 Methodology
 Open Access
A modification to geographically weighted regression
 YinYee Leong^{1} and
 Jack C. Yue^{1}Email author
https://doi.org/10.1186/s1294201700859
© The Author(s) 2017
 Received: 12 October 2016
 Accepted: 20 March 2017
 Published: 31 March 2017
Abstract
Background
Geographically weighted regression (GWR) is a modelling technique designed to deal with spatial nonstationarity, e.g., the mean values vary by locations. It has been widely used as a visualization tool to explore the patterns of spatial data. However, the GWR tends to produce unsmooth surfaces when the mean parameters have considerable variations, partly due to that all parameter estimates are derived from a fixed range (bandwidth) of observations. In order to deal with the varying bandwidth problem, this paper proposes an alternative approach, namely Conditional geographically weighted regression (CGWR).
Methods
The estimation of CGWR is based on an iterative procedure, analogy to the numerical optimization problem. Computer simulation, under realistic settings, is used to compare the performance between the traditional GWR, CGWR, and a local linear modification of GWR. Furthermore, this study also applies the CGWR to two empirical datasets for evaluating the model performance. The first dataset consists of disability status of Taiwan’s elderly, along with some socialeconomic variables and the other is Ohio’s crime dataset.
Results
Under the positively correlated scenario, we found that the CGWR produces a better fit for the response surface. Both the computer simulation and empirical analysis support the proposed approach since it significantly reduces the bias and variance of data fitting. In addition, the response surface from the CGWR reviews local spatial characteristics according to the corresponded variables.
Conclusions
As an explanatory tool for spatial data, producing accurate surface is essential in order to provide a first look at the data. Any distorted outcomes would likely mislead the following analysis. Since the CGWR can generate more accurate surface, it is more appropriate to use it exploring data that contain suspicious variables with varying characteristics.
Keywords
 Geographically weighted regression
 Modifiable areal unit problem (MAUP)
 Generalized additive model
 Computer simulation
 Cross validation
Background
The data collected nowadays are diversified and many of them possess the records of locations, namely spatial data. Spatial regression is a popular tool for analysing the spatial data [2, 35] and the firstorder stationarity is a common assumption, which means that the expected (mean) values are fixed at different locations. The error terms of spatial regression are usually not independent and, like in time series analysis, their covariance is assumed to follow some spatial models, such as the simultaneous autoregressive (SAR) and moving average (MA) models [12, 30, 34]. However, the firstorder stationarity is a questionable assumption in practice and the modifiable areal unit problem (MAUP) often occurs [5, 13, 22]. The MAUP is a spatial version of the Simpson paradox, where the trends appearing in individual groups of data are different to those in the aggregate data. The biased estimates might be a consequence of the parameter values not being identical in the study area and the inclusion of data with different attributes.
Because the parameter values are not identical at different locations, estimation via the ordinary least squares (OLS) with all observations would likely distort the local distinctness. One possible solution is to include only the locations of data with similar attributes (i.e., homogeneity). However, it is difficult to decide the number of groups with different attributes and identify the locations of data in each group. Moreover, the mean value of a nonstationary 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 possibility 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].
The GWR allows the regression coefficients to vary across space, and the coefficient estimates of all variables are obtained from a moving data window, which is analogous to kernel regression for obtaining a smoothing estimate. It is also a popular tool for exploratory data analysis (EDA) on spatial data [19, 32]. In particular, the GWR is often a popular visualization tool in geographical information system, to explore possible patterns of a study region and acquire valuable information for further data analysis (such as clusters detection) [11, 36]. Note that the optimal width (or bandwidth) of the moving windows in a GWR is determined by crossvalidation (CV) or Akaike’s information criterion (AIC) [16]. The OLS can be treated as a special case of the GWR with a window of infinite width (although the local distinctness is likely to be lost by averaging all observations).
Most of the modifications to the GWR are on the selection and testing of the bandwidth. For example, using the CV and AIC for the bandwidth selection is a datadriven method, similar to the kernel regression method, wherein the estimates are sensitive to outliers [16]. In addition, the data variations are not necessarily the same and a fixed bandwidth is likely to create discrepancy in parameters’ estimates at different locations. On the other hand, the hypothesis testing of the parameters depends on the bandwidth as well. For example, Leung et al. [21] proposed goodnessoffit tests and found that the degree of freedom of GWR residuals is a function of the bandwidth, and this makes bandwidth selection somewhat subjective.
Determining the bandwidth probably is the focus of modifying GWR over the years. Brunsdon et al. [7] introduced a mixed GWR model with vector bandwidths, allowing the coefficients having different bandwidths (via a backfitting algorithm) and the bandwidths being functions of data density. Shi et al. [27] suggested the weight of data determined by their attributes, rather than by the distance between observations. Furthermore, Farber and Páez [16] found that it is possible to reduce the bias by modifying the CV procedure. Subsequently, Wang et al. [31] introduced local linear estimation, or a polynomial fitting technique, to reduce the bias in parameters’ estimates.
In this study, our focus is also on the bandwidth selection for each variable, using the correlations between independent variables. The idea of the proposed approach is to use the correlations to improve the estimation via an iteration algorithm, similar to the method of control variate in variance reduction [24]. The empirical analyses of GWR showed that correlations often exist between GWR coefficients. For example, Bivand and Brunstad [4] found the coefficients to be highly correlated in a case study. We found the rough coefficient surfaces can be smoother (Fig. 1) if the coefficients are positively correlated.
For the rest of this manuscript, we first introduce the GWR and the proposed modification of GWR, conditional GWR (CGWR), and its theoretical results. Then, we use simulation to evaluate the proposed method and compare it with the basic GWR and the local linear method proposed by Wang et al. [31]. In addition to simulations, we also apply the proposed method to two data sets for empirical study. Finally, we conclude with discussions on the limitations and future applicability of the proposed method.
Methods
As mentioned earlier, the bandwidth selection is generally calibrated by minimizing the CV score or the AIC. However, if the data locations are sparse in the study area, the distanceweighted kernel might not be appropriate due to insufficient information. Brunsdon et al. [8] introduced rankbased and knearest neighbourhood methods to deal with sparse data. In addition to the GWR, we also consider one of its modifications by Wang et al. [31]. It is a local linear approach, or a Taylor expansion version of the GWR, and is expected to have better fitting if the geographical surface is linearshaped.
Using a single bandwidth in GWR is likely to create unsatisfactory estimates if the attributes of independent variables are not similar. For example, independent variables with larger variations require a larger bandwidth (and more observations). On the other hand, locally sampling (i.e. narrower bandwidth) is preferable over global sampling (i.e. wider bandwidth) for the areas with larger gradient changes. The concept is similar to widely recognized importance sampling [25], which assigns more sampling weight on informative area. Therefore, allowing different bandwidths for each independent variable seems to be a desirable modification to the GWR. Unfortunately, the varying bandwidths cannot be managed by the weighted least squares or Eq. (2). Brunsdon et al. [7] proposed a backfitting algorithm for selecting different bandwidths, but the selection of bandwidths is somewhat objective and it usually require lots of computation time.

Step 1. Set the initial solution of \(\varvec{f}_{k}\) to be zero, i.e. \(\varvec{f}_{k} \{ 0\} = {\mathbf{0}}\), where \(k = 1, \ldots ,p\), and let \(l = 1\).

Step 2. For each element \(\varvec{f}_{k} \{ l\}\), apply the basic GWR model with only one independent variable, \(x_{k} .\) The value of the dependent variable is \(\varvec{y}^{*} = \varvec{y}  \sum\nolimits_{\begin{subarray}{l} j = 1 \\ j \ne k \end{subarray} }^{p} {\varvec{f}_{j} \{ \varvec{l}  {\mathbf{1}}\} }\), i.e. we regress \(\left( {\varvec{y}  \sum\nolimits_{\begin{subarray}{l} j = 1 \\ j \ne k \end{subarray} }^{p} {\varvec{f}_{j} \{ l  1\} } } \right)\) on the variable \(\varvec{x}_{k}\) without a fitting intercept. The bandwidth is obtained by minimizing the crossvalidated sum of squares (CVSS) or the AIC.

Step 3. Repeat Step 2 until the given stopping criterion is reached.
There are at least two reasons for finding optimal bandwidth solutions individually using the Jacobi iteration. First, although more complex numerical methods (such as the quasiNewton method) could be used, the Jacobi iteration usually requires less computation time. Second, although there are algorithms that converge faster than the Jacobi iteration, they are likely to produce biased estimates. For example, in the Gauss–Seidel iteration process, the estimate of one variable is updated based on the simultaneous estimates of other variables. If the estimates of some variables have severe biases, it might contaminate the estimates of other variables.
We think that the proposed estimation process can guarantee the convergence of the CGWR. In particular, if the bandwidth is predetermined during the iteration, then the GWR coefficients will converge to a constant for each location. We should use the case of two coefficients to demonstrate the convergence and the outline proof is given in Appendix A in Additional File 1. Note that the method of Brunsdon et al. [7] can be treated as a special case of the CGWR method when the bandwidths are never updated. In the next section, we will use computer simulation to evaluate the stability of the CGWR and compare it with basic GWR and its local linear modification by Wang et al. [31].
Results and discussion
Simulated data
The computer simulation is separated into two parts: scenarios without clusters and with a cluster. For the latter scenario, a cluster is added into the intercept to exhibit the mean shift intervention. The cluster scenario is to evaluate the performance of estimation methods under the influence of a systematic change (or hot spots) in space, such as sources of pollution. Moreover, the coefficients are assumed to follow one of the following four surfaces: linear, quadratic, ridge, or hillside, and these settings are to check which would cause raggedness in the estimated surfaces. For the former scenario, we also examine two types of surfaces: singletype and mixedtype. The difference between these two types of surfaces is whether the coefficients follow same type of surfaces (singletype) or different types of surfaces (mixedtype). We want to know if the coefficients follow different types of surface would cause biased estimation of coefficients.
For a scenario without clusters, several cases are tested under different S/N ratios. The first case is the singletype surface, where both intercept and independent variable follow the same type of surface. The second case, namely mixedtype surface, assumes that the intercept and the independent variable follow different types of surfaces. For the second case, we only examine two combinations: linearquadratic (a polynomial surface) and ridgehillside (a nonpolynomial surface).
The scenario settings of simulation study
Mean shift intervention  

No cluster  Cluster  
Surfaces type  
I. Singletype surface  1. Linear, 2. quadratic, 3. ridge, 4. hillside  1. Linear, 2. hillside 
II. Mixedtype surface  1. (Linear, quadratic) 2. (ridge, hillside)  – 
Fitting method  
Twostage fitting procedure  1. Linear, 2. hillside  – 
For CGWR, the Gaussian kernel is chosen, and the optimal bandwidth is the one with the minimum CVSS. Furthermore, we require a reasonable range of bandwidth to prevent the estimation from being too localized or globalized for extremely small or large bandwidths, respectively. The upper bound of the range is the maximum length on the map, and the lower bound has to have at least five data points each of 1/5th weight. The preceding setting is also used in the ‘spgwr’ package [3] (version 0.5–4) of R, a free statistical software.
We will first show the simulation results of the scenario without clusters. In particular, we will compare three GWR estimations with the smoothness of the mean surface, the average discount rate, average bandwidth, average variance, and average bias of estimates. The stopping criterion for the CGWR is reached when the average absolute relative change rate of \(\beta_{0}\) and \(\beta_{1}\) is less than 0.005% of the previous step. The simulation results are similar if adopting smaller stopping criteria.
To simplify the notation, we will use \(\beta_{0}\) and \(\beta_{1}\) to denote the coefficients of the intercept and slope of the independent variable x, respectively. In the case of singletype surfaces and mixedtype surfaces, the two coefficients are assumed to be perfectly positively correlated and close to uncorrelated, respectively. We have not considered the negative correlation because the proposed algorithm does not work when the coefficients are not positively correlated. Nonetheless, we will consider a twostage modification for CGWR when the coefficients are not positively correlated.
Singletype surfaces
Average discount rates on a linear surface (singletype)
S/N ratio  \(\beta_{ 0}\)  \(\beta_{1}\)  

\(\beta_{ 0}\)  1  3  5  1  3  5  
\(\beta_{1}\)  
Basic GWR  1  1.064  0.312  0.158  1.815  1.589  1.328 
3  1.348  0.321  0.173  0.598  0.663  0.755  
5  1.775  0.382  0.199  0.329  0.397  0.459  
Local linear  1  1.317  0.192  0.078  2.111  1.033  0.686 
3  1.107  0.164  0.069  0.403  0.347  0.284  
5  1.299  0.188  0.059  0.197  0.179  0.126  
CGWR  1  0.664  0.195  0.082  1.076  0.775  0.412 
3  0.880  0.202  0.086  0.400  0.366  0.286  
5  0.899  0.227  0.092  0.198  0.207  0.178 
Average discount rates on a quadratic surface (singletype)
S/N ratio  \(\beta_{ 0}\)  \(\beta_{1}\)  

\(\beta_{ 0}\)  1  3  5  1  3  5  
\(\beta_{1}\)  
Basic GWR  1  1.955  0.791  0.510  2.877  7.729  7.542 
3  3.217  0.896  0.570  1.418  2.332  3.650  
5  4.334  1.108  0.639  0.844  1.316  1.819  
Local linear  1  3.279  0.914  0.535  5.159  8.900  7.941 
3  5.177  1.119  0.599  2.039  2.634  3.481  
5  6.147  1.196  0.602  1.001  1.169  1.488  
CGWR  1  1.168  0.324  0.146  1.529  1.612  0.928 
3  1.491  0.380  0.166  0.675  0.713  0.730  
5  1.709  0.429  0.183  0.374  0.451  0.430 
Average discount rates on a ridge surface (singletype)
S/N ratio  \(\beta_{ 0}\)  \(\beta_{1}\)  

\(\beta_{ 0}\)  1  3  5  1  3  5  
\(\beta_{1}\)  
Basic GWR  1  1.740  0.571  0.383  2.642  2.325  2.249 
3  2.623  0.728  0.401  1.150  1.375  1.321  
5  2.892  0.824  0.433  0.591  0.810  0.894  
Local linear  1  2.890  1.023  0.621  4.745  4.374  3.672 
3  4.340  1.206  0.618  1.960  2.255  2.023  
5  5.385  1.386  0.677  1.081  1.326  1.344  
CGWR  1  0.884  0.203  0.111  1.029  0.428  0.258 
3  1.048  0.239  0.116  0.529  0.385  0.250  
5  0.854  0.221  0.107  0.253  0.208  0.159 
Average discount rates on a hillside surface (singletype)
Signal/noise ratio  \(\beta_{ 0}\)  \(\beta_{1}\)  

\(\beta_{ 0}\)  1  3  5  1  3  5  
\(\beta_{1}\)  
Basic GWR  1  1.075  0.315  0.153  1.697  1.905  1.359 
3  1.353  0.340  0.173  0.583  0.603  0.668  
5  1.152  0.343  0.185  0.234  0.278  0.360  
Local linear  1  1.339  0.243  0.149  2.187  1.549  1.239 
3  1.403  0.289  0.188  0.617  0.529  0.722  
5  1.113  0.339  0.209  0.266  0.307  0.421  
CGWR  1  0.726  0.170  0.077  1.098  0.761  0.473 
3  0.926  0.199  0.079  0.384  0.348  0.239  
5  0.578  0.192  0.079  0.159  0.158  0.140 
The CGWR and the local linear method again outperform the basic GWR method in the case of a quadratic surface. However, the CGWR appears to be the best, and the advantage is more obvious when the S/N is increased. For the nonlinear surfaces, the CGWR continues to work satisfactorily, whereas the local linear model does not. In fact, the local linear model might even produce worse results than the basic GWR. The CGWR is still reliable for the nonlinear surfaces, and it performs much better than the other two methods.
Intuitively, we expect that the bandwidth to be small if the S/N is large because distant observations can be very different and cause biased estimations. In general, all three GWR methods have significant drops in bandwidth when the S/N ratio increases from one to three. Moreover, the bandwidths for a linear surface should be larger than those for a nonlinear surface under the same S/N ratio because the surface change is quite homogenous in any direction.
The average bandwidths for linear and hillside surfaces (singletype)
S/N  Linear surface  Hillside surface  

Method  Basic GWR  Local linear  CGWR  Basic GWR  Local linear  CGWR  
1  2.35  9.73  4.16; 7.24  2.25  9.78  4.72; 7.45  
3  1.46  10.8  1.65; 2.31  1.47  9.01  1.57; 2.80  
5  1.2  10.34  1.25; 1.47  1.24  6.04  1.25; 1.66 
The bandwidths of the CGWR seem to be related to the signal strength. For example, if the S/N ratio is small, the bandwidth is expected to be large in order to provide a stable estimate. If we fix the S/N ratio of \(\beta_{1}\), then the \(\beta_{0}\) bandwidth of the CGWR decreases as the S/N ratio of \(\beta_{0}\) increases for all four surfaces (Appendix B in Additional file 1). Similar results hold for the \(\beta_{1}\) bandwidths if we fix the S/N ratio of \(\beta_{0}\). The simulation results of CGWR match our expectations.
Average variances and biases of \(\beta_{ 0}\) and \(\beta_{1}\) on a linear surface (singletype)
S/N of \(\beta_{ 0}\):  OLS  Basic GWR  Local linear  CGWR  

1  5  1  5  1  5  1  5  
(i) \(\beta_{ 0}\)  
Conditional on S/N of \(\beta_{1}\) = 1  
Variance  0.009  0.011  0.039  0.067  0.051  0.052  0.024  0.036 
Bias  0.029  0.028  0.002  0.004  0.0003  0.0002  0.001  0.0002 
Conditional on S/N of \(\beta_{1}\) = 5  
Variance  0.010  0.009  0.090  0.107  0.057  0.043  0.048  0.057 
Bias  0.723  0.719  0.026  0.038  0.0005  0.0002  0.012  0.009 
(ii) \(\beta_{1}\)  
Conditional on S/N of \(\beta_{1}\) = 1  
Variance  0.031  0.036  0.113  0.190  0.146  0.145  0.071  0.117 
Bias  0.038  0.695  0.013  0.051  0.001  0.0005  0.004  0.028 
Conditional on S/N of \(\beta_{1}\) = 5  
Variance  0.030  0.027  0.248  0.294  0.152  0.114  0.088  0.152 
Bias  0.191  0.880  0.048  0.124  0.001  0.0002  0.004  0.010 
Average variances and biases of \(\beta_{ 0}\) and \(\beta_{1}\) on a hillside surface (singletype)
S/N of \(\beta_{ 0}\)  OLS  Basic GWR  Local linear  CGWR  

1  5  1  5  1  5  1  5  
(i) \(\beta_{ 0}\)  
Conditional on S/N of \(\beta_{1}\) = 1  
Variance  0.008  0.008  0.031  0.060  0.044  0.053  0.026  0.040 
Bias  0.028  0.702  0.003  0.033  0.002  0.033  0.002  0.009 
Conditional on S/N of \(\beta_{1}\) = 5  
Variance  0.009  0.008  0.053  0.080  0.048  0.061  0.025  0.046 
Bias  0.027  0.703  0.003  0.042  0.003  0.041  0.001  0.008 
(ii) \(\beta_{1}\)  
Variance  0.023  0.021  0.090  0.183  0.115  0.178  0.054  0.084 
Bias  0.034  0.129  0.013  0.107  0.007  0.031  0.004  0.009 
Conditional on S/N of \(\beta_{1}\) = 5  
Variance  0.024  0.022  0.168  0.239  0.159  0.192  0.110  0.106 
Bias  0.068  0.811  0.049  0.179  0.045  0.097  0.013  0.033 
The results of the linear surface are in Table 7. As mentioned earlier, the average bandwidths of the local linear method are the largest, which possibly indicates the smallest variances. In addition, the local linear method has the smallest bias and the smallest discount rates for linear surfaces (Table 2). Although the CGWR has a larger bias than the local linear method in the case of linear surface, it dominates the basic GWR method with respect to both the variance and bias. The CGWR performs the best with ridge surfaces, outperforming the basic GWR and the local linear method with respect to both variance and bias.
Mixedtype surfaces
Next, we repeat the same comparisons for the three GWR estimation methods with the mixedtype surfaces. The results are similar to those in the singletype surfaces, and thus we will only show the results of the discount rates. As mentioned earlier, there are two cases in this scenario: linearquadratic (a polynomial surface) and ridgehillside (a nonpolynomial surface). In the first case, the underlying surface of intercept is linear, and the slope is quadratic. In the second case, all surfaces are of nonpolynomial type, and it is more complex than the first one.
The average discount rates of a linearquadratic surface (mixedtype)
\(\beta_{ 0}\)  \(\beta_{1}\)  

\(\beta_{ 0}\)  1  3  5  1  3  5  
\(\beta_{1}\)  
Basic GWR  1  1.062  0.279  0.155  1.934  1.643  1.420 
3  2.110  0.407  0.181  1.048  1.067  0.935  
5  3.119  0.537  0.231  0.688  0.687  0.711  
Local linear  1  1.492  0.225  0.080  2.888  1.514  0.905 
3  3.100  0.534  0.169  1.405  1.314  0.954  
5  5.357  0.775  0.268  0.928  0.871  0.753  
CGWR  1  0.764  0.183  0.100  1.301  0.802  0.538 
3  1.132  0.345  0.179  0.708  0.849  0.785  
5  1.257  0.388  0.217  0.404  0.523  0.584 
The average discount rates of a ridgehillside surface (mixedtype)
\(\beta_{ 0}\)  \(\beta_{1}\)  

\(\beta_{ 0}\)  1  3  5  1  3  5  
\(\beta_{1}\)  
Basic GWR  1  1.457  0.545  0.337  2.160  3.268  2.868 
3  1.623  0.582  0.349  0.622  1.373  1.786  
5  1.511  0.604  0.344  0.251  0.586  0.943  
Local linear  1  2.105  0.853  0.536  3.312  5.602  4.755 
3  2.236  0.903  0.497  0.896  2.348  2.595  
5  1.999  0.980  0.540  0.333  1.076  1.522  
CGWR  1  0.956  0.245  0.116  1.166  0.750  0.351 
3  1.279  0.361  0.151  0.544  0.677  0.443  
5  1.338  0.485  0.204  0.245  0.434  0.415 
Twostage fitting procedure
We found that the CGWR works well when there is a positive correlation. However, in reality, there is a great likelihood of variables not being positively correlated. To overcome this difficulty, the CGWR can be modified into a twostage process. In the first stage, we divide the variables into two groups. In both groups, the variables are nonnegatively (or positively) correlated within the group. Any two variables are nonpositively (or negatively) correlated if they are from different groups. We choose one group of variables and apply the basic GWR method to this group. In the second stage, we apply the CGWR method to the other group of variables by treating the first group of variables (chosen in the first stage) as constants.
SingleType Surfaces with Clusters
The average discount rates of \(\beta_{ 0}\) on surfaces with cluster
Linear  Hillside  

\(\beta_{ 0}\)  1  3  5  1  3  5  
\(\beta_{1}\)  
Basic GWR  1  0.975  0.395  0.214  0.929  0.386  0.198 
3  1.116  0.421  0.240  0.964  0.416  0.196  
5  1.202  0.473  0.257  1.007  0.412  0.197  
Local linear  1  1.166  0.402  0.207  1.142  0.543  0.284 
3  1.246  0.45  0.210  1.238  0.593  0.325  
5  1.287  0.419  0.225  1.358  0.619  0.288  
CGWR  1  0.714  0.226  0.098  0.625  0.216  0.106 
3  0.943  0.302  0.133  0.852  0.281  0.112  
5  1.146  0.368  0.150  1.000  0.303  0.123 
From the preceding computer simulations studies, we found that the proposed CGWR method makes a significant improvement over the basic GWR method. Although the locallinear method behaves well in the linear surface, if the coefficient surfaces are nonlinear, the CGWR also outperforms the local linear method. In the following discussion, we will use two realworld datasets to compare the CGWR and other two methods and provide further evidence in support of the CGWR.
Empirical data
We apply the CGWR to two empirical data sets: the first is from the 2000 Taiwan Census and the other is the Ohio crime data provided by Anselin [1]. These two examples are designed to demonstrate that the CGWR would yield better estimation results. For the Taiwan data, our goal is to explore the relationship between the proportion of elderly disability and social factors. The elderly population in Taiwan have been increasing rapidly all around the country, while the medical resources still concentrate in the metropolitan areas (or northern Taiwan). Hu and Yue [18] applied spatial regression model to the elderly disability data of township level and found they are spatially autocorrelated. Brunsdon [7] argued that the spatial autocorrelation seems to be caused by spatial nonstationarity (i.e., identifiability). His claim motivates us to reexamine the data using the GWRbased model.
Taiwan data
Taiwan 2000 Census includes data of 350 townships and their proportions of disabled elderly are set as the dependent variable. Since this variable appears to be rightskewed, a log transformation (i.e.\(y_{i}^{*} = \log (y_{i} + 1)\)) is applied. Four independent variables are selected: the population density (POP), proportion of elderly (ELD), elderly mortality rate (EMR), and education level (EDU). These independent variables are standardized into the [0, 1] interval. Before applying the GWR, we first test spatial nonstationarity with the F test suggested by Leung et al. [21]. The F test shows that the model is spatially nonstationary with p value <0.001. This confirms the hypothesis of Brunsdon [7] and creates incentive for plugging the GWRtype analysis.
Correlations of regression coefficients from the disability data (INT and POP) → group 1, (ELD, EMR, EDU) → group 2
INT  POP  ELD  EMR  EDU  

INT  1.000  0.463  −0.949  −0.951  −0.971 
POP  0.463  1.000  −0.281  −0.281  −0.653 
ELD  −0.949  −0.281  1.000  0.914  0.875 
EMR  −0.951  −0.281  0.914  1.000  0.868 
EDU  −0.971  −0.653  0.875  0.868  1.000 
Ohio Data
The bandwidths of different models
Method  Variable  

Intercept  x1 (average income value)  2 (average housing cost)  
Basic GWR  2.27  2.27  2.27 
Local linear  13.81  13.81  13.81 
CGWR  999  999  1.77 
Pseudo R^{2} and p values of Kolmogorov–Smirnov normality test
Method  OLS  Basic GWR  Local linear  CGWR 

Pseudo R^{2}  0.743  0.953  0.837  0.937 
K–S test p value  0.906  0.625  0.870  0.772 
Conclusions
The GWR has become a popular tool for explanatory data analysis and detecting spatial nonstationarity ever since its introduction. The GWR provides useful information for data analysis, especially helpful in deciding important explanatory variables. This technique allows regression coefficients to vary across space and obtains their estimates from a bandwidth of observations according the data attribute. However, the GWR tends to produce ragged surfaces (as shown in Fig. 1), and a fixed bandwidth may not be appropriate since the independent variables are necessary to be homogeneous (e.g., their variations can be quite different). In this study, we proposed a modification to the GWR, namely CGWR, which allows the group of positively correlated independent variables to have its own bandwidth via an iterative calibration process.
We used computer simulated and empirical data to compare the proposed method with the GWR and its local linear modification by Wang et al. [31]. Based on the simulation results, we found that the CGWR outperforms other two methods, with respect to the bias and variance, when the regression coefficients are positively correlated. The advantage is especially noticeable in the case of nonlinear surfaces. In particular, the clusters have little influences on the estimation of CGWR. The results of empirical studies also support the CGWR and it generally has larger Rsquare and has fewer extreme outliers (e.g., the absolute value of standardized residual larger than 2 or 3) than the GWR and the local linear method.
However, the proposed method has its limitations. First, probably the most critical limitation, the current setting of CGWR only works for the case if there are independent variables with positive correlation. Although not shown here, we found that the CGWR does not work well in the case of independent variables with negative correlation. It is like the antithetic variate in variance reduction of Monte Carlo Integration. Antithetic variate is one of the popular variance reduction methods but it only works when two variables are negatively correlated [25]. Thus, we suggest first calculating the correlation coefficients between independent variables. Then, form a group of variables which are pairwise positively correlated and apply the CGWR only to this group of variables. Another possibility is that the independent variables often can be separated into two groups and variables within/between groups are positively/negatively correlated, as seen in Taiwan 2000 Census data. We can apply the twostage CGWR to two groups of variables.
Second, the CGWR is a computerintensive method and its computing time increases rapidly as the number of variables increases, although the convergence of coefficients can be speed up by using the moving average method. Third, the CGWR is not guaranteed to work if there are many variables, and so far it is effective for the case up to four variables. A possible modification to the case with more variables would be to separate the variables into two groups and use double iteration. Then, the CGWR can be applied to each group of variables forming the inner loop and the process reiterated between the two groups forming the outer loop until both groups of variables converge. To demonstrate the feasibility of this idea, we also conducted an experiment with six variables, separating them into two groups of three variables each. We found that the estimation did converge and produce satisfactory estimates.
In addition to the fixed bandwidth, it seems that there is still room for improvement about the GWR. In particular, when the S/N ratio is small, the estimated coefficient surfaces would be nonlinear (i.e., ragged surfaces), even when the true surfaces are linear. In addition, the variance reduction of the CGWR over the GWR is more obvious than that for bias reduction. This indicates that the GWR estimates have large variance when the S/N ratio is small. In other words, if the variances of GWR estimates are reduced, the bias can also be further reduced, producing more stable estimates.
Declarations
Authors’ contributions
Leong developed the estimation algorithm, conducted simulation studies, as well as empirical data analysis, and drafted the manuscript. Yue proposed the general methodology, designed the simulation, interpreted the results and helped to draft the manuscript. Both authors read and approved the final manuscript.
Acknowledgements
This research was supported in part by a grant from the National Science Council in Taiwan, NSC 1042410H004 023 MY2. We greatly appreciate the insightful comments from the editor and two anonymous reviewers, which helped us to clarify the context of our work.
Competing interests
The authors declare that they have no competing interests.
Availability of data and materials
The Columbus Ohio crime data is prepared by Luc Anselin, Spatial Analysis Laboratory, Department of Agricultural and Consumer Economics, University of Illinois, UrbanaChampaign. The dataset can be downloaded at following link: https://spatial.uchicago.edu/sampledata, or available at the ‘spgwr’ package in R statistical software.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.
Authors’ Affiliations
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