## 3 Materials and Methods
### Stage 3: Point Estimation and Complexity Index  
In this stage, the top-performing models from Stage 2 were used to generate point estimates for materialized cost contingency. To assess the influence of the complexity index, two versions of each model were developed: one excluding the complexity index and one incorporating it. A feature importance analysis was conducted to quantify the contribution of each input variable, particularly the complexity index, in enhancing predictive accuracy. The methodology for measuring the complexity index and computing feature importance is detailed in the following sections.  
Project scope is a crucial determinant of project complexity [15]. Unlike other complexity indices, such as expert interviews to measure technological and environmental complexity [57], which rely heavily on subjective data, scope complexity can be qualified with objective project data. This makes it a readily measurable metric for categorizing construction projects. Therefore, scope complexity was selected to quantify the complexity index used in this study. [PERSON] et al. [58] identified project size (in terms of capital), scope ambiguity, and project duration as key determinants of scope complexity. Due to data availability, this study focuses on two measurable factors for the complexity index: project size (capital) and project duration. Each factor is weighted equally (50:50) in the index calculation. The construction of the complexity index is outlined as follows:  
To classify project complexity by size and duration, a quantile-based approach was applied. This approach facilitated a straightforward division of projects based on their relative capital size and duration, ensuring objectivity grounded in the dataset distribution. Projects were divided into five categories according to their position within predefined quantile ranges, ensuring objective categories according to the distribution of the dataset. Specifically, a scale of 1 to 5 was used, where projects falling within the [0.8, 1.0] quartile range were assigned a complexity score of 5, those within the [0.6, 0.8] range were assigned a score of 4, etc. Projects within the lowest quartile were assigned a score of 1. The final complexity index is calculated as the average of the two scores, providing an objective and scalable measure of project complexity. This approach standardized the complexity index, enabling consistent comparisons across diverse project cases.  
Second, feature importance quantifies the contribution each input feature makes to the predictions made by an ML model [28]. It identifies the most significant features for accurate predictions by evaluating their impact on the model outcome. Depending on the algorithm, various techniques can be used to determine feature importance. As the ANN was the contingency estimation model with the most promising predictive performance in Stage 2, permutation feature importance was conducted at this stage [59]. This works by randomly shuffling the values of a single feature and then measuring the change in the model's performance [60]. If the performance drops significantly when a feature is permuted, that feature is considered important because it has a critical effect on the prediction model. Conversely, if shuffling a feature results in little to no performance change, the feature is considered less important. The permutation importance of a feature is computed as follows:  
\[I(F_{j})=P_{baseline}-P_{perm} \tag{7}\]  
where \(I(F_{j})\) is the permutation importance of feature \(F_{j}\), and \(P_{baseline}\) and \(P_{perm}\) are the model performance before and after shuffling, respectively. The baseline and permuted performance can be assessed with any of the metrics discussed in Stage 2, including \(R^{2}\), MAE, MSE, and RMSE. In this study, MSE was used to evaluate changes in performance.