## 4 Methodology
### Implementation of the Radial Basis Function Method (RBF)  
Beginning with the RBF and with one neuron in the hidden layer, the analysis proceeds on by adding one neuron for each additional time and analysis. The hidden layer activation function refers to the normalized radial basis function, used to connect the values of the units in a single layer to the values in the one that follows. The identity function roles as the output layer's activation function, and thus the output units are simply the weighted sums of the hidden units. Within the current design of the model, the normalized radial basis function serves as the activation function for the hidden layer. This function utilizes the SoftMax activation function to normalize all concealed unit activations in order to ensure their sum equals 1. The overlapping factor is the multiplier of the breadth of the radial basis functions. The overlapping factors were computed automatically. The amount of input data, denoted by d, is 1 + 0.1d.  
The hyperbolic tangent sigmoid transfer function, which specifies the behavior of the network model and produces output values within \(-1\) and 1, was chosen as the function for the current model. Equation (1) defines the hyperbolic tangent sigmoid transfer function, which is the function that has been adopted [58]:  
\[\mathrm{Tansig(x)}=(2/(1+\exp(-2\mathrm{x})))-1 \tag{1}\]  
The search finally arrived at a hidden layer ANN with 50 neurons. The research essentially yielded fifty models. According to the study, the ANN with 43 neurons in the hidden layer produced the most accurate outcomes based on relative error and sum of squares error.  
The data collected by the trained model showed that the method using ANN was effective in predicting the buildings' cost estimate with the least amount of construction data available and without requiring a more detailed architecture. We carefully selected attributes to maximize the model's ability to learn from the sparse input. To accomplish such a feat, the most pertinent independent variables had to be chosen. Additionally, in an attempt to eliminate errors or variations, we verified using high-quality data that were as accurate as possible. Ensuring the accuracy and consistency of our data source was the initial goal. When a building renovation construction company with decades of expertise provides high-quality data, it may significantly improve the performance of models, especially when the dataset is small to prevent overfitting. Furthermore, we employed regularization approaches such as early stopping, notably in the MLP approach. The proposed methodological approach's flowchart is shown in Figure 1:  
Figure 1: The recommended methodological procedure. Note: * 29 independent variables for final cost, estimated completion time, actual completion time, delay time, initial demolition-drainage costs, final demolition-drainage costs, cost of expenses (fuel and parking), initial plumbing costs, final plumbing costs, initial heating costs, final heating costs, initial electrical costs, final electrical costs, initial masonry costs, final masonry costs, initial construction costs of plasterboard construction, final cost of plasterboard construction, initial cost of bathrooms, final cost of bathrooms, initial cost of flooring, final cost of flooring, initial cost of frames, final cost of frames, initial cost of doors, final cost of doors, initial cost of paint, final cost of paint, initial cost of kitchen construction, final cost of kitchen; RBF for radial basis function; MLP for multi-layer perceptron.