Simulation Study
4.2 Single type surfaces:
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weight). The preceding setting is also used in the “spgwr” package (Bivard and Yu, 2010, ver0.5-4) in R, a free statistical software.
The rest of this section discusses the simulation results for the single type surfaces, followed by those for the mixed type surfaces, and finally the random effect models. Specifically, we will compare the performances of three GWR estimations based on the average discount rate, the average bandwidth and the average variance and average bias of the estimates. In addition, the stopping criteria in following simulations are average absolute change rate (
0 &
1) less than 0.005, which is 0.01% of the noise. The settings is different from the usual settings because the convergence times to 10-3 is much greater than 0.005. Due to the mass simulations considered in this paper, we sacrifice a little precision in exchange of the simulation time and the results would be similar for smaller stopping criteria.To simplify the notation, we will use
0 and
1 to denote the coefficients of the intercept and slope of independent variable x in the following tables, respectively.4.2 Single type surfaces:
Table 1-1 shows the results of the discount rates in the case where B0 and B1 satisfy a linear surface. We can see that both the proposed CGWR and the local linear method have significant improvements over the basic CGWR.
Interestingly, the local linear method is better (with respect to smaller discount rates) than the GWR when the S/N ratio is large, but the basic GWR is better when the S/N is small. The reason might be that larger noises produce larger fluctuations, and thus the average tangent line in the local linear method is not accurate or stable. Similar patterns also appear for the other three surfaces (Tables
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The CGWR and the local linear method again outperform the basic GWR in the case of a quadratic surface. However, the CGWR appears to be the best and the edge is more obvious when the S/N increased. For the nonlinear surfaces, the CGWR continues to work satisfactorily, but the local linear model does not. It is possible to create worse results than the basic GWR. The CGWR is still reliable in the cases of non- linear surfaces, performing much better than the other two methods.
Table 1-1. The average discount rates on a linear surface (single-type)
B0 B1
Table 1-2. The average discount rates on a quadratic surface (single-type)
B0 B1
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Table 1-3. The average discount rates on a ridge surface (single-type)
B0 B1
Table 1-4. The average discount rates on a hillside surface (single-type)
B0 B1
To further evaluate the performances of the three different GWR methods, we can consider the hillside surface (Table 1-4) and the S/N ratio of B0 = B1 = 5 as a demonstration. We can see that the CGWR produces the best fit and the mean
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but looks like a linear surface.M ethod 1: Basic GWR M ethod 2: Local liner estimation M ethod 3: CGWR
Figure 2: TheB1 mean surface of each estimation method fro m 100 simu lations. The simulation scenario is a hillside, and the S/N ratio of B0 = B1 = 5.
Table 2. The average bandwidths for linear and hillside surfaces (single-type) For CGW R, the first and second values are the average bandwidths of B0 and B1.
Line ar surface Hillside surface
Method
Intuitively, we expect that the bandwidth is small if the S/N is large, since distant observations can be very different and cause biased estimation. Basically, all three GWR methods have significant drops in bandwidth if we increase the S/N ratio from 1 to 3. Moreover, the bandwidths in the case of a linear surface should be larger than those of a nonlinear surface under the same S/N ratio, since the surface change is quite homogenous in any direction.
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observations within a larger bandwidth and thus ha ve smaller variances than those in the case of a nonlinear surface. Since the shape of a hillside is close to linear, S/N ratio of B0. This is exactly why we want to consider a different bandwidth for each coefficient, since the numbers of observations needed can be different.Table 3-1. The average variances and biases of B0 and B1 on a linear surface (sin gle-type) (i) B0
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Table 3-2. The average variances and biases of B0 and B1 on a ridge surface (single-type) (i) B0
(ii) B1
The variances and biases of the estimates from the three GWR methods can
OLS Basic GWR Local line ar CGWR
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ridge surfaces. Because there are many combinations for the S/N ratios of B0 and B1, we will only show the results of S/N = 1 and 5 (Tables 3-1 to 3-2). Unlike in the previous comparisons, we will also provide the variances and biases of the OLS estimates. In general, a larger S/N ratio tend s to produce a larger bias.
Moreover, the OLS estimates fail to capture the spatial trend causing the largest bias, but it uses all of the observations in the estimation (i.e., infinity bandwidth) and thus has the smallest variance. As for the three GWR estimations, the variances of the estimates are generally larger than the biases of the estimates.
The results for a linear surface are shown in Table 3-1. As mentioned earlier, the average bandwidths of the local linear method were the largest, possibly indicating the smallest variances. In addition, the local linear method has the smallest bias, and also the smallest discount rates in the case of a linear surface (Table 1-1). Although the CGWR has a larger bias than the local linear method in the case of a linear surface, it dominates the basic GWR with respect to both variance and bias. The CGWR has the best performance in the case of a ridge surface, and it also outperforms both the basic GWR and the local linear method for both the variance and bias.