# Exploring Analytical Hierarchy Process for Multicriteria Assessment of Reinforced Concrete Slabs
## 2 Materials and Methods
### Principal Component Analysis  
Datasets sometimes have many features that make it difficult to analyze and obtain information contributing to decision-making processes. One way to minimize this is using dimensional reduction techniques such as Principal Component Analysis (PCA). The purpose of PCA is to represent the multidimensional dataset in a set of coordinates in a new dimensional space. The central idea is to adapt the coordinate systems into sets that are easy to analyze and discard unnecessary ones (those that do not present an advantage for data analysis). One of the main goals of PCA is dimensionality reduction using the first principal components that reflect the structure of the data [22]. The decomposition of the original data into subspaces (master and residual) can be performed according to Equation (3). \[X=\sum_{i=1}^{m}t_{i}p_{i}^{T}=TP^{T}+E \tag{3}\]  
where \(t_{i}\) and \(p_{i}\) are the \(i\)th principal component and the corresponding load vector, respectively; \(P\) and \(T\) are the load matrix and score matrix of the master subspace, respectively; \(E\) represents the residual matrix; K represents the number of retained principal components [23] (p. 4). The statistics used to monitor the two subspaces can be calculated according to Equation (4). \[T^{2}=x^{T}P\Lambda^{-1}P^{T}x\]  
\[Q=x^{T}\Big{(}I-PP^{T}\Big{)}x \tag{4}\]  
where \(\Lambda\) is the covariance matrix of \(T\). The control limits (Equation (5)) are utilized to assess if the observed process operates within the expected parameters, thus indicating that the process is functioning normally. \[T_{lim}^{2}\leq\ \frac{d(n-1)}{n-k}F_{k,(n-k),\rho}\]  
\[Q_{\rho}\leq g\chi_{h,\rho}^{2};g=\frac{\
u}{2a},\ h=\frac{2a^{2}}{\
u} \tag{5}\]  
\"Among them, \(F_{k,(n-k),\rho}\) is the F distribution of degrees of freedom \(k\) and \(n-k\) with the confidence \(\rho\). \(a\) and \(\
u\) are the mean and variance estimated from the Q statistic.\" [23] (p. 4). The use of PCA in the context of the Architecture, Engineering, and Construction (AEC) industry can be found in the most diverse areas, from application in demolition waste generation to material characteristics, as presented by [24], who used this method to design modern concrete mixes. This work was based on a sample of 550 types of mixture, which were later categorized and divided according to the compressive strength of the concrete. The main objective is to verify the possibility of using PCA to create the amount of each ingredient needed to reach a determined compressive strength. The author concluded that its use for this purpose was unsatisfactory. Reference [25] studied the application of PCA for the concrete mix design process, using 38 different recipes to achieve a refined one. For [26], the PCA was used to transform the correlated properties of the concrete to obtain combinations of parameters for multiple performance characteristic indexes and consequently optimize the properties of the fiber-reinforced geopolymer composites. In this paper, PCA was combined with the Taguchi Method. In addition to the use of the method in the material subject, works such as [27] present the possibility of using PCA for purposes of material strength. In the case of [27], this method was used to reduce the features of the original dataset and, from this reduction, apply them to Artificial Neutral Network (ANN) to predict the shear strength of reinforced concrete beams. The combination of ANN with PCA can also be seen in the work of [28], who used the dimensionality reduction of the data for application in the machine learning algorithm to obtain a predictive model of the slump and compressive strength of concrete with mineral additives. Other applications can be found in [29], which develops a prediction model based on machine learning algorithms combined with PCA for demolition waste; of the algorithms combined with PCA, the k-nearest neighbors proved to be the ones with the best results.