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.     

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.     

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 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 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.     

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.     

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.     

Destination Choice of Athenians: An Application of Geographically Weighted Versions of Standard and Zero Inflated Poisson Spatial Interaction Models

The main aim of this article is to combine recent developments in spatial interaction modeling to better model and explain spatial decisions. The empirical study refers to migration decisions made by internal migrants from Athens, Greece. To achieve this, geographically weighted versions of standard and zero inflated Poisson (ZIP) spatial interaction models are defined and fit. In the absence of empirical studies for the effect of potential determinants on internal migration decisions in Greece and the presence of an excessive number of zero migration flows among municipalities in Greece, this article provides empirical evidence for the power of the proposed Geographically Weighted ZIP regression method to better explain destination choices of Athenian internal migrants. We also discuss statistical inference issues in relation to the application of the proposed regression techniques.     

Rethinking Remoteness: A Simple and Objective Approach

This paper re‐examines the characteristics and assumptions of current remoteness/accessibility classifications in Australia and proposes a simple and easily understandable alternative measure for remoteness. In this study, remoteness is redefined simply as the average distance between two nearest people within an appropriate spatial unit where population distribution is assumed to be homogenous. By definition, the most straightforward remoteness and incapacity index (RII) would be remoteness times a measure of the incapacity for social and commercial interaction, where remoteness is gauged by the square root of the area divided by the population, and incapacity is measured by the reciprocal of population. Australian Bureau of Statistics Statistical Local Area (SLA) level population data and digital boundaries have been utilised for assessment of this index. The utility of the RII is demonstrated with two examples of activity measures for general practitioner services and businesses. At the State/Territory level, RIIs are negatively related to both general practitioner services per person (Pearson correlation coefficient r=?0.873), and the number of businesses per person (r=?0.546). The correlation can be further enhanced by normalising the distributions of the remoteness scores with a simple logarithmic function. The strong correlations confirm that remoteness has a substantial inverse impact on daily activities. Greater distance means longer time and higher costs for travelling, diseconomy of scale, and higher personnel costs. The RII provides an alternative measure of remoteness that is both intuitive and statistically straightforward and, at an SLA level, closely coincides with the commonly used but complex Accessibility/Remoteness Index of Australia Plus (ARIA+). Significantly, the RII is free of the service specific and policy sensitive adjustments justified by accessibility that have been introduced into existing measures.