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.     

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.     

Modeling spatially non-stationary land use/cover change in the lower Connecticut River Basin by combining geographically weighted logistic regression and the CA-Markov model

Simulating land use/cover change (LUCC) and determining its transition rules have been a focus of research for several decades. Previous studies used ordinary logistic regression (OLR) to determine transition rules in cellular automata (CA) modeling of LUCC, which often neglected the spatially non-stationary relationships between driving factors and land use/cover categories. We use an integrated geographically weighted logistic regression (GWLR) CA-Markov method to simulate LUCC from 2001–2011 over 29 towns in the Connecticut River Basin. Results are compared with those obtained from the OLR-CA-Markov method, and the sensitivity of LUCC simulated by the GWLR-CA-Markov method to the spatial non-stationarity-based suitability map is investigated. Analysis of residuals indicates better goodness of fit in model calibration for geographically weighted regression (GWR) than OLR. Coefficients of driving factors indicate that GWLR outperforms OLR in depicting the local suitability of land use/cover categories. Kappa statistics of the simulated maps indicate high agreement with observed land use/cover for both OLR-CA-Markov and GWLR-CA-Markov methods. Similarity in simulation accuracy between the methods suggests that the sensitivity of simulated LUCC to suitability inputs is low with respect to spatial non-stationarity. Therefore, this study provides critical insight on the role of spatial non-stationarity throughout the process of LUCC simulation.     

Exploring population spatial concentrations in Northern Ireland by community background and other characteristics: an application of geographically weighted spatial statistics

Information on how populations are spatially concentrated by different characteristics is a key means of guiding government policies in a variety of contexts, in addition to being of substantial academic interest. In particular, to reduce inequalities between groups, it is necessary to understand the characteristics of these groups in terms of their composition and their geographical structure. This article explores the degree to which the population of Northern Ireland is spatially concentrated by a range of characteristics. There is a long history of interest in residential segregation by religion in Northern Ireland; this article assesses population concentration not only by community background (‘religion or religion brought up in’) but also by housing tenure, employment and other socioeconomic and demographic characteristics. The spatial structure of geographical variables can be captured by a range of spatial statistics including Moran's I. Such approaches utilise information on connections between observations or the distances between them. While such approaches are conceptually an improvement on standard aspatial statistics, a logical further step is to compute statistics on a local basis on the grounds that most real-world properties are not spatially homogenous and, therefore, global measures may mask much variation. In population geography, which provides the substantive focus for this article, there are still relatively few studies that assess in depth the application of geographically weighted statistics for exploring population characteristics individually and for exploring relations between variables. This article demonstrates the value of such approaches by using a variety of geographically weighted statistical measures to explore outputs from the 2001 Census of Population of Northern Ireland. A key objective is to assess the degree to which the population is spatially divided, as judged by the selected variables. In other words, do people cluster more strongly with others who share their community background or others who have a similar socioeconomic status in some respect? The analysis demonstrates how geographically weighted statistics can be used to explore the degree to which single socioeconomic and demographic variables and relations between such variables differ at different spatial scales and at different geographical locations. For example, the results show that there are regions comprising neighbouring areas with large proportions of people from the same community background, but with variable unemployment levels, while in other areas the first case holds true but unemployment levels are consistently low. The analysis supports the contention that geographical variations in population characteristics are the norm, and these cannot be captured without using local methods. An additional methodological contribution relates to the treatment of counts expressed as percentages.     

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.     

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.

     

Evaluation and comparison of landslide susceptibility mapping methods: a case study for the Ulus district,Bartın,northern Turkey

The purpose of this study was to investigate the capabilities of different landslide susceptibility methods by comparing their results statistically and spatially to select the best method that portrays the susceptibility zones for the Ulus district of the Bart?n province (northern Turkey). Susceptibility maps based on spatial regression (SR), linear discriminant analysis (LDA), quadratic discriminant analysis (QDA), logistic regression (LR) method, and artificial neural network method (ANN) were generated, and the effect of each geomorphological parameter was determined. The landslide inventory map digitized from previous studies was used as a base map for landslide occurrence. All of the analyses were implemented with respect to landslides classified as rotational, active, and deeper than 5 m. Three different sets of data were used to produce nine explanatory variables (layers). The study area was divided into grids of 90 m × 90 m, and the ‘seed cell’ technique was applied to obtain statistically balanced population distribution over landslide inventory area. The constructed dataset was divided into two datasets as training and test. The initial assessment consisted of multicollinearity of explanatory variables. Empirical information entropy analysis was implemented to quantify the spatial distribution of the outcomes of these methods. Results of the analyses were validated by using success rate curve (SRC) and prediction rate curve (PRC) methods. Additionally, statistical and spatial comparisons of the results were performed to determine the most suitable susceptibility zonation method in this large-scale study area. In accordance with all these comparisons, it is concluded that ANN was the best method to represent landslide susceptibility throughout the study area with an acceptable processing time.     

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。     

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.     

Spatial land use regression of nitrogen dioxide over a 5-year interval in Calgary,Canada

We analysed the spatial distribution of nitrogen dioxide over Calgary (Canada) in summer 2010 and winter 2011 and in summer 2015 and winter 2016, and estimated land use regressions for 2015–16 (2010–11 models were estimated previously). As nitrogen dioxide exhibited spatial clustering, we evaluated the following spatial specifications against a linear model: spatially autoregressive (lag), spatially autoregressive (error), and geographically weighted regression. The spatially autoregressive (lag) specification performed best, achieving goodness-of-fit aligned with or greater than values reported in the literature. We compared the 2015–16 spatially autoregressive models with the 2010–11 models and reparametrized them on the 2010–11 and the 2015–16 data. Finally, we identified a single set of predictors to best fit the data. Nitrogen dioxide concentration decreased over the 5 years, retaining consistent spatial and seasonal patterns, with higher concentrations over traffic corridors and industrial areas, and greater variation in summer than winter. The multi-temporal analysis suggested that spatial land use regressions were robust over the time interval, despite moderate land use change. Multi-temporal spatial land use regressions yielded consistent predictors for each season over time, which can aid estimation of air pollution at fine spatial resolution over an extended time period.     

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.     

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.     

An exploratory spatial analysis of western medical services in Republican Beijing

In this paper, we reconstructed the spatial organization of Western medical services in Beijing city during the Republican period using a recently completed Republican Beijing GIS dataset. The primary objective is to explore the utility of spatial analytical methods, such as hotspot analysis and geographically weighted regression (GWR), in studying the spatial patterns of Western medical services. Our study is successful in depicting the spatial structure of Western medical services in the city. In addition, our analysis offers a preliminary but holistic view of the spatial relationships between Western medical services in the city and traditional Chinese medicine, population distribution, temple locations and industry-commerce patterns.     

Concrete evidence & geographically weighted regression: A regional analysis of wealth and the land cover in Massachusetts

Several studies indicate that there is a positive relationship between green vegetation land cover and wealthy socio-economic conditions in urban areas. The purpose of this research is to test for and explore spatial variation in the relationship between socio-economic and green vegetation land cover across urban, suburban, and rural areas, using geographically weighted regression (GWR). The analysis was conducted at the census block group level for Massachusetts, using Census 2000 data and impervious surface data at 1-m resolution. To explore regional variations in the relationship, four scenarios were generated by regressing each of the following socio-economic variables – median household income, percentage of poverty, percentage of minority population, and median home value – against two environmental variables – percent of impervious surface and population density. GWR results show that there is a considerable spatial variation in the character and the strength of the relationship for each model. There are two main conclusions in this study. First, the impervious surface is generally a strong predictor of the level of wealth as measured by four variables included in the analysis, at the scale of census block group; however, the strength of the relationship varies geographically. Second, GWR, not ordinary least squares technique, should be used for regional scale spatial analysis because it is able to account for local effects and shows geographical variation in the strength of the relationship.     

Testing spatial heterogeneity in geographically weighted principal components analysis

We propose a method to evaluate the existence of spatial variability in the covariance structure in a geographically weighted principal components analysis (GWPCA). The method, that is extensive to locally weighted principal components analysis, is based on performing a statistical hypothesis test using the eigenvectors of the PCA scores covariance matrix. The application of the method to simulated data shows that it has a greater statistical power than the current statistical test that uses the eigenvalues of the raw data covariance matrix. Finally, the method was applied to a real problem whose objective is to find spatial distribution patterns in a set of soil pollutants. The results show the utility of GWPCA versus PCA.     

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.     

Sensitivity analysis of a decision tree classification to input data errors using a general Monte Carlo error sensitivity model

We analysed the sensitivity of a decision tree derived forest type mapping to simulated data errors in input digital elevation model (DEM), geology and remotely sensed (Landsat Thematic Mapper) variables. We used a stochastic Monte Carlo simulation model coupled with a one‐at‐a‐time approach. The DEM error was assumed to be spatially autocorrelated with its magnitude being a percentage of the elevation value. The error of categorical geology data was assumed to be positional and limited to boundary areas. The Landsat data error was assumed to be spatially random following a Gaussian distribution. Each layer was perturbed using its error model with increasing levels of error, and the effect on the forest type mapping was assessed. The results of the three sensitivity analyses were markedly different, with the classification being most sensitive to the DEM error, than to the Landsat data errors, but with only a limited sensitivity to the geology data error used. A linear increase in error resulted in non‐linear increases in effect for the DEM and Landsat errors, while it was linear for geology. As an example, a DEM error of as small as ±2% reduced the overall test accuracy by more than 2%. More importantly, the same uncertainty level has caused nearly 10% of the study area to change its initial class assignment at each perturbation, on average. A spatial assessment of the sensitivities indicates that most of the pixel changes occurred within those forest classes expected to be more sensitive to data error. In addition to characterising the effect of errors on forest type mapping using decision trees, this study has demonstrated the generality of employing Monte Carlo analysis for the sensitivity and uncertainty analysis of categorical outputs that have distinctive characteristics from that of numerical outputs.     

An initial classification of 10-km squares in Great Britain from a land characteristic data bank

In a pilot classification of 282 10-km squares in Great Britain, data on physiography, climate and geology were extracted. Parallel classifications were run using these variables and also using spatial location. Two classification methods were compared: minimum within-group variance and indicator species analysis. Similarities between the resulting classifications were considered, and the groups were assessed for geographic coherence. Their validity for use as stratifications were tested using analysis of variance and also by matching the classifications with known distributions of a number of bird and plant species.Classifications using spatial variables were geographically more coherent than those without. The different methods resulted in different groupings of the squares which were partly a result of the differences in weightings applied to the four types of variable. However, the analysis of variance showed that either classification method provided a good stratification of the country, in particular with respect to altitude and rainfall. Some bird and plant species distributions correlated well with the classifications, but others did not, dependent on the factors limiting those distributions.     

A Review of “Turf Wars: Discourse,Diversity, and the Politics of Place”

The spatial correlation, or colocation, of two or more variables is a fundamental issue in geographical analysis but has received much less attention than the spatial correlation of values within a single variable, or autocorrelation. A recent paper by Leslie and Kronenfeld (2011) contributes to spatial correlation analysis in its development of a colocation statistic for categorical data that is interpreted in the same way as a location quotient, a frequently used measure in human geography and other branches of regional analysis. Geographically weighted colocation measures for categorical data are further developed in this article by generalizing Leslie and Kronenfeld's global measure as well as specifying a local counterpart for each global statistic using two different types of spatial filters: fixed and adaptive. These geographically weighted colocation quotients are applied to the spatial distribution of housing types to demonstrate their utility and interpretation.     

The comap as a diagnostic tool for non-stationary kriging models

In this study, we demonstrate a novel use of comaps to explore spatially the performance, specification and parameterisation of a non-stationary geostatistical predictor. The comap allows the spatial investigation of the relationship between two geographically referenced variables via conditional distributions. Rather than investigating bivariate relationships in the study data, we use comaps to investigate bivariate relationships in the key outputs of a spatial predictor. In particular, we calibrate moving window kriging (MWK) models, where a local variogram is found at every target location. This predictor has often proved worthy for processes that are heterogeneous, and most standard (global variogram) kriging algorithms can be adapted in this manner. We show that the use of comaps enables a better understanding of our chosen MWK models, which in turn allows a more informed choice when selecting one MWK specification over another. As case studies, we apply four variants of MWK to two heterogeneous example data sets: (i) freshwater acidification critical load data for Great Britain and (ii) London house price data. As both of these data sets are strewn with local anomalies, three of our chosen models are robust (and novel) extensions of MWK, where at least one of which is shown to perform better than a non-robust counterpart.