# An Integrated Fuzzy C-Means Method for Missing Data Imputation Using Taxi GPS Data
## 4 Results and Discussions
### Comparisons and Analyses of the Results
The integrated imputation algorithm based on the FCM and the GA with the Manhattan and the Euclidean distance also resulted in the two highest values in Figure 7a-d.  
To reflect the superiority of the integrated imputation method in depth, the RA was calculated according to two scenarios, including \(\pm\)5% and \(\pm\)10%. From Figure 8a-d, when the threshold was set strictly as \(\pm\)5%, the integrated imputation method outperformed the other methods significantly. Meanwhile, the performance of the integrated imputation method in Figure 8a-d was consistently stable. Moreover, the RA obtained by the FCM-GA with the Euclidean distance type had the highest value in Figure 8a-d. As seen in Figure 9a-d, when the threshold was set less strictly as \(\pm\)10%, the integrated imputation method still outperformed the other methods. On the other hand, although the performance of the conventional FCM in Figure 9b with a 15% missing ratio was remarkably enhanced, its imputation performance was still unstable under other scenarios. In other words, the conventional FCM was sensitive to the time interval and missing ratio. With respect to the integrated imputation method, the integrated imputation method was less sensitive to the time interval and missing ratio. Therefore, although the phenomenon from Figure 9 indicated that the conventional FCM was a possible substitute when the condition was relaxed, and the integrated imputation method was still a stable and reliable choice for estimating missing values. Figure 6: Root mean squared error (RMSE) comparisons under the condition of different data aggregation intervals and different missing ratios. Another interesting phenomenon was also derived from the results presented in Figures 8 and 9. When the threshold was strictly set as \(\pm\)5%, the average relative accuracy (RAs) obtained by the EDFCMGA were, respectively, 0.532, 0.535, 0.509, and 0.576 under the 5, 10, 15, and 20 min aggregation time intervals. Thus, the best aggregation time interval should be set as 20 min under this scenario. In addition, when the threshold was less strictly set as \(\pm\)10%, the average RAs obtained by the EDFCMGA were, respectively, 0.785, 0.742, 0.718, and 0.735 under the 5, 10, 15, and 20 min aggregation time intervals. Thus, the best aggregation time interval should be set as five minutes under this scenario. In general, a longer aggregation time interval would remove randomness, which is typically viewed as a certain smoother filter. However, a longer aggregation time interval may conceal available variation information as well. In fact, the accuracy of imputation is highly associated with the detailed information in a dataset. Thus, how to determine the aggregation time interval in the imputation is another hot topic [47; 48]. Figure 7: Correlation coefficient comparisons under the condition of different data aggregation intervals and different missing ratios. Figure 8: Relative accuracy (RA) comparisons under the condition of different data aggregation intervals, different missing ratios, and the \(\pm 5\%\) tolerance error range. Figure 9: _Cont._  
To reinforce our interpretation of the superiority of the proposed integrated FCM imputation method, an example that covers the time series of observed curve and estimated values is presented in Figure 10. In Figure 10, five scenarios with different missing ratios are respectively given. By comparing the estimated time series with the original time series, we found that the estimated time series under different scenarios was close to the original one. Thus, in terms of different missing ratios, our proposed integrated imputation method could be applicable to estimating missing values in the time series. In Table 4, the average computation time in seconds is described. It should be noted that all experiments were conducted on a personal computer with an Intel Core i5-6200U (Bayan Lepas, Malaysia) and 12 GB RAM. Meanwhile, ten repeated tests of each scenario were carried out, and the average computation time of each scenario was obtained.