Geographically Weighted Regression (GWR)

Abstract: Ecological influences on health outcomes are associated with the spatial stratification of health. However, the majority of studies that seek to understand these ecological influences utilise aspatial methods. Geographically weighted regression (GWR) is a spatial statistics tool that expands standard regression by allowing for spatial variance in parameters. This study contributes to the urban health literature, by employing GWR to uncover geographic variation in Limiting Long Term Illness (LLTI) and area level effects at the small area level in a relatively small, urban environment. Using GWR it was found that each of the three contextual covariates, area level deprivation scores, the percentage of the population aged 75 years plus and the percentage of residences of white ethnicity for each LSOA exhibited a non-stationary relationship with LLTI across space. Multicollinearity among the predictor variables was found not to be a problem. Within an international policy context, this research indicates that even at the city level, a “one-size fits all” policy strategy is not the most appropriate approach to address health outcomes. City “wide” health polices need to be spatially adaptive, based on the contextual characteristics of each area.

The global modeling techniques, such as the ordinary least squares regression (OLS), linear and other non- linear models cannot detect spatial variation and rela- tionships within geographic entities. As a result, intrinsic relationships may be obscured and spatial association between variables in a region is concealed. Such incom- plete information (derived from global statistics), when adopted for addressing policy issues, may be counter- productive. To strengthen this weakness, statistical ge- ographers ([8] Brunsdon et al., 1996 and [1] Fothering- ham et al., 2002) recently came up with geographically weighted regression (GWR)—a technique designed to explore spatial non-stationarity or heterogeneity in geo- graphic dataset. Spatial non-stationarity is a scenario in which global statistical models cannot explain the rela- tionship between sets of variables ([8] Brunsdon et al., 1996).

The purpose of this paper is to analyze the spatially varying impacts of some classical regressors on per capita household income in Spanish provinces. The authors model this distribution following both a traditional global regression and a local analysis with Geographically Weighted Regression (GWR). Several specifications are compared, being the adaptive bisquare weighting function the more efficient in terms of goodness-of-fit. We test for global and local spatial instability using some F-tests and other statistical measures. We find some evidence of spatial instability in the distribution of this variable in relation to some explanatory variables, which cannot be totally solved by spatial dependence specifications. GWR has revealed as a better specification to model per capita household income. It highlights some facets of the relationship completely hidden in the global results and forces us to ask about questions we would otherwise not have asked. Moreover, the application of GWR can also be of help to further exercises of micro-data spatial prediction.

Geographically Weighted Regression (GWR), as first described in Brunsdon et al. [5], is a commonly used approach in spatial analysis. It has at its core the idea that global or whole map statistical models may make unreasonable assumptions of spatial non-stationarity amongst the processes under investigation [33]. The intention of GWR was to provide an exploratory approach to investigate the spatial nature of relationships between response and predictor variables, and in so doing, to provide a better understanding of the process under consideration. It has conceptual elegance; local regression models are constructed at different locations using data under a moving window or kernel, which are weighted by the distance to the kernel center such that data furthest away contribute less to the overall model. Because of this, the geographically weighted (GW) framework has been extended to include different types of models including GW principal components analysis [20], GW summary statistics [4], GW discriminant analysis [6], GW variograms [22], GW Structural Equation Models [11], and has been applied in domains with little tradition of local statis- tical approaches such as remote sensing (eg [9, 12, 15]). The fundamental aims of GWR and GW frameworks are thus to explore spatial relationships in data and processes.

Geographically Weighted Regression (GWR) is a local technique that models spatially varying relationships, where Euclidean distance is traditionally used as default in its calibration. However, empirical work has shown that the use of non-Euclidean distance metrics in GWR can improve model performance, at least in terms of predictive fit. Furthermore, the relationships between the dependent and each independent variable may have their own distinctive response to the weighting computation, which is reflected by the choice of distance metric. Thus, we propose a back-fitting approach to calibrate a GWR model with parameter-specific distance metrics. To objectively evaluate this new approach, a simple simulation experiment is carried out that not only enables an assessment of prediction accuracy, but also parameter accuracy. The results show that the approach can provide both more accurate predictions and parameter estimates, than that found with standard GWR. Accurate localised parameter estimation is crucial to GWR’s main use as a method to detect and assess relationship non-stationarity.

Land cover is of fundamental importance to many environmental applications and serves as critical baseline information for many large scale models e.g. in developing future scenarios of land use and climate change. Although there is an ongoing movement towards the development of higher resolution global land cover maps, medium resolution land cover products (e.g. GLC2000 and MODIS) are still very useful for modelling and assessment purposes. However, the current land cover products are not accurate enough for many applications so we need to develop approaches that can take existing land covers maps and produce a better overall product in a hybrid approach. This paper uses geographically weighted regression (GWR) and crowdsourced validation data from Geo-Wiki to create two hybrid global land cover maps that use medium resolution land cover products as an input. Two different methods were used: a) the GWR was used to determine the best land cover product at each location; b) the GWR was only used to determine the best land cover at those locations where all three land cover maps disagree, using the agreement of the land cover maps to determine land cover at the other cells. The results show that the hybrid land cover map developed using the first method resulted in a lower overall disagreement than the individual global land cover maps. The hybrid map produced by the second method was also better when compared to the GLC2000 and GlobCover but worse or similar in performance to the MODIS land cover product depending upon the metrics considered. The reason for this may be due to the use of the GLC2000 in the development of GlobCover, which may have resulted in areas where both maps agree with one another but not with MODIS, and where MODIS may in fact better represent land cover in those situations. These results serve to demonstrate that spatial analysis methods can be used to improve medium resolution global land cover information with existing products.

Geographically Weighted Regression (GWR) is a local modeling approach that explicitly allows parameter estimates to vary over space (Brunsdon et al. 1996). Rather than specifying a single model to characterize the entire housing market, GWR estimates a separate model for each sale point and weights observations by their distance to this point, thus allowing unique marginal-price estimates at each location. This method is appealing because it mimics to some extent the “sales comparison” approach to valuation used by appraisers in that only sales within close proximity to the subject property are considered, and price adjustments are made based on differences in characteristics within this subset of properties.

The applications of standard regression analysis on spatial data are not appropriate because of the characteristics of the spatial data. Spatial data has two characteristics are spatial dependence and spatial heterogeneity. Modeling spatial data using standard linear regression model leads to bias, inconsistency and inefficient results. Several models have been developed to accommodate the characteristics of the spatial data. However, the models generally developed to solve only one problem of the spatial data (e.g., spatial dependence or spatial heterogeneity). Four kinds of spatial econometrics models usually used to accommodate spatial dependence are spatial autoregressive (SAR), spatial lagged exogenous variables (SLX), spatial error model (SEM), and spatial Durbin model (SDM). To accommodate the spatial heterogeneity, geographically weighted regression (GWR) or varying coefficient model (VCM) is usually used. Our research proposed to develop a new model to accommodate two characteristics of the spatial data. The model is developed based on the combination SAR and GWR model. We call the model as Spatial Autoregressive Geographically Weighted Regression (SAR-GWR). We used Instrumental Variables (IV) approach and Two Stage Least Square (TSLS) to estimate the parameters of the model. We have done the simulation study by mean Monte Carlo simulation to check the bias and efficiency of the parameter estimates. SAR-GWR model provides better results with small bias and Root Mean Square Error (RMSE) rather than standard GWR. We also found that our method relative robust to the multicollinearity problem. We also applied SAR-GWR model in modeling prevalence rate of the Tuberculosis (TB + ) disease in Bandung and we found the healthy house index gives serious effect in increasing theprevalence rate of TB + in Bandung City.

Geographically weighted regression (GWR) is a technique used to explore spatially-varying data relationships (Brunsdon et al. 1996, Fotheringham et al. 2002). Its original conception reflected a desire to move beyond from global, Ôwhole mapÕ (Openshaw 1996) and Ôone-size- fits-allÕ (Fotheringham and Brunsdon 1999) statistics to ones that captured and reflected local process heterogeneity. This was reflected in GoodchildÕs (2004) proposal for a second law of geography, the principle of spatial heterogeneity or nonstationarity, in which he noted the lack of a Ôconcept of an average place on the Earth's surface comparable, for example, to the concept of an average humanÕ (Goodchild 2004, p302). For regression, and from a nonstationary relationship viewpoint, a number of localised approaches have been developed including the expansion method (Casetti 1972), weighted spatial adaptive filtering model (Gorr and Olligschlaeger 1994), GWR, Bayesian space-varying coefficient (SVC) models (Gelfand et al. 2003; Assun•‹o 2003) and re-focused versions of eigenvector spatial filtering (ESF) (Griffith 2008; Murakiami et al. 2017). Other aspects of nonstationarity in regression can also be considered, such as those centred around the error term (e.g. Paez et al. 2002a, 2002b; Harris et al. 2010; 2011a). In particular, GWR has been the most widely applied localised regression in geographical analyses. It has conceptual simplicity: as geographers, we implicitly expect processes and relationships to vary locally and not to be the same everywhere. Rather, we acknowledge that the relationship among predictor and response variables may change over space. GWR provides a tool to identify and explore these varying relationships. The originators have long supported different implementations, either as standalone (e.g. GWR3.x Charlton et al. 2003) or as packages (e.g. R package GWmodel by Lu et al. 2014a, Gollini et al. 2015). It has been also incorporated as a tool in the most popular GIS software (ESRI 2009).

Although various schemes have been developed, rainfall merging is still a complex and important issue. The results of rainfall merging are influenced by the kind of merging scheme, the quality of satellite rainfall data, the density of raingauges and so on. Motivated by this, the objective of this paper is to develop a residual-based method for merging satellite and raingauge rainfall using geographically weighted regression (GWR). Theoretically, this novel method is capable of simultaneously blending various satellite rainfall data with gauge measurements and could describe the non-stationary influences of geographical and terrain factors on rainfall spatial distribution. Using the proposed method, an experimental study on merging the rainfall from CMOROH (Joyce et al., 2004) and gauge measurements was conducted for the Ganjiang River basin, in southeast China. The capability of our merging scheme for constructing daily rainfall fields under different gauge densities is investigated and discussed. The accuracy gain achieved by rainfall merging relative to traditional interpolation merely only raingauge measurements is analysed.

Geographically Weighted Regression (GWR), as first described in Brunsdon et al [1], is a commonly used approach in spatial analysis. It has at its core the idea that global or whole map statistical models may make unreasonable assumptions of spatial non-stationarity amongst the processes under investigation [32]. The intention of GWR was to provide an exploratory approach to explore the spatial nature of relationships between (response and predictor) variable and, in so doing to provide a better understanding of the process under consideration. It conceptual elegance: local regression models are constructed at different locations using data under a moving window or kernel, which are weighted by the distance to the kernel center such that data furthest away contribute less to the overall model. Because of this, the geographically weighted (GW) framework has been extended to include different types of models including GW principal components analysis [23], GW summary statistics [4], GW discriminant analysis [5], GW variograms [20], GW Structural Equation Models [10] and applied in domains with little tradition of local statistical ap- proaches (eg [11, 12, 15]). The fundamental aims of GWR and GW frameworks are thus to explore spatial relationships in data and processes.

influence (Gilbert, 2011) & (Robinson, 2011). Spatial regression method frequently used is Geographically Weighted Regresssion (GWR), which is a regression method involving the effect of the location into the predictor (Fotheringham, et al., 2002). In the linear regression model generated only parameter estimator that apply globally, while in the GWR models generated model parameter estimator that is local to each observation location. Mixed geographically Weighted Regression (MGWR) is a combination of global linear regression model with the GWR model. So that the model will be generated MGWR estimator parameters are global and some others are localized in accordance with the location of observations (Purhadi and Yasin, 2012).

distinctness. One possible solution is to include only the locations of data with similar attributes (i.e., homo- geneity). However, it is difficult to decide the number of groups with different attributes and identify the loca- tions of data in each group. Moreover, the mean value of a non-stationary process is usually a step function [8] or is continuous across space, and it is difficult to find the exact boundary of appropriate locations. The other possi- bility is to use the varying coefficient model [10], allowing the coefficient terms to vary according to locations. Then, the model is a form of local linear models [15] and can be used to explore the dynamic property of spatial data. Based on the concept of the varying coefficient model, geographically weighted regression (GWR) is modified to solve the MAUP [6].

Constructing mathematical models of processes is a common theme in analytical research in a wide range of disciplines. “Researchers search for variables to identify various dimensions of phenomena and for relationships among the variables to interpret or change the real world.” (Casetti, 1972, P.82). Traditional analytical methods, however, tend to assume constant relationships among variables; that is, models with constant parameters are constructed to describe the relationships between variables. This is usually achievable and acceptable in physical sciences, where the investigated phenomena are determined by certain natural laws, for example, Newton’s law of universal gravitation involves a constant parameter, the gravitational constant. In social and environmental sciences, however, the relationships between variables may not be constant. For example, the relationship between elevation and precipitation may change according to complex geographical phenomena. Traditional global models can mask this non-stationarity in relationships (Fotheringham, 1997, Brunsdon et al., 1999a, Brunsdon et al., 1999b, Farber and Yeates, 2006). Models that allow parameters to vary across geographical space, over time, or according to other contexts are thus useful. Typical forms of such models are regression models that have spatially varying coefficients. This thesis focuses on a widely applied model of this type, namely Geographically Weighted Regression (GWR).

Despite growing research for residential crowding effects on housing market and public health perspectives, relatively little attention has been paid to explore and model spatial patterns of res- idential crowding over space. This paper focuses upon analyzing the spatial relationships between residential crowding and socio-demographic variables in Alexandria neighborhoods, Egypt. Global and local geo-statistical techniques were employed within GIS-based platform to identify spatial variations of residential crowding determinates. The global ordinary least squares (OLS) model assumes homogeneity of relationships between response variable and explanatory variables across the study area. Consequently, it fails to account for heterogeneity of spatial relationships. Local model known as a geographically weighted regression (GWR) was also employed using the same response variable and explanatory variables to capture spatial non-stationary of residential crowding. A comparison of the outputs of both models indicated that OLS explained 74 percent of residential crowding variations while GWR model explained 79 percent. The GWR improved strength of the model and provided a better goodness of fit than OLS. In addition, the findings of this analysis revealed that residential crowding was significantly associated with different struc- tural measures particularly social characteristics of household such as higher education and illi- teracy. Similarly, population size of neighborhood and number of dwelling rooms were found to have direct impacts on residential crowding rate. The spatial relationship of these measures dis- tinctly varies over the study area.

Nonresponse can undermine the quality of social survey data. Understanding who does/does not respond to surveys is important for those involved in the collection and analysis of these data. Levels of nonresponse are known to vary geographically. However, there has been little consideration of how the predictors of survey nonresponse might vary geographically within countries. This study examines the possibility of spatial variation in response behavior using regional interactions and geographically weighted regression. Our results suggest that there is geographical variation in response behavior. Relying on “one size fits all” global models in nonresponse modelling might, therefore, be insufficient.

Geographically weighted regression models with the measurement error are a modeling method that combines the global regression models with the measurement error and the weighted regression model. The assumptions used in this model are a normally distributed error with that the expectation value is zero and the variance is constant. The purpose of this study is to estimate the parameters of the model and find the properties of these estimators. Estimation is done by using the Weighted Least Squares (WLS) which gives different weighting to each location. The variance of the meas-

This study has limitations. First, the lack of data on the latest district demarcations means that this study could not offer insights into outcomes in the recently created districts. Related to the above data limitation is the challenge of using predictors from different measurement occasions. Our preference for predictors measured in the same year was not feasible given the non-availability of data. Nonetheless, the temporal explanation of our results is valid because all predictors used in this study preceded the outcome variable. Second, because of the high correlation between the explanatory variables, we had to use recommended data reduction techniques to reduce the data. Although this approach is rigorous and recommended to address multicolinearity problems, it limited our ability to have more information about the direct connection between speci fi c indicators and our outcome variable. Despite these limitations, the study's strengths are noteworthy. The use of the GWR analytical method improved the study's explanatory po- wer, beyond that offered by traditional regression models. In addition, the use of thematic mapping made it possible to highlight geographical disparities in academic achievement in Ghana that have been ignored in many studies.

where there is usually high fire occurrence (check Fig. 1a). However, these maps are too oversmoothed in capturing lo- cal variations, especially in the logistic GWR model. The trend of the residuals of the logistic model towards cluster- ing did not significantly decrease from the ordinary model to the GWR model according to the Average Nearest Neighbour Distance Analysis (Table 2), and there was only a minor im- provement, especially in the overestimation errors. Although in the GW logistic model there were fewer errors, the spatial distribution was very similar to the ordinary logistic model (Fig. 2). Also, analysis of the linear GW regression model residuals revealed similar characteristics to the global OLS model, with a mean value of 0.01 and a SD of 0.51, accept- ably following the shape of the normal curve (Fig. 5b). The Kolmogorov-Smirnov test value was low (0.03) but still sig- nificant (p = 0.000), showing that the normal fit was poor. The residuals fitted properly except for low fire density val- ues according to the normal Q − Q plot (Fig. 5c). However, the scatterplot was more compact along the tendency line and the standardized residual map (Fig. 5d) showed a more dis- persed distribution through the study area in comparison to the OLS model (Fig. 3d), without any evident systematic pat- tern. These analyses indicated a slightly better performance of the GWR model.

Even though GWR was used as a more robust method for accounting for spatial dependence, the final model explained a relatively small amount of the variation. The relationships identified were complex with regression coefficients switching between negative and positive val- ues in different locations, which indicates that while focusing interventions on a particular risk factor might be beneficial in one area, it may actually be detrimental in another area. As with any regression method, they cannot prove causality, and other drivers may not have been retained as a result of the deliberately stringent method of covariate selection. The fact that the most important vari- ables occur in blocks greater than half the GWR bandwidth rather than random scattering suggests the regional heterogeneity is real, with demonstrably different factors associated with the spread of bTB in different areas where spread has occurred, and it is this that is perhaps the most important finding of this study.