Where is an eigenvalue matrix and its components are . Then matrix A by eigenvalue decomposition is (2.8) Here, the property was used from the fact that is orthonormal. If a covariance PT=P-1 P matrix of is a matrix the data manipulation involves decomposition of the data S X A, matrix into two matrices and and is orthonormal, X V U, V (2.9) The columns of are known as scores and those of are called loadings. PCA is a U V technique to decompose eigenvalues of a covariance matrix, of a given data matrix. S, The loadings can be understood as the weights for each original variable when calculating the principal components. The matrix contains the original data in a rotated coordinate U system. The mathematical analysis involves finding these new matrices and U V. “data” The dimensions of (i.e. its rank) that capture all the information of the entire data set of U (i.e. # of variables) is far less than that of (ideally 2 or 3). One now compresses the N X X dimensional plot of the data matrix into 2 or 3 dimensional plot of and While the X U V. eigenvalues geometrically represent the length of each of the principal axes (i.e. scores), the eigenvectors of the covariance matrix represent the orientation of principal axes of the ellipsoid (i.e. loadings). By using just a few latent variables, the dimensionality of the original multivariate data sets are reduced and visualized by their projections in 2D or 3D with a minimal loss of information. Therefore, PCA is a process of dimensionally reduced mapping of a multivariate data set [2-6]. Figure 2.1. A graphical representation of the data points and their eigenvalues
Where is an eigenvalue matrix and its components are . Then matrix A by eigenvalue decomposition is (2.8) Here, the property PT=P-1 was used from the fact that P is orthonormal. If a covariance matrix S of X is a matrix A, the data manipulation involves decomposition of the data matrix X into two matrices V and U, and V is orthonormal, (2.9) The columns of U are known as scores and those of V are called loadings. PCA is a technique to decompose eigenvalues of a covariance matrix, S, of a given data matrix. The loadings can be understood as the weights for each original variable when calculating the principal components. The matrix U contains the original data in a rotated coordinate system. The mathematical analysis involves finding these new “data” matrices U and V. The dimensions of U (i.e. its rank) that capture all the information of the entire data set of X (i.e. # of variables) is far less than that of X (ideally 2 or 3). One now compresses the N dimensional plot of the data matrix X into 2 or 3 dimensional plot of U and V. While the eigenvalues geometrically represent the length of each of the principal axes (i.e. scores), the eigenvectors of the covariance matrix represent the orientation of principal axes of the ellipsoid (i.e. loadings). By using just a few latent variables, the dimensionality of the original multivariate data sets are reduced and visualized by their projections in 2D or 3D with a minimal loss of information. Therefore, PCA is a process of dimensionally reduced mapping of a multivariate data set [2-6]. Figure 2.1. A graphical representation of the data points and their eigenvalues
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