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Chapter 6.
Discussion and Concluding Remarks
Beginning from past decade, the GWR has acted as a new modelling technique used to deal with spatial non-stationarity. This technique allows regression coefficients to vary across space and obtains their estimates from a fixed bandwidth of observations. However, using a fixed bandwidth might not be appropriate since the independent variables would behave differently. In this paper, we proposed a method (CGWR) for modifying the GWR by relaxing the restriction of a fixed bandwidth. We compared the proposed method to the GWR and a local linear estimation, which was shown to be effective at reducing the bias. Based on simulation results, we found that the CGWR has the best performances given that the regression coefficients are positively correlated, and this advantage is especially noticeable in the cases of non-linear surfaces.
Of course, the improvements in the CGWR are due to the fact that it allows the bandwidth to vary for each coefficient. If the coefficients are positively correlated, the CGWR can reduce the bias, as well as the variance. In the empirical study, if the coefficients are not always positively correlated, the CGWR can still be modified and applied to a set of variables that are positively correlated for any pairs of variables.
However, the proposed method also has limitations. First, the most critical one is probably that the current settings for CGWR do not work well in the case of negative correlation, allowing the basic GWR to outperform the proposed CGWR.
As a solution, we suggest calculating the correlation coefficients of the variables before applying the CGWR. Nevertheless, we can perform a two-stage fitting as we did at previous empirical study. In order to verify the feasibility of this
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components in this model, which is intercept and two independent variables. Here, we redo the simulation by fixing second variable at first stage. By ratio of average discount rate, the outcome is similar to previous result. CGWR seems to fit well by this approach. Second, the CGWR is a computer intensive method, and would become extremely time-consuming if there are more variables. To increase the iteration speed, a moving average method could be used to increase the convergence speed. Third, we have not shown that the CGWR will converge if there are many variables, although we found that it would for cases up to four variables. A possible modification to a case with more variables would be to separate the variables into two groups and use double iteration. Then, CGWR could be used on each group of variables (inner loop ), and the process re- iterated between the two groups (outer loop) until both groups of variables converge. To show that this idea is feasible, we conduct an experiment with six variables. The variables are separated into two groups of three variables each, and the estimation does converge.
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Figure 5: Average discount rate of each method by fixing variab le x2, The baseline of the ratio of average discount rate is the GW R.
In this study, we also found some potential problems in applying the GWR. The GWR still has room for further improvement, especially when the S/N ratio is small, the surfaces of the coefficients are non- linear and the coefficients are very different. In addition, the variance reduction of CGWR over GWR is much more obvious than that for the bias reduction. This indicates that the GWR estimates have large variances. In other words, if the variances of the GWR estimates could be reduced, the bias could also be further reduced, and thus it is likely that the estimates would be more stable.
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and k* is an arbitrary natural number.If the largest eigenvalue in the absolute values of
S S
1 2,
, is smaller than 1, then the remainderR
1 converges to 0 ask
.The smoothing matrices in the proof are similar to those used in the GAM.
According to Hastie and Tishibirani (1990), the smoother matrices should satisfy the bounded condition (p. 121). Moreover, this assumption seems to be true in practice and it is always the case in our simulation and empirical studies.
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Appendix B.
Detailed results forbandwidths (fixed effect models, single-type)Note that the CGWR has two bandwidths for two coefficients, B0 and B1,
Ri dge surface Hillside surface
1 4.2 1.01 0.83 2.25 1.67 1.41