EEG Signal Processing and Machine Learning. Saeid Sanei. Читать онлайн. Newlib. NEWLIB.NET

Автор: Saeid Sanei
Издательство: John Wiley & Sons Limited
Серия:
Жанр произведения: Программы
Год издания: 0
isbn: 9781119386933
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      All suboptimal transforms such as the DFT and DCT decompose the signals into a set of coefficients, which do not necessarily represent the constituent components of the signals. Moreover, the transform kernel is independent of the data hence they are not efficient in terms of both decorrelation of the samples and energy compaction. Therefore, separation of the signal and noise components is generally not achievable using these suboptimal transforms.

      Expansion of the data into a set of orthogonal components certainly achieves maximum decorrelation of the signals. This enables separation of the data into the signal and noise subspaces.

Schematic illustration of the general application of PCA.

      (4.117)equation

      where Φ = {ϕk } is the set of orthogonal basis functions. The weights wi, k are then calculated as:

      (4.118)equation

      4.9.1 Singular Value Decomposition

      Singular value decomposition (SVD) is often used for solving the least‐squares (LS) problem. This is performed by decomposition of the M × M square autocorrelation matrix R into its eigenvalue matrix Λ = diag1, λ2, … λ M ) and an M × M orthogonal matrix of eigenvectors V, i.e. R = VΛVH , where (.) H denotes Hermitian (conjugate transpose) operation. Moreover, if A is an M × M data matrix such that R = AH A then there exist an M × M orthogonal matrix U, an M × M orthogonal matrix V, and an M × M diagonal matrix with diagonal elements equal to images, such that:

      (4.119)equation

      Hence 2 = Λ. The columns of U are called left singular vectors and the rows of VH are called right singular vectors. If A is rectangular N × M matrix of rank k then U will be N × N and will be:

      (4.120)equation

      where S = diag1, σ2, … σ k ), where σ i = images. For such a matrix the Moore–Penrose pseudo‐inverse is defined as an M × N matrix A defined as:

      (4.121)equation

      (4.122)equation

      A has a major role in the solutions of least‐squares problems, and S −1 is a k × k diagonal matrix with elements equal to the reciprocals of the singular values of A, i.e.

      (4.123)equation

      In order to see the application of the SVD in solving the LS problem consider the error vector e defined as:

      (4.125)equation

      or equivalently

      (4.126)equation

      Since U is a unitary matrix, ‖e 2‖ = ‖UH e2. Hence, the vector h that minimizes ‖e 2‖ also minimizes ‖UH e2. Finally, the unique solution as an optimum h (coefficient vector) may be expressed as [43]:

      (4.127)equation

      where k is the rank of A. Alternatively, as the optimum least‐squares coefficient vector:

      (4.128)equation

      Performing PCA is equivalent to performing an SVD on the covariance matrix. PCA uses the same concept as SVD and orthogonalization to decompose the data into its constituent uncorrelated orthogonal components such that the autocorrelation matrix is diagonalized. Each eigenvector represents a principal component and the individual eigenvalues are numerically related to the variance they capture in the direction of the principal components.