# An Integrated Fuzzy C-Means Method for Missing Data Imputation Using Taxi GPS Data
## 4 Results and Discussions  
### Evaluation Criteria of Imputation Performance  
Before presenting the comparisons of the results, some optimal parameters obtained by the integrated imputation algorithm are shown in Table 3. In particular, the data were aggregated at four different levels, namely 5, 10, 15, and 20 min, respectively. Correspondingly, the number of time interval was 288, 144, 96, and 72, respectively. Moreover, five kinds of missing ratios, 5%, 10%, 15%, 20%, and 25% are respectively given, and the data were randomly deleted according to the missing ratio.  
In this line of research, three evaluation criteria were presented to evaluate the estimation accuracy, which included the root mean squared error (RMSE), the correlation coefficient (\(R\)), and the relative accuracy (\(RA\)). Specifically, the RMSE represented the error between the actual values \(x_{jk}\) and the  
Figure 4: Flowchart of the integrated imputation algorithm.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Time Interval (min)**} & \multicolumn{2}{c}{**Manhattan Distance**} & \multicolumn{2}{c}{**Euclidean Distance**} \\ \cline{2-5}  & \(c\) & \(m\) & \(c\) & \(m\) \\ \hline
5 min & 12 & 1.1083 & 14 & 1.1936 \\
10 min & 12 & 1.1123 & 14 & 1.1183 \\
15 min & 13 & 1.1016 & 12 & 1.1145 \\
20 min & 8 & 1.1426 & 9 & 1.1211 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Optimized parameters obtained by the integrated imputation method.  
estimated values \(x^{*}_{jk}\), and the RMSE was calculated by Formula (8). In addition to the error estimation, the R is typically utilized to present the approximation degree between the actual values and the estimated values, and the R was obtained by Formula (10). It should be noted that \(\overline{x^{*}}\) is the average value of all missing values, and \(\overline{x}\) is the average value of all actual values corresponding to missing values. Meanwhile, the RA reflects the number of estimated results falling within a specified tolerance level, and it was calculated by Formulas (11) and (12). In this study, the specified tolerance level was set as 5% and 10%, respectively. The exact formulas are illustrated as follows:  
\[R=\frac{\sum\limits_{k=1}^{n}\sum\limits_{j=1}^{p}I_{jk}(\overline{x^{*}}-x^{* }_{jk})\cdot(\overline{x}-x_{jk})}{\sqrt{\left(\sum\limits_{k=1}^{n}\sum \limits_{j=1}^{p}\left[I_{jk}(\overline{x^{*}}-x^{*}_{jk})\right]^{2}\right) \cdot\{\sum\limits_{k=1}^{n}\sum\limits_{j=1}^{p}\left[I_{jk}(\overline{x}-x_{ jk})\right]^{2}\}}} \tag{10}\]  
\[APE_{jk}=\frac{\left|I_{jk}(x_{jk}-x^{*}_{jk})\right|}{x_{jk}} \tag{11}\]  
\[RA=\frac{n_{APE}}{\sum\limits_{k=1}^{n}\sum\limits_{j=1}^{p}I_{jk}} \tag{12}\]  
APE is the absolute percentage error, and \(n_{APE}\) represents the number of \(APE_{jk}\) in the \(\pm\)5% and \(\pm\)10% error range.  
Moreover, based on the condition of 25% missing loss and the five minutes aggregation interval, the actual values of the missing points and the estimated points obtained by the integrated imputation method with the Euclidean distance are shown in Figure 5 to represent their differences.