Mixed Geographically Weighted Regression Research Articles

Assessing spatial inequalities in accessing community pharmacies: a mixed geographically weighted approach.

Geographical accessibility is an important determinant for the utilisation of community pharmacies. The present study explored patterns of spatial accessibility with respect to pharmacies in Liguria, Italy, a region with particular geographical and demographic features. Municipal density of pharmacies was proxied as the number of pharmacies per capita and per km2, and spatial autocorrelation analysis was performed to identify spatial clusters. Both non-spatial and spatial models were constructed to predict the study outcome. Spatial autocorrelation analysis showed a highly significant clustered pattern in the density of pharmacies per capita (I=0.082) and per km2 (I=0.295). Potentially under-supplied areas were mostly located in the mountainous hinterland. Ordinary least-squares (OLS) regressions established a significant positive relationship between the density of pharmacies and income among municipalities located at high altitudes, while no such association was observed in lower-lying areas. However, residuals of the OLS models were spatially auto-correlated. The best-fitting mixed geographically weighted regression (GWR) models outperformed the corresponding OLS models. Pharmacies per capita were best predicted by two local predictors (altitude and proportion of immigrants) and two global ones (proportion of elderly residents and income), while the local terms population, mean altitude and rural status and the global term income functioned as independent variables predicting pharmacies per km2. The density of pharmacies in Liguria was found to be associated with both socio-economic and landscape factors. Mapping of mixed GWR results would be helpful to policy-makers.

Mapping soil organic matter concentration at different scales using a mixed geographically weighted regression method

The present regression models in digital soil mapping usually assume that relationships between soil properties and environmental variables are always fixed (as in MLR) or varying (as in GWR) in geographical space. In reality, some of the environmental variables may be fixed in affecting soil property variation and some are local varying. In this study, a mixed geographically weighted regression (MGWR) method which can deal with fixed and varying spatial relationships between a target variable and its environmental variables were proposed and used to predict topsoil soil organic matter (SOM) concentration in two study areas (Heshan, Heilongjiang province and Xuancheng, Anhui province, China) at two scales. Three groups of sample sets were created based on the total samples in the study areas to evaluate the robustness and stability of the model. Multiple linear regression (MLR), geographically weighted regression (GWR), GWR-kriging (GWRK), local regression-kriging (LRK), kriging with an external drift (KED), and ordinary kriging (OK) were used for comparison with MGWR. The validation results showed that the use of MGWR reduced the RMSE of GWR by 10.5% and 7.6% on average, reduced the RMSE of MLR by 12.8% and 9.9% on average for Heshan and Xuancheng study areas respectively. MGWR also showed a good competitiveness when compared with GWRK, LRK, KED and OK. In Heshan study area, the influence of flow length, relative position index, foot slope and distance to the nearest drainage were constant, whereas the elevation, topographic wetness index and valley index showed different influence in different regions. In Xuancheng study area, the fixed environmental variables were profile curvature, topographic wetness index and slope, whereas the varying environmental variables were precipitation, temperature, elevation, and limestone. The results indicate that the accuracy of predictions can be improved by adaptive coefficient according to the variation of environmental variables as implemented in MGWR compared with others considering only the local or global relationships. It was concluded that mixed geographically weighted regression model could be a potential method for digital soil mapping.

Exploring the spatially varying innovation capacity of the US counties in the framework of Griliches’ knowledge production function: a mixed GWR approach

Griliches’ knowledge production function has been increasingly adopted at the regional level where location-specific conditions drive the spatial differences in knowledge creation dynamics. However, the large majority of such studies rely on a traditional regression approach that assumes spatially homogenous marginal effects of knowledge input factors. This paper extends the authors’ previous work (Kang and Dall’erba in Int Reg Sci Rev, 2015. doi: 10.1177/0160017615572888 ) to investigate the spatial heterogeneity in the marginal effects by using nonparametric local modeling approaches such as geographically weighted regression (GWR) and mixed GWR with two distinct samples of the US Metropolitan Statistical Area (MSA) and non-MSA counties. The results indicate a high degree of spatial heterogeneity in the marginal effects of the knowledge input variables, more specifically for the local and distant spillovers of private knowledge measured across MSA counties. On the other hand, local academic knowledge spillovers are found to display spatially homogenous elasticities in both MSA and non-MSA counties. Our results highlight the strengths and weaknesses of each county’s innovation capacity and suggest policy implications for regional innovation strategies.

Spatial Heterogeneity in Hedonic House Price Models: The Case of Austria

Modelling spatial heterogeneity (SH) is a controversial subject in real estate economics. Single-family-home prices in Austria are explored to investigate the capability of global and locally weighted hedonic models. Even if regional indicators are not fully capable to model SH and technical amendments are required to account for unmodelled SH, the results emphasise their importance to achieve a well-specified model. Due to SH beyond the level of regional indicators, locally weighted regressions are proposed. Mixed geographically weighted regression (MGWR) prevents the limitations of fixed effects by exploring spatially stationary and non-stationary price effects. Besides reducing prediction errors, it is concluded that global model misspecifications arise from improper selected fixed effects. Reported findings provide evidence that the SH of implicit prices is more complex than can be modelled by regional indicators or purely local models. The existence of both stationary and non-stationary effects implies that the Austrian housing market is economically connected.

On the estimation and testing of mixed geographically weighted regression models

Mixed geographically weighted regression (MGWR) model is a useful technique to explore spatial non-stationarity by allowing that some coefficients of the explanatory variables are constant and others are spatially varying, but its estimation and inference have not been systematically studied. This paper is concerned with estimation and testing of the model when there are certain linear constraints on the elements of constant coefficients. We propose a constrained two-step technique for estimating the constant coefficients and spatial varying coefficients, and develop a test procedure for the validity of the linear constraints. Finally, some simulations are conducted to examine the performance of our proposed procedure and the results are satisfactory.

A SPATIAL HEDONIC APPROACH ON LAND USE CHANGE ANTICIPATIONS*

ABSTRACT This paper estimates land use conversion anticipation when zoning is the primary tool of land use regulation. Extending the Capozza and Helsey model (1989) to the case of uncertainty in future land use zoning, a spatialized hedonic model is proposed to estimate such anticipation phenomena at a fine level (cadastral unit). Estimations use Mixed Geographically Weighted Regression (MGWR) techniques with a two-stage model that links agricultural and developable land markets. This allows for mapping varying spatial parameters that measure anticipation effects within the theoretical framework. Results confirm the influence of anticipation on agricultural land prices in the Provence region. Moreover, the level of data spatialization allows us to take into account intra-municipalities' heterogeneity of land use conversion anticipation.

Testing the Importance of the Explanatory Variables in a Mixed Geographically Weighted Regression Model

A mixed geographically weighted regression (MGWR) model is a kind of regression model in which some coefficients of the explanatory variables are constant, but others vary spatially. It is a useful statistical modelling tool in a number of areas of spatial data analysis. After an MGWR model is identified and calibrated, which has been well studied recently, one of the important inference problems is to evaluate the influence of the explanatory variables in the constant-coefficient part on the response of the model. This is useful in the selection of the variables and for the purpose of explanation. In this paper, a statistical inference framework for this issue is suggested and, besides the F-approximation, which has been frequently used in the literature of the geographically weighted regression technique, a bootstrap procedure for deriving the p-value of the test is also suggested. The performance of the test is investigated by extensive simulations. It is demonstrated that both the F-approximation and the bootstrap procedure work satisfactorily.

A Note on the Mixed Geographically Weighted Regression Model*

Abstract. A mixed, geographically weighted regression (GWR) model is useful in the situation where certain explanatory variables influencing the response are global while others are local. Undoubtedly, how to identify these two types of the explanatory variables is essential for building such a model. Nevertheless, It seems that there has not been a formal way to achieve this task. Based on some work on the GWR technique and the distribution theory of quadratic forms in normal variables, a statistical test approach is suggested here to identify a mixed GWR model. Then, this note mainly focuses on simulation studies to examine the performance of the test and to provide some guidelines for performing the test in practice. The simulation studies demonstrate that the test works quite well and provides a feasible way to choose an appropriate mixed GWR model for a given data set.