Mixed Geographically Weighted Regression Models Research Articles

Modeling the Flood Disaster in South Kalimantan Using Geographically Weighted Regression and Mixed Geographically Weighted Regression

The flood disaster in South Kalimantan is a crucial problem that needs to be addressed because the impact is relatively severe. So, this study aims to model flood disasters in South Kalimantan based on factors suspected to be the cause, including population density, rainfall, residential area, and forest area. This study uses two methods of spatial statistics, namely the Geographically Weighted Regression (GWR) and Mixed Geographically Weighted Regression (MGWR) methods. The weighting used is Adaptive Gaussian. The modeling results show that the GWR model is superior in explaining the causes of flood events in South Kalimantan, which is indicated by the highest coefficient of determination value of 95.62% compared to the regression and MGWR models. Nonetheless, the MGWR model can explain the causes of flooding in Kalimantan. The GWR and MGWR models show that the area that is vulnerable to flooding is Balangan District. The results of this study contribute to providing alternative information for disaster mitigation to minimize losses.

Exploring associations between built environment and crash risk of children in school commuting

Understanding how built environment are associated with crash risk (CR) in school commuting is essential to improving travel safety through land use and transportation policies. Scholars often assume that this relationship is consistent across space, but this may lead to inconsistent estimates. To address this issue, using data in Shenzhen, China, the data covers traffic accident data of children taken from police incident reports and supplemented with local land use, transportation network and specific school information. The measurement model of crash scale was conducted to represent crash severity, and the CR was further quantified. The study applies three models, spatial dubin model (SDM), geographically weighted regression (GWR), and mixed GWR (MGWR), to explore spatio-temporal heterogeneity relationships between built environment attributes and CR of children in school commuting. The findings reveal that the crash scale can better represent crash severity of school commuting than a single indicator. Policy interventions should be targeted at specific spatial scales, school types, and time windows to effectively improve travel safety. However, there are some common findings. It is recommended to use a scale of 200 m to explain the relationship between the variables. The MGWR model outperforms the other two models. To reduce CR, it is important to consider lower road network density, a reasonable layout of educational facilities, fewer bus routes, and more on-street parking spaces. Our findings can help to enrich the understanding of associations between land use and CR of children, as well as offer local planning and operating guidance for creating child-friendly environment.

Spatial Distribution Characteristics and Influencing Factors of Soil Organic Carbon Density in Yellow River Basin Based on MGWR Model

As the largest terrestrial carbon pool, the spatial distribution characteristics and influencing factors of soil organic carbon have important implications for global carbon cycle processes. Soil organic carbon density (SOCD) and influencing factors were predicted in the Yellow River basin using a mixed geographically weighted regression (MGWR) model based on soil organic carbon density data and environmental factors. The results showed that:① the SOCD ranged from 0-14.82 kg·m-2 and 0-32.39 kg·m-2 for the soil depths of 0-20 cm and 0-100 cm, with mean values of 3.48 kg·m-2 and 8.07 kg·m-2 and reserves of 2.76 Pg and 6.48 Pg, respectively. The high SOCD value areas were mainly located in the southern part of the Qinghai-Tibet Plateau and Loess Plateau, and the low value areas were located in the eastern part of the upper Yellow River and the inland flow area. ②Among the ecosystem types, the SOCD of soil depth in 0-20 cm was in the descending order of:forest>water body and wetland>other>grassland>farmland>settlement>desert, with mean values of 4.52, 4.31, 3.84, 3.73, 2.89, 2.78, and 2.22 kg·m-2, respectively, and the SOCD of the 0-100 cm soil depth was in the descending order of:water bodies and wetlands>forest>other>grassland>farmland>settlement>desert, with mean values of 9.58, 9.58, 8.85, 8.66, 7.07, 6.81, and 5.29 kg·m-2, respectively. The SOCR in descending order was:grassland>farmland>forest>desert>water bodies and wetlands>settlement>others, with 1.40, 0.60, 0.47, 0.11, 0.07, 0.06, and 0.05 Pg at a soil depth of 0-20 cm and 3.31, 1.49, 0.99, 0.26, 0.17, 0.14, and 0.12 Pg at a soil depth of 0-100 cm, respectively. ③ The main factors affecting the SOCD distribution were intercept, profile curvature, NDVI, and precipitation; in addition, curvature and silt also had important effects on the deep SOCD distribution in the Yellow River basin. Among the ecosystem types, precipitation and NDVI were the main factors affecting the SOCD distribution. The intercept also had important effects on the SOCD distribution in the all ecosystems except forests, whereas curvature and silt only had important effects on deserts and other ecosystems. These results revealed the spatial distribution of SOCD, influencing factors, and SOCR in the Yellow River basin and can provide a scientific basis for carbon balance, soil quality evaluation, and ecological management restoration and consolidation in the region.

Combining APHRODITE Rain Gauges-Based Precipitation with Downscaled-TRMM Data to Translate High-Resolution Precipitation Estimates in the Indus Basin

Understanding the pixel-scale hydrology and the spatiotemporal distribution of regional precipitation requires high precision and high-resolution precipitation data. Satellite-based precipitation products have coarse spatial resolutions (~10 km–75 km), rendering them incapable of translating high-resolution precipitation variability induced by dynamic interactions between climatic forcing, ground cover, and altitude variations. This study investigates the performance of a downscaled-calibration procedure to generate fine-scale (1 km × 1 km) gridded precipitation estimates from the coarser resolution of TRMM data (~25 km) in the Indus Basin. The mixed geographically weighted regression (MGWR) and random forest (RF) models were utilized to spatially downscale the TRMM precipitation data using high-resolution (1 km × 1 km) explanatory variables. Downscaled precipitation estimates were combined with APHRODITE rain gauge-based data using the calibration procedure (geographical ratio analysis (GRA)). Results indicated that the MGWR model performed better on fit and accuracy than the RF model to predict the precipitation. Annual TRMM estimates after downscaling and calibration not only translate the spatial heterogeneity of precipitation but also improved the agreement with rain gauge observations with a reduction in RMSE and bias of ~88 mm/year and 27%, respectively. Significant improvement was also observed in monthly (and daily) precipitation estimates with a higher reduction in RMSE and bias of ~30 mm mm/month (0.92 mm/day) and 10.57% (3.93%), respectively, after downscaling and calibration procedures. In general, the higher reduction in bias values after downscaling and calibration procedures was noted across the downstream low elevation zones (e.g., zone 1 correspond to elevation changes from 0 to 500 m). The low performance of precipitation products across the elevation zone 3 (>1000 m) might be associated with the fact that satellite observations at high-altitude regions with glacier coverage are most likely subjected to higher uncertainties. The high-resolution grided precipitation data generated by the MGWR-based proposed framework can facilitate the characterization of distributed hydrology in the Indus Basin. The method may have strong adoptability in the other catchments of the world, with varying climates and topography conditions.

Spatial non-stationary characteristics between grass yield and its influencing factors in the Ningxia temperate grasslands based on a mixed geographically weighted regression model

Spatial models are effective in obtaining local details on grassland biomass, and their accuracy has important practical significance for the stable management of grasses and livestock. To this end, the present study utilized measured quadrat data of grass yield across different regions in the main growing season of temperate grasslands in Ningxia of China (August 2020), combined with hydrometeorology, elevation, net primary productivity (NPP), and other auxiliary data over the same period. Accordingly, non-stationary characteristics of the spatial scale, and the effects of influencing factors on grass yield were analyzed using a mixed geographically weighted regression (MGWR) model. The results showed that the model was suitable for correlation analysis. The spatial scale of ratio resident-area index (PRI) was the largest, followed by the digital elevation model, NPP, distance from gully, distance from river, average July rainfall, and daily temperature range; whereas the spatial scales of night light, distance from roads, and relative humidity (RH) were the most limited. All influencing factors maintained positive and negative effects on grass yield, save for the strictly negative effect of RH. The regression results revealed a multiscale differential spatial response regularity of different influencing factors on grass yield. Regression parameters revealed that the results of Ordinary least squares (OLS) (Adjusted R2 = 0.642) and geographically weighted regression (GWR) (Adjusted R2 = 0.797) models were worse than those of MGWR (Adjusted R2 = 0.889) models. Based on the results of the RMSE and radius index, the simulation effect also was MGWR > GWR > OLS models. Ultimately, the MGWR model held the strongest prediction performance (R2 = 0.8306). Spatially, the grass yield was high in the south and west, and low in the north and east of the study area. The results of this study provide a new technical support for rapid and accurate estimation of grassland yield to dynamically adjust grazing decision in the semi-arid loess hilly region.

PEMODELAN MIXED GEOGRAPHICALLY WEIGHTED REGRESSION DENGAN ADAPTIVE BANDWIDTH UNTUK ANGKA HARAPAN HIDUP (Studi Kasus : Angka Harapan Hidup di Jawa Tengah)

Life expectancy at birth (AHH) is an estimate of the years a person will take from birth. AHH is used as an indicator of public health and welfare. These two indicators are of concern to the government in relation to human development. It is hoped that the AHH value will continue to increase so that the quality of human development will also increase. Modeling of the factors that influence AHH needs to be done so that efforts to increase AHH become more effective.The AHH value for Central Java (Central Java) in 2020 is 74.37. Factors thought to influence AHH in Central Java are the percentage of poor people (X1), the percentage of households with proper sanitation (X2), the percentage of children under five who are fully immunized (X3) and the open unemployment rate (X4). The assumption of homoscedasticity in AHH modeling in Central Java using linear regression was not fulfilled, meaning that there was spatial heterogeneity between districts/cities, so the Geographically Weighted Regression (GWR) method was used. The weighting function used is the Bisquare and Tricube kernels with adaptive bandwidth. The GWR method will encounter problems if not all independent variables are local, so the Mixed Geographically Weighted Regression (MGWR) method is used. The results of the GWR analysis for the two weighting functions are that the X1 variable is not local, so the MGWR method is used. The results of MGWR modeling for the two weighting functions are that local variables and global variables have a significant effect. The best model is the MGWR model with Kernel Tricube weighting because it has the smallest AICc value. Keyword : AHH, GWR, MGWR, Adaptive Kernel Bisquare, Adaptive Kernel Tricube, AICc

Spatially Varying Relation between Built Environment and Station-Level Subway Passenger-Distance

As a sustainable mode of transportation, subways bring great convenience to the society. Although there have been many studies examining the relationship between the built environment and the station-level ridership, those studies focused mainly on the ridership, which is defined as the number of trips for each station. While ridership is an important indicator for evaluating subway demand, passenger-distance is another critical indicator that incorporates distance into demand evaluation, which has not yet been fully explored. To fill this gap, this paper investigates the relationship between the built environment around stations and the station-level passenger-distance (SLPD). As noted in previous studies, the relationship between the built environment and travel demand can vary by space. Therefore, a geographically weighted regression (GWR) model and a mixed geographically weighted regression (MGWR) model have been used to explore this spatially varying relationship using Chengdu, China, as an example case. The results were compared with that of an ordinary least squares (OLS) model. The comparison shows that the MGWR model that considers both global and local variables has the best goodness of fit. Results also show that 11 of the 25 potential variables are significantly related to SLPD. The accessibility of the station, station type, such as transfer or terminal, number of bus stops, number of restaurants, density of building area, density of the national road network, and density of the provincial road network, all have a positive correlation with SLPD. Meanwhile, the variables, whether it is a newly opened subway station, density of living points of interest (POIs), and density of railroad network, are all negatively correlated with SLPD. Ten of the eleven significant variables (except accessibility) have spatially varying relationships with SLPD. These findings can serve a useful reference for transportation planners for the demand evaluation.

Characteristics of parameter mixed geographically weighted regression model : global (a-group) and local (b-group)

Mixed Geographically Weighted Regression (MGWR) is a combination of a linear regression model and a Geographically Weighted Regression (GWR) model that considers several variables to be constant and some to vary spatially. The MGWR model also considers location characteristics as indicated by global and local variables. The MGWR model is more flexible than the GWR model because its characteristics are the same for each location. These characteristics are shown in the variables contained in the MGWR model, namely global and local variables. Global variables produce global a-group parameters that have the same effect for all observation locations and local variables produce local b-group parameters that have different effects for each observation location. This article reviews the characteristics of global (a-group) and local (b-group) parameters in the MGWR model. The research shows that the two parameters can be predicted by different methods, estimating global parameters in the regression model by the least squared (LS) method and estimating local parameters by the weighted least squared method (WLS). The difference in this parameter estimation method is due to local parameters influenced by the presence of observational characteristics so that the assumption does not occur heteroscedasticity cannot be obtained.

Pengujian Hipotesis Parameter Model Mixed Geographically Weighted Regression Data Indeks Pembangunan Manusia di Kalimantan Tahun 2016

Mixed Geographically Weighted Regression (MGWR) model is a Geographically Weighted Regression (GWR) model with some parameters are global (have the same value) and several other parameters are local (have different values) for each observation location. The purpose of this study is to obtain a MGWR model on the Human Development Index (HDI) data and find out the factors that influence the HDI of each district (city) in the provinces of East Kalimantan, Central Kalimantan and South Kalimantan in 2016. The parameter estimation method is carried out through two stages (backshift), namely local parameter estimation by using the Weighted Least Square (WLS) method and global parameter estimation by using the Ordinary Least Square (OLS) method. Spatial weighting on local parameter estimation is obtained by using an adaptive Bisquare weighting functions, where optimum bandwidth determination uses Generalized Cross-Validation (GCV) criterion. Based on the result of MGWR parameter testing, it was concluded that the school enrollment rates (SMP) affected the HDI of all districts (cities) in East Kalimantan, Central Kalimantan and South Kalimantan, while the population density affects the HDI only in a few districts (cities), namely East Kutai, Balikpapan, Samarinda and Bontang.

Penerapan Model Mixed Geographically Weighted Regression dengan Fungsi Pembobot Adaptive Tricube pada IPM 30 Kabupaten/Kota di Provinsi Kalimantan Timur, Kalimantan Tengah dan Kalimantan Selatan Tahun 2016

Mixed Geographically Weighted Regression (MGWR) model is a Geographically Weighted Regression (GWR) model which has global (equal value) and local (inequal value) parameters at every different observation location. The goal of this study is to obtain MGWR model of the Human Development Index (HDI) data and find out significant factors influencing the HDI in each district (city) East Kalimantan, Central Kalimantan and South Kalimantan province in 2016. Parameter estimation method is conducted in two stages namely local parameter estimation and global parameter estimation. Local parameter estimation method is Maximum Likelihood Estimation (MLE), with spatial weighting is calculated by adaptive tricube weighting function and optimum bandwidth determination uses the Akaike Information Criteria (AIC). Global parameter estimation method is Ordinary Least Square (OLS). Based on the result of MGWR parameter testing, it was concluded that the school enrollment rates (SMP) and poor people percentage affected the HDI of 30 districts (cities) in East Kalimantan, Central Kalimantan and South Kalimantan. Meanwhile the population density affected the HDI of two districts namely HDI of Samarinda and Bontang.

The Economics of Blockchain-Based Virtual Worlds: A Hedonic Regression Model for Virtual Land

Virtual worlds have become increasingly more important, and are being used for a large variety of recreational and business activities. Various companies have opened virtual offices and brand ambassador spaces, and the recent surge in online interactions has further increased the interest in these social platforms. In this paper, we analyze the first large-scale virtual world that has been built on a public blockchain network. We compile a comprehensive data set that allows us to study the economics of blockchain-based virtual property. We then use this data set to estimate hedonic regression models for virtual land and employ a semi-parametric Mixed Geographically Weighted Regression (MGWR) model that allows for spatially flexible coefficients. In particular, we expect the central business district (CBD) gradient and the street distance coefficients to vary across the virtual world. Conversely, we consider the coefficients of binary variables to be more meaningful in a global context. For inference purposes, we employ spatial robust heteroskedasticity and autocorrelation consistent (HAC) standard errors. We find evidence that prices of virtual land parcels are primarily driven by factors that are of particular importance in business applications. Land parcels in close proximity to popular landmarks, such as the city center, main streets, plazas and commercially-driven districts, as well as parcels with easily memorable address-like features were claimed at substantially higher prices. We used different model specifications and conducted various robustness checks, including a separate estimation for price segments and investor-specific effects. All specifications have led to similar results and support the evidence from our MGWR model.

Efficient estimation of heteroscedastic mixed geographically weighted regression models

Mixed geographically weighted regression (MGWR) models are a useful tool to model a regression relationship where the impact of some explanatory variables on the response variable is global and that of the others is spatially varying. The existing estimation methods for MGWR models assume that the model errors are homoscedastic. However, heteroscedasticity is very common in geo-referenced data and ignoring heteroscedasticity may cause efficiency loss on the coefficient estimates. In this paper, we propose a re-weighting estimation method for heteroscedastic MGWR models, in which the variance function of the model errors is estimated by the kernel method with an adaptive bandwidth and the coefficients are re-estimated based on the weighted observations. The simulation study shows that the proposed method can substantially improve the estimation efficiency especially for the constant coefficients. A real-world example based on the Dublin voter turnout data is given to demonstrate the application of the proposed method.

Scale-adaptive estimation of mixed geographically weighted regression models

Mixed geographically weighted regression (GWR) models, a combination of linear and spatially varying coefficient models, have found their applications in a variety of disciplines including economic modelling for geo-referenced data analysis. Generally, different explanatory variables may operate at different spatial scales, leading to different levels of spatial heterogeneity of the varying coefficients. To deal with such a multiscale problem, we propose a scale-adaptive method to calibrate mixed GWR models, in which a different bandwidth is separately assumed for each spatially varying coefficient and is selected based on the backfitting procedure. Extensive simulations with different spatial layouts and a real-world example based on the Dublin voter turnout data demonstrate that the scale-adaptive method can not only significantly improve the estimation accuracy of the spatially varying coefficients, but also provide valuable information on the scale at which each explanatory variable operates.

Direct modeling of subway ridership at the station level: a study based on mixed geographically weighted regression

Station-level ridership modeling is one of the ways to forecast metro ridership and reveal how factors influence ridership. Previous studies assumed that the relationships between the dependent variable and independent variables are either global or local, as indicated by the global model or the geographically weighted regression (GWR) model. This study explores the possibility that some independent variables have spatially varying relationships with metro ridership while others have constant relationships by employing the mixed GWR model. Data from the Chicago metro system were used. To establish an effective forecasting model, possible influencing factors are collected. OLS model results indicate that the proportion of recreational jobs to total jobs, number of bus stops, employment density, number of high-income workers, and the type of station (transfer or terminal) are significant variables influencing station-level metro ridership. By using the mixed GWR model, we find that the proportion of recreational jobs to total jobs is a global variable while the others are local variables. By comparing the results of mixed GWR, full GWR, and OLS models, we find that mixed GWR fits the data better and the residuals are less correlated. However, results of cross-validation indicate that the prediction power of the OLS model is better than that of the full and mixed GWR models.

Geographically Weighted Regression Models in Estimating Median Home Prices in Towns of Massachusetts Based on an Urban Sustainability Framework

Housing is a key component of urban sustainability. The objective of this study was to assess the significance of key spatial determinants of median home price in towns in Massachusetts that impact sustainable growth. Our analysis investigates the presence or absence of spatial non-stationarity in the relationship between sustainable growth, measured in terms of the relationship between home values and various parameters including the amount of unprotected forest land, residential land, unemployment, education, vehicle ownership, accessibility to commuter rail stations, school district performance, and senior population. We use the standard geographically weighted regression (GWR) and Mixed GWR models to analyze the effects of spatial non-stationarity. Mixed GWR performed better than GWR in terms of Akaike Information Criterion (AIC) values. Our findings highlight the nature and spatial extent of the non-stationary vs. stationary qualities of key environmental and social determinants of median home price. Understanding the key determinants of housing values, such as valuation of green spaces, public school performance metrics, and proximity to public transport, enable towns to use different strategies of sustainable urban planning, while understanding urban housing determinants—such as unemployment and senior population—can help modify urban sustainable housing policies.

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

Geographically weighted regression (GWR) models have been employed in previous studies regarding vehicular travel demands, but few studies have locally modeled walking travel demands at intersections to address the issue of spatially varying relationships. Harnessing a comprehensive collection of walking and bicycling traffic counts over 10years in Chittenden County, Vermont, US, along with socioeconomic characteristics, transit accessibility indices, land use attributes and characteristics of intersections and roadway networks, this study utilizes GWR models to identify whether there are spatially varying relationships between active mode travel demands and ambient built-environment attributes. One Ordinary Least Square (OLS) model and two GWR models were parametrically calibrated: a full GWR model of all local variables and a mixed GWR model of both global and local variables. K-fold cross-validation method is used to select variables that significantly influence the volume of active travel modes in the OLS model. The uniform set of variables is investigated in two GWR models. Only residuals of the mixed GWR model exhibit spatial independence. The prediction accuracy of the three models is respectively compared by means of the k-fold cross-validation method. Results show that the mixed GWR model has higher prediction accuracy, while the other two models have roughly the same level of performance. We find that not all independent variables possess a spatially varying relationship with active mode volumes. The flexibility of the mixed GWR model that allows some independent variables to be global strengthens its prediction power. With these findings, transportation planners can dynamically estimate bicycle and pedestrian volumes at widespread intersections, and this geographical realism would facilitate local transportation planning, facility design, safety enhancement and operation analysis, as well as instilling new insights into interdisciplinary spatial research domain.

MIXED GEOGRAPHICALLY WEIGHTED REGRESSION USING ADAPTIVE BANDWIDTH TO MODELING OF AIR POLLUTER STANDARD INDEX

Air pollution is one of the most concerned problems on earth today. It is closely related with and mostly generated from the transportation and industrialization sectors, as well as from the environmentally degrading effect of the urban physical development. Air pollution promotes the lower level of air quality, which in turn promotes the greater risk on health, especially that of the human being. This research aims to aid the government in the policy making process related to air pollution mitigation by developing a standard index model for air polluter (Air Polluter Standard Index - APSI) based on the Mixed Geographically Weighted Regression (MGWR) approach using the adaptive bandwidth. The adaptive bandwidth kernel has different bandwidth value in each observation location. Akaike Information Criterion-corrected (AICc) value is used to choose the most optimum bandwidth. The Monte Carlo Simulation is used to tests for regression coefficient non-stationarity. In this research, we also consider seven variables that are directly related to the air pollution level, which are the traffic velocity, the population density, the business center aspect, the air humidity, the wind velocity, the air temperature, and the area size of the urban forest. Based on AICc and MSE value it is know that the MGWR model with adaptive bisquare kernel is the best bandwidth to analyze this model.

Mixed geographically weighted regression (MGWR) model with weighted adaptive bi-square for case of dengue hemorrhagic fever (DHF) in Surakarta

ABSTRACTMGWR model is combination of linear regression model and geographically weighted regression (GWR) model, therefore, MGWR model could produce parameter estimation that had global parameter estimation, and other parameter that had local parameter in accordance with its observation location. The linkage between locations of the observations expressed in specific weighting that is adaptive bi-square. In this research, we applied MGWR model with weighted adaptive bi-square for case of DHF in Surakarta based on 10 factors (variables) that is supposed to influence the number of people with DHF. The observation unit in the research is 51 urban villages and the variables are number of inhabitants, number of houses, house index, many public places, number of healthy homes, number of Posyandu, area width, level population density, welfare of the family, and high-region. Based on this research, we obtained 51 MGWR models. The MGWR model were divided into 4 groups with significant variable is house index as a global variable, an area width as a local variable and the remaining variables vary in each. Global variables are variables that significantly affect all locations, while local variables are variables that significantly affect a specific location.

A Mixed Geographically and Temporally Weighted Regression: Exploring Spatial-Temporal Variations from Global and Local Perspectives

To capture both global stationarity and spatiotemporal non-stationarity, a novel mixed geographically and temporally weighted regression (MGTWR) model accounting for global and local effects in both space and time is presented. Since the constant and spatial-temporal varying coefficients could not be estimated in one step, a two-stage least squares estimation is introduced to calibrate the model. Both simulations and real-world datasets are used to test and verify the performance of the proposed MGTWR model. Additionally, an Akaike Information Criterion (AIC) is adopted as a key model fitting diagnostic. The experiments demonstrate that the MGTWR model yields more accurate results than do traditional spatially weighted regression models. For instance, the MGTWR model decreased AIC value by 2.7066, 36.368 and 112.812 with respect to those of the mixed geographically weighted regression (MGWR) model and by 45.5628, −38.774 and 35.656 with respect to those of the geographical and temporal weighted regression (GTWR) model for the three simulation datasets. Moreover, compared to the MGWR and GTWR models, the MGTWR model obtained the lowest AIC value and mean square error (MSE) and the highest coefficient of determination (R2) and adjusted coefficient of determination (R2adj). In addition, our experiments proved the existence of both global stationarity and spatiotemporal non-stationarity, as well as the practical ability of the proposed method.

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.