# Functional Data Analysis in Sport Science: Example of Swimmers' Progression Curves Clustering
## 3 Results  
### Results on Simulated Data  
Table 2 below provides results on the comparison between the six algorithms of the _funcy_ package. These results are mainly illustrative and one should be aware that the quality of a clustering algorithm cannot be addressed through simulation. However, it can give some clues on the type of situations where algorithms seem to perform properly or not. Sample 1 was designed to be easy to cluster and most model based algorithms perform well. Nevertheless, they are outperformed by the only distance based method _distclust_ that gives almost perfect results. As Sample 2 is simply a noisier version of Sample 1, the problem becomes harder and results slightly decrease. One can note that, although the stochastic processes we sampled from are the same as in Sample 1, the \"hierarchy\" between methods \begin{table}
\begin{tabular}{l l l l} \hline \hline
**Data Set** & **Functions** & **Noise** & **Observations on t-Axis** \\ \hline \multirow{3}{*}{**Sample 1**} & \(t\longmapsto t-1\) & \multirow{3}{*}{\(\varepsilon\sim\mathcal{N}(0,0.05)\)} & \multirow{3}{*}{10 points at _regular_ instants} \\  & \(t\longmapsto t\) & & \\  & \(t\longmapsto t\) & & \\ \hline \multirow{3}{*}{**Sample 2**} & \(t\longmapsto t-1\) & \multirow{3}{*}{\(\varepsilon\sim\mathcal{N}(0,0.1)\)} & \multirow{3}{*}{10 points at _regular_ instants} \\  & \(t\longmapsto t^{2}\) & & \\  & \(t\longmapsto t^{3}\) & & \\ \hline \multirow{3}{*}{**Sample 3**} & \(t\longmapsto t-1\) & \multirow{3}{*}{\(\varepsilon\sim\mathcal{N}(0,0.5)\)} & \multirow{3}{*}{\(\leq 10\) points at _irregular_ instants} \\  & \(t\longmapsto sin(2\pi t)\) & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Details on the simulated samples. Processes are defined as \(f(t)+\varepsilon\) with four different functions \(f\) in each sample and a varying noise parameter \(\varepsilon\). changes. This might indicate differences at noise robustness between the methods. For example, performances of the _fcsm_ algorithm decrease only slightly compared to _distclust_. Finally, as expected, the results fall on the fuzzy situation of Sample 3. Only three methods achieve moderate performances, and one can note that there is an algorithm of both families among them. Although Table 2 gives information about the performances of these algorithms, it does not give information on the ability of the methods to retrieve the actual shape of the underlying functions. The following graphs will add some visual evidence to judge the quality of the results. Figure 2 gives one representation of the Sample 1 curves. In addition, the curves of each clusters' centres of the best performing algorithm are drawn. One can see that Sample 1 is quite simple to deal with, since curves of different groups are well separated. Not surprisingly, the _distclust_ clustering algorithm satisfyingly figures out the actual shape of each process. One can see in Figure 3 that, if the noisier situation of Sample 2 affects the good clustering rate, the shapes of the underlying functions remain correctly approximated by clusters' centres of _fscm_. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Method** & **Sample 1** & **Sample 2** & **Sample 3** & **Running Speed** \\ \hline fitflclust & 0.945 (0.14) & 0.857 (0.01) & 0.307 (0.06) & 2.8 \\ distclust & **0.996 (0.01)** & 0.888 (0.05) & 0.523 (0.07) & 19.2 \\ iterSubspace & 0.938 (0.14) & 0.850 (0.12) & **0.527 (0.07)** & **1** \\ funclust & 0.450 (0.17) & 0.418 (0.16) & 0.084 (0.07) & **1** \\ fscm & 0.948 (0.12) & **0.902 (0.01)** & **0.527 (0.07)** & 7 \\ waveclust & 0.920 (0.12) & 0.810 (0.01) & 0.324 (0.13) & 34 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Mean Rand Index and (Standard Deviation) on 100 simulations of the tree samples. Each algorithm run in at most a few seconds on our simulated data sets. Comparison in speed between algorithms is given as a multiple of the fastest which is set arbitrarily to 1.