# Improving the Robustness of DTW to Global Time Warping Conditions in Audio Synchronization
## 3 Results
### Experimental Setup
The end result of this process is a set of seven different versions of the Mazurka Dataset, which we refer to as median_x0.500, median_x0.630, median_x0.794, median_x1.000, median_x1.260, median_x1.588, and median_x2.000. Table 2 provides an overview of these modified datasets. In each of these seven datasets, all of the recordings of the same Mazurka have exactly the same duration. So, for example, in the median_x2.000 dataset, all of the Op. 30 No. 2 recordings are approximately 174.6 seconds in duration. We then ran experiments by considering pairs of modified datasets, as shown in Table 3. Each pairing of datasets constitutes a benchmark with a fixed average global time warp factor, where the global time warp factors range between 1.000 and 4.000. Note that there are multiple different pairings of modified datasets that could achieve the same fixed global time warp factor. Among the possible pairings, we adopted the strategy of pairing a dataset that is \"sped up\" with a dataset that is \"slowed down\", in order to minimize artifacts from performing TSM with extreme factors. For each pairing of datasets, we then ran an experiment exactly as before: we considered every possible pair of audio recordings \((A,B)\) of the same Mazurka (where \(A\) is taken from one modified dataset and \(B\) is taken from the other modified dataset), aligned the pair of recordings, and then evaluated the accuracy of the predicted alignment. Note that, in each of the seven benchmarks described above, we have exactly the same number of recording pairs as in the original Mazurka Dataset benchmark: \(\binom{64}{2}+\binom{88}{2}=5844\) pairs for training and \(\binom{64}{2}+\binom{34}{2}+\binom{51}{2}=3852\) pairs for testing. The difference is that in each pairing of recordings \((A,B)\), the durations of \(A\) and \(B\) have been modified via TSM to achieve a fixed global time warping factor. For example, when pairing the median_x2.000 and median_x0.500 datasets, every pair of recordings from Opus 17 No 4 will have a global average time warping factor of \(\frac{509.96}{127.55}=4.0\), and every pair of recordings from Opus 24 No 2 will also have a global average time warping factor of \(\frac{273.46}{68.36}=4.0\), etc. We evaluate alignment accuracy in the following manner. Given a predicted alignment between two recordings \(A\) and \(B\), we determine the predicted timestamp in \(B\) that corresponds to each ground truth beat timestamp in \(A\). The difference between the predicted timestamp in \(B\) and the actual ground truth beat timestamp in \(B\) is the alignment error. We can then calculate the percentage of predictions that are incorrect (error rate), where we define an incorrect prediction as one that has an alignment error greater than a fixed threshold value (error tolerance). Thus, there is a tradeoff between error rate and error tolerance. Figure 2 shows the process of calculating the error rate for a predicted alignment with two different error tolerances. With a small error tolerance (left), two out of six beat timestamps are considered incorrect, resulting in an error rate of \(\frac{2}{6}=33.3\%\). With a larger \begin{table}
\begin{tabular}{l c c c c c c c c} \hline \hline \multirow{2}{*}{**Piece**} & \multirow{2}{*}{**Files**} & \multicolumn{8}{c}{**Durations**} \\  & & \(\times\)**0.500** & \(\times\)**0.630** & \(\times\)**0.794** & \(\times\)**1.000** & \(\times\)**1.260** & \(\times\)**1.588** & \(\times\)**2.000** \\ \hline Opus 17, No 4 & 64 & 127.5 & 160.6 & 202.4 & 254.9 & 321.2 & 404.8 & 509.9 \\ Opus 63, No 3 & 88 & 64.1 & 80.8 & 101.8 & 128.2 & 161.6 & 203.6 & 256.4 \\ \hline Opus 24, No 2 & 64 & 68.3 & 86.1 & 108.5 & 136.7 & 172.2 & 217.0 & 273.4 \\ Opus 30, No 2 & 34 & 43.7 & 55.0 & 69.3 & 87.3 & 110.0 & 138.6 & 174.6 \\ Opus 68, No 3 & 51 & 49.8 & 62.8 & 79.1 & 99.6 & 125.6 & 158.2 & 199.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Overview of the modified Mazurka Datasets.