naginterfaces.library.correg.corrmat_​nearest_​kfactor

naginterfaces.library.correg.corrmat_nearest_kfactor(g, k, errtol=0.0, maxit=0)[source]

corrmat_nearest_kfactor computes the factor loading matrix associated with the nearest correlation matrix with -factor structure, in the Frobenius norm, to a given square, input matrix.

For full information please refer to the NAG Library document for g02ae

https://www.nag.com/numeric/nl/nagdoc_27.1/flhtml/g02/g02aef.html

Parameters
gfloat, array-like, shape

, the initial matrix.

kint

, the number of factors and columns of .

errtolfloat, optional

The termination tolerance for the projected gradient norm. See references for further details. If , is used. This is often a suitable default value.

maxitint, optional

Specifies the maximum number of iterations in the spectral projected gradient method.

If , is used.

Returns
xfloat, ndarray, shape

Contains the matrix .

iteraint

The number of steps taken in the spectral projected gradient method.

fevalint

The number of evaluations of .

nrmpgdfloat

The norm of the projected gradient at the final iteration.

Raises
NagValueError
(errno )

On entry, .

Constraint: .

(errno )

On entry, and .

Constraint: .

(errno )

Spectral gradient method fails to converge in iterations.

Notes

A correlation matrix with -factor structure may be characterised as a real square matrix that is symmetric, has a unit diagonal, is positive semidefinite and can be written as , where is the identity matrix and has rows and columns. is often referred to as the factor loading matrix.

corrmat_nearest_kfactor applies a spectral projected gradient method to the modified problem such that , for , where is the th row of the factor loading matrix, , which gives us the solution.

References

Birgin, E G, Martínez, J M and Raydan, M, 2001, Algorithm 813: SPG–software for convex-constrained optimization, ACM Trans. Math. Software (27), 340–349

Borsdorf, R, Higham, N J and Raydan, M, 2010, Computing a nearest correlation matrix with factor structure, SIAM J. Matrix Anal. Appl. (31(5)), 2603–2622