# Improving the Robustness of DTW to Global Time Warping Conditions in Audio Synchronization
## 3 Results
### Experimental Results
We see that both systems have good performance for low global time warp factors, but that performance is terrible for global time warp factors of 2.0 and higher. This makes intuitive sense, since the set of transitions \((1,1)\), \((1,2)\), \((2,1)\) imposes a maximum allowable time warping factor of 2. Thus, we can summarize the performance of our baselines in the following way: they perform well with moderate global time warping (\(<\)2), but they cannot handle severe global time warping (\(\geq\)2) at all. Second, consider the three blue systems (DTW3, DTW4, DTW5). What these three systems have in common is that they allow \((0,1)\) and \((1,0)\) transitions, which theoretically should be able to handle an infinite amount of time warping. Their performance thus addresses the question, \"Can we handle severe global time warping by including \((0,1)\) and \((1,0)\) transitions in standard DTW?\" We can see in Figure 3 that the performance of these systems is relatively invariant to the global time warp factor. For example, the error rate of the DTW3 system only degrades from about 32% to 37% as the global time warp factor increases from 1.0 to 4.0. However, the error rates of all three systems is relatively high, and they do not have competitive performance compared to the other systems under any global time warp conditions. So, the short answer to the question above is: no, adding \((0,1)\) and \((1,0)\) transitions is not an effective method of handling severe global time warping. Third, consider the two orange systems (DTW1-add3, DTW1-add4). These two systems add additional transitions like \((3,1)\) and \((4,1)\) to the standard set \(\{(1,1)\), \((1,2)\), \((2,1)\}\). These systems address the question, \"Can we handle more severe global time warping conditions by adding more extreme transitions to standard DTW?\" We can see in Figure 3 that the performance is degraded for low global time warp factors, but that the two systems are indeed able to cope with more extreme time warping. For example, DTW1-add3 has reasonably good performance for global time warp factors \(<\) 3.0, and DTW1-add4 has reasonably good performance for global time warp factors \(<\) 4.0. However, we must also keep in mind that these systems require more computation, since each step in the dynamic programming stage must consider more possibilities (e.g., seven for DTW1-add4 compared to three for DTW1). So, the short answer to the question above is that adding more extreme transitions can make DTW more robust to severe global time warping, but it comes at a heavy computational cost and a moderate degradation in accuracy. Fourth, consider the four purple systems (DTW2-downsampleNN, DTW2-downsampleLin, DTW2-upsampleNN, DTW2-upsampleLin). These four systems all perform some form of sequence length normalization before applying DTW with standard settings. We can see in Figure 3 that these four systems have the best performance among all systems across all seven global time warp factors. Furthermore, the performance is largely invariant to the global time warp factor. Interestingly, all four systems have very similar accuracy, with the upsampling variants showing a slight performance benefit at high global time warp factors. Given that upsampling results in much longer sequences (and therefore more computation in performing DTW), the most practical option is DTW2-downsampleNN due to its simplicity. In summary, the sequence length normalization technique is the clear winner among the four methods--it has the best performance across all conditions and is largely invariant to the global time warping factor. Fifth, consider the two green systems (AdaptiveWeight1, AdaptiveWeight2). These systems have \((0,1)\) and \((1,0)\) transitions with weights that are computed at run time based on the lengths of the two sequences. We can see in Figure 3 that these systems have roughly the same behavior as the other systems that include \((0,1)\) and \((1,0)\) transitions.