naginterfaces.library.correg.corrmat_​nearest_​kfactor

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

corrmat_nearest_kfactor computes the factor loading matrix associated with the nearest correlation matrix with $k$-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://support.nag.com/numeric/nl/nagdoc_30/flhtml/g02/g02aef.html

Parameters

gfloat, array-like, shape $(n,n)$

$G$, the initial matrix.

kint

$k$, the number of factors and columns of $X$.

errtolfloat, optional

The termination tolerance for the projected gradient norm. See references for further details. If $errtol\le 0.0$, $0.01$ is used. This is often a suitable default value.

maxitint, optional

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

If $maxit\le 0$, $40000$ is used.

Returns

xfloat, ndarray, shape $(n,k)$

Contains the matrix $X$.

iteraint

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

fevalint

The number of evaluations of ${\Vert G-X{X}^T+diag(X{X}^T-I)\Vert }_F$.

nrmpgdfloat

The norm of the projected gradient at the final iteration.

Raises

NagValueError

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n>0$.

(errno $1$)

On entry, $k=⟨value⟩$ and $n=⟨value⟩$.

Constraint: $0<k\le n$.

(errno $2$)

Spectral gradient method fails to converge in $⟨value⟩$ iterations.

Notes

A correlation matrix $C$ with $k$-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 $C=X{X}^T+diag(I-X{X}^T)$, where $I$ is the identity matrix and $X$ has $n$ rows and $k$ columns. $X$ is often referred to as the factor loading matrix.

corrmat_nearest_kfactor applies a spectral projected gradient method to the modified problem $min{\left(\Vert G-X{X}^T+diag(X{X}^T-I)\Vert \right)}_F$ such that ${\Vert x_{\text{i}}^T\Vert }_2\le 1$, for $\text{i}=1,2,\dots ,n$, where $x_i$ is the $i$th row of the factor loading matrix, $X$, 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