svd

Singular value decomposition (SVD).

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Usage 1

svd(0, A, trn)

A

the NxM matrix to be transformed

trn

transformation to be applied to A

trn	transformed matrix T (NxM)	description	trn(T)*T (MxM)
0	T[i,j] = A[i,j]	no transformation	"Streumatrix"
1	T[i,j] = A[i,j]-avr(A)	subtract matrix mean
2	T[i,j] = A[i,j]-avr(A[*,j])	subtract column mean (center columns)	covariance matrix
3	T[i,j] = (A[i,j]-avr(A[*,j]))/dev(A[*,j])	subtract column mean, devide by column deviation (center and standardize columns)	correlation matrix
4	T[i,j] = A[i,j]-(avr(A[i,*])+avr(A[*,j]))+avr(A)	subtract row and column mean, add matrix mean (center rows and columns)

Result 1

The transformed matrix T.

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Usage 2

svd(1, A, trn)

Result 2

Returns the matrix C=trn(T)*T. C is a square matrix with M rows and columns (MxM).

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Usage 3

svd(2, A, trn, U, S, V)

U

contains on return the NxM matrix U

S

contains on return the vector S with M elements; Note: the returned vector contains the diagonal elements of the matrix Σ, which are called the singular values.

V

contains on return the MxM matrix V

Result 3

Computes the SVD of the transformed input matrix T.

solve: T = U * Σ * trn(V)

The results are stored in the (optional) numerical tables (references) U (NxM), S (Mx1, diagonal of Σ) and V (MxM). The return value is the NxM matrix PC=U * Σ.

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Usage 4

svd(3, A, trn, U, S, V)

S

contains on return the vector S with M elements; Note: the returned vector contains the diagonal elements of the matrix Σ (singular values).

V

contains on return the MxM matrix V

Result 4

Computes the SVD of the transformed and squared input matrix T.

C = trn(T) * T

solve: C = V * Σ * trn(V)

The results are stored in the (optional) numerical tables (references) S (Mx1, diagonal of Σ) and V (MxM). The return value is the NxM matrix PC=A * V.

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Usage 5

svd(4, C, S, V)

C

the MxM input data matrix

S

contains on return the vector S with M elements; Note: the returned vector contains the diagonal elements of the matrix Σ (singular values).

V

contains on return the MxM matrix V

Result 5

Computes the SVD of the matrix C. It is assumed that C is (optional) transformed and squared matrix derived from a NxM data matrix.

solve: C = V * Σ * trn(V)

The results are stored in the (optional) numerical tables (references) S (Mx1, diagonal of Σ) and V (MxM). The return value is always 0.

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See also

var, corr, dist, haclust, modclust

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