# Functional Outlier Detection by Means of h-Mode Depth and Dynamic Time Warping
## 5 Industrial Test-Case Study
### Sensitivity Analysis on Outliers  
In this kind of numerical simulations, the detection of outputs that present a globally anomalous behavior is of critical importance, and characterizing what are the physical phenomena which have an actual influence on it can have the same importance, if not more. A way of performing this analysis is to establish some kind of dependence measure between the inputs of the simulation code and the outlying score \(\theta\). However, the high dimensionality of the problem and the possible correlations between the input variables  
Figure 3: IBLOCA transient curves presenting the highest degree of outlyingness (red) and the least outlying curve (blue). Figure 2: Examples of IBLOCA transients simulated with CATHARE2. Only 40 curves are displayed for clarity. of the code can make this a difficult task. The Hilbert Schmidt Independence Criterion (HSIC) [43] can be a useful tool in this context to test the dependence between the scalar input variables of the code and the outlying score. This is a first step to understand which physical variables are the ones that actually influence the anomalous behavior of the outputs. By performing statistical tests on the HSIC values of the couples \((X_{i},Y)\) in the design of experiments it is possible to quantify their dependence. Without going into the technical details of the procedure (see [44] ), the HSIC represents a dependence measure between both variables, and can be used to build a statistical test with null hypothesis: \(\mathcal{H}_{0}\): the variable \(X_{i}\) and \(Y\) are independent. The hypothesis is rejected if the associated \(p\)-value of the test is inferior to a significance level threshold \(\alpha\). If \(\mathcal{H}_{0}\) is rejected, then the existence of a dependence structure between the input variable \(X_{i}\) and the output exists. In this case, we apply this measure to perform a Target Sensitivity Analysis, i.e., sensitivity analysis but quantifying the influence of the considered parameters in a restricted domain \(\mathcal{M}\) of the possible output values (\(\mathcal{M}\subset\Theta\) such that \(\mathcal{M}=\{\theta\in\Theta|\theta>0.9\}\)). This application to the set of input data and the obtained values of \(\theta\) yields several influential variables that are shown in Table 4.  
All these input variables can be considered to have an influence on the outlyingness of the resulting simulated curve, which represent the evolution of the maximum cladding temperature during the transient. Variable \(X_{16}\) represents the friction between the injected water in the primary circuit by the accumulators and the injection line. This parameter has been found to be influential in other similar studies since the compensation of the lost water during the transient is mainly guaranteed by this system (the accumulators), and therefore the line connecting it to the primary is of crucial importance. If this value of friction increases, the water flow will be reduced, with the consequent increase in the average temperatures of the fuel. Regarding variables \(38,45\) and \(64\), they are representative of physical phenomena occurring in the reactor pressure vessel during the transient. They model respectively: the heat transfer coefficient between the nuclear fuel and the surrounding coolant; the increase in pressure drop due to the deformation of the nuclear fuel due to the thermal-mechanical stress, prevents the coolant from ascending easily to the top of the reactor pressure vessel; and the friction between the steam and water in the core during the reflood phase of the transient. These elements are relevant since their evolution greatly influences the rewetting dynamic of the fuel, and the heat extraction in the short-term phase of the transient. Finally, variables \(X_{62}\) and \(X_{68}\) are representative of phenomena which occur between the steam and coolant water in the downcomer (the annular part which links the injection line of the accumulators and the nuclear core).