Bandwidth Selection in Geographically Weighted Regression Models via Information Complexity Criteria

Geographically Weighted Regression (GWR) is increasingly used in spatial analyses of social and environmental data. It allows spatial heterogeneities in processes and relationships to be investigated through a series of local regression models rather than a single global one. Standard GWR assumes that relationships between the response and predictor variables operate at the same spatial scale, which is frequently not the case. To address this, several GWR variants have been proposed. This paper describes a route map to decide whether to use a GWR model or not, and if so which of three core variants to apply: a standard GWR, a mixed GWR or a multiscale GWR (MS‐GWR). The route map comprises 3 primary steps that should always be undertaken: (1) a basic linear regression, (2) a MS‐GWR, and (3) investigations of the results of these in order to decide whether to use a GWR approach, and if so for determining the appropriate GWR variant. The paper also highlights the importance of investigating a number of secondary issues at global and local scales including collinearity, the influence of outliers, and dependent error terms. Code and data for the case study used to illustrate the route map are provided.

Soil erosion is one of the most common types of land degradation. To provide useful information for proper management, quantitative soil erosion evaluation and identifying influential factors are needed. However, rare studies have been reported on spatial modeling of soil erosion in connection with affective factors to prioritize the locality and the type of erosion control measures. Hence, the aim of this study was to (1) assess erosion-prone areas in the Talar watershed, Iran, using the revised universal soil loss equation (RUSLE) model and (2) investigate the relationship between soil erosion variability and land-use changes. Toward that, the ordinary least squares (OLS), geographically weighted regression (GWR) models, and principal component analysis (PCA) were used to analyze spatial relationships between soil erosion, land-use, and the RUSLE factors. The results of the OLS and GWR models indicated that these relationships are spatially non-stationary. GWR models had a good predictive performance than OLS with lower Akaike’s Information Criterion (from 254.31 to 276.81 in OLS and from 247.87 to 269.42 in GWR) and higher adjusted R2 values (from 0.12 to 0.54 in OLS, and from 0.36 to 0.66 in GWR). Among the variables mentioned above, LS factor, P factor, forest, and irrigated land were the most influential variables in GWR models. The results of PCA showed that PC1 and PC2 explained 66.2% of the variation in soil erosion concerning land-use and the RUSLE factors. These results provided appropriate references for managers and experts properly planning the study watershed.Graphical abstract

Understanding the spatial distribution of land surface temperature (LST) and its impact factors is crucial for mitigating urban heat island effect. However, few studies have quantitatively investigated the spatial non-stationarity and spatial scale effects of the relationships between LST and its impact factors at multi-scales. The main purposes of this study are as follows: (1) to estimate the spatial distributions of urban heat island (UHI) intensity by using hot spots analysis and (2) to explore the spatial non-stationarity and scale effects of the relationships between LST and related impact factors at multiple resolutions (30–1200 m) and to find appropriate scales for illuminating the relationships in a plain city. Based on the LST retrieved from Landsat 8 OLI/TIRS images, the Geographically-Weighted Regression (GWR) model is used to explore the scale effects of the relationships in Zhengzhou City between LST and six driving indicators: The Fractional Vegetation Cover (FVC), the Impervious Surface (IS), the Population Density (PD), the Fossil-fuel CO2 Emission data (FFCOE), the Shannon Diversity Index (SHDI) and the Perimeter-area Fractal Dimension (PAFRAC),which indicate the vegetation abundance, built-up, social-ecological variables and the diversity and shape complexity of land cover types. Our findings showed that the spatial patterns of LST show statistically significant hot spot zones in the center of the study area, partly extending to the western and southern industrial areas, indicating that the intensity of the urban heat island is significantly spatial clustering in Zhengzhou City. In addition, compared with the Ordinary Least Squares (OLS) model, the GWR model has a better ability to characterize spatial non-stationarity and analyze the relationships between the LST and its impact factors by considering the space-varying relationships of different variables, especially at the fine spatial scales (30–480 m). However, the strength of GWR model has become relatively weak with the increase of spatial scales (720–1200 m). This reveals that the GWR model is recommended to be applied in the analysis of UHI problems and related impact factors at scales finer than 480 m in the plain city. If the spatial scale is coarser than 720 m, both OLS and GWR models are suitable for illustrating the correct relationships between UHI effect and its influence factors in the plain city due to their undifferentiated performance. These findings can provide valuable information for urban planners and researchers to select appropriate models and spatial scales seeking to mitigate urban thermal environment effect.

A forest plot with a clustered spatial pattern of tree locations was used to investigate the impacts of different kernel functions (fixed vs. adaptive) and different sizes of bandwidth on model fitting, model performance, and spatial characteristics of the geographically weighted regression (GWR) coefficient estimates and model residuals. Our results indicated that (i) the GWR models with smaller bandwidths fit the data better, yielded smaller model residuals across tree sizes, significantly reduced spatial autocorrelation and heterogeneity for model residuals, and generated better spatial patterns for model residuals; however, smaller bandwidth sizes produced a high level of coefficient variability; (ii) the GWR models based on the fixed spatial kernel function produced smoother spatial distributions for the model coefficients than those based on the adaptive kernel function; and (iii) the GWR cross-validation or Akaike’s information criterion (AIC) optimization process may not produce an “optimal” bandwidth for model fitting and performance. It was evident that the selection of spatial kernel function and bandwidth has a strong impact on the descriptive and predictive power of GWR models.

Regional disparity of economic development in China is always greatly concerned by investigators domestic and abroad, and sets of models have been used in the analysis. Spatial dependence, which is hidden in the data with spatial attributes, usually is not taken into account in classical statistics methods, such as ordinary linear regression(OLR) model. Along with the development of spatial statistics, more and more attentions are paid on spatial interactions between observations in the study of regional disparity. Geographically weighted regression (GWR) is a simple but effective model to recognize spatial variation and local difference, which considers the influence of the spatially non-stationarity of the variables. In this study, GWR model and Geo-Informatic Tupu were used to analyze the disparity of regional economic development in China, taking GDP per capita in 1999 and 2003 as a case, which is usually used to measure level of economic development. GDP per capita in the 338 cities in 1999 were selected to simulate GDP per capita in 2003 and to analysis non-stationarity of the growth of GDP per capita. Using Geo-Informatic Tupu, A series of maps were processed to display patterns of local parameter estimates, such as local r-squares, the residual sum of squares, local residual and so on, to gain a better understanding of the degree of spatial non-stationarity in a relationship over space. We tested for geographic heterogeneity in the parameters and compare them to estimates obtained from global regression approaches. The results suggested there was heterogeneity in the regression coefficients across broad regions of China, and a one-size fits all approach to describe growth processes appeared simplistic. The GWR model improved over the OLR model, and it was able to better explain the variation in the data and to simulate GDP per capita with smaller errors than the OLR models.

Newtown Business of Seoul had been performed to reduce deterioration of Gangbuk and economic inequality between Gangnam and Gangbuk. According to this, Eunpyeong-gu was set as test-bed for Newtown business and Newtown business had been completed until 2013. This study aims to analyze the influence of social and economical factors which affect land price using GWR (Geographically Weighted Regression) considered spatial effect. As a result of analysis, GWR model demonstrated a better goodness-of-fit than OLS (Ordinary least square) model typically used in most study. Furthermore, AIC value and Moran’s I of residual prove that GWR model is more suitable than OLS model. GWR model enable to explain more detailed than global regression model as coefficient and sign show different value locally. In future, this research will be helpful to develop Eunpyeong-gu considering spatial characters and strength effectiveness of development.Keywords : Eunpyeong Newtown, Newtown Projects, Land Value, Geographically Weighted Regression

Spatial regression analysis is a form of regression model that considers spatial effects. Geographically weighted regression (GWR) is the spatial regression methods that can be used to deal with the problem of spatial diversity. This method generates local model parameter estimates for each observation location. The application of spatial statistics can be done in all areas such as the problem of poverty. Poverty can be influenced by factors of proximity between regions, so that in determining the poverty factor, the proximity factor of the region cannot be ignored. West Java Province is a province with the largest population, so this study aims to model the poverty data in West Java Province by incorporating spatial effects. The weighting function used for the GWR model is the function of the fixed and adaptive kernels. The analysis results show that the fixed exponential kernel function has the smallest cross validation (CV) value, so the weighting matrix used in the model is determined by the exponential kernel function. The largest value and the smallest AIC value are owned by the GWR model with an exponential kernel function. Based on the results obtained by the the ANOVA table to test GWR's global goodness, the GWR model is more effective than global regression. Therefore, the GWR model is the best model when it used in West Java’s poverty cases. The effect of each explanatory variable on the percentage of poverty varies in each district/city in West Java Province.

BackgroundAngiostrongyliasis is a food-borne parasitic zoonosis. Human infection is caused by infection with the third-stage larvae of Angiostrongylus cantonensis. The life cycle of A. cantonensis involves rodents as definitive hosts and molluscs as intermediate hosts. This study aims to investigate on the infection status and characteristics of spatial distribution of these hosts, which are key components in the strategy for the prevention and control of angiostrongyliasis.MethodsThree villages from Nanao Island, Guangdong Province, China, were chosen as study area by stratified random sampling. The density and natural infection of Pomacea canaliculata and various rat species were surveyed every three months from December 2015 to September 2016, with spatial correlations of the positive P. canaliculata and the infection rates analysed by ArcGIS, scan statistics, ordinary least squares (OLS) and geographically weighted regression (GWR) models.ResultsA total of 2192 P. canaliculata specimens were collected from the field, of which 1190 were randomly chosen to be examined for third-stage larvae of A. cantonensis. Seventy-two Angiostrongylus-infected snails were found, which represents a larval infection rate of 6.1% (72/1190). In total, 110 rats including 85 Rattus norvegicus, 10 R. flavipectus, one R. losea and 14 Suncus murinus were captured, and 32 individuals were positive (for adult worms), representing an infection rate of 29.1% of the definitive hosts (32/110). Worms were only found in R. norvegicus and R. flavipectus, representing a prevalence of 36.5% (31/85) and 10% (1/10), respectively in these species, but none in R. losea and S. murinus, despite testing as many as 32 of the latter species. Statistically, spatial correlation and spatial clusters in the spatial distribution of positive P. canaliculata and positive rats existed. Most of the spatial variability of the host infection rates came from spatial autocorrelation. Nine spatial clusters with respect to positive P. canaliculata were identified, but only two correlated to infection rates. The results show that corrected Akaike information criterion, R2, R2 adjusted and σ2 in the GWR model were superior to those in the OLS model.ConclusionsP. canaliculata and rats were widely distributed in Nanao Island and positive infection has also been found in the hosts, demonstrating that there was a risk of angiostrongyliasis in this region of China. The distribution of positive P. canaliculata and rats exhibited spatial correlation, and the GWR model had advantage over the OLS model in the spatial analysis of hosts of A. cantonensis.

Two popular data science analyses are spatial analysis and machine learning. The most popular spatial analysis is geographically weighted regression (GWR), and the most popular of machine learning is random forest (classification dan regression). The paper used GWR and Random Forest Regression (RFR) to find the variables that significantly impact life expectancy and compare two methods using RMSE. Knowledge of life expectancy and determining factors on life expectancy is essential because the increase in life expectancy is linked to a country's economic and social prosperity. The researcher employed life expectancy data as the dependent variable (Y) and some independent variables. At each observation location in Sumatra Region, the GWR model and significant independent variables are different. The RMSE value in the GWR model is 64.99. There are three variables with significant influence in most areas are X3 (percentage of proper sanitation households), X5 (number of doctors), and X7 (average years of schooling). The RMSE value in the RFR model is 84.04 with three variables importance: X3 (percentage of proper sanitation households), X5 (number of doctors), and X7 (average years of schooling) have the most influence. Keywords: GWR, Life Expectancy, RFR, RMSE.

With the fast urbanization process, how does the vegetation environment change in one of the most economically developed metropolis, Shanghai in East China? To answer this question, there is a pressing demand to explore the non-stationary relationship between socio-economic conditions and vegetation across Shanghai. In this study, environmental data on vegetation cover, the Normalized Difference Vegetation Index (NDVI) derived from MODIS imagery in 2003 were integrated with socio-economic data to reflect the city’s vegetative conditions at the census block group level. To explore regional variations in the relationship of vegetation and socio-economic conditions, Ordinary Least Squares (OLS) and Geographically Weighted Regression (GWR) models were applied to characterize mean NDVI against three independent socio-economic variables, an urban land use ratio, Gross Domestic Product (GDP) and population density. The study results show that a considerable distinctive spatial variation exists in the relationship for each model. The GWR model has superior effects and higher precision than the OLS model at the census block group scale. So, it is more suitable to account for local effects and geographical variations. This study also indicates that unreasonable excessive urbanization, together with non-sustainable economic development, has a negative influence of vegetation vigor for some neighborhoods in Shanghai.

Detailed information about spatial distribution of soil properties is important in ecological modelling, environmental prediction, precision agriculture, and natural resources management, as well as land-use planning. In the present study, a recently developed method called geographically weighted regression (GWR) is applied to predict spatial distribution of soil properties (pH, soil organic matter, available nitrogen, available potassium) based on topographical indicators, climate factors, and geological stratum at a regional scale. In total, 1914 soil samples collected from a depth of 0–20 cm were used to calibrate and validate the models. Performances of the GWR models were compared with the traditional, ordinary least-squares (OLS) regression. The results indicated that the GWR models made significant improvements to model performances over OLS regression, based on F-test, coefficient of determination, and corrected Akaike information criterion. GWR models also improved the reliability of the soil–environment relationships by reducing the spatial autocorrelations in model residuals. Meanwhile, the use of GWR models disclosed that the relationships between soil properties and environmental variables were not invariant over space but exhibited significant spatial non-stationarity. Accordingly, the GWR models remarkably improved the prediction accuracies over the corresponding OLS models. The results demonstrated that GWR could serve as a useful tool for digital soil mapping in areas with complex terrain.

This study identifies drivers of deforestation in Mexico by applying Geographically Weighted Regression (GWR) models to cartographic and statistical data. A wall-to-wall multitemporal GIS database was constructed incorporating digital data from Global Forest Change (2000-2012); along with ancillary data (road network, settlements, topography, socio-economical parameters and government policies). The database analysis allowed assessing the spatial distribution of deforestation at the municipal level. The statistical analysis of deforestation drivers presented here was focused on the rate of deforestation during the period 2007-2011 as dependent variable. In comparison with the global model, the use of GWR increased the goodness-of-fit (adjusted R2) from 0.46 (global model) to 0.58 (average R2 of GWR local models), with individual GWR models ranging from 0.52 to 0.64. The GWR model highlighted the spatial variation of the relationship between the rate of deforestation and its drivers. Factors identified as having a major impact on deforestation were related to topography (slope), accessibility (road and settlement density) and marginalization. Results indicate that some of the drivers explaining deforestation vary over space, and that the same driver can exhibit opposite effects depending on the region.

Research using a global regression model might not be appropriate to find out the factors that influence strategic food prices based on spatial characteristics. To analyze the spatial effect, Geographically Weighted Regression (GWR) was employed. GWR models are better than OLS, which is indicated by the higher R2 GWR and lower AIC values. The GWR analysis provides the following findings: (1) the wholesale price most influential on the retail price of medium rice and red chili both during the main harvest and non-harvest periods; (2) the harvest pattern results in the effect of production and producer prices on the retail prices of the major harvest and non-harvest periods. Management of inter-regional distribution must be carried out to maintain supply stability and disparity in food prices between regions; (3) producer prices are integrated with trader prices in the district of production centers and surrounding areas while the integration of food prices at the consumer level occurs in the main economic center area of the region. These aspects have different effects for each region and district because the estimated parameters can be positive or negative. Testing during the harvest season (April) and non-harvest can also produce estimates that vary according to the specific characteristics of each location. Keywords: spatial analysis, Geographically Weighted Regression (GWR), retail prices, wholesale price, spatial distribution patterns

Spatial variations in active mode trip volume at intersections: a local analysis utilizing geographically weighted regression

PurposeThe purpose of this study is to enhance the estimation of vertical and horizontal inequity within property valuation. Property taxation is a crucial source of finance for local government around the world – based on a presumptive tax base underpinned by estimates of property value, inaccurate real estate valuations used for such ad valorem or value-based property tax calculations potentially lead to a variety of costs, both financial and other, for tax payers and governments alike. More common are increased costs in time, staff and, in some cases, legal fees. Some governments are even bound by acceptability thresholds to promote fairness, equitability and overall government accountability with respect to valuation.Design/methodology/approachThere exist a number of vertical inequity measurements that have undergone academic testing and scrutiny within the property tax industry since the 1970s. While these approaches have proved successful in detecting horizontal and vertical inequity, one recurring disadvantage pertains to measurement error/omitted variable bias, stemming largely from a failure to accurately account for location. A natural progression within property tax research is the application of a more spatially local weighted modelling approach to examine vertical and horizontal inequity. This research, therefore, specifies a geographically weighted regression (GWR) methodology to detect and measure vertical inequity in property valuations.FindingsThe findings show the efficacy of using more applied spatial approaches for vertical tax estimation and indeed the limitations of employing conditional mean estimates coupled with delineated boundaries for assessing property tax inequity. The GWR model findings highlight the more fluctuating nature of vertical inequity across the Belfast market for the apartment sector both in a progressive and regressive sense and at different magnitudes. Moreover, the results reveal spatial clustering in the effects and are indicative of systematic inequities related to location inferring that spatial (horizontal) tax inequities are not random. The findings further show increased GWR model predictability overall.Originality/valueThis research adds to the existing literature base for evaluating both vertical and horizontal inequity in value-based property taxation at the intra-neighbourhood level. This is accomplished by modifying the Birch–Sunderman approach by transforming the traditional OLS model architecture to a GWR model, thereby allowing coefficient estimates of inequity to vary not only across a jurisdiction, but also at a more local level, while incorporating property characteristic variables. This arguably allows assessors to identify specific geographical areas of concern, saving them money, time and resources on identifying, addressing and correcting for inequity.