## 3 Methodology
For ensemble averaging and generalized averaging, the combination of \(\hat{y}_{k}\) values is based on a linear combination:  
\[\hat{y}=\sum_{k=1}^{K}\alpha_{k}f_{k}(x_{j})=\sum_{k=1}^{K}\alpha_{k}\hat{y}_{k} \tag{5}\]  
where:  
\(\hat{y}\)--level-1 cost prediction made by the ensemble of neural networks;  
\(\hat{y}_{k}\)--level-0 cost predictions by the \(k\)-th ANN--an expert belonging to the ensemble;  
\(f_{k}\)--mapping function implemented by \(k\)-th member network;  
\(\alpha_{k}\)--linear combination weight for \(k\)-th expert;  
under the condition:  
\[\sum_{k=1}^{K}\alpha_{k}=1 \tag{6}\]  
In the case of ensemble averaging, also referred to as simple averaging, the assumption is that the weights \(\alpha_{k}\) are equal. In other words, cost predictions \(\hat{y}_{k}\) provided by all ensemble members equally contribute to the cost prediction of \(\hat{y}\) provided by the ensemble:  
\[\alpha_{k}=\frac{1}{K} \tag{7}\]  
In the case of generalized averaging, the weights \(\alpha\)k are optimized based on the training results and training errors of the ANN member networks. In this case:  
\[\alpha_{k}=\frac{\sum_{l=1}^{K}\left(C^{-1}\right)_{kl}}{\sum_{k=1}^{K}\sum_{ l=1}^{K}\left(C^{-1}\right)_{kl}} \tag{8}\]  
Figure 2: General idea of the EoNN approach. where:  
\(C\)--correlation matrix of errors produced by the members of an ensemble;  
\(k\), \(h\), \(l\)--are indexes of ANN member networks (used for clarity). Matrix \(C\) elements, denoted as \(c_{kl}\) are computed using a finite-sample approximation:  
\[y=h(x)+\varepsilon \tag{2}\]0  
where:  
\(p\)--denotes the sample for which predicted values \(\hat{y}_{k}\) and expected values \(\hat{y}\);  
\(N\)--cardinality of a set of samples;  
\(k\), \(l\)--are indexes of ANN member networks (used for clarity). Below are the performance and error measures employed throughout the research, used for assessing both individual trained ANNs and EoNN cost predictions (\(R\)--Pearson's correlation coefficient, \(\frac{1}{2}MSE\)--half of mean squared error, \(RMSE\)--root mean squared error, \(MAPE\)--mean absolute percentage error, \(PE\)--percentage error). \[y=h(x)+\varepsilon \tag{2}\]1  
\[y=h(x)+\varepsilon \tag{2}\]2  
\[y=h(x)+\varepsilon \tag{2}\]3  
\[y=h(x)+\varepsilon \tag{2}\]4  
\[y=h(x)+\varepsilon \tag{2}\]5  
\[y=h(x)+\varepsilon \tag{2}\]6  
where:  
\(p\)--index of a sample;  
\(M\)--cardinality of subsets used in the course of the training process: for learning (subset \(L\)), validating (subset \(V\)), and testing process (subset \(T\)). (Division into L, V, and T subsets is explained in Section 5). The proposed EoNN-based models are intended to assist in what are commonly known as early cost estimates for construction projects, often referred to as conceptual estimates. Based on the literature [59; 60; 61] and practical considerations, certain expectations regarding prediction accuracy were established. In the literature, expected accuracy for conceptual estimates typically varies, primarily between \(\pm 20\%\) and \(\pm 30\%\). For the purpose of this study, it was cautiously assumed that the model's cost predictions should not deviate by more than \(\pm 30\%\) from actual contract costs. The results of the development of EoNN-based models are presented in Section 5. To assess the proposed models, benchmark models based on the linear regression approach were constructed. The comparison between the EoNN-based and benchmark models is presented, along with a discussion of the research results, in Section 6.