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Principal Component Analysis
Published in N.C. Basantia, Leo M.L. Nollet, Mohammed Kamruzzaman, Hyperspectral Imaging Analysis and Applications for Food Quality, 2018
Cristina Malegori, Paolo Oliveri
Remembering that PCA is based on the assumption that a high variability is synonymous with a high amount of information and that covariance matrix resumes information regarding data variability of a given data matrix X (Section 6.2), one of the most important approaches for computing PCs starts from the diagonalization of the covariance matrix. Diagonalizing means transforming a square matrix into a diagonal matrix, which has non-zero elements only along its leading diagonal. When a covariance matrix is diagonalized, all the covariance terms become zeros while the leading diagonal contains variance values, referred to eigenvalues. The sum of these eigenvalues is the so-called trace of the diagonalized matrix and corresponds to the total variance of the data.
Matrix Analysis
Published in Ramin S. Esfandiari, Bei Lu, Modeling and Analysis of Dynamic Systems, 2018
We say that B is obtained from A through a similarity transformation. Eigenvalues of a matrix are preserved under similarity transformation; that is, λ(A)=λ(B). Similarity transformations are often utilized to transform a matrix into a diagonal matrix, with eigenvectors playing a key role in that process.
Review of Basic Laws and Equations
Published in Pradip Majumdar, Computational Methods for Heat and Mass Transfer, 2005
where index i represents the row number and varies from 1,2,…,n. The index j represents the column number and varies from 1,2,…,m, and m=n+1, respectively. A square matrix is a matrix where the number of rows is the same as the number of columns, i.e., m=n. A diagonal matrix is a square matrix where all elements off the main diagonal are equal to zero.
A New Fast Approach for an EEG-based Motor Imagery BCI Classification
Published in IETE Journal of Research, 2023
Mohammad Ali Amirabadi, Mohammad Hossein Kahaei
In this paper, both rejecting the low-quality trials by separating independent components of the recorded signal and reducing the high computation of matrix inversion in ICA are addressed. Generally speaking, the diagonalization of a matrix is the best way to reduce the complexity of its inversion. Because the inverse of a diagonal matrix is just the inverse of its diagonal elements. However, the diagonalization of a matrix is not so easy. Therefore, this paper presents an approximate joint diagonalization algorithm [20]. Generally speaking, the diagonalization of symmetric matrixes is easier than non-symmetric matrixes. Therefore, instead of, its covariance matrix is used for diagonalization. So, the proposed method would be faster than common ICA algorithms, because, in addition to using diagonalization before matrix inversion, the proposed fast diagonalization algorithm (see Appendix A) reduces the complexity of diagonalization.
An unsupervised learning algorithm for computer vision-based blind modal parameters identification of output-only structures from video measurements
Published in Structure and Infrastructure Engineering, 2022
Vishal Allada, T. Jothi Saravanan
The EVD of covariance of the motion matrix gives the stable eigenvalue decay, which gives the number of the principal components. The stable point of Eigen decay occurs when the eigenvalue is greater than or equal to 50% of the previous value and less than 15% of the maximum eigenvalue. The EVD figures show the decaying nature of the eigenvalues and optimum order selection. The rank r of the matrix is equal to the number of non-zero singular values, i.e. apart from every value in the diagonal matrix is zero. is directly related to the ith principal direction vector According to Yang and Nagarajaiah (2013), the principal directions will converge to the mode shape direction for a lightly damped structure if the mass matrix is proportional to its uniform mass distribution identity matrix. The structure’s active modes, under broadband excitation, are projected onto the r principal components. Empirically, it is observed that principal active components are less compared with the spatial dimension of the matrix. So, PCA can significantly reduce the dimension of the motion matrix by projecting data linearly onto a small number of principal components: where is a matrix containing principal components and represents r rows of the matrix. PCA also reserves the matrix by using: