# Means and Issues for Adjusting Principal Component Analysis Results
## 1 Introduction  
Much of the data we handle consist of numerous measurement items, requiring multivariate analysis. Among the various methods available, principal component analysis (PCA) is particularly well-suited for scientific studies due to its limited methodological flexibility and high reproducibility [1, 2, 3, 4, 5, 6, 7, 8, 9]. The simplicity of PCA ensures that its results are consistent, regardless of the individual performing the analysis. This property is highly valuable in scientific research as it guarantees objectivity. In PCA, we use singular value decomposition (SVD) to separate the matrix into unitary matrices (\(U\) and \(V\)) and a diagonal matrix \(D\),  
\[M=UDV^{\star},\]  
where \(V^{\star}\) denotes the conjugate transpose of \(V\), and \(M\) represents the data to be analyzed, often centred or even scaled. The diagonal elements of \(D\) are sorted in descending order. Using this decomposition, the principal components (PCs) can be derived as follows:  
\[Y=MV=UD\text{ and }Z=M^{\star}U=VD\]  
where the column vector of \(Y\) is PC for the samples and that of \(Z\) represents the PCs for the measurement items (e.g., PC1, PC2, etc.) [8]. Since \(U\) and \(V\) are unitary matrices, they satisfy the properties \(U^{\star}U=I\) and \(UU^{\star}=I\). This ensures that the row and column vectors are orthogonal and have a Euclidean norm of 1. Specifically, the inner product of avector with itself equals 1, while the inner product with other vectors equals 0, confirming orthogonality. As a result, the column vectors of \(Y\) and \(Z\) are also independent. These vectors can be interpreted as rotations of the original matrix \(M\), with the unitary matrices defining the axes and the diagonal elements of \(D\) determining their lengths. Since the diagonal elements of \(D\) are sorted, PC1, PC2, and subsequent components account for the decreasing proportions of the total data variance. PCA thus identifies common directions within \(M\) and summarizes them in order of their significance. This ability to effectively reduce the dimensionality of multivariate data is a key advantage of PCA. The advantage of PCA as a scientific method lies in its objectivity. Due to the limited degree of choice involved in its application, the results remain consistent regardless of who conducts the analysis. Given that science is inherently a collaborative endeavour, it is essential that participants are interchangeable at all times and that the implications of the data are clear and accessible to all. While there are other methods of multivariate analysis, such as cluster analysis and multiple regression analysis, none appear to be as well suited to scientific research as PCA in terms of objectivity. For instance, cluster analysis is inherently less suitable for scientific purposes due to the abundance of options available, which can lead to varying results depending on the choices made. This is because cluster analysis typically presents only one possible interpretation of the data. Additionally, PCA can be viewed as an upwardly compatible method, as it is capable of revealing multiple regression analysis outcomes. Furthermore, PCA has the ability to condense a vast amount of information into a well-defined and interpretable form. To illustrate this, consider a study examining changes in the mammary glands of rodents during pregnancy and childbirth. This study involved dozens of sample measurements and a microarray analysis of tens of thousands of genes [8]. Despite the sheer volume of data, PCA facilitated a clear and concise summary, elucidating the directional trends associated with pregnancy and childbirth, as well as the relationships between individual genes. However, one of the drawbacks of PCA is its lack of robustness. Even a single outlier can cause errors in SVD. Naturally, noise affects the results. It is common for the identified directions to be slightly off. For instance, group positions or individual \(Y\) values may deviate slightly from the axes due to rotation.