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.     

Modified Kaplan–Meier Estimator and Nelson–Aalen Estimator with Geographical Weighting for Survival Data

The Kaplan–Meier and Nelson–Aalen estimators are universally used methods in clinical studies. In a public health study, people often collect data from different locations of the medical services provider. When some studies need to consider survival curves from different locations, traditional estimators simply estimate the marginal survival curves using stratification. In this article, we use the idea from geographically weighted regression to add geographical weights to the observations to get modified versions of the Kaplan–Meier and Nelson–Aalen estimators which can represent the local survival curve and cumulative hazard. We use counting process methods to derive these modified estimators and to estimate their variances. In addition, we discuss some general spatial weighting functions which can be used in computing these estimators. Furthermore, we present simulation results to illustrate the performance of the modified estimators. Finally, we apply our method to prostate cancer data from the SEER cancer registry for the state of Louisiana.     

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.     

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 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.     

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.     

Bayesian Spatial Filtering for Hedonic Models: An Application for the Real Estate Market

This article presents a Bayesian method based on spatial filtering to estimate hedonic models for dwelling prices with geographically varying coefficients. A Bayesian Adaptive Sampling algorithm for variable selection is used, which makes it possible to select the most appropriate filters for each hedonic coefficient. This approach explores the model space more systematically and takes into account the uncertainty associated with model estimation and selection processes. The methodology is illustrated with an application for the real estate market in the Spanish city of Zaragoza and with simulated data. In addition, an exhaustive comparison study with a set of alternatives strategies used in the literature is carried out. Our results show that the proposed Bayesian procedures are competitive in terms of prediction; more accurate results are obtained in the estimation of the regression coefficients of the model, and the multicollinearity problems associated with the estimation of the regression coefficients are solved.     

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.     

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.     

Spatially varying relationships between municipal operating expenditure and its determinants: The case of South Africa

We apply the geographically weighted regression to investigate spatial variability (nonstationarity) in the relationships between municipal operating expenditure (in total and for a separate component thereof) and the determinants of this expenditure, in South Africa. The empirical findings indicate that some of these relationships are spatially varying for the period under consideration. The global model (i.e., least square regression) cannot take account of the unequal environment in which South African municipalities operate. One implication of our findings is that a “one-size-fits-all” approach in the design of policies targeting municipal finances may not be appropriate for municipalities in South Africa, and indeed in other contexts in which such heterogeneity is found. Instead, policy formulation should explicitly consider relevant differences in local conditions.     

交通路网中心性对土地利用强度的影响研究——以青岛市中心城区为例

采取青岛市交通路网矢量数据和 POI 数据，综合运用核密度估计法、地理加权回归模型和多中心性评价模型，从中心性角度解析了青岛市中心城区交通路网与土地利用强度的相关关系。研究表明：青岛市中心城区交通路网的临近中心性与直线中心性结构呈现“中心—外围”模式，中介中心性具有显著的点轴模式；商业、公共服务以及居住用地布局均呈现多中心结构；交通路网中心性与土地利用强度具有较强的相关性且相关关系具有空间异质性；商业用地和公共服务用地与交通路网中心性的相关性高于居住用地。研究揭示了交通路网与土地利用强度之间具有较强的依赖性，为城市规划提供可靠的理论依据，有助于进一步优化城市空间布局。     

Modelling spatial heterogeneity and nonstationarity in artifact-rich landscapes

In this paper we consider a crucial issue for survey archaeology: how we identify and make sense of the heterogeneous and often inter-dependent behaviours and processes responsible for apparent archaeological patterns across the landscape. We apply two spatial statistical tools, kriging and geographically weighted regression, to develop a model that addresses the spatial heterogeneity and spatial nonstationarity present in the pottery distributions identified by our intensive survey of the Greek island of Antikythera. Our modelling results highlight a clear spatial structure underlying different scales of pottery density as well as locally varying relationships between pottery densities and several environmental variables. This allows us to develop further testable hypotheses about long-term settlement and land-use patterns on Antikythera, including more explicit models of community organisation, and of the relationship between the island's geomorphological structure and its history of past human activity.     

Exploring scale-dependent correlations between cancer mortality rates using factorial kriging and population-weighted semivariograms

This article presents a geostatistical methodology that accounts for spatially varying population size in the processing of cancer mortality data. The approach proceeds in two steps: (1) spatial patterns are first described and modeled using population-weighted semivariogram estimators, (2) spatial components corresponding to nested structures identified on semivariograms are then estimated and mapped using a variant of factorial kriging. The main benefit over traditional spatial smoothers is that the pattern of spatial variability (i.e., direction-dependent variability, range of correlation, presence of nested scales of variability) is directly incorporated into the computation of weights assigned to surrounding observations. Moreover, besides filtering the noise in the data, the procedure allows the decomposition of the structured component into several spatial components (i.e., local versus regional variability) on the basis of semivariogram models. A simulation study demonstrates that maps of spatial components are closer to the underlying risk maps in terms of prediction errors and provide a better visualization of regional patterns than the original maps of mortality rates or the maps smoothed using weighted linear averages. The proposed approach also attenuates the underestimation of the magnitude of the correlation between various cancer rates resulting from noise attached to the data. This methodology has great potential to explore scale-dependent correlation between risks of developing cancers and to detect clusters at various spatial scales, which should lead to a more accurate representation of geographic variation in cancer risk, and ultimately to a better understanding of causative relationships.     

Elimination of Source A and B Errors in p-Median Location Problems

The p-median problem is a powerful tool in analyzing facility location options when the goal of the location scheme is to minimize the average distance that demand must traverse to reach its nearest facility. It may be used to determine the number of facilities to site, as well as the actual facility locations. Demand data are frequently aggregated in p-median location problems to reduce the computational complexity of the problem. Demand data aggregation, however, results in the loss of locational information. This loss may lead to suboptimal facility location configurations (optimality errors) and inaccurate measures of the resulting travel distances (cost errors). Hillsman and Rhoda (1978) have identified three error components: Source A, B, and C errors, which may result from demand data aggregation. In this article, a method to measure weighted travel distances in p-median problems which eliminates Source A and B errors is proposed. Test problem results indicate that the proposed measurement scheme yields solutions with lower optimality and cost errors than does the traditional distance measurement scheme.     

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.     

Modeling agricultural potential in Chaco Canyon during the Bonito phase: a predictive geospatial approach

This study presents a geospatial analysis of surficial hydrology and geomorphology and their relationship to potential agricultural productivity in order to better understand the economic role of water in Chaco Canyon during the Bonito Phase (ca. AD 850–1150). Defined as the Natural Agricultural Suitability Analysis, the foundation of this study is a hierarchical geospatial analysis that integrates six key natural factors: slope, soil texture, soil depth, non-catastrophic overbank flooding potential, drainage flow length, and drainage proximity and flow potential. These factors are combined through a raster weighted overlay function to generate composite suitability map that offers a testable proxy for variability in relative agricultural potential during the Bonito Phase at Chaco. The rationale for including this set of natural factors is based largely on ethnographic and modern agricultural studies, but the predictive model differs from previous studies of agricultural potential in that it is independent of the specific archaeological distribution of evidence of agriculture in the study area. The results of this analysis suggest that previous models of Chacoan agricultural productivity have underestimated local production capacity. Previous studies have focused solely on floodplain contexts, whereas this study points to a more comprehensive and geographically distributed use of the landscape.     

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 Simulation Study on Specifying a Regression Model for Spatial Data: Choosing between Autocorrelation and Heterogeneity Effects

In this simulation study, regressions specified with autocorrelation effects are compared against those with relationship heterogeneity effects, and in doing so, provides guidance on their use. Regressions investigated are: (1) multiple linear regression, (2) a simultaneous autoregressive error model, and (3) geographically weighted regression. The first is nonspatial and acts as a control, the second accounts for stationary spatial autocorrelation via the error term, while the third captures spatial heterogeneity through the modeling of nonstationary relationships between the response and predictor variables. The geostatistical‐based simulation experiment generates data and coefficients with known multivariate spatial properties, all within an area‐unit spatial setting. Spatial autocorrelation and spatial heterogeneity effects are varied and accounted for. On fitting the regressions, that each have different assumptions and objectives, to very different geographical processes, valuable insights to their likely performance are uncovered. Results objectively confirm an inherent interrelationship between autocorrelation and heterogeneity, that results in an identification problem when choosing one regression over another. Given this, recommendations on the use and implementation of these spatial regressions are suggested, where knowledge of the properties of real study data and the analytical questions being posed are paramount.