A response to ‘A comment on geographically weighted regression with parameter-specific distance metrics’

In this article, we respond to ‘A comment on geographically weighted regression with parameter-specific distance metrics’ by Oshan et al. (2019), published in this journal, where several concerns on the parameter-specific distance metric geographically weighted regression (PSDM GWR) technique are raised. In doing so, we review the developmental timeline of the multiscale geographically weighed regression modelling framework with related and equivalent models, including flexible bandwidth GWR, conditional GWR and PSDM GWR. In our response, we have tried to answer all the concerns raised in terms of applicability, veracity, interpretability and computational efficiency of the PSDM GWR model.     

A comment on geographically weighted regression with parameter-specific distance metrics

A recent paper in this journal proposed a form of geographically weighted regression (GWR) that is termed parameter-specific distance metric geographically weighted regression (PSDM GWR). The central focus of the PSDM generalization of the GWR framework is that it allows the kernel function that weights nearby data to be specified with a distinct distance metric. As with the recent paper on Multiscale GWR (MGWR), the PSDM framework presents a form of GWR that also allows for parameter-specific bandwidths to be computed. As a result, a secondary focus of the PSDM GWR framework is to reduce the computational overhead associated with searching a massive parameter space to find a set of optimal parameter-specific bandwidths and parameter-specific distance metrics. In this comment, we discuss several concerns with the PSDM GWR framework in terms of model interpretability, complexity, and computational efficiency. We also recommend some best practices when using these models, suggest how to more holistically assess model variations, and set out an agenda to constructively focus future research endeavors.     

Geographically weighted regression with a non-Euclidean distance metric: a case study using hedonic house price data

Geographically weighted regression (GWR) is an important local technique for exploring spatial heterogeneity in data relationships. In fitting with Tobler’s first law of geography, each local regression of GWR is estimated with data whose influence decays with distance, distances that are commonly defined as straight line or Euclidean. However, the complexity of our real world ensures that the scope of possible distance metrics is far larger than the traditional Euclidean choice. Thus in this article, the GWR model is investigated by applying it with alternative, non-Euclidean distance (non-ED) metrics. Here we use as a case study, a London house price data set coupled with hedonic independent variables, where GWR models are calibrated with Euclidean distance (ED), road network distance and travel time metrics. The results indicate that GWR calibrated with a non-Euclidean metric can not only improve model fit, but also provide additional and useful insights into the nature of varying relationships within the house price data set.     

Multi-scale analysis of spatially varying relationships between agricultural landscape patterns and urbanization using geographically weighted regression

Scientific interpretation of the relationships between agricultural landscape patterns and urbanization is important for ecological planning and management. Ordinary least squares (OLS) regression is the primary statistical method in previous studies. However, this global regression lacks the ability to uncover some local-specific relationships and spatial autocorrelation in model residuals. This study employed geographically weighted regression (GWR) to examine the spatially varying relationships between several urbanization indicators (urbanization intensity index, distance to urban centers and distance to road) and changes in metrics describing agricultural landscape patterns (total area, patch density, perimeter area ratio distribution and aggregation index) at two block scales (5 km and 10 km). Results denoted that GWR was more powerful than OLS in interpreting relationships between agricultural landscape patterns and urbanization, since GWR was characterized by higher adjust R², lower Akaike Information Criterion values and reduced spatial autocorrelations in model residuals. Character and strength of the relationships identified by GWR varied spatially. In addition, GWR results were scale-dependent and scale effects were particularly significant in three aspects: kernel bandwidth of weight determination, block scale of pattern analysis, and window size of local variance analysis. Homogeneity and heterogeneity in the relationships between agricultural landscape patterns and urbanization were subject to the coupled influences of the three scale effects. We argue that the spatially varying relationships between agricultural landscape patterns and urbanization are not accidental but nearly universal. This study demonstrated that GWR has the potential to provide references for ecological planners and managers to address agricultural landscapes issues at all scales.     

The Minkowski approach for choosing the distance metric in geographically weighted regression

In this study, the geographically weighted regression (GWR) model is adapted to benefit from a broad range of distance metrics, where it is demonstrated that a well-chosen distance metric can improve model performance. How to choose or define such a distance metric is key, and in this respect, a ‘Minkowski approach’ is proposed that enables the selection of an optimum distance metric for a given GWR model. This approach is evaluated within a simulation experiment consisting of three scenarios. The results are twofold: (1) a well-chosen distance metric can significantly improve the predictive accuracy of a GWR model; and (2) the approach allows a good approximation of the underlying ‘optimal distance metric’, which is considered useful when the ‘true’ distance metric is unknown.     

基于地理加权回归的漫湾库区景观破碎化及影响因子分析

应用地理加权回归模型分析漫湾库区景观破碎化指数——有效筛网大小与相关因子之间的空间关系。选取的解释变量分别是距道路的距离、距乡村的距离、距河流的距离、坡度。结果表明：大坝修建后4种解释变量与有效筛网大小呈现较显著的正相关性。与线性回归模型相比,地理加权回归模型的拟合效果显著提高。1974~1988年,有效筛网大小对各影响因子最敏感的区域面积呈现显著的时空变化这为确定水电站建设及其他因素对景观破碎化影响的大小,并进一步改善库区景观破碎化的现状提供了依据。     

Local Spatiotemporal Modeling of House Prices: A Mixed Model Approach

The real estate market has long provided an active application area for spatial–temporal modeling and analysis and it is well known that house prices tend to be not only spatially but also temporally correlated. In the spatial dimension, nearby properties tend to have similar values because they share similar characteristics, but house prices tend to vary over space due to differences in these characteristics. In the temporal dimension, current house prices tend to be based on property values from previous years and in the spatial–temporal dimension, the properties on which current prices are based tend to be in close spatial proximity. To date, however, most research on house prices has adopted either a spatial perspective or a temporal one; relatively little effort has been devoted to situations where both spatial and temporal effects coexist. Using ten years of house price data in Fife, Scotland (2003–2012), this research applies a mixed model approach, semiparametric geographically weighted regression (GWR), to explore, model, and analyze the spatiotemporal variations in the relationships between house prices and associated determinants. The study demonstrates that the mixed modeling technique provides better results than standard approaches to predicting house prices by accounting for spatiotemporal relationships at both global and local scales.     

基于3种地理加权回归方法的安徽省土壤pH空间预测制图对比研究

基于安徽省140个采样点的土壤pH数据,综合考虑土壤、地形、气候、生物等因子对土壤pH的影响,采用地理加权回归(Geographically Weighted Regression, GWR)、主成分地理加权回归(Principal Component Geographically Weighted Regression, PCA-GWR)和混合地理加权回归(Mixed Geographically Weighted Regression, M-GWR)3种模型对安徽省土壤pH空间分布进行建模预测,揭示环境因子对土壤pH的影响在空间上的差异,最后以多元线性回归模型(Multiple Linear Regression, MLR)为基准比较3种GWR模型的精度。研究表明：(1)安徽省土壤pH具有空间异质性,且集聚特征明显。(2) 3种GWR模型中M-GWR模型略优,GWR、PCA-GWR和M-GWR的建模集调整后决定系数(R_adj²)分别为0.59、0.62和0.63;对比MLR模型,3种GWR模型的R_adj^2<...     

Geographically and temporally weighted regression for modeling spatio-temporal variation in house prices

By incorporating temporal effects into the geographically weighted regression (GWR) model, an extended GWR model, geographically and temporally weighted regression (GTWR), has been developed to deal with both spatial and temporal nonstationarity simultaneously in real estate market data. Unlike the standard GWR model, GTWR integrates both temporal and spatial information in the weighting matrices to capture spatial and temporal heterogeneity. The GTWR design embodies a local weighting scheme wherein GWR and temporally weighted regression (TWR) become special cases of GTWR. In order to test its improved performance, GTWR was compared with global ordinary least squares, TWR, and GWR in terms of goodness-of-fit and other statistical measures using a case study of residential housing sales in the city of Calgary, Canada, from 2002 to 2004. The results showed that there were substantial benefits in modeling both spatial and temporal nonstationarity simultaneously. In the test sample, the TWR, GWR, and GTWR models, respectively, reduced absolute errors by 3.5%, 31.5%, and 46.4% relative to a global ordinary least squares model. More impressively, the GTWR model demonstrated a better goodness-of-fit (0.9282) than the TWR model (0.7794) and the GWR model (0.8897). McNamara's test supported the hypothesis that the improvements made by GTWR over the TWR and GWR models are statistically significant for the sample data.     

土地利用变化的多尺度空间自相关分析——以内蒙古翁牛特旗为例

用传统统计学方法模拟和解释土地利用变化的前提条件是研究分析的数据在统计上必须独立且均匀分布。但是空间数据相互之间通常具有依赖性 (即空间自相关),某一变量的值随着测定距离的缩小而变得更相似或更为不同。由于经典线性回归方法未能抓住数据的空间自相关特征,而空间自相关包含一些有用的信息,为了克服这一缺点,利用Moran的I系数自相关图来描述研究区土地利用变化的空间自相关,并且建立了不仅考虑回归而且又考虑空间自相关的混合回归-空间自相关回归模型 (即空间滞后模型)。研究得到：① 研究区土地利用变化模型中不但自变量之间而且因变量之间存在空间正自相关,这表明土地利用变化数据的空间自相关很强;② Moran的I系数随着尺度的变粗而减小,这是由于数据平均时的滤波特性和Moran的I系数对距离的非线性特征造成的;③ 经典线性回归模型的残差也表现出正相关,这表明标准的多元线性回归模型未能考虑土地利用数据所存在的空间依赖性;④ 混合回归-空间自相关回归模型 (即空间滞后模型) 的残差未存在空间自相关,并且有更好的拟合度;⑤ 相对于经典线性回归模型,混合回归-空间自相关回归模型 (即空间滞后模型) 对于存在空间自相关性的数据来说有着统计上的合理性,而经典线性回归模型未能考虑这些因素。     

基于空间扩展模型和地理加权回归模型的城市住房价格空间分异比较

鉴于已有研究主要集中探讨住房价格的空间依赖性,较少涉及空间异质性对住房特征价格的影响,也很少尝试构建不同计量模型来比较模型间刻画住房价格影响因素空间分异的准确性,以长沙市中心城区为研究区,采用空间扩展模型和地理加权回归模型比较分析城市住房价格影响因素的空间分异,结果表明：① 空间扩展模型和地理加权回归模型都表明,长沙市中心城区的住房属性边际价格随着区位的变化而变化,揭示住房价格影响因素具有显著的空间异质性;小区环境、交通条件、教育配套、生活设施等因素对住房价格的影响强度存在明显的空间分异。② 地理加权回归模型和空间扩展模型都能对传统特征价格模型进行改进,但地理加权回归模型在解释能力和精度方面都超过空间扩展模型;对属性系数估计空间模式的分析,地理加权回归模型形成的结果比采用坐标多义扩展的空间扩展模型更为复杂和直观。     

中国省域犯罪率影响因素的空间非平稳性分析

收入差距和流动人口是影响犯罪率的两个重要因素, 以往研究基于OLS模型, 在假设地域空间为均质的前提下分析其对犯罪率的影响, 但现实世界的空间单元往往难以满足“均质”的假设, 多数表现为“空间异质”。以OLS计量空间异质会造成计量结果出现偏差, 同时无法了解不同空间单元的不同影响。而地理加权回归模型通过将空间结构嵌入线性回归模型中, 很好的解决了空间异质的计量问题。利用地理加权回归模型研究2008 年中国大陆省域单元犯罪率的影响因素, 结果表明:① 犯罪率的影响因素表现出空间非平稳性, 流动人口与犯罪率显著相关, 但各个省份相关程度并不相同, 影响关系随空间位置变化而变化;② 地理加权回归模型的计量精度和拟合度比OLS模型有大幅提高     

Spatial non-stationary characteristics between grass yield and its influencing factors in the Ningxia temperate grasslands based on a mixed geographically weighted regression model

Spatial models are effective in obtaining local details on grassland biomass, and their accuracy has important practical significance for the stable management of grasses and livestock. To this end, the present study utilized measured quadrat data of grass yield across different regions in the main growing season of temperate grasslands in Ningxia of China (August 2020), combined with hydrometeorology, elevation, net primary productivity (NPP), and other auxiliary data over the same period. Accordingly, non-stationary characteristics of the spatial scale, and the effects of influencing factors on grass yield were analyzed using a mixed geographically weighted regression (MGWR) model. The results showed that the model was suitable for correlation analysis. The spatial scale of ratio resident-area index (PRI) was the largest, followed by the digital elevation model, NPP, distance from gully, distance from river, average July rainfall, and daily temperature range; whereas the spatial scales of night light, distance from roads, and relative humidity (RH) were the most limited. All influencing factors maintained positive and negative effects on grass yield, save for the strictly negative effect of RH. The regression results revealed a multiscale differential spatial response regularity of different influencing factors on grass yield. Regression parameters revealed that the results of Ordinary least squares (OLS) (Adjusted R² = 0.642) and geographically weighted regression (GWR) (Adjusted R² = 0.797) models were worse than those of MGWR (Adjusted R² = 0.889) models. Based on the results of the RMSE and radius index, the simulation effect also was MGWR > GWR > OLS models. Ultimately, the MGWR model held the strongest prediction performance (R² = 0.8306). Spatially, the grass yield was high in the south and west, and low in the north and east of the study area. The results of this study provide a new technical support for rapid and accurate estimation of grassland yield to dynamically adjust grazing decision in the semi-arid loess hilly region.     

基于时空窗口改进的时空加权回归分析——以湖北省黄石市住房价格为例

针对地理加权回归(GWR)模型不能有效处理样本数据空间自相关性这一问题,构造局部时空窗口统计量,尝试改进时空加权回归(GTWR)模型。定义多时空窗口的概念,给出其选取、计算和验证方法;计算时空窗口包含的各样本点的被解释变量平均值,与样本拟合点的被解释变量值的比值,作为新的解释变量,构建改进的时空加权回归(IGTWR)模型。以土地稀缺、多中心、资源型城市——湖北省黄石市为例,收集2007~2012年商品住宅成交价格1.93万个数据和398个楼栋样本点,选取小区等级、绿化率、楼栋总层数、容积率、距区域中心距离和销售年份6个解释变量,分别利用常规线性回归(OLS)、GWR、GTWR和IGTWR方法进行回归分析。模型结果表明：计算Moran’s I指数和分析时间序列的自相关性,能确定时空窗口的大小和数量的选取;IGTWR模型和各变量的回归统计均通过0.05的显著性水平检验,有关解释变量的系数估计值在空间分布上能合理解释;GWR拟合结果优于OLS,GTWR优于GWR,而IGTWR拟合精度最好。与GTWR模型分析相比, IGTWR模型R²从0.877提升到0.919,而AICc、残差方(RSS)和均方差(MSE)分别从6 226、49 996 201和354.427下降到6 206、32 327 472和284.969。案例研究表明：IGTWR能够表达一定时空范围的时空自相关特征,减小了估计误差,提高了回归拟合精度。     

都市新区住宅地价空间异质性驱动因素研究 ——基于空间扩展模型和GWR模型的对比

以南京市江宁区为例,基于2004~2011年住宅用地出让数据,利用空间扩展模型和GWR模型对都市新区住宅地价空间异质性及其驱动因素进行研究。结果表明：① 空间扩展模型与GWR模型分别可解释采样区63%、61%的住宅地价变化,较全局回归模型(47%)有显著提升,更有利于研究土地市场的空间异质性。② 空间扩展模型可有效表征各解释变量及其交互项对住宅地价作用的空间结构总体趋势,其拟合效果相对较优。GWR模型则在局部参数估计方面存在优势,借助GIS可将各变量的地价作用模式可视化,从而比空间扩展模型更能有效刻画住宅地价影响因素的空间非平稳性特征,各因素对地价的平均边际贡献排序为水域> 地铁> 大学园区> CBD> 商业网点> 医院,且商业网点、 医院系数值具有方向差异性。③ 距地铁站点、水域、大学园区以及CBD的距离是研究区住宅地价的关键驱动因素,各自存在特有的地价空间作用模式,可为研究区住宅土地市场细分提供科学依据。     

Exploring Spatially Non-stationary Relationships in the Determinants of Mineralization in 3D Geological Space

Exploring the spatial relationships between various geological features and mineralization is not only conducive to understanding the genesis of ore deposits but can also help to guide mineral exploration by providing predictive mineral maps. However, most current methods assume spatially constant determinants of mineralization and therefore have limited applicability to detecting possible spatially non-stationary relationships between the geological features and the mineralization. In this paper, the spatial variation between the distribution of mineralization and its determining factors is described for a case study in the Dingjiashan Pb–Zn deposit, China. A local regression modeling technique, geological weighted regression (GWR), was leveraged to study the spatial non-stationarity in the 3D geological space. First, ordinary least-squares (OLS) regression was applied, the redundancy and significance of the controlling factors were tested, and the spatial dependency in Zn and Pb ore grade measurements was confirmed. Second, GWR models with different kernel functions in 3D space were applied, and their results were compared to the OLS model. The results show a superior performance of GWR compared with OLS and a significant spatial non-stationarity in the determinants of ore grade. Third, a non-stationarity test was performed. The stationarity index and the Monte Carlo stationarity test demonstrate the non-stationarity of all the variables throughout the area. Finally, the influences of the degree of non-stationary of all controlling factors on mineralization are discussed. The existence of significant non-stationarity of mineral ore determinants in 3D space opens up an exciting avenue for research into the prediction of underground ore bodies.

     

美国俄亥俄州土壤有机碳密度空间分布(英文)

Historical database of National Soil Survey Center containing 1424 geo-referenced soil profiles was used in this study for estimating the organic carbon(SOC) for the soils of Ohio,USA.Specific objective of the study was to estimate the spatial distribution of SOC density(C stock per unit area) to 1.0-m depth for soils of Ohio using geographically weighted regression(GWR),and compare the results with that obtained from multiple linear regression(MLR).About 80% of the analytical data were used for calibration and 20% for validation.A total of 20 variables including terrain attributes,climate data,bedrock geology,and land use data were used for mapping the SOC density.Results showed that the GWR provided better estimations with the lowest(3.81 kg m 2) root mean square error(RMSE) than MLR approach.Total estimated SOC pool for soils in Ohio ranged from 727 to 742 Tg.This study demonstrates that,the local spatial statistical technique,the GWR can perform better in capturing the spatial distribution of SOC across the study region as compared to other global spatial statistical techniques such as MLR.Thus,GWR enhances the accuracy for mapping SOC density.     

Residential Burglaries and Neighborhood Socioeconomic Context in London,Ontario: Global and Local Regression Analysis*

The main aim of this article is to analyze the relationships between the spatial patterns of residential burglaries and the socioeconomic characteristics of neighborhoods in London, Ontario. Relative risk ratios are applied as a measure of the intensity of residential burglary. The variation in the risks of burglary is modeled as a function of contextual neighborhood variables. Following a conventional (global) regression analysis, spatial variations in the relationships are examined using geographically weighted regression (GWR). The GWR results show that there are significant local variations in the relationships between the risk of residential burglary victimization and the average value of dwellings and percentage of the population in multifamily housing. The results are discussed in the context of four hypotheses, which may explain geographical variations in residential burglary. The practical implication of the GWR analysis is that different crime prevention policies should be implemented in different neighborhoods of the city.     

Using landscape characteristics to define an adjusted distance metric for improving kriging interpolations

Interpolation of point measurements using geostatistical techniques such as kriging can be used to estimate values at non-sampled locations in space. Traditional geostatistics are based on the spatial autocorrelation concept that nearby things are more related than distant things. In this study, additional information was used to modify the traditional Euclidean concept of distance into an adjusted distance metric that incorporates similarity in terms of quantifiable landscape characteristics such as topography or land use. This new approach was tested by interpolating soil moisture content, pH and carbon-to-nitrogen (C:N) ratio measured in both the mineral and the organic soil layers at a field site in central Sweden. Semivariograms were created using both the traditional distance metrics and the proposed adjusted distance metrics to carry out ordinary kriging (OK) interpolations between sampling points. In addition, kriging with external drift (KED) was used to interpolate soil properties to evaluate the ability of the adjusted distance metric to incorporate secondary data into interpolations. The new adjusted distance metric typically lowered the nugget associated with the semivariogram, thereby better representing small-scale variability in the measured data compared to semivariograms based on the traditional distance metric. The pattern of the resulting kriging interpolations using KED and OK based on the adjusted distance metric were similar because they represented secondary data and, thus, enhanced small-scale variability compared to traditional distance OK. This created interpolations that agreed better with what is expected for the real-world spatial variation of the measured properties. Based on cross-validation error, OK interpolations using the adjusted distance metric better fit observed data than either OK interpolations using traditional distance or KED.     

A bootstrap test for constant coefficients in geographically weighted regression models

Statistical tests for whether some coefficients really vary over space play an important role in using the geographically weighted regression (GWR) to explore spatial non-stationarity of the regression relationship. In view of some shortcomings of the existing inferential methods, we propose a residual-based bootstrap test to detect the constant coefficients in a GWR model. The proposed test is free of the assumption that the model error term is normally distributed and admits some useful extensions for identifying more complicated spatial patterns of the coefficients. Some simulation with comparison to the existing test methods is conducted to assess the test performance, including the accuracy of the bootstrap approximation to the null distribution of the test statistic, the power in identifying spatially varying coefficients and the robustness to collinearity among the explanatory variables. The simulation results demonstrate that the bootstrap test works quite well. Furthermore, a real-world data set is analyzed to illustrate the application of the proposed test.