Comparing spatially varying coefficient models: a case study examining violent crime rates and their relationships to alcohol outlets and illegal drug arrests

In this paper, we compare and contrast a Bayesian spatially varying coefficient process (SVCP) model with a geographically weighted regression (GWR) model for the estimation of the potentially spatially varying regression effects of alcohol outlets and illegal drug activity on violent crime in Houston, Texas. In addition, we focus on the inherent coefficient shrinkage properties of the Bayesian SVCP model as a way to address increased coefficient variance that follows from collinearity in GWR models. We outline the advantages of the Bayesian model in terms of reducing inflated coefficient variance, enhanced model flexibility, and more formal measuring of model uncertainty for prediction. We find spatially varying effects for alcohol outlets and drug violations, but the amount of variation depends on the type of model used. For the Bayesian model, this variation is controllable through the amount of prior influence placed on the variance of the coefficients. For example, the spatial pattern of coefficients is similar for the GWR and Bayesian models when a relatively large prior variance is used in the Bayesian model.      

Multicollinearity and correlation among local regression coefficients in geographically weighted regression

Present methodological research on geographically weighted regression (GWR) focuses primarily on extensions of the basic GWR model, while ignoring well-established diagnostics tests commonly used in standard global regression analysis. This paper investigates multicollinearity issues surrounding the local GWR coefficients at a single location and the overall correlation between GWR coefficients associated with two different exogenous variables. Results indicate that the local regression coefficients are potentially collinear even if the underlying exogenous variables in the data generating process are uncorrelated. Based on these findings, applied GWR research should practice caution in substantively interpreting the spatial patterns of local GWR coefficients. An empirical disease-mapping example is used to motivate the GWR multicollinearity problem. Controlled experiments are performed to systematically explore coefficient dependency issues in GWR. These experiments specify global models that use eigenvectors from a spatial link matrix as exogenous variables.This study was supported by grant number 1 R1 CA95982-01, Geographic-Based Research in Cancer Control and Epidermiology, from the National Cancer Institute. The author thank the anonymous reviewers and the editor for their helpful comments.     

An assessment of coefficient accuracy in linear regression models with spatially varying coefficients

The realization in the statistical and geographical sciences that a relationship between an explanatory variable and a response variable in a linear regression model is not always constant across a study area has led to the development of regression models that allow for spatially varying coefficients. Two competing models of this type are geographically weighted regression (GWR) and Bayesian regression models with spatially varying coefficient processes (SVCP). In the application of these spatially varying coefficient models, marginal inference on the regression coefficient spatial processes is typically of primary interest. In light of this fact, there is a need to assess the validity of such marginal inferences, since these inferences may be misleading in the presence of explanatory variable collinearity. In this paper, we present the results of a simulation study designed to evaluate the sensitivity of the spatially varying coefficients in the competing models to various levels of collinearity. The simulation study results show that the Bayesian regression model produces more accurate inferences on the regression coefficients than does GWR. In addition, the Bayesian regression model is overall fairly robust in terms of marginal coefficient inference to moderate levels of collinearity, and degrades less substantially than GWR with strong collinearity.     

Exploring the Impact of Non-normality on Spatial Non-stationarity in Geographically Weighted Regression Analyses: Tobacco Outlet Density in New Jersey

The principal rationale for applying geographically weighted regression (GWR) techniques is to investigate the potential spatial non-stationarity of the relationship between the dependent and independent variables—i.e., that the same stimulus would provoke different responses in different locations. The calibration of GWR employs a geographically weighted local least squares regression approach. To obtain meaningful inference, it assumes that the regression residual follows a normal or asymptotically normal distribution. In many classical econometric analyses, the assumption of normality is often readily relaxed, although it has been observed that such relaxation might lead to unreliable inference of the estimated coefficients' statistical significance. No studies, however, have examined the behavior of residual non-normality and its consequences for the modeled relationships in GWR. This study attempts to address this issue for the first time by examining a set of tobacco-outlet-density and demographic variables (i.e., percent African American residents, percent Hispanic residents, and median household income) at the census tract level in New Jersey in a GWR analysis. The regression residual using the raw data is apparently non-normal. When GWR is estimated using the raw data, we find that there is no significant spatial variation of the coefficients between tobacco outlet density and percentage of African American and Hispanics. After transforming the dependent variable and making the residual asymptotically normal, all coefficients exhibit significant variation across space. This finding suggests that relaxation of the normality assumption could potentially conceal the spatial non-stationarity of the modeled relationships in GWR. The empirical evidence of the current study implies that researchers should verify the normality assumption prior to applying GWR techniques in analyses of spatial non-stationarity.     

The pyrogeography of sub-Saharan Africa: a study of the spatial non-stationarity of fire–environment relationships using GWR

This study analyses the relationship between fire incidence and some environmental factors, exploring the spatial non-stationarity of the phenomenon in sub-Saharan Africa. Geographically weighted regression (GWR) was used to study the above relationship. Environment covariates comprise land cover, anthropogenic and climatic variables. GWR was compared to ordinary least squares, and the hypothesis that GWR represents no improvement over the global model was tested. Local regression coefficients were mapped, interpreted and related with fire incidence. GWR revealed local patterns in parameter estimates and also reduced the spatial autocorrelation of model residuals. All the covariates were non-stationary and in terms of goodness of fit, the model replicates the data very well (R ² = 87%). Vegetation has the most significant relationship with fire incidence, with climate variables being more important than anthropogenic variables in explaining variability of the response. Some coefficient estimates exhibit locally different signs, which would have gone undetected by a global approach. This study provides an improved understanding of spatial fire–environment relationships and shows that GWR is a valuable complement to global spatial analysis methods. When studying fire regimes, effects of spatial non-stationarity need to be incorporated in vegetation-fire modules to have better estimates of burned areas and to improve continental estimates of biomass burning and atmospheric emissions derived from vegetation fires.     

Alleviating the effect of collinearity in geographically weighted regression

Geographically weighted regression (GWR) is a popular technique to deal with spatially varying relationships between a response variable and predictors. Problems, however, have been pointed out (see Wheeler and Tiefelsdorf in J Geogr Syst 7(2):161–187, 2005), which appear to be related to locally poor designs, with severe impact on the estimation of coefficients. Different remedies have been proposed. We propose two regularization methods. The first one is generalized ridge regression, which can also be seen as an empirical Bayes method. We show that it can be implemented using ordinary GWR software with an appropriate choice of the weights. The second one augments the local sample as needed while running GWR. We illustrate both methods along with ordinary GWR on an example of housing prices in the city of Bilbao (Spain) and using simulations.     

混合地理加权回归模型算法研究

以迭代算法为基础，推导出混合地理加权回归模型的常系数（全局参数）和变系数（局域参数）的计算方法，并以上海市住宅小区楼盘销售平均价格为例进行验证。结果表明，混合地理加权回归模型的计算量略大于地理加权回归模型，但对样本数据的拟合更好，局域参数估计更稳健。     

SPOT地面场定标与星上定标结果的比较分析

本文研究是在遥感辐射定标场选择的基础。利用６Ｓ大气辐射传输模型进行ＳＰＯＴ遥感数据的定标和地物的光谱反射率反演,即在遥感器飞越辐射定标场上空,在定标场选择若干像元区,测量遥感器对应的各波段地物的光谱反射率和大气光谱参量,并利用大气辐射传输模型给出遥感器人瞳处各光谱带的辐射亮度,最后确定它与遥感器对应输出的数字量化的数量关系,求解定标系数。然后,对相应的研究训练区的遥感数据进行大气辐射校正,进而反演训练区内的地物光谱反射率。最后,通过将反演值与实地测量的地物光谱反射率进行对比分析,来估算定标不确定度,并比较说明两种不同方式定标差异及优势和限制。     

《风云二号》静止气象卫星

《风云二号》静止气象卫星１９９７年６月１０日２０点０１分,从我国西昌卫星发射中心,由长征三号运载火箭成功地将我国自己研制的《风云二号》静止气象卫星发射入轨（封面）。６月１７日,《风云二号》卫星成功地定点在东经１０５°赤道上空。６月２１日１４时０１分,...     

顾及空间非平稳特征的遥感干旱监测

遥感技术具备实时快速、时空连续、广覆盖尺度等独特优势,在全球气候恶化大背景下,利用遥感干旱监测方法相比于传统地面监测手段,能够提供实时、准确、稳定的旱情信息,辅助科学决策。目前常用遥感旱情监测方法大多依赖全域性数学模型建模,假定了旱情模式的空间平稳特性,因而难以准确反映旱情模式的局部差异特征。本文提出利用地理加权回归模型GWR (Geographically Weighted Regression),考虑旱情模式的空间非平稳特性,综合多种遥感地面旱情监测指数,以实现传统全域旱情监测模型的局部优化。以美国大陆为研究区,监测2002年—2011年共10年的旱情状态。研究表明,GWR模型能够提供空间变化的局部最佳估计模型参数,监测结果更加吻合标准美国旱情监测USDM (U.S Drought Monitor)验证数据,且与地面实测值的最高相关系数R达到0.8552,均方根误差RMSE达到0.972,显著优于其他遥感旱情监测模型。GWR模型具备空间非平稳探测优势,实现了旱情模式的局部精细探测,能够显著提升遥感旱情监测精度,具备较好的应用前景。     

大尺度土地覆盖数据集在中国及周边区域的精度评价

大尺度土地覆盖数据是全球陆地表层过程研究、生态系统评估、环境建模等科学研究的重要基础,研究现有数据集的特点对数据使用者及生产新的数据集都具有指导意义。本研究以中国及周边区域为研究区,根据不同分类体系对地物的定义,研究不同分类体系中对应地物的相关系数,并将所有分类体系转换为IGBP分类体系;然后,从定性和定量两方面分析现有5种土地覆盖数据集(IGBP DISCover、UMD、GLC2000、MOD12Q1和GlobCover 2005)的空间一致性;并利用Google Earth高分影像选取两期验证样本评价5种土地覆盖数据集的精度。结果表明:同种地物在不同土地覆盖数据集之间的空间分布格局差异较大,且不同土地覆盖数据集之间的总体一致性系数较低;5种土地覆盖数据集中,GLC2000的总体精度和Kappa系数均最高,GlobCover 2005的总体精度和Kappa系数均最低。     

Drivers of land cover and land use changes in St. Louis metropolitan area over the past 40 years characterized by remote sensing and census population data

In this study, we explored the spatial and temporal patterns of land cover and land use (LCLU) and population change dynamics in the St. Louis Metropolitan Statistical Area. The goal of this paper was to quantify the drivers of LCLU using long-term Landsat data from 1972 to 2010. First, we produced LCLU maps by using Landsat images from 1972, 1982, 1990, 2000, and 2010. Next, tract level population data of 1970, 1980, 1990, 2000, and 2010 were converted to 1-km square grid cells. Then, the LCLU maps were integrated with basic grid cell data to represent the proportion of each land cover category within a grid cell area. Finally, the proportional land cover maps and population census data were combined to investigate the relationship between land cover and population change based on grid cells using Pearson's correlation coefficient, ordinary least square (OLS), and local level geographically weighted regression (GWR). Land cover changes in terms of the percentage of area affected and rates of change were compared with population census data with a focus on the analysis of the spatial-temporal dynamics of urban growth patterns. The correlation coefficients of land cover categories and population changes were calculated for two decadal intervals between 1970 and 2010. Our results showed a causal relationship between LCLU changes and population dynamics over the last 40 years. Urban sprawl was positively correlated with population change. However, the relationship was not linear over space and time. Spatial heterogeneity and variations in the relationship demonstrate that urban sprawl was positively correlated with population changes in suburban area and negatively correlated in urban core and inner suburban area of the St. Louis Metropolitan Statistical Area. These results suggest that the imagery reflects processes of urban growth, inner-city decline, population migration, and social spatial inequality. The implications provide guidance for sustainable urban planning and development. We also demonstrate that grid cells allow robust synthesis of remote sensing and socioeconomic data to advance our knowledge of urban growth dynamics from both spatial and temporal scales and its association with population change.     

Dynamic land use change simulation using cellular automata with spatially nonstationary transition rules

The dynamic relationships between land use change and its driving forces vary spatially and can be identified by geographically weighted regression (GWR). We present a novel cellular automata (GWR-CA) model that incorporates GWR-derived spatially varying relationships to simulate land use change. Our GWR-CA model is characterized by spatially nonstationary transition rules that fully address local interactions in land use change. More importantly, each driving factor in our GWR model contains effects that both promote and resist land use change. We applied GWR-CA to simulate rapid land use change in Suzhou City on the Yangtze River Delta from 2000 to 2015. The GWR coefficients were visualized to highlight their spatial patterns and local variation, which are closely associated with their effects on land use change. The transition rules indicate low land conversion potential in the city’s center and outer suburbs, but higher land conversion potential in the inner near suburbs along the belt expressway. Residual statistics show that GWR fits the input data better than logistic regression (LR). Compared with an LR-based CA model, GWR-CA improves overall accuracy by 4.1% and captures 5.5% more urban growth, suggesting that GWR-CA may be superior in modeling land use change. Our results demonstrate that the GWR-CA model is effective in capturing spatially varying land transition rules to produce more realistic results, and is suitable for simulating land use change and urban expansion in rapidly urbanizing regions.     

GIS-based analysis of obesity and the built environment in the US

This paper examines the statewide relationship between built environment and obesity at the county scale by using the Geographically Weighted Regression (GWR) method. The independent variables include three built environment factors – street connectivity, walk score, fast-food/full-service restaurants ratio – and two sociodemographic variables, race heterogeneity and poverty rate. The urban influence is considered as a covariate in the analysis. Through the regression model we found that walk score and street connectivity are negatively related to obesity, poverty rate and metro are positively related, and the fast-food/full-service restaurants ratio is not significant. A regionalization method is used to group US counties to regions based on their GWR coefficients. Qualitative inferences of policies are made available to facilitate better understanding of the obesity problem associated with the built environment in these regions.     

时序变差函数特征驱动下的月平均降水空间分布模拟

高精度降水场是水文、气象以及环境分析的重要数据支撑,直接影响相关服务的准确性。传统降水分布模拟大多依赖站点空间维的驱动因素,而忽略了降水时序变化特征对其空间分布的影响。使用2015—2017年中国湖北省83个国家气象观测站点和28个省级观测站点近3 a月平均累积降水资料,通过相关性分析,引入站点降水时序理论变差函数模型的拱高值(C)和块金值(C₀)作为影响因素,使用地理加权回归(geographically weighted regression, GWR)建立湖北省月平均降水分布模型。结果表明:（1）各站点降水的时序变差函数曲线与降水的季节性基本吻合。站点时序理论变差函数模型中,有25.3%能够在4个月内达到平稳,36.14%在6个月内达到平稳。（2）站点降水时序理论变差函数模型的C和C₀与逐年12月平均累积降水在0.01水平（双侧）上显著相关,平均相关系数分别为0.745和0.526,大于地理位置和高程对降水的影响。（3）引入C和C₀ 有助于提升GWR模型的整体拟合优度和插值精度。对比仅使用经纬度的GWR模型和引入时序理论变差函数特征的GWR模型,3 a平均整体拟合优度从0.852提升至0.912。验证集站点插值精度评价显示,3 a绝对误差、均方根误差和平均绝对百分误差下降幅度均大于60%。因此,引入时序理论变差函数特征的时空GWR模型能够获得较高精度的降水模拟结果,更适合具有丰富历史降水资料地区的降水空间分布估算。     

Geographically weighted regression‐based determinants of malaria incidences in northern China

Geographically weighted regression (GWR) is an important local method to explore spatial non‐stationarity in data relationships. It has been repeatedly used to examine spatially varying relationships between epidemic diseases and predictors. Malaria, a serious parasitic disease around the world, shows spatial clustering in areas at risk. In this article, we used GWR to explore the local determinants of malaria incidences over a 7‐year period in northern China, a typical mid‐latitude, high‐risk malaria area. Normalized difference vegetation index (NDVI), land surface temperature (LST), temperature difference, elevation, water density index (WDI) and gross domestic product (GDP) were selected as predictors. Results showed that both positively and negatively local effects on malaria incidences appeared for all predictors except for WDI and GDP. The GWR model calibrations successfully depicted spatial variations in the effect sizes and levels of parameters, and also showed substantially improvements in terms of goodness of fits in contrast to the corresponding non‐spatial ordinary least squares (OLS) model fits. For example, the diagnostic information of the OLS fit for the 7‐year average case is R² = 0.243 and AICc = 837.99, while significant improvement has been made by the GWR calibration with R² = 0.800 and AICc = 618.54.     

Full Positional Accuracy Analysis of Spatial Data by Means of Circular Statistics

The positional error in spatial data is defined as a vector by comparing the coordinates between the true position and the measured position. The standard tests to assess the positional accuracy use only the magnitude of the vector and omit the azimuth. This article suggests that the use of both values allows a much more complete analysis of the positional error. A set of tests is proposed that are relevant for this purpose and demonstrate that some important features are not identified by the common procedures. The test samples come from two datasets. The first is obtained from the comparison of 100 homologous points in two conventional maps, and the second one comes from the geometric calibration of a photogrammetric scanner. The results are analyzed and discussed, showing that important issues such as error anisotropy are detected only by means of the circular statistics tests and density maps of distribution. Therefore, tests that assess the goodness of fit for uniform distribution in azimuths, such as Rayleigh and Rao tests, give low probabilities (P = 0 and P > 0.01). Moreover, density maps working with both magnitude and angle can locate the outlier candidate and offer more information about the spatial distribution of error.     

Increasing the accuracy of nitrogen dioxide (NO2) pollution mapping using geographically weighted regression (GWR) and geostatistics

Nitrogen dioxide (NO₂) is known to act as an environmental trigger for many respiratory illnesses. As a pollutant it is difficult to map accurately, as concentrations can vary greatly over small distances. In this study three geostatistical techniques were compared, producing maps of NO₂ concentrations in the United Kingdom (UK). The primary data source for each technique was NO₂ point data, generated from background automatic monitoring and background diffusion tubes, which are analysed by different laboratories on behalf of local councils and authorities in the UK. The techniques used were simple kriging (SK), ordinary kriging (OK) and simple kriging with a locally varying mean (SKlm). SK and OK make use of the primary variable only. SKlm differs in that it utilises additional data to inform prediction, and hence potentially reduces uncertainty. The secondary data source was oxides of nitrogen (NO_x) derived from dispersion modelling outputs, at 1 km × 1 km resolution for the UK. These data were used to define the locally varying mean in SKlm, using two regression approaches: (i) global regression (GR) and (ii) geographically weighted regression (GWR). Based upon summary statistics and cross-validation prediction errors, SKlm using GWR derived local means produced the most accurate predictions. Therefore, using GWR to inform SKlm was beneficial in this study.     

利用Minkowski距离逼近道路网络距离算法研究

道路网络背景下的距离度量（如道路网络距离、旅行时间）是在空间分析或空间统计过程中常用的距离度量,但在科研过程中由于道路数据的可获得性和精度等方面的限制,该类距离的计算可能较为困难。Minkowski距离函数是欧氏空间中的广义距离函数,其参数p值的不同代表着对空间不同的度量。利用Minkowski的通用性和灵活性（参数p不同的取值）,研究如何更好地逼近道路网络距离。同时,探索不同道路网络的部分计量特征（如密度、弯曲度等）与最优p值之间的关系。实验证明,相对于最常用的欧氏距离度量,优选p值后的Minkowski距离函数能够更大程度上逼近道路距离。而通过对道路网络计量特征与最优p值之间的关系的分析,指出了弯曲度与最优p值之间的对应关系,它对于p值的选择具有重要的指导意义。此外,为了验证Minkowski距离逼近算法的可行性,以地理加权回归分析为例,通过对比传统的欧氏距离度量、最优Minkowski距离度量和道路网络距离（旅行时间）对模型解算结果的影响,指出优选后Minkowski距离一定程度上更接近于采用旅行时间对模型解算的结果。     

Choosing pasture maps: An assessment of pasture land classification definitions and a case study of Brazil

Pasture land occupies extensive areas and is increasingly of interest for sustainable intensification, land use diversification, greenhouse gas emission mitigation, and bioenergy expansion. Accurate maps of pasture and other managed land covers are needed for monitoring, intercomparison, assessing potential uses, and planning. Yet, land maps can be generated from different types of classification datasets – i.e. as a land use or land cover type – as well as different sources. In this study our aim was to assess and compare land use and land cover definitions for pasture, and examine variability in the resulting pasture land classification maps. First, we conducted a review of pasture definitions in commonly used mapping databases. We then performed a case study involving Brazil, a dominant global producer of pasture-based livestock. Six geospatial databases were harmonized and compared to each other and to MODIS land cover for Brazil including the Cerrado and Amazon biomes, which are internationally recognized for their ecological value. Total pasture area estimates for Brazil ranged by a factor greater than four, from about 430,000 km² to over 1.7 million km². Our analysis showed high variability in pasture land maps depending on the definitions, methods and underlying datasets used to generate them. The results are illustrative of a symptomatic problem for all manage land datasets, demonstrating the need for land categories studies and geospatial data resources that fully define land terms and describe measurable management attributes. Additionally, the suitability of individual geospatial datasets for different types of land mapping must be better described and reported. These recommendations would help bring more consistency in the consideration of managed lands in research, reporting, and policy development, as demonstrated here for pasture land using six case study datasets from multiple sources.