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 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。     

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.     

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.     

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.     

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.     

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.     

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.     

Gaussian Process Regression and Bayesian Model Averaging: An Alternative Approach to Modeling Spatial Phenomena

Gaussian Process Regression (GPR) is a nonparametric technique that is capable of yielding reliable out‐of‐sample predictions in the presence of highly nonlinear unknown relationships between dependent and explanatory variables. But in terms of identifying relevant explanatory variables, this method is far less explicit about questions of statistical significance. In contrast, more traditional spatial econometric models, such as spatial autoregressive models or spatial error models, place rather strong prior restrictions on the functional form of relationships, but allow direct inference with respect to explanatory variables. In this article, we attempt to combine the best of both techniques by augmenting GPR with a Bayesian Model Averaging (BMA) component that allows for the identification of statistically relevant explanatory variables while retaining the predictive performance of GPR. In particular, GPR‐BMA yields a posterior probability interpretation of model‐inclusion frequencies that provides a natural measure of the statistical relevance of each variable. Moreover, while such frequencies offer no direct information about the signs of local marginal effects, it is shown that partial derivatives based on the mean GPR predictions do provide such information. We illustrate the additional insights made possible by this approach by applying GPR‐BMA to a benchmark BMA data set involving potential determinants of cross‐country economic growth. It is shown that localized marginal effects based on partial derivatives of mean GPR predictions yield additional insights into comparative growth effects across countries.     

基于DEA的浙江省饭店企业效率分析

饭店企业的效率分析一直受到行业和学术研究的共同关注。本文以问卷形式对浙江省不同管理模式下的饭店企业进行调查,根据问卷数据,运用数据包络分析（Data Envelopment Analysis,DEA）方法计算其相对效率,最后对计算结果进行定性分析,以期分析结论能为浙江省饭店企业的发展提供有益参考。     

探源文明 追迹历史——李伯谦先生专访

中国考古学经过了近一百年的建设与发展历程,其理论与实践均取得了长足进步。著名考古学家李伯谦先生认为,考古学要与文献史学、社会学组成三个认知系统,考古学文化研究要将区域研究与长时段研究结合起来,要重视考古遗存所蕴含的精神内涵,更全面地研究、复原古代社会;考古学研究要掌握地层学、类型学和文化因素分析方法,其中地层学在使用中应该与埋藏学相结合,而文化因素分析则是从考古学研究过渡到历史学研究的桥梁;江苏的商周考古研究应该把握江北徐淮夷文化和江南吴文化两条主线;要正确处理考古与文化遗产保护的关系,充分发挥考古学的作用,使考古真正成为人民的事业。     

Scandinavia and the Huns: An Interdisciplinary Approach to the Migration Era

The aim of this paper is to discuss the early Migration period as a particular period of ‘short term history’ and its formative impact on the Scandinavian longue durée in the first millenium. During this particular period of time, the object world of Scandinavia demonstrates radical changes in symbolic representation, followed by long term continuity and social/mental resistance to change. It is argued that the Huns, as a historical fact, were present in Scandinavia in the early fifth century. Their impact was to generate an ‘episodic transition’ that opened up a whole new set of social, religious and political strategies, in Scandinavia in particular as well as in Barbarian Europe in general, and gave rise to a new Germanic identity in the aftermath of the Roman Empire.     

Views into the Fragments: An Approach from a Microhistorical Perspective

The approach I call the “singularization of history,” which I have been developing in recent years within the methodological structure of microhistory, is the main subject of this article. It has the precise aim of defining the ways in which scholars can use sources to enter into the past in as detailed and varied a way as possible without becoming trapped within the received channels of the grand narratives. I will make an attempt to demonstrate what the Icelandic School of Microhistory (ISM) is all about and its connection to the scribal culture in the country, as well as the importance of ego-documents for microhistorical analysis. The central element in the analysis of this paper will be the sources themselves—their creation, their context within the events they describe, the opportunities they present for analysis, and in what kind of academic context they have become a subject for enquiry.     

Prehispanic Dugout Canoes in Mexico: A Typology Based on a Multidisciplinary Approach

Navigation implies a deep knowledge of the environment in which it is practised as well as the development of correlated technologies and techniques. In fact, sailing in lakes, rivers, along coasts or in high seas requires specific skills. This is why watercraft reflect the specific expertise of cultures in relation to the available resources and materials. Navigation as a means of movement and boats as modes of transportation is a well-documented topic in Europe, Asia and the Pacific Ocean, whether focusing on antiquity or the present. Nonetheless, in the New World, especially in Mesoamerica where the Aztec, Tarascan and Mayan Empires flourished, indigenous maritime history has been thoroughly under-investigated. This article explores the first prehispanic dugout canoes of the region, based on a multidisciplinary approach for the Mexico and Pátzcuaro Basins, in the Late Postclassic period (AD 1325-1521). For the first time, a typology of these dugout types is proposed, with a discussion of the importance of dugout canoes as a means of transportation in Mesoamerican civilizations, where wheels and draught animals did not exist.