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NAG Toolbox

NAG Toolbox: nag_correg_corrmat_nearest_kfactor (g02ae)

Purpose

nag_correg_corrmat_nearest_kfactor (g02ae) computes the factor loading matrix associated with the nearest correlation matrix with k k -factor structure, in the Frobenius norm, to a given square, input matrix.

Syntax

[g, x, iter, feval, nrmpgd, ifail] = g02ae(g, k, 'n', n, 'errtol', errtol, 'maxit', maxit)
[g, x, iter, feval, nrmpgd, ifail] = nag_correg_corrmat_nearest_kfactor(g, k, 'n', n, 'errtol', errtol, 'maxit', maxit)

Description

A correlation matrix C C  with k 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 = XXT + diag(IXXT) C=XXT+diag (I-XXT) , where I I  is the identity matrix and X X  has n n  rows and k k  columns. X X  is often referred to as the factor loading matrix.
nag_correg_corrmat_nearest_kfactor (g02ae) applies a spectral projected gradient method to the modified problem min GXXT + diag(XXTI)F min G-XXT+diag (XXT-I) F  such that xiT21 xiT 21 , for i = 1,2,,n i=1,2,,n , where xi xi  is the i i th row of the factor loading matrix, X 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

Parameters

Compulsory Input Parameters

1:     g(ldg,n) – double array
ldg, the first dimension of the array, must satisfy the constraint ldgn ldgn .
G G , the initial matrix.
2:     k – int64int32nag_int scalar
k k , the number of factors and columns of X X .
Constraint: 0 < kn 0<kn .

Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The first dimension of the array g and the second dimension of the array g. (An error is raised if these dimensions are not equal.)
n n , the order of the matrix G G .
Constraint: n > 0 n>0 .
2:     errtol – double scalar
The termination tolerance for the projected gradient norm. See references for further details. If errtol0.0 errtol0.0  then 0.01 0.01  is used. This is often a suitable default value.
Default: 0.0 0.0  
3:     maxit – int64int32nag_int scalar
Specifies the maximum number of iterations in the spectral projected gradient method.
If maxit0 maxit0 , 40000 40000  is used.
Default: 0 0  

Input Parameters Omitted from the MATLAB Interface

ldg ldx

Output Parameters

1:     g(ldg,n) – double array
ldgn ldgn .
A symmetric matrix (1/2)(G + GT) 12 (G+GT) with the diagonal elements set to unity.
2:     x(ldx,k) – double array
ldxn ldxn .
Contains the matrix X X .
3:     iter – int64int32nag_int scalar
The number of steps taken in the spectral projected gradient method.
4:     feval – int64int32nag_int scalar
The number of evaluations min GXXT + diag(XXTI)F min G-XXT+diag (XXT-I) F .
5:     nrmpgd – double scalar
The norm of the projected gradient at the final iteration.
6:     ifail – int64int32nag_int scalar
ifail = 0 ifail=0 unless the function detects an error (see [Error Indicators and Warnings]).

Error Indicators and Warnings

Errors or warnings detected by the function:
  ifail = 1 ifail=1
Constraint: 0 < kn 0<kn .
Constraint: ldgn ldgn .
Constraint: ldxn ldxn .
Constraint: n > 0 n>0 .
  ifail = 2 ifail=2
Spectral gradient method fails to converge in _ _  iterations.
  ifail = 999 ifail=-999
Dynamic memory allocation failed.

Accuracy

The returned accuracy is controlled by errtol and limited by machine precision.

Further Comments

Arrays are internally allocated by nag_correg_corrmat_nearest_kfactor (g02ae). The total size of these arrays is n × n + 4 × n × k + (nb + 3) × n + n + 50 n×n+4×n×k+ (nb+3) ×n+n+50  double elements and 6 × n 6×n  integer elements. Here nb nb  is the block size required for optimal performance by nag_lapack_dsytrd (f08fe) and nag_lapack_dormtr (f08fg) which are called internally. All allocated memory is freed before return of nag_correg_corrmat_nearest_kfactor (g02ae).
See nag_mv_factor (g03ca) for constructing the factor loading matrix from a known correlation matrix.

Example

function nag_correg_corrmat_nearest_kfactor_example
g = [2, -1,  0,  0;
    -1,  2, -1,  0;
     0, -1,  2, -1;
     0,  0, -1,  2];
k = int64(2);
% Calculate nearest correlation matrix
[g, x, iter, feval, nrmpgd, ifail] = nag_correg_corrmat_nearest_kfactor(g, k);

if (ifail == 0)
  fprintf('\n Factor Loading Matrix x:\n');
  disp(x);
  fprintf('\n Number of Newton steps taken:   %d\n', iter);
  fprintf(' Number of function evaluations: %d\n', feval);

  % Generate Nearest k factor correlation matrix

  fprintf('\n Nearest Correlation Matrix:\n');
  disp(x*transpose(x) + diag(diag(eye(4)-x*transpose(x))));
end
 

 Factor Loading Matrix x:
    0.7665   -0.6271
   -0.4250    0.9052
   -0.4250   -0.9052
    0.7665    0.6271


 Number of Newton steps taken:   5
 Number of function evaluations: 6

 Nearest Correlation Matrix:
    1.0000   -0.8935    0.2419    0.1943
   -0.8935    1.0000   -0.6388    0.2419
    0.2419   -0.6388    1.0000   -0.8935
    0.1943    0.2419   -0.8935    1.0000


function g02ae_example
g = [2, -1,  0,  0;
    -1,  2, -1,  0;
     0, -1,  2, -1;
     0,  0, -1,  2];
k = int64(2);
% Calculate nearest correlation matrix
[g, x, iter, feval, nrmpgd, ifail] = g02ae(g, k);

if (ifail == 0)
  fprintf('\n Factor Loading Matrix x:\n');
  disp(x);
  fprintf('\n Number of Newton steps taken:   %d\n', iter);
  fprintf(' Number of function evaluations: %d\n', feval);

  % Generate Nearest k factor correlation matrix

  fprintf('\n Nearest Correlation Matrix:\n');
  disp(x*transpose(x) + diag(diag(eye(4)-x*transpose(x))));
end
 

 Factor Loading Matrix x:
    0.7665   -0.6271
   -0.4250    0.9052
   -0.4250   -0.9052
    0.7665    0.6271


 Number of Newton steps taken:   5
 Number of function evaluations: 6

 Nearest Correlation Matrix:
    1.0000   -0.8935    0.2419    0.1943
   -0.8935    1.0000   -0.6388    0.2419
    0.2419   -0.6388    1.0000   -0.8935
    0.1943    0.2419   -0.8935    1.0000



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Chapter Contents
Chapter Introduction
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