# Functional Outlier Detection by Means of h-Mode Depth and Dynamic Time Warping
## 5 Industrial Test-Case Study  
### Presentation  
In this section, the outlier detection methodology is applied to a real industrial dataset of time-dependent numerical simulations. We consider an Intermediate Break Loss of Coolant Accident (commonly called _IBLOCA_ or simply _LOCA_) in a nuclear power plant, simulated with the CATHARE2 code. CATHARE2 is a _best estimate_ computer code capable of recreating the main physical phenomena that may occur in the different systems involved in nuclear reactors, in particular in the 900 MW French Pressurized Water Reactors. It embeds two-phase modeling to calculate the thermal-hydraulic behavior of the coolant fluid in the reactor. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**N = 100, \(p\) = 1\%** & **Model 1** & **Model 2** & **Model 3** & **Model 4** \\ \hline Algorithm & 98.36 & 97.80 & 99.57 & 93.06 \\ Sequential Transformations & 98.14 & 97.34 & 99.97 & 99.88 \\ Modified Band Depth & 84.00 & 61.39 & 98.49 & 1 \\ Integrated Depth & 83.15 & 59.81 & 98.42 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Average ranking of the outlier curve across the 100 replications of the experiments for the selected models. The Sequential transformations procedure is presented in [35]. The Modified Band Depth is presented in [36], and the standard Integrated Depth appears in [41]. In this case, the closer the value of a method is to 100, the more outlying it will be according to the corresponding ranking measure. A LOCA accident is originated by a breach in the primary circuit, which is designed to evacuate the heat generated by the nuclear core. The sudden loss of large quantities of coolant implies a fast increase of the water temperature nearby the nuclear fuel rods, due to the residual power generated by the core during the accidental transient. This power discharge and the subsequent temperature elevation must be compensated by the injection of water through a dedicated safety system. This is supposed to ensure that fuel rods temperature would remain below the fusion point at all times. Hence, the main safety criterion in LOCA regarding the confinement of the fuel concerns its Peak Cladding Temperature (PCT), i.e., its maximum cladding temperature over the duration of the LOCA (here, the time-dependent cladding temperature is the maximum cladding temperature for all the fuel rods, whatever its localization in the core). The particular statistical model under study involves many scalar input parameters (\(p=97\)), which specify all sorts of physical phenomena whose relative influences in the PCT are difficult to assess a priori. These input variables can be classified into various categories, such as (i) initial conditions and limit states for the system (Primary pressure, starting thermal power, the primary pumps inertia...), (ii) some parameters of specific physical models and correlations that are used (thermal exchanges between the components, friction between fluid phases or some geometric parameters of the installation), as well as (iii) some scenario variables (existence of blockages in the heat exchangers, fuel use in its life-cycle or initial temperature of the safety water injection). All of these scalar inputs of the simulation code are uncertain, and are hence represented by the random vector \(X=(X_{1},X_{2},...,X_{97})\), whereas the output of interest would be \(Y\), which is normally a critical safety parameter in this case, such as the aforementioned PCT. The total number of simulations that can be performed is relatively limited for such a high-dimensional input vector, since each run of the code takes around one hour to finish. For this reason, the use of classical multivariate statistical techniques is not straightforward. This explains the widely spread use of space-filling design methods to maximize the coverage of the input space [42], as well as metamodeling techniques to better exploit the number of code runs available for the physical model.