# Improving the Robustness of DTW to Global Time Warping Conditions in Audio Synchronization
## 3 Results
### Experimental Results
\begin{table}
\begin{tabular}{l l} \hline \hline
**System** & **Description** \\ \hline DTW1 & Standard DTW with \((1,1)\), \((1,2)\), \((2,1)\) transitions and weights \(2\), \(3\), \(3\). \\ DTW2 & Standard DTW with \((1,1)\), \((1,2)\), \((2,1)\) transitions and weights \(1\), \(2\), \(2\). \\ DTW3 & Standard DTW with \((0,1)\), \((1,0)\), \((1,1)\) transitions and weights \(1\), \(1\), \(2\). \\ DTW4 & Standard DTW with \((0,1)\), \((1,0)\), \((1,1)\) transitions and weights \(1\), \(1\). \\ DTW5 & Standard DTW with \((0,1)\), \((1,0)\) transitions and weights \(1\), \(1\). \\ DTW1-add3 & Standard DTW with \((1,1)\), \((1,2)\), \((2,1)\), \((1,3)\), \((3,1)\) transitions and weights \(2\), \(3\), \(3\), \(4\), \(4\). \\ DTW1-add4 & Standard DTW with \((1,1)\), \((1,2)\), \((2,1)\), \((1,3)\), \((3,1)\), \((1,4)\), \((4,1)\) transitions and weights \(2\), \(3\), \(3\), \(4\), \(4\), \(5\), \(5\). \\ \hline DTW2-downsampleNN & Downsample the longer sequence to match the shorter sequence using nearest neighbor interpolation, then use standard DTW with \((1,1)\), \((1,2)\), \((2,1)\) transitions and weights \(1\), \(2\), \(2\). \\ DTW2-downsampleLin & Downsample the longer sequence to match the shorter sequence using linear interpolation, then use standard DTW with \((1,1)\), \((1,2)\), \((2,1)\) transitions and weights \(1\), \(2\), \(2\). \\ DTW2-upsampleNN & Upsample the shorter sequence to match the longer sequence using nearest neighbor interpolation, then use standard DTW with \((1,1)\), \((1,2)\), \((2,1)\) transitions and weights \(1\), \(2\), \(2\). \\ DTW2-upsampleLin & Upsample the shorter sequence to match the longer sequence using linear interpolation, then use standard DTW with \((1,1)\), \((1,2)\), \((2,1)\) transitions and weights \(1\), \(2\), \(2\). \\ \hline AdaptiveWeight1 & Standard DTW with \((0,1)\), \((1,0)\) transitions and adaptive weights \(w_{1}=1\), \(w_{2}=\frac{L_{max}}{L_{min}}\), where the second sequence is assumed to be longer. \\ AdaptiveWeight2 & Standard DTW with \((0,1)\), \((1,0)\), \((1,1)\) transitions and adaptive weights \(w_{1}=1\), \(w_{2}=\frac{L_{max}}{L_{min}}\), \(w_{3}=1+\frac{L_{max}}{L_{min}}\), where the second sequence is assumed to be longer. \\ \hline SelectiveTransitions-max2 & Modified DTW with \((0,1)\), \((1,0)\), \((1,1)\) nominal transitions and weights \(1\), \(1\), \(2\), but with \((1,0)\) transitions removed from locations \((i,j)\) where \(i\%W_{max}==0\) and \((0,1)\) transitions removed from locations \((i,j)\) where \(j\%W_{max}==0\), with \(W_{max}=2\). \\ SelectiveTransitions-max3 & Same as above, but with \(W_{max}=3\). \\ SelectiveTransitions-max4 & Same as above, but with \(W_{max}=4\). \\ SelectiveTransitions-max5 & Same as above, but with \(W_{max}=5\). \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Explanation of the 17 systems whose performance is compared in Figure 3. Systems are grouped into four different categories: standard DTW, sequence normalization, adaptive weighting, and selective transitions. Figure 3: Assessing the robustness of various alignment methods to global time warping conditions. The seven groups along the horizontal axis correspond to seven benchmarks with different average global time warping factors. Colored bars indicate the error rate at 200 ms error tolerance, and the black horizontal lines indicate error rate at error tolerances of 100 ms (above) and 400 ms (below). Note that the bar colors are grouped by category, so that similar algorithms are shown as different shades of the same hue. There are many things to notice about Figure 3. We will unpack and interpret the results for each color group in the paragraphs below. First, consider the two yellow systems (DTW1, DTW2). These two systems have allowable transitions \((1,1)\), \((1,2)\), \((2,1)\) with two different weighting schemes. This set of allowable transitions is often used in audio-audio alignment, so these systems can be interpreted as a type of baseline.