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.     

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.     

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.     

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.     

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.     

The Saddlepoint Approximation of Moran's I's and Local Moran's Ii's Reference Distributions and Their Numerical Evaluation

Global Moran's I and local Moran's I_i are the most commonly used test statistics for spatial autocorrelation in univariate map patterns or in regression residuals. They belong to the general class of ratios of quadratic forms for whom a whole array of approximation techniques has been proposed in the statistical literature, such as the prominent saddlepoint approximation by Offer Lieberman (1994). The saddlepoint approximation outperforms other approximation methods with respect to its accuracy and computational costs. In addition, only the saddlepoint approximation is capable of handling, in analytical terms, reference distributions of Moran's I that are subject to significant underlying spatial processes. The accuracy and computational benefits of the saddlepoint approximation are demonstrated for a set of local Moran's Ii statistics under either the assumption of global spatial independence or subject to an underlying global spatial process. Local Moran's I_i is known to have an excessive kurtosis and thus void the use of the simple approximation methods of its reference distribution. The results demonstrate how well the saddlepoint approximation fits the reference distribution of local Moran's I_i. Furthermore, for local Moran's I_i under the assumption of global spatial independence several algebraic simplifications lead to substantial gains in numerical efficiency. This makes it possible to evaluate local Moran's I_i's significance in large spatial tessellations.     

Navigating the Methodological Landscape in Spatial Analysis: A Comment on “A Route Map for Successful Applications of Geographically-Weighted Regression”

The development of “route maps” for spatial analytical methods is a pursuit with important ramifications. Comber et al. propose a route map to guide applications of geographically weighted regression consisting of a three-step primary pathway and a series of secondary arterials. This comment first highlights some concerns about the underlying “map” (i.e., experimental setup and assumptions) and then with the proposed “route” (i.e., core decisions and evaluation criteria). It closes by suggesting a more general focus on identifying modeling issues with the highest impact and facilitating consensus-building, which could improve the future production of route maps for navigating the methodological landscape in spatial analysis.     

Local Forms of Spatial Analysis

Local forms of spatial analysis focus on exceptions to the general trends represented by more traditional global forms of spatial analysis. There is currently a rapid expansion in the development of such techniques but their history almost exactly parallels that of Geographical Analysis, with the first examples of local analysis appearing in the late 1960s. Indeed, Geographical Analysis has published many of the significant contributions in this field. This paper reviews the development of local forms of spatial analysis and assesses the current situation. Following a discussion on the nature and importance of local analysis, examples are given of local forms of point pattern analysis; local graphical approaches; local measures of spatial dependency; the spatial expansion method; adaptive filtering; multilevel modeling; geographically weighted regression; random coefficients models; autoregressive models; and local forms of spatial interaction models.     

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.     

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.     

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.     

Railways as a Factor of Change in the Distribution of Population in Spain, 1900–1970

Abstract

The central focus of this work is to test a new methodology to measure the impact of the railway on the distribution of population, in this case in Spain. To achieve this, it was necessary to previously integrate data relating to population and railway lines into a geographical information system. The result was a spatial database that includes population data from homogeneous census series obtained for the municipal scale and the evolution of the railway network in service at corresponding points in time. This allowed the authors to apply spatial-temporal analysis. By so doing, this work constitutes an analysis of a new methodology, as they used exploratory spatial data analysis and geographically weighted regression to detect spatial patterns and estimate the influence of the railway and distance from the coast on population change. The results obtained show that the influence of the railway was very pronounced in some areas, while in others it was just one of the factors that could explain major changes in population distribution.     

乡村振兴视角下秦巴山区农村路网通达性评价与分区优化研究——以陕西省山阳县为例

交通是经济发展的命脉，路网通达性是衡量交通发展水平的有效评价指标。以位于秦巴山区的陕西省山阳县为例，采用网络分析、空间自相关与地理加权回归等方法，从分项评价、综合评价两方面研究农村路网通达性，并进行通达性分区；提取各区中对经济发展影响较大的分项因子，有针对性地提出了优化策略。结果表明：①山阳县路网通达性呈北高南低、由高通达性极核向外围递减的“核心-外围”空间分异特征。②通达性水平高的区域地形平坦，区内外交通条件优良，是山阳县的经济、政治及文化中心，对周边村镇有一定辐射带动作用。通达性水平低的区域区位偏远，地形复杂，经济发展滞后。③路网通达性水平与社会经济的相关系数在不同分区有空间异质性。最后，提出分区优化建议，以促进经济发展，助力乡村振兴。     

南京市住宅价格时空分异格局及其影响因素分析——基于地理加权回归模型的实证研究

以2009-2017年南京市“一主三副”商品房社区为基本研究单元,运用GIS地统计分析中的普通Kriging插值法对“一主三副”住宅价格空间分布进行模拟和估计,并利用地理加权回归（GWR）模型探究社区属性、商业区位、交通区位、服务区位和景观区位等类型变量对住宅价格的影响规律。研究结果表明：①南京市房价总体上呈现主城向副城递减的中心外围模式,“一主三副”住宅价格空间结构呈现出同心圆和扇形融合的混合模型。②中心位势对主城住宅价格影响相对下降,对副城影响相对提升,交通位势表现出相反的趋势,住宅房龄、绿化环境对住宅价格的影响由主城向副城递减,山水景观的影响由长江沿岸向外围递减。③主副城住宅价格影响因素具有空间异质性,其中主城受距CBD距离、住宅建筑年代和绿化率的影响较大,而副城主要受距地铁站距离、距景观资源距离的影响。     

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.     

A Local Model for Multivariate Analysis: Extending Wartenberg’s Multivariate Spatial Correlation

Local analysis can provide specific information about individual observations that is often useful in understanding nonstationary interactions among variables. This paper extends the application of Wartenberg’s Multivariate Spatial Correlation (MSC) method to a local setting. The original MSC can be considered as an adaptation of Principal Component Analysis for spatial effects with respect to spatial autocorrelation. The extended MSC method described in this paper, however, further incorporates another spatial effect, spatial heterogeneity, by the addition of geographic weights in standardizing the data and in calculating the spatial association weight matrix. The extension allows more local analysis and facilitates additional visualization of the results. The geographically weighted MSC is illustrated and justified using the classic dataset collected by André-Michel Guerry on moral statistics in 1830s France.     

STATISTICAL ASPECTS OF POTTERY QUANTIFICATION FOR THE DATING OF SOME ARCHAEOLOGICAL CONTEXTS*

This paper describes some statistical analyses of a particular archaeological material (pottery) originating at some sites in the city of Tours. An important part of the archaeological study of pottery is the comparison of ceramic assemblages to establish the absolute dates of contexts. In this paper, a statistical model is built to assess this comparison. The statistical procedure uses classical tools (correspondence analysis, linear regression and resampling methods) in an iterative scheme. Archaeologists may find in the paper a useful set of known statistical methods, while statisticians can learn a method of ‘arranging’ well‐known techniques. No method is new, but their combination is characteristic of this application.