## 3 Methodology Concept
### Mathematical Framework  
The distributions of monitoring data are denoted as \(S_{j}^{\mathit{mon}}\), where \(j=1,\ 2\ldots n^{\mathit{mon}}\) represents individual measured quantities, with a total of \(n^{\mathit{mon}}\). These distributions must be statistically defined and modeled before the calculation starts, as they represent input data for the probabilistic analysis, which can be referred to as a \"weighted\" probabilistic analysis. Modeling of the distribution functions of the monitoring data is also performed via a statistical analysis of the available measured quantities. The \"weighted\" analysis begins in the same way as the baseline analysis, with the sampling of input variables (N sampled combinations) for the computational model. Then, for each parameter combination, the FEM calculation is conducted. After the calculation, results in each simulation are verified against corresponding monitoring data probability distributions. In this context, the calculation results \(S_{ij}\) are being weighted (i = 1, 2, \ldots, N\) represents a single simulation, and \(j=1,\ 2\ldots n^{\mathit{mon}}\) represents one of the results that have a monitoring data distribution). Results that better match the monitoring data \(S_{j}^{\mathit{mon}}\) are given a higher impact by introducing weight coefficients \(f_{i}\) for each simulation \(i\). The coefficient \(f_{i}\) is obtained by aggregating the individual weight coefficients \(f_{ij}\), which correspond to individual expected distributions of monitoring data. The procedure for calculating the individual weight coefficients \(f_{ij}\) is described as follows:  
* Once each FEM run is finished, the relevant response values \(S_{ij}\)--those matched by available monitoring data--are logged for every simulation \(i\). * Each result \(S_{ij}\) is subsequently evaluated in the monitoring distribution \(S_{j}^{mon}\) yielding a probability density \(PDF_{ij}\), that quantifies how likely that simulation outcome is relative to the observed data. * The individual weight coefficients \(f_{ij}\) for each monitoring data distribution \(S_{j}^{mon}\) are calculated as the ratio of the probability density of the simulation result \(PDF_{ij}\) to the maximum probability density of the function \(S_{j}^{mon}\), which is denoted as \(PDF_{j,max}\): \[f_{ij}=\frac{PDF_{ij}}{PDF_{j,max}}\cdot I_{j}^{w}\] (2) where the coefficient \(I_{j}^{w}\) represents the importance weight of each expected distribution \(S_{j}^{mon}\) (between 0 and 1). If all results have the same importance, \(I_{j}^{w}=1\) is adopted for all values of \(j\). The total weight coefficient \(f_{i}\) (which refers to simulation \(i\)) is obtained using the individual weight coefficients \(f_{ij}\) by utilizing the chosen aggregation function [35]. The choice of method for combining individual weight coefficients depends on various factors that characterize the specific analysis, such as the complexity and cost of the FEM analysis, the number of monitoring individual distributions and the assigned importance coefficients. In this paper, the product of weights method, given by Equation (3), is chosen as an example. This method calculates the product of the individual weight coefficients within a single observed simulation. \[f_{i}=\prod\
olimits_{j=1}^{n^{mon}}f_{ij} \tag{3}\]  
Next, the histogram of the calculation results is created by considering the weight coefficients of the simulations \(f_{i}\). The frequency of results \(F_{k}^{weighted}\) for the observed bin \(k\) in the case of a weighted histogram is equal to the sum of the weight coefficients of those simulations whose solutions fall within the range of the bin, that is:  
\[F_{k}^{weighted}=\sum\limits_{i=1}^{n_{k}}f_{i} \tag{4}\]  
where \(n_{k}\) represents the number of simulations whose solutions fall within the range of the bin. Weighted histograms take into account the weight coefficient of each simulation when calculating the frequency, which allows for a more accurate estimation of the probability density when the data vary significantly.