# Functional Outlier Detection by Means of h-Mode Depth and Dynamic Time Warping
## 6 Discussion and Conclusions  
This paper has dealt with a particular branch of unsupervised anomaly detection: the outlier detection problem in functional data. The main aspects to take into account when dealing with functional data or high-dimensional objects in general have also been developed, exposing its main challenges and advantages. A new time-dependent outlier detection methodology based on the use of non-parametric features has been proposed, assessed with synthetic data, and illustrated on thermal-hydraulic simulations, aiming at capturing the outlyingness of the elements of any considered functional dataset both in the magnitude and the shape senses. This is done via the joint use of the h-mode depth and the dynamic time warping, and by defining outliers as data which do not belong to the minimum volume set of a chosen probability. The underlying distribution in the feature space is modeled through a GMM whose estimation is adapted to the outlier detection framework by means of modifications of the EM algorithm. The maximal probability for which a datum is not regarded as an outlier anymore is used as a score \(\theta\) of its outlyingness. An original detection algorithm has been proposed, effectively allowing the trimming of functional data. This methodology, based on the use of two features, benefits from the notion of level set to treat real industrial problems based on time-dependent data even if the available data are scarce. Several features have been compared in this framework based  
Figure 4: Scatter plots of two input variables of CATHARE2 and the outlyingness score \(\theta\). The points correspond to the bivariate plot of the values of the selected variable and the corresponding \(\theta\). The red dots correspond to the simulations that have been retained as outliers. on some toy examples and two scores related to the outlyingness of functional data. Based on the results of these application cases, both the notions of dynamic time warping, and the h-mode depth have proven their efficiency when compared to other features such as the \(L^{2}\) norm and the Modified Band Depth. Finally, the analysis of simulations of the thermal-hydraulic behavior of a nuclear reactor during a Loss Of Coolant Accident has been carried out to illustrate the benefits of the method. This was achieved thanks to the use of sensitivity analysis tools (HSIC-based) capable of accounting for the dependence of the input variables of the numerical simulator. Regarding the perspectives of this work, a primary objective would be the in-depth quantification of the causes for the detected anomalous characteristics of certain functions in real physical cases. In the case of numerical simulators, the identification of the inputs of the code that actually present an influence on the anomalous outputs can help engineers to detect possible defects of the code or finding physical phenomena of interest. This is also relevant to ensure the quality of the datasets that are used in the assessment reports of critical systems such as nuclear power plants. The following are available online at [[https://www.mdpi.com/article/10.390/app112311475/s1](https://www.mdpi.com/article/10.390/app112311475/s1)]([https://www.mdpi.com/article/10.390/app112311475/s1](https://www.mdpi.com/article/10.390/app112311475/s1)), file SI: Functional outlier detection by means of h-mode depth and dynamic time warping. Conceptualization, A.R.d.P. ; Methodology, A.R.d.P. ; Supervision, M.C., B.I., N.M., A.M., E.M. and R.S. ; Writing--original draft, A.R.d.P. All authors have read and agreed to the published version of the manuscript. This research received no external funding. Not applicable. Research data are not shared due to confidentiality reasons. Part of this work has been done in connection with the ACIDITHY project (French research program NEEDS, including CNRS, CEA, IRSN and EDF). We acknowledge two anonymous reviewers for their helpful remarks. Many thanks to [PERSON] for his help with the industrial use case. The authors declare no conflict of interest.