## 2 Literature Review and Cost Estimation Methods
### Analysis of Gray Theory in Cost Estimation  
This theory dates back over 30 years from the first test, being structural and stable in terms of mathematical principles. Used in the case of gray equations, gray algebra, matrices, vectors, and other axioms, it generates gray sequences within some systems and gives life to some sub-systems. Therefore, it performs correlations and analyses in models with small samples and poor information, only by processing legible data. This model can be a significant predictor in the field of constructions, at the base being a first-order equation that contains a variable. We can say that the main idea of prediction takes into account the sequence of initial numbers, iterated several times in order to generate a sequence of numbers, which has a linear character of the first order. Therefore, the generated equation is differential and shows an aspect related to regularity.  
\[x^{0}=\left(x^{0}(1),x^{0}(2),x^{0}(n)\right) \tag{7}\]  
The equation expresses the set of accumulated original data that can later generate a new cumulative sequence and is also characterized as an AGO-type operator. The process of selecting the characteristic factors is conducted depending on the analysis of the estimated cost of the real one, and information comes from other projects. When analyzing the composition of the costs of a project and the changes that will be made later, the type of foundation, the kind of structure, the number of layers of the project, the interior and exterior decoration, and the location of the project must be taken into account. Following the process of quantifying these factors, the use of this system is used as a variable of the cost control model within the neural network, and an output variable defines the final unit cost. We can say that the quantitative processing of eigenvectors can affect the cost of the project. The definition and representation of the eight factors used are thus made U1 \(\approx\) U8, being significant input vectors of the network model in the final representation of the project as output vectors. The first result of the quantifications performed is presented in Table 1.  
[PERSON] and other collaborators have treated the problem of cost estimation since the 1970s with developments in the MATLAB application [33]. They designed an interface-type program with EISPACK and LINKPACK using the combination of matrices to analyze values using advanced programming. These methods proved at that time extremely convenient to use in terms of training data and the set of tools provided. The most important aspect was that it could be used in different fields such as economics, mathematics, and even engineering. Another important aspect was that the MATLAB tool reduced the complexity of programming and improvement; only the use of a single template and the introduction of real data and parameters managed to provide the data they needed at that time.  
\begin{table}
\begin{tabular}{c c c} \hline
**ID Factors** & **Influence Factors** & **Quantitative Indicators** \\ \hline \(U_{1}\) & Standard type & Foundation type: complete, prefabricated, belt \\ \(U_{2}\) & Structure type & Frame structure: structure with concrete frames, walls, or bricks \\ \(U_{3}\) & Number of layers of project & Height levels: multistory, high, super high \\ \(U_{4}\) & Exterior decoration & Decorative exterior, ceramic tile, stone dust \\ \(U_{5}\) & Window and door & Laminated double glazed windows, wooden doors, steel hardware, secure door \\ \(U_{6}\) & Interior decoration & Rough elements, cement with mortar, paint and mixed mortar \\ \(U_{7}\) & Floor & Mortar, concrete or fine stone flooring, ceramic wall, and floor tiles \\ \(U_{8}\) & Project location & Interior, exterior, or exterior exposure design on urban or rural levels and areas \\ \hline \end{tabular}
\end{table}
Table 1: Exposure of the vector quantization index of the technical type characteristics.