# Efficient Time-Series Clustering through Sparse Gaussian Modeling
## 5 Experimental Evaluation
### Parameter Selection and Tuning  
We run our experiments with a logarithmic (in the length \(T\) of the original time series) number \(m\) of inducing points. More specifically, we run our experiments with \(m=\gamma\log_{2}T\) (rounded to the closest integer), for \(\gamma\in\{1,2,3,4,5\}\). For the Sparse Gaussian Process Regression, we choose a constant-mean prior for the Gaussian Process and use the Matern kernel with parameter \(\
u=1.5\). We use Adam [30] to optimize the parameters of SGPR. We use a learning rate of \(1/10\) for optimizing the parameters of the Gaussian Process (i.e., the hyperparameters of the Matern kernel and the standard deviation of the noise) and a learning rate of \(1/(10\gamma)\) for optimizing the locations of the inducing points. For the initialization of SGPR's inducing point location optimization, we divide the time horizon of the dataset into \(m\) equally sized intervals. Thus, we avoid ending up with quite different inducing point locations for different initializations, which may happen due to the non-convexity of SGPR's objective and its sensitivity to different initializations. \begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Dataset** & **Size** & **Length** &
\begin{tabular}{c} **No. of** \\ **Classes** \\ \end{tabular} & **Type** \\ \hline
**Meat** & 120 & 448 & 3 & SPECTRO \\
**MedicalImages** & 1141 & 99 & 10 & IMAGE \\
**MiddlePhalanxOutlineAgeGroup** & 554 & 80 & 3 & IMAGE \\
**MiddlePhalanxOutlineCorrect** & 891 & 80 & 2 & IMAGE \\
**MiddlePhalanxTW** & 553 & 80 & 6 & IMAGE \\
**MoteStrain** & 1272 & 84 & 2 & SENSOR \\
**OliveOil** & 60 & 570 & 4 & SPECTRO \\
**OSULeaf** & 442 & 427 & 6 & IMAGE \\
**PhalangesOutlinesCorrect** & 2658 & 80 & 2 & IMAGE \\
**Plane** & 210 & 144 & 7 & SENSOR \\
**ProximalPhalanxOutlineAgeGroup** & 605 & 80 & 3 & IMAGE \\
**ProximalPhalanxOutlineCorrect** & 891 & 80 & 2 & IMAGE \\
**ProximalPhalanxTW** & 605 & 80 & 6 & IMAGE \\
**RefrigationDevices** & 750 & 720 & 3 & DEVICE \\
**ShapeletSim** & 200 & 500 & 2 & SIMULATED \\
**ShapesAll** & 1200 & 512 & 60 & IMAGE \\
**SmallKitchenAppliances** & 750 & 720 & 3 & DEVICE \\
**SonyAIBORobotSurface1** & 621 & 70 & 2 & SENSOR \\
**SonyAIBORobotSurface2** & 980 & 65 & 2 & SENSOR \\
**Strawberry** & 983 & 235 & 2 & SPECTRO \\
**SwedishLeaf** & 1125 & 128 & 15 & IMAGE \\
**Symbols** & 1020 & 398 & 6 & IMAGE \\
**SyntheticControl** & 600 & 60 & 6 & SIMULATED \\
**ToeSegmentation1** & 268 & 277 & 2 & MOTION \\
**ToeSegmentation2** & 166 & 343 & 2 & MOTION \\
**Trace** & 200 & 275 & 4 & SENSOR \\
**TwoLeadECG** & 1162 & 82 & 2 & ECG \\
**TwoPatterns** & 5000 & 128 & 4 & SIMULATED \\
**Wine** & 111 & 234 & 2 & SPECTRO \\
**WordSynonyms** & 905 & 270 & 25 & IMAGE \\
**Worms** & 258 & 900 & 5 & MOTION \\ \hline \hline \end{tabular}
\end{table} 
Table 1: _Cont._We should highlight that SGPR's objective function is highly sensitive to [PERSON]'s learning rate. If the learning rate is too large, we end up with inducing points outside the time horizon of the original time series, while if the learning rate is too low, the learning becomes local, and inducing points tend to reflect the behavior of the time series in a small interval around them. Hence, we had to carefully select the learning rates for our experiments, with the value of \(1/10\) proven to be a good and consistent choice. A summary of the parameters used in the experimental evaluation, with a short description of their role and their values, can be found in Table 2.  
For the \(k\)-means algorithm, we use the \(\beta\)-DTW distance with \(\beta=\alpha T/m\) (which we often denote \((\alpha T/m)\)-DTW), for \(\alpha\in\{0,10^{-4},10^{-3},10^{-2}\}\) and the DTW Barycenter Averaging (DBA) [19] for updating the cluster representatives in each iteration. We run \(k\)-means and DBA for a maximum of 100 iterations each.