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.     

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 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 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 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.     

Spatial Autoregressive Models for Geographically Hierarchical Data Structures

This article discusses how standard spatial autoregressive models and their estimation can be extended to accommodate geographically hierarchical data structures. Whereas standard spatial econometric models normally operate at a single geographical scale, many geographical data sets are hierarchical in nature—for example, information about houses nested into data about the census tracts in which those houses are found. Here we outline four model specifications by combining different formulations of the spatial weight matrix W and of ways of modeling regional effects. These are (1) groupwise W and fixed regional effects; (2) groupwise W and random regional effects; (3) proximity‐based W and fixed regional effects; and (4) proximity‐based W and random regional effects. We discuss each of these model specifications and their associated estimation methods, giving particular attention to the fourth. We describe this as a hierarchical spatial autoregressive model. We view it as having the most potential to extend spatial econometrics to accommodate geographically hierarchical data structures and as offering the greatest coming together of spatial econometric and multilevel modeling approaches. Subsequently, we provide Bayesian Markov Chain Monte Carlo algorithms for implementing the model. We demonstrate its application using a two‐level land price data set where land parcels nest into districts in Beijing, China, finding significant spatial dependence at both the land parcel level and the district level.     

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.     

A Spatial Prior for Bayesian Vector Autoregressive Models

In this paper we develop a Bayesian prior motivated by cross-sectional spatial autoregressive models for use in time-series vector autoregressive forecasting involving spatial variables. We compare forecast accuracy of the proposed spatial prior to that from a vector autoregressive model relying on the Minnesota prior and find a significant improvement. In addition to a spatially motivated prior variance as in LeSage and Pan (1995) we develop a set of prior means based on spatial contiguity. A Theil-Goldberger estimator may be used for the proposed model making it easy to implement.     

北京房地产中介空间格局演化特征及驱动因素分析

本文基于微观企业数据,采用泰森多边形、核密度及空间自相关方法,多尺度考察2001-2010年北京房地产中介门店空间格局演化特征,通过构建地理加权回归模型（GWR）,探究2010年房地产中介空间格局驱动因素。研究表明：①房地产中介集中分布在城六区,数量增速最快功能区是城市发展新区。②空间格局不均衡性增大,空间集聚态势加强。③集聚主核心点出现由"东城"向"朝阳"方向的东推位移,集聚程度加深,主核心区团块面积扩大,次级核心点增多,"两轴、两带、多中心"基本版图形成。④空间集聚现象明显,热点区域扩张,但始终集中在区域中心。⑤驱动效应最大的因素为常住人口,多种驱动因素效应均比较强的街区（乡镇）基本上属于各区县经济水平相对发达的区域。此外,政府政策对其影响也较大。     

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。     

Beyond Moran's I: Testing for Spatial Dependence Based on the Spatial Autoregressive Model

The statistic known as Moran's I is widely used to test for the presence of spatial dependence in observations taken on a lattice. Under the null hypothesis that the data are independent and identically distributed normal random variates, the distribution of Moran's I is known, and hypothesis tests based on this statistic have been shown in the literature to have various optimality properties. Given its simplicity, Moran's I is also frequently used outside of the formal hypothesis-testing setting in exploratory analyses of spatially referenced data; however, its limitations are not very well understood. To illustrate these limitations, we show that, for data generated according to the spatial autoregressive (SAR) model, Moran's I is only a good estimator of the SAR model's spatial-dependence parameter when the parameter is close to 0. In this research, we develop an alternative closed-form measure of spatial autocorrelation, which we call APLE , because it is an approximate profile-likelihood estimator (APLE) of the SAR model's spatial-dependence parameter. We show that APLE can be used as a test statistic for, and an estimator of, the strength of spatial autocorrelation. We include both theoretical and simulation-based motivations (including comparison with the maximum-likelihood estimator), for using APLE as an estimator. In conjunction, we propose the APLE scatterplot, an exploratory graphical tool that is analogous to the Moran scatterplot, and we demonstrate that the APLE scatterplot is a better visual tool for assessing the strength of spatial autocorrelation in the data than the Moran scatterplot. In addition, Monte Carlo tests based on both APLE and Moran's I are introduced and compared. Finally, we include an analysis of the well-known Mercer and Hall wheat-yield data to illustrate the difference between APLE and Moran's I when they are used in exploratory spatial data analysis.     

Loglinear Residual Tests of Moran's I Autocorrelation and their Applications to Kentucky Breast Cancer Data

This article bridges the permutation test of Moran's I to the residuals of a loglinear model under the asymptotic normality assumption. It provides the versions of Moran's I based on Pearson residuals ( I _PR) and deviance residuals ( I _DR) so that they can be used to test for spatial clustering while at the same time account for potential covariates and heterogeneous population sizes. Our simulations showed that both I _PR and I _DR are effective to account for heterogeneous population sizes. The tests based on I _PR and I _DR are applied to a set of log-rate models for early-stage and late-stage breast cancer with socioeconomic and access-to-care data in Kentucky. The results showed that socioeconomic and access-to-care variables can sufficiently explain spatial clustering of early-stage breast carcinomas, but these factors cannot explain that for the late stage. For this reason, we used local spatial association terms and located four late-stage breast cancer clusters that could not be explained. The results also confirmed our expectation that a high screening level would be associated with a high incidence rate of early-stage disease, which in turn would reduce late-stage incidence rates.     

Model Uncertainty in Matrix Exponential Spatial Growth Regression Models

This article considers the most important aspects of model uncertainty for spatial regression models, namely, the appropriate spatial weight matrix to be employed and the appropriate explanatory variables. We focus on the spatial Durbin model (SDM) specification in this study that nests most models used in the regional growth literature, and develop a simple Bayesian model‐averaging approach that provides a unified and formal treatment of these aspects of model uncertainty for SDM growth models. The approach expands on previous work by reducing the computational costs through the use of Bayesian information criterion model weights and a matrix exponential specification of the SDM model. The spatial Durbin matrix exponential model has theoretical and computational advantages over the spatial autoregressive specification due to the ease of inversion, differentiation, and integration of the matrix exponential. In particular, the matrix exponential has a simple matrix determinant that vanishes for the case of a spatial weight matrix with a trace of zero. This allows for a larger domain of spatial growth regression models to be analyzed with this approach, including models based on different classes of spatial weight matrices. The working of the approach is illustrated for the case of 32 potential determinants and three classes of spatial weight matrices (contiguity‐based, k‐nearest neighbor, and distance‐based spatial weight matrices), using a data set of income per capita growth for 273 European regions.     

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。     

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.     

Dynamics of Land Surface Temperature in Response to Land‐Use/Cover Change

In this study, we employed Geographical Information Systems and remote sensing techniques to investigate the impact of land‐use/cover change on land surface temperature (LST) in a rapidly urbanisation city, Kunming in south‐west China. Spatial patterns of LST and land use for 1992 and 2006 were derived from Landsat images to examine how LST responded to urban growth. Remote sensing indices were used to quantify land‐use types and employed as explanatory variables in LST modelling. The geographically weighted regression (GWR), a location dependent model, was performed to explore the influences of the spatially varied land‐use conditions on the LST patterns. Results revealed that rapid urbanisation in Kunming altered the local thermal environment, particularly in increasing the LST in the zone surrounding the urban core. Remote sensing indices demonstrated that water and vegetation played an important role in mitigating the urban heat island effect, while built‐up and barren land accounted for the increase in LST. The GWR improved the goodness‐of‐fit for LST modelling and provided insights into the spatially varied relationship between LST and land‐use conditions.     

Fitting Constrained Poisson Regression Models to Interurban Migration Flows

This paper demonstrates the effects of fitting singly and doubly constrained spatial interaction models using the Poisson regression approach. A large data set containing migration flows between labor market areas in Great Britain in 1970–71 is used. The results of fitting unconstrained, singly constrained, and doubly constrained models are compared with respect to goodness of fit and the interpretability of parameter estimates. The addition of other explanatory variables to the model is also explored.     

Calibration and Test of a Discrete Choice Model with Endogenous Choice Sets

Conventional discrete choice models assume implicitly that the choice set is independent of the decisionmaker's preferences conditional on the explanatory variables of the models. This assumption is implausible in many choice situations where the decisionmaker selects his or her choice set. This paper estimates and tests a discrete choice model with endogenous choice sets based on Horowitz' theoretical work. To calibrate the model, a new probability simulator is introduced and a sequential estimation procedure is developed. The model and calibration methods are tested in an empirical application as well as Monte Carlo simulations. The empirical results are used to test the theory of endogenous choice sets and to examine the differences between the new model and a conventional choice model in parameter estimates and predicted choice probabilities. The empirical results strongly suggest that ignoring the endogeneity of choice sets in choice modeling can have serious consequences in applications.     

INFERENCE BASED ON ALTERNATIVE BOOTSTRAPPING METHODS IN SPATIAL MODELS WITH AN APPLICATION TO COUNTY INCOME GROWTH IN THE UNITED STATES*

ABSTRACT This study examines aggregate county income growth across the 48 contiguous states from 1990 to 2005. To control for endogeneity, we estimate a two‐stage spatial error model and implement a number of spatial bootstrap routines to infer parameter significance. Among the results, we find that outdoor recreation and natural amenities favor positive growth in rural counties and property taxes correlate negatively with rural growth. Comparing bootstrap inference with other models, including the recent General Moment heteroskedastic‐robust spatial error estimator, we find similar conclusions suggesting bootstrapping can be effective in spatial models where asymptotic results are not well established.