# Functional Data Analysis in Sport Science: Example of Swimmers' Progression Curves Clustering
## 4 Discussion  
As mentioned in the simulated data set context, we shall emphasise that no objective criterion might reflect correctly the quality of a clustering procedure. The authors of [41] recall that all clustering algorithms are some way subjective regarding how they gather individuals or which metric they use. Thus, the resulting clusters should be judged and analysed according to the context. Like many other statistical tools, a clustering procedure does not give any quantitative certainty, but rather a new point of view on the data. One should consider as good results any useful perspective hidden in the raw data. Thus, we worked closely with sport experts, not only to analyse the results but throughout the entire analysis. All choices of parameters and/or methods were driven both by mathematical and sport considerations. In this work, we provide enlightenment on some classical methods and useful practical packages as well as provide some clues on the particularities of the different algorithms. One can note that distance-based methods are generally easy to use and give rather good results for simple problems. On the other hand, model-based methods rely on more complicated design but often give good results for a wider range of problems. It explains why they are often recommended by experts of the field [28] and form most of the algorithms implemented in _funcy_. Algorithms using Gaussian mixtures are naturally more flexible than methods like k-means, since they might be considered as a generalisation with elliptic clusters rather than circular ones. However, one should also keep in mind that this flexibility often requires longer computational time. Indeed, even if the EM algorithm is really efficient to solve the mixture of Gaussian problem, the multiplicity of models and the number of clusters to test might take non-negligible time to run (few hours in our case). For our purpose, which is to help a swimming federation with the detection of young promising athletes, computational time was not an issue since the aim was more long-term decision-making. Nevertheless, many sport-related problems today need to be solved quickly or even live, and our methodological choices would have been different under such constraints. Regarding the results on the swimming data set, we observe consistent outcomes from both mathematical and sport point of views. If our work does not give any certainty about the progression phenomenon of young swimmers, it sheds some light on its general pattern and provide a practical tool to gather similar profiles. Moreover, using FDA, we were able to figure out information from uneven  
Figure 6: Cluster centres’ curves of swimmers (**left**) and derivatives (**right**) coloured by clusters, obtained with the multivariate clustering _funHDDC_ algorithm. Similar information as in Figure 4 can be seen on this graph, in a clearer way with only the centre of clusters displayed. time series. Using smooth functions instead of raw data points provides a first understanding of the main trends and the continuous nature of the progression phenomenon. However, one should always pay attention to the random fluctuations of the data that serve to fit the studied functions. In order to improve the quality of the approximation and decrease the influence of the noise, we would like to collect more data on swimmers, with training performances for example. Nevertheless, these results might help the detection of promising young athletes with both a better understanding and graphical outcomes to support the decision process. Note that this work remains descriptive and thus preliminary, but one can think of it as a first step for a further predictive analysis. If we do not discuss here findings about any particular swimmers for confidentiality concerns, we can highlight some points that seem interesting to swimming experts. First, as mentioned in [3; 24], it seems difficult to precisely detect young talents before 16 years old because of the fast evolution before this age.