## 3 Methodology
In the Czech Republic, this procedure is commonly used for indexing the prices of construction works between different time periods and is also permissible for the needs of court evidence. Outlier analysis was applied to the updated values of contract net costs for sewerage project construction works. The concept and fundamentals of outlier analysis can be found in the statistical literature, such as [56, 57, 58]. The purpose was to eliminate data points that deviated significantly from others, essentially excluding unusual cost values from the dataset. The approach used in this study relied on the concept of the interquartile range (_IQR_). The _IQR_ was calculated as the difference between the values of the third quartile (_Q3_) and the first quartile (_Q1_), representing the range of values between these quartiles: \(IQR=Q3-Q1\). The rule below allowed us to identify and eliminate outlying values, as well as entire records for certain project cases from the dataset:  
* 1.5\(\cdot\)_IQR_; _Q3 + 1.5\(\cdot\)_IQR_\(>\) \(\rightarrow\) eliminate the \(j\)-th record from the dataset. The rationale for this approach, based on the authors' prior experiences, is that outliers can be problematic when developing cost prediction models for various construction projects, facilities, and structures, as they often represent specific, high-cost projects that are sparsely represented in datasets. Such data can distort the predictive performance of a developed model. The collected data underwent further pre-processing. Numerical values of cost predictors were linearly scaled, while descriptive values were processed differently based on their nature; they were pseudo-fuzzy scaled, coded as one-of-n values, or converted into binary values. Detailed information and outcomes of this step are presented in Section 4. The next stage of the research involved computations and simulations of artificial neural networks (ANN), as well as the combination of these networks to create an ensemble. The development of models based on ensembles of neural networks (EoNN) designed to predict costs involves solving regression problems. Let the dependent variable \(y\) of such models represent the construction cost for a sewerage project. Also, let the independent variables be the cost predictors, with the vector of these variables denoted as \(x\). The EoNN-based models are expected to approximate the mapping \(x\to y\). Importantly, in the case of employing EoNN, the approximation function \(h\) is implicitly defined as follows:  
\[y=h(x)+\varepsilon \tag{2}\]  
where \(\varepsilon\) corresponds to the prediction error. Consequently, the formal notation for predicting costs \(\hat{y}\) can be expressed as follows:  
\[\hat{y}=h(x) \tag{3}\]  
The utilization of EoNN as the core of a cost estimation model relies on combining a set of trained neural networks to form an ensemble. As outlined in [28], this set might encompass various types of networks or similar networks trained to different local minima. The two methods built on this premise, which were applied during research presented herein, are (1) ensemble averaging and (2) generalized averaging. A summary of the core principles behind the two ensemble-based methods mentioned is provided based on [28, 30]. The general idea is schematically depicted in Figure 2.  
At level 0, the tools consist of \(K\) experts, which are individually trained artificial neural networks (ANN), each providing cost predictions:  
\[\hat{y}_{k}=f_{k}(\mathbf{x}) \tag{4}\]  
The cost predictions \(\hat{y}_{k}\) serve as inputs for the level-1 tool, which combines the predictions generated by the \(K\) experts to obtain a more objective final prediction of construction costs \(\hat{y}\).