regression: A Method for Exploring Spatial Nonstationarity

Spatial nonstationarity is a condition in which a simple “global” model cannot explain the relationships between some sets of variables. The nature of the model must alter over space to reflect the structure within the data. In this paper, a technique is developed, termed geographically weighted regression, which attempts to capture this variation by calibrating a multiple regression model which allows different relationships to exist at different points in space. This technique is loosely based on kernel regression. The method itself is introduced and related issues such as the choice of a spatial weighting function are discussed. Following this, a series of related statistical tests are considered which can be described generally as tests for spatial nonstationarity. Using Monte Carlo methods, techniques are proposed for investigating the null hypothesis that the data may be described by a global model rather than a non-stationary one and also for testing whether individual regression coefficients are stable over geographic space. These techniques are demonstrated on a data set from the 1991 U.K. census relating car ownership rates to social class and male unemployment. The paper concludes by discussing ways in which the technique can be extended.     

Geographically Weighted Discriminant Analysis

In this article, we propose a novel analysis technique for geographical data, Geographically Weighted Discriminant Analysis. This approach adapts the method of Geographically Weighted Regression (GWR), allowing the modeling and prediction of categorical response variables. As with GWR, the relationship between predictor and response variables may alter over space, and calibration is achieved using a moving kernel window approach. The methodology is outlined and is illustrated with an example analysis of voting patterns in the 2005 UK general election. The example shows that similar social conditions can lead to different voting outcomes in different parts of England and Wales. Also discussed are techniques for visualizing the results of the analysis and methods for choosing the extent of the moving kernel window.     

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.     

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.     

Prediction of Bike-sharing Trip Counts: Comparing Parametric Spatial Regression Models to a Geographically Weighted XGBoost Algorithm

Regression models are commonly applied in the analysis of transportation data. This research aims at broadening the range of methods used for this task by modeling the spatial distribution of bike-sharing trips in Cologne, Germany, applying both parametric regression models and a modified machine learning approach while incorporating measures to account for spatial autocorrelation. Independent variables included in the models consist of land use types, elements of the transport system and sociodemographic characteristics. Out of several regression models with different underlying distributions, a Tweedie generalized additive model is chosen by its values for AIC, RMSE, and sMAPE to be compared to an XGBoost model. To consider spatial relationships, spatial splines are included in the Tweedie model, while the estimations of the XGBoost model are modified using a geographically weighted regression. Both methods entail certain advantages: while XGBoost leads to far better values regarding RMSE and sMAPE and therefore to a better model fit, the Tweedie model allows an easier interpretation of the influence of the independent variables including spatial effects.     

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。     

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.     

Guns and Homicides: A Multiscale Geographically Weighted Instrumental Variables Approach

This article assesses the locally varying effects of gun ownership levels on total and gun homicide rates in the contiguous United States using cross-sectional county data for the period 2009–2015. Employing a multiscale geographically weighted instrumental variables regression that takes into account spatial nonstationarity in the processes and the endogenous nature of gun ownership levels, estimates show that gun ownership exerts spatially monotonically negative effects on total and gun homicide rates, indicating that there are no counties supporting the “more guns, more crime” hypothesis for these two highly important crime categories. The number of counties in the contiguous United States where the “more guns, less crime” hypothesis is confirmed is limited to at least 1258 counties (44.8% of the sample) with the strongest total homicide-decreasing effects concentrated in southeastern Texas and the deep south. On the other hand, stricter state gun control laws exert spatially monotonically negative effects on gun homicide rates with the strongest effects concentrated in the southern tip of Texas extending toward the deep south.     

Geographically Weighted Extreme Learning Machine: A Method for Space–Time Prediction

Spatial heterogeneity has been regarded as an important issue in space–time prediction. Although some statistical methods of space–time predictions have been proposed to address spatial heterogeneity, the linear assumption makes it difficult for these methods to predict geographical processes accurately because geographical processes always involve complicated nonlinear characteristics. An extreme learning machine (ELM) has the advantage of approximating nonlinear relationships with a rapid learning speed and excellent generalization performance. However, determining how to incorporate spatial heterogeneity into an ELM to predict space–time data is an urgent problem. For this purpose, a new method called geographically weighted ELM (GWELM) is proposed to address spatial heterogeneity based on an ELM in this article. GWELM is essentially a locally varying ELM in which the parameters are regarded as functions of spatial locations, and geographically weighted least squares is applied to estimate the parameters in a local model. The proposed method is used to analyze two groups of different data sets, and the results demonstrate that the GWELM method is superior to the comparative method, which is also developed to address spatial heterogeneity.     

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.     

The Assessment of Spatial Autocorrelation through Constrained Multiple Regression

Most published measures of spatial autocorrelation (SA) can be recast as a (normalized) cross-product statistic that indexes the degree of relationship between corresponding entries from two matrices—one specifying the spatial connections among a set of n locations, and the other reflecting a very explicit definition of similarity between the set of values on some variable x realized over the n locations. We first give a very brief sketch of the basic cross-product approach to the evaluation of SA, and then generalize this strategy to include less restrictive specifications for the notion of similarity between the values on x. Using constrained multiple regression, the characterization of variate similarity basic to any assessment of SA can itself be framed according to the information present in the measure of spatial separation. These extensions obviate the inherent arbitrariness in how SA is usually evaluated, which now results from the requirement of a very restrictive definition of variate similarity before a cross-product index can be obtained.