# A Boundary Distance-Based Symbolic Aggregate Approximation Method for Time Series Data
## 4 Experimental Validation
### Result Analysis  
To make the table fit all the data, we abbreviate SAX-TD for SAXTD and SAX-BD for SAXBD. The overall classification results are listed in Table 3, where entries with the lowest classification error rates are highlighted. SAX-BD has the lowest error in the most of the data sets (69/100), followed by SAX-TD (22/100), EU (15/100). In some cases, it performs much better than the other two methods, and even achieves a 0 classification error rate. We use the Wilcoxon signed ranks test to test the significance of our method against other methods. The test results are displayed in Table 4. Where \(n^{+}\), \(n\)-, and \(n^{0}\) denote the numbers of data sets where the error rates of the SAX-BD are lower, larger than and equal to those of another method respectively. The \(p\)-values (the smaller a p-value, the more significant the improvement) demonstrate that our distance measure achieves a significant improvement over the other four methods on classification accuracy. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**ID** & **Type** & **Name** & **Train** & **Test** & **Class** & **Length** \\ \hline
91 & Sensor & Trace & 100 & 100 & 4 & 275 \\
92 & ECG & TwoLeadECG & 23 & 1139 & 2 & 82 \\
93 & Simulated & TwoPatterns & 1000 & 4000 & 4 & 128 \\
94 & Simulated & UMD & 36 & 144 & 3 & 150 \\
95 & Sensor & Wafer & 1000 & 6164 & 2 & 152 \\
96 & Spectro & Wine & 57 & 54 & 2 & 234 \\
97 & Image & WordSynonyms & 267 & 638 & 25 & 270 \\
98 & Motion & Worms & 181 & 77 & 5 & 900 \\
99 & Motion & WormsTwoClass & 181 & 77 & 2 & 900 \\
100 & Image & Yoga & 300 & 3000 & 2 & 426 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: _Cont._  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**ID**} & **EU** & **SAX** & **SAX** & **SAX** & **SAX** & **SAX** & **ESAX** & **ESAX** & **ESAX** & **SAXTD** & **SAXTD** & **SAXTD** & **SAXTD** & **SAXTD** &  
To provide a more intuitive illustration of the performance of the different measures compared in Tables 3 and 4, we use scatter plots for pairwise comparisons. In a scatter plot, the accuracy rates of two measures under comparison are used as the \(x\) and \(y\) coordinates of a dot, where a dot represents a data set. When a dot above the diagonal line, the '\(y\)' method performs better than the '\(x\)' method. In addition, the further a dot is from the diagonal line, the greater the margin of an accuracy improvement. The region with more dots indicates a better method than the other. In the following, we explain the results in Figure 6.  
We illustrate the performance of our distance measure against the Euclidean distance, SAX distance, ESAX distance, SAX-TD distance in Figure 6a-d, respectively. Our method outperforms the other four methods by a large margin, both in the number of points and the distance of these points from the diagonals. From these figures, we can see that most of the points are far away from the diagonals, which indicates that our method has much lower error rates on most of the data sets. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Methods** & \(n^{*}\) & \(n\) & \(n^{0}\) & \(p\)**-Value** \\ \hline SAX-BD vs. Euclidean & 79 & 15 & 6 & \(p<0.05\) \\ SAX-BD vs. SAX & 83 & 11 & 6 & \(p<0.05\) \\ SAX-BD vs. ESAX & 87 & 10 & 3 & \(p<0.05\) \\ SAX-BD vs. SAX-TD & 69 & 22 & 10 & \(p<0.05\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The Wilcoxon signed ranks test results of the SAX-BD vs. other methods. A \(p\)-value less than or equal to 0.05 indicates a significant improvement. \(n^{*}\) means positive, n means equal and \(n^{0}\) means negative. The larger the value of \(n^{*}\), the better performance of SAX-TD. Figure 6: The SAX-BD algorithm is compared with other algorithms for accuracy. (**a–d**) represents a comparison between SAX-BD with Euclidean, SAX, ESAX, SAX-TD. The more dots above the red slash, the better performs of SAX-BD. To show the continuity performance of our method and other three methods, we run the experiments on data set Yoga.