# Improving the Robustness of DTW to Global Time Warping Conditions in Audio Synchronization
## 3 Results
### Experimental Results  
We evaluated the performance of 17 alignment algorithms. The first seven systems explore the use of different transition types and weights for standard DTW, as described in Section 2.1. The next four systems explore the use of sequence length normalization, as described in Section 2.2. The next two systems explore the use of adaptive weighting schemes, as described in Section 2.3. The last four systems explore the use of non-uniform transition patterns, as described in Section 2.4. Table 4 defines each of these 17 systems, grouped into the four categories described above. We ran experiments with many other  
Figure 2: Example showing how we evaluate alignment accuracy. Red points indicate ground truth beat timestamps, the blue trajectory indicates a predicted alignment, and the black vertical bars indicate error tolerance. With a small error tolerance (**left**), the error rate is \(\frac{2}{6}=33.3\%\). With a larger error tolerance (**right**), the error rate for the same predicted alignment is 0%. We report error rates at multiple error tolerances. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Global Time Warping Ratio** & **Dataset 1** & **Dataset 2** \\ \hline
1.000 & median\_x1.000 & median\_x1.000 \\
1.260 & median\_x1.260 & median\_x1.000 \\
1.588 & median\_x1.260 & median\_x0.794 \\
2.000 & median\_x1.588 & median\_x0.794 \\
2.521 & median\_x1.588 & median\_x0.630 \\
3.175 & median\_x2.000 & median\_x0.630 \\
4.000 & median\_x2.000 & median\_x0.500 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The pairings of modified Mazurka Datasets used to study fixed global time warping ratios. For example, to study performance at a global time warping ratio of 2, we consider pairs of audio recordings \((A,B)\) where \(A\) is selected from the median_x1.588 dataset and where \(B\) is selected from the median_x0.794 dataset. variants and only included the best-performing ones in the 17 listed in the table. For example, we tried applying sequence length normalization to other transition and weighting schemes, but only present results with DTW2 since it yielded the best results. Likewise, we considered several other weighting schemes with standard DTW, but only present the best-performing ones. Figure 3 shows the alignment accuracy of all 17 alignment methods on our benchmarks. The seven groups along the horizontal axis correspond to the seven different benchmarks, where each benchmark has an average global time warping factor between 1.0 and 4.0. Within each group, the colored bars correspond to the 17 different alignment methods. Note that bar colors have been chosen to group similar alignment methods together, so that variations of the same method are indicated with a gradient of the same hue. Each individual bar indicates the error rate at an error tolerance of 200 ms when projected onto the median_x1.000 time scale. On top of each bar, we have also overlaid two horizontal black lines indicating the error rates at tolerances of 100 ms (above) and 400 ms (below). The y-axis indicates error rate, so lower is better.