## 2 The Proposed Method for Integration of Uncertain Expert Knowledge to the System Model
### Pre-Processing of Uncertain Expert Knowledge  
Expert knowledge is a set of subjective assessments of several professionals who created their judgments independently from each other. If they can give their judgment about the unknown variable (\(x_{i}\), that can be a model parameter, input or output) as an exact value, the uncertainty of the assessment can be modeled well with normal distribution. Figure 1: Parameter estimation and evaluation of expert knowledge based on the integration of expert-based information into the system model. The expert knowledge is available in the form of probability distributions of the parameters and outputs. Outputs are estimated by Monte Carlo (MC) simulation based on the system model and the sampled probability distributions of parameters, and parameters are deduced and refined according to the expert-based outputs by importance sampling (IS), iteratively. Expert knowledge can also be evaluated according to the results. \(\mathbf{u}\), \(\mathbf{\Theta}\), and \(\mathbf{y}\) are represent the input, parameter, and output vector of the system, respectively. They need to be estimated gathered to vector \(\mathbf{x}\), and its elements are marked by \(x_{i}\). However, sometimes experts are aware of the inaccuracy of their knowledge, and are not able to give an exact estimation for the unknown values, but they provide an interval by minimum \(x_{i,min}\) and maximum \(x_{i,max}\) bounds, e.g., in the case of the \(e\)th expert:  
\[x_{i}\in[x_{i,min}^{e},x_{i,max}^{e}] \tag{3}\]  
This information is usually handled in the form of uniform distribution mathematically, if there is no reason to assume that the experts consider some elements of the given interval more likely than others [25]:  
Figure 2: The steps of the procedure used in this paper to integrate expert knowledge into the system model. The initial information from the experts is represented by uniform distributions (green), which are aggregated for each estimable \(x_{i}\) variable (red). Sample elements are drawn from the aggregated distributions of the parameters and are weighted (\(w_{i}\)) according to those of the outputs (\(y_{i}\)). Thereby, empirical distributions (blue) are generated and compared to the original expert-based distributions (green), thus creating goodness values (\(g_{i,e}\)) of expert judgments. \[p_{e}(x_{i})= \begin{cases}\dfrac{1}{x_{i,max}^{e}-x_{i,min}^{e}}&\text{for}\quad x _{i,min}^{e}\leq x_{i}\leq x_{i,max}^{e}\\ 0&\text{for}\quad x_{i}<x_{i,min}^{e}\quad\text{or}\quad x_{i}>x_{i,max}^{e} \end{cases}  \tag{4}\]  
where \(p_{e}(x_{i})\) represents the opinion of the \(e\)th expert about \(x_{i}\). If multiple expert assessments are available, the individual distributions should be aggregated for every \(x_{i}\), as seen in the second row of Figure 2 by forming their mean. Thereby, all expert judgments were handled with the same weights:  
\[p(x_{i})=\frac{1}{E}\sum_{e=1}^{E}p_{e}(x_{i}) \tag{6}\]  
where \(E\) denotes the number of experts. If some variables or parameters of the system are available as a probability distribution instead of an exact number, a sample that represents them has to be generated. Monte Carlo simulation is a good solution for the problem. According to the technique, samples are drawn from the probability distributions of uncertain inputs (\(p(u_{i})\)) and parameters (\(p(\Theta_{i})\)):  
\[p(x_{i})\approx\{x_{i}^{j}\}_{j=1}^{N} \tag{7}\]  
where \(x_{i}^{j}\) is the \(j\)th sample element from \(p(x_{i})\), and \(N\) is the number of the samples. Thus, the original continuous probability distribution is represented by a discrete sample, as seen in Figure 3.  
Then, the output of the model can be calculated for every sample element according to Equation (1). If the sample size is large enough, the resulting probability distributions of the outputs will consequently be the same for every run of the simulation.