A two-stage estimation method with bootstrap inference for semi-parametric geographically weighted generalized linear models

Semi-parametric geographically weighted generalized linear models (S-GWGLMs) are a useful tool in modeling a regression relationship where the impact of certain explanatory variables on a non-Gaussian distributed response variable is global while that of others is spatially varying. In this article, we propose for S-GWGLMs a new estimation method, called two-stage geographically weighted maximum likelihood estimation, and further develop a likelihood ratio statistic-based bootstrap test to determine constant coefficients in the models. The performance of the estimation and test methods is then evaluated by simulations. The results show that the proposed estimation method performs as well as the existing method in estimating both constant and spatially varying coefficients but it is more efficient in terms of computation time; the bootstrap test is of accurate size under the null hypothesis and satisfactory power in identifying spatially varying coefficients. A real-world data set is finally analyzed to demonstrate the application of the proposed estimation and test methods.     

Local least absolute deviation estimation of spatially varying coefficient models: robust geographically weighted regression approaches

The geographically weighted regression (GWR) has been widely applied to many practical fields for exploring spatial non-stationarity of a regression relationship. However, this method is inherently not robust to outliers due to the least squares criterion in the process of estimation. Outliers commonly exist in data sets and may lead to a distorted estimate of the underlying regression relationship. Using the least absolute deviation criterion, we propose two robust scenarios of the GWR approaches to handle outliers. One is based on the basic GWR and the other is based on the local linear GWR (LGWR). The proposed methods can automatically reduce the impact of outliers on the estimates of the regression coefficients and can be easily implemented with modern computer software for dealing with the linear programming problems. We then conduct simulations to assess the performance of the proposed methods and the results demonstrate that the methods are quite robust to outliers and can retrieve the underlying coefficient surfaces satisfactorily even though the data are seriously contaminated or contain severe outliers.     

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.     

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     

Predictive mapping of soil total nitrogen at a regional scale: A comparison between geographically weighted regression and cokriging

Accurately mapping the spatial distribution of soil total nitrogen is important to precision agriculture and environmental management. Geostatistical methods have been frequently used for predictive mapping of soil properties. Recently, a local regression method, geographically weighted regression (GWR), got the attention of environmentalists as an alternative in spatial modeling of environmental attributes, due to its capability of incorporating various auxiliary variables with spatially varied correlation coefficients. The objective of this study is to compare GWR and ordinary cokriging (OCK) in predictive mapping of soil total nitrogen (TN) using multiple environmental variables. 353 soil Samples within the surface horizon of 0–20 cm in a study area were collected, and their TN contents were measured for calibrating and validating the GWR and OCK interpolations. The environmental variables finally chosen as auxiliary data include elevation, land use types, and soil types. Results indicate that, although OCK is slightly better than GWR in global accuracy of soil TN prediction (the adjusted R² for GWR and OCK are 0.5746 and 0.6858, respectively), the soil TN map interpolated by GWR shows many details reflecting the spatial variations of major auxiliary variables while OCK smoothes out almost all local details. Geographically weighted regression could account for both the spatial trend and local variations, whilst OCK had difficulties to capture local variations. It is concluded that GWR is a more promising spatial interpolation method compared to OCK in predicting soil TN and potentially other soil properties, if a suitable set of auxiliary variables are available and selected.     

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。     

基于GWR模型的中国区域旅行社业效率空间分异及动力机制分析

基于DEA模型对中国31个省域的旅行社业效率空间分异特征进行了分析,首次运用GWR模型探索交通、资本、人才、信息化和经济动力对区域旅行社业效率影响的空间差异。结果表明：旅行社业效率在空间上具有正相关性和集聚特征,空间格局从“川”字型向“山”字型转变。各动力因子的系数均存在空间非平稳性。资本和人才动力的回归系数在空间分布上从南向北依次递减;经济动力的分布趋势为从北向南依次递减;交通动力对中西部地区旅行社效率提升的促进作用显著于东部地区;信息化动力则在东部地区表现出较强的促进作用。     

Geographically weighted regression with parameter-specific distance metrics

Geographically weighted regression (GWR) is an important local technique to model spatially varying relationships. A single distance metric (Euclidean or non-Euclidean) is generally used to calibrate a standard GWR model. However, variations in spatial relationships within a GWR model might also vary in intensity with respect to location and direction. This assertion has led to extensions of the standard GWR model to mixed (or semiparametric) GWR and to flexible bandwidth GWR models. In this article, we present a strongly related extension in fitting a GWR model with parameter-specific distance metrics (PSDM GWR). As with mixed and flexible bandwidth GWR models, a back-fitting algorithm is used for the calibration of the PSDM GWR model. The value of this new GWR model is demonstrated using a London house price data set as a case study. The results indicate that the PSDM GWR model can clearly improve the model calibration in terms of both goodness of fit and prediction accuracy, in contrast to the model fits when only one metric is singly used. Moreover, the PSDM GWR model provides added value in understanding how a regression model’s relationships may vary at different spatial scales, according to the bandwidths and distance metrics selected. PSDM GWR deals with spatial heterogeneities in data relationships in a general way, although questions remain on its model diagnostics, distance metric specification, and computational efficiency, providing options for further research.     

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。     

中国亚热带丘陵山区植被沿海拔梯度分布格局(英文)

Knowledge of vegetation distribution patterns is very important.Their relationships with topography and climate were explored through a geographically weighted regression(GWR) framework in a subtropical mountainous and hilly region,Minjiang River Basin of Fujian in China.The HJ-1 satellite image acquired on December 9,2010 was utilized and NDVI index was calculated representing the range of vegetation greenness.Proper analysis units were achieved through segregation based on small sub-basins and altitudinal bands.Results indicated that the GWR model was more powerful than ordinary linear least square(OLS) regression in interpreting vegetation-environmental relationship,indicated by higher adjusted R 2 and lower Akaike information criterion values.On one side,the OLS analysis revealed dominant positive influence from parameters of elevation and slope on vegetation distribution.On the other side,GWR analysis indicated that spatially,the parameters of topography had a very complex relationship with the vegetation distribution,as results of the various combinations of environmental factors,vegetation composition and also anthropogenic impact.The influences of elevation and slope generally decreased,from strongly positive to nearly zero,with increasing altitude and slope.Specially,most rapid changes of coefficients between NDVI and elevation or slope were observed in relatively flat and low-lying areas.This paper confirmed that the non-stationary analysis through the framework of GWR could lead to a better understanding of vegetation distribution in subtropical mountainous and hilly region.It was hoped that the proposed scale selection method combined with GWR framework would provide some guidelines on dealing with both spatial(horizontal) and altitudinal(vertical) non-stationarity in the dataset,and it could easily be applied in characterizing vegetation distribution patterns in other mountainous and hilly river basins and related research.     

Geographically neural network weighted regression for the accurate estimation of spatial non-stationarity

ABSTRACT

Geographically weighted regression (GWR) is a classic and widely used approach to model spatial non-stationarity. However, the approach makes no precise expressions of its weighting kernels and is insufficient to estimate complex geographical processes. To resolve these problems, we proposed a geographically neural network weighted regression (GNNWR) model that combines ordinary least squares (OLS) and neural networks to estimate spatial non-stationarity based on a concept similar to GWR. Specifically, we designed a spatially weighted neural network (SWNN) to represent the nonstationary weight matrix in GNNWR and developed two case studies to examine the effectiveness of GNNWR. The first case used simulated datasets, and the second case, environmental observations from the coastal areas of Zhejiang. The results showed that GNNWR achieved better fitting accuracy and more adequate prediction than OLS and GWR. In addition, GNNWR is applicable to addressing spatial non-stationarity in various domains with complex geographical processes.     

省域经济增长与电力消费的局域空间计量经济分析

中国各个地区经济发展对电力消费需求量大且存在地域差异,不同区域间的电力需求与经济增长之间的关系十分复杂,并非能由常系数的普通最小二乘回归分析所解释.采用电力消费模型,利用局域空间计量经济学模型方法--空间变系数的地理加权回归模型,对中国省域电力消费与经济增长之间的关系进行了局域空间计量经济分析.结果发现,中国大陆30个省域的电力消费和经济增长之间表现为一种非均衡的联动关系和局域性特征,制定差异化的区域电力消费调控政策是非常必要的.     

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.

     

空间依赖、空间异质与京津冀都市地区经济收敛

基于探索性空间数据分析技术(ESDA)划分出京津冀都市地区的中心区域和外围区域,并在传统经济收敛模型基础上,运用空间俱乐部收敛模型和局部空间回归模型对京津冀都市地区经济收敛情况进行研究。结果表明,首先,京津冀都市地区已形成了以北京、天津和唐山为核心的中心区域和以张家口市、保定市为核心的环绕京津的外围区域,京津冀都市地区整体上存在微弱的经济收敛。其次,京津冀都市地区中心地区由于经济发展水平较高,空间外溢效应较大,加之中心地区接受知识、技术扩散的能力较强,存在经济收敛,并且收敛速度较快,而外围区域则不存在经济收敛。再次,中心地区和外围地区内部存在经济收敛系数结构的不稳定性。     

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.     

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<...     

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.     

东北地区旅游经济影响因素时空特征研究

从中观层面以东北地区41个市域为分析单元,选择东北振兴战略实施以来2004、2009和2015年截面数据,采用ESDA法分析旅游经济发展的空间关联特征,运用OLS和GWR模型分析旅游经济和旅游产业因子、消费因子、投资因子之间的关系,以此挖掘旅游经济影响因素的时空结构信息。结果表明：① 旅游经济发展呈显著空间正相关,相关性逐渐增大;② OLS回归结果表明,旅游产业因子对旅游经济发展的影响强度最大,在旅游产业发展中始终起基础性决定作用,其次为消费因子和投资因子,后两者差别不大;③ GWR回归结果显示,模型3 a拟合优度均比OLS有所提高,回归系数均为正值,但分布规律不同;旅游产业因子回归系数高值区经历了西南部-中南部-东北部转移,向外围圈层递减;消费因子回归系数高值区由东北部向东南部转移,向外围逐渐递减;投资因子回归系数高值区则由东北部-中南部-东北部转移,向外围逐渐递减。