# Efficient Time-Series Clustering through Sparse Gaussian Modeling
## 5 Experimental Evaluation
### Dataset Level Assessment  
Our experimental evaluation indicates that Algorithm 1 (Sparse Time-Series Clustering), with sufficiently many (but still logarithmic in \(T\)) inducing points and for relatively small values of \(\alpha\) in \((\alpha m/T)\)-DTW, computes clusterings with ARI metrics quite similar to the ARI metrics of the baseline, computed by applying Algorithm 3 (i.e., standard \(k\)-means) to the original datasets. However, the asymptotic running time of Algorithm 1 is \(\Omega(T^{2}/\log^{2}T)\) times faster than the asymptotic running time of Algorithm 3. Following the practices described in [22], in our experimental evaluation, we report the average ARI score over 10 runs of Algorithms 1 and 3 and also rank the performance (according to the average ARI in decreasing order) of Algorithms 1 and 3 with different parameter configurations in all the datasets of Table 1. A comprehensive summary of the average ARI and the average rank for each method (over all Table 1 UCR datasets for each parameter configuration) can be found in Table 3. \begin{table}
\begin{tabular}{l l c} \hline \hline
**Parameter** & **Algorithm—Role** & **Value** \\ \hline learning rate & Adam, parameter optimization SGPR & \(1/10\) \\ learning rate & Adam, optimization of inducing point & \(1/(10\gamma)\), for \(\gamma\in\{1,2,3,4,5\}\) \\  & locations in SGPR & \\ \(\
u\) & Matérn kernel, SGPR & \(3/2\) \\ \(m\) & \# inducting points in SGPR & \(\gamma\log_{2}T\), for \(\gamma\in\{1,2,3,4,5\}\) \\ \(\alpha\) & \((\alpha T/m)\)-DTW distance for clustering & \(\in\{0,10^{-4},10^{-3},10^{-2}\}\) \\ \(I\) & max \# iterations of \(k\)-means and DBA & \(100\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Collection of parameters (with their role in our approach) and their corresponding values used in experimental evaluation. We recall that \(T\) denotes the length of the time series, \(N\) is the number of time series in the instance, and \(k\) is the number of clusters. In each of the first four lines of Table 3, we compare the average ARI of the standard \(k\)-means (baseline) with the average ARI of our framework for a different number of inducing points while maintaining the parameter \(\alpha\) in \((\alpha T/m)\)-DTW constant. In the fifth (resp. the sixth) line, we take the best (resp. the mean) ARI over all values of \(\alpha\), for each different number of inducing points \(\gamma\log_{2}T\), for \(\gamma\in\{1,2,3,4,5\}\). In Table 3, we observe that average ARI values improve as the number of inducing points increases (for every fixed value of \(\alpha\), see also the spread diagrams in Figure 3, where we use \(\alpha=10^{-4}\)). Moreover, for \(m\leq 4\log_{2}T\) inducing points, average ARI values slightly deteriorate as the value of \(\alpha\) increases (see also the spread diagrams in Figure 4, where we use \(m=2\log_{2}T\)), while for \(m=5\log_{2}T\) inducing points, average ARI values marginally improve as \(\alpha\) increases (see also the spread diagrams in Figure 5, where we use \(m=5\log_{2}T\)). We note that as the number \(m\) of inducing points increases, \((\alpha T/m)\)-DTW becomes less sensitive in an increase in \(\alpha\). Then, a combination of many inducing points and \(\alpha\in[10^{-3},10^{-2}]\) produces quite satisfactory results (see also the spread diagrams in Figure 5c,d. For \(m=5\log_{2}T\) inducing points, keeping the best ARI over all values of \(\alpha\) for each dataset gives clusterings of marginally better quality on average compared against the clusterings produced by the standard \(k\)-means algorithm (see the best ARI for \(m=5\log_{2}T\) in the best line of Table 3 and the spread diagram in Figure 5e). On the other hand, even for \(m=5\log_{2}T\) inducing points, any fixed value of \(\alpha\) results in clusterings of slightly worse quality on average compared against the clusterings produced by the standard \(k\)-means algorithm (see the mean ARI for \(m=5\log_{2}T\) in the mean line of Table 3 and the spread diagram in Figure 5f).