## 2 Literature Review
### Integration of PCA to Address QFD Limitation: The Complexity of Matrices
#### 2.3.2 Integration of PCA with QFD  
PCA, involving a reconstructed dataset \(\mathbb{\hat{X}}\) approximating \(\mathbb{X}\) through a scores matrix (T) and a loadings matrix (P), was utilized in this study for unsupervised feature selection to determine the importance of feature requirements in meeting customer expectations as the algorithm represented in Figure 1. It outlines the algorithm of feature selection using PCA for feature requirements.  
The algorithm begins by normalizing the dataset \(\mathbb{X}\) to have zero mean and unit variance, which is crucial for PCA analysis. Next, the covariance matrix \(C_{x}\) is computed using the normalized dataset \(\mathbb{X}\), given by Equation (1), where \(\mathbb{m}\) is the number of samples and \(\mathbb{X}^{T}\) is the transpose of \(\mathbb{X}\).  
\[C_{x}=\frac{1}{m-1}XX^{T} \tag{1}\]  
Following this, the eigenvectors (\(\mathbb{p_{i}}\)) and eigenvalues (\(\mathbb{\lambda}\)i) of the covariance matrix \(\mathbb{C}_{x}\) are calculated by Equation (2).  
\[C_{x}p_{i}=\lambda_{i}p_{i} \tag{2}\]  
The eigenvectors represent the principal components, and the eigenvalues indicate the amount of variance explained by each principal component. The eigenvectors are then sorted in descending order based on their corresponding eigenvalues to identify the most significant principal components that capture the maximum variance in the data, or utilizing singular value decomposition (SVD) to automatically rank principal components (PCs) based on their singular values following Equation (3).  
\[\mathbb{X}=\mathbb{U}\Sigma\mathbb{V}^{T} \tag{3}\]  
The delineation of the SVD of a matrix (3) serves as a pivotal tool for dimensionality reduction in the context of this algorithm. It elucidates the decomposition of the original matrix \(X\) into its constituent components as follows: \(U\), the matrix of left singular vectors; \(\Sigma\), the diagonal matrix of singular values; and either \(V\) or its transpose \(\tau\), representing the right singular vectors matrix.  
Then, the loadings matrix (P) is determined by minimizing the reconstruction error, and the expected value of the reconstruction error is calculated as the sum of minor eigenvalues [48].  
\[P=\text{argmin}\parallel X-XPP^{T}\parallel^{2} \tag{4}\]  
Figure 1: The algorithm of feature selection using PCA for feature requirements.  
Equation (4) indicates that \(P=V\) since \(PP^{T}\) is the orthogonal projection of \(X\) onto the subspace spanned by the columns of \(V\). Thus, the loadings, scores, and reconstructed dataset are derived through the singular value decomposition of X following Equation (5).  
\[P=V \tag{5}\]  
The algorithm involves steps such as normalizing the dataset, calculating the covariance matrix, computing the eigenvectors and eigenvalues, sorting the eigenvectors in descending order of eigenvalues, and selecting the top eigenvectors as principal components. By selecting the principal components with the largest eigenvalues, the algorithm aims to capture the maximum variance in the data, thus identifying the most critical FRs for meeting customer requirements. The algorithm's objective is to reduce the dimensionality of the dataset while retaining as much variance as possible, simplifying the feature requirements selection process in QFD matrices. Through PCA, the algorithm provides a systematic approach to identifying the most influential feature requirements, facilitating efficient decision-making in the conceptual design phase of product and service.