# An Integrated Fuzzy C-Means Method for Missing Data Imputation Using Taxi GPS Data
## 4 Results and Discussions
### Comparisons and Analyses of the Results  
With the aim of evaluating the integrated imputation algorithm, five imputation methods were compared to demonstrate the effectiveness of the proposed integrated imputation method. The detailed information is proposed as follows:  
Figure 5: Comparisons between the missing values and the estimated values. * The history imputation method. Considering the similarity in data obtained at the same time among different weekdays, in particular, the missing value was replaced by the mean value of the existing values on the same row in the matrix of Table 2 (History). This method also belongs to one of the regression imputation methods. * The conventional FCM using the Manhattan distance (MDFCM). * The conventional FCM using the Euclidean distance (EDFCM). * The integrated FCM-GA using the Manhattan distance (MDFCMGA). * The integrated FCM-GA using the Euclidean distance (EDFCMGA). With respect to the RMSE, as shown in Figure 6a-d, the overall RMSE increased as the missing ratio increased. The reason for this was that when the missing ratio increased with the fixed time interval, the amount of data that could be used in imputation would decrease as well. Thus, the performance of imputation deteriorated. Analogously, as the aggregation interval increased from 5 min to the larger intervals, the overall RMSE would increase simultaneously. The reason for this was that the taxi demand that aggregated from a longer interval was normally greater than that from a shorter interval. Thus, with the same APE, the absolute error between the missing value and the estimated value was larger as well when the time interval was longer. Thus, with a longer interval, the overall RMSE increased as well. In particular, as for the history imputation method, when the aggregation interval was five minutes and the missing ratio was set as 5% (Figure 6a), the gap of the RMSE among the five methods was narrow. However, when the aggregation interval and the missing ratio became larger in Figure 6a, the conventional FCM and the integrated FCM-GA outperformed the history imputation method greatly. The superiority of the FCM and the FCM-GA is also ascertained from Figure 6b-d. Meanwhile, from Figure 6a-d, the RMSE produced by the history imputation method in each figure varied drastically, which indicated that the history imputation method was sensitive to the historical data. In contrast, as for the FCM and the FCM-GA, the RMSE in each figure varied on a small scale, which indicated that the performance of the FCM and the FCM-GA was stable. From Figure 6a-d, we can also see that no matter what the Manhattan or the Euclidean distance was, the RMSE produced by the integrated FCM-GA was smaller than the conventional FCM. The RMSE calculated for the Euclidean distance was smaller than that of the Manhattan distance. It can be seen from Figure 7a-d that as the missing ratio increased, the overall correlation coefficient decreased. This phenomenon was similar to the changes in Figure 6. Evidently, with a higher missing ratio, a decrease in the amount of useful data would also result in a lower correlation coefficient. As seen in Figure 7a, when the aggregation interval was five minutes and the missing ratio were 5%, 10%, and 15%, respectively, the correlation coefficients of the MDFCM, the EDFCM, and the history imputation method were similar. However, when the aggregation interval and the missing ratio became larger in Figure 7a, the conventional FCM and the FCM-GA outperformed the history imputation method substantially. The effectiveness of the FCM and the FCM-GA is also shown in Figure 7b-d. Meanwhile, as seen in Figure 7a-d, the correlation coefficient obtained from the FCM and the FCM-GA remained stable from 0.98 to one, which demonstrated the fact that the missing values had a close relationship with the existing values. Moreover, this phenomenon indicated that missing values could be estimated by using the existing values, which was coherent with our proposed assumption.