nag_rand_corr_matrix (g05pyc) (PDF version)
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NAG Library Function Document

nag_rand_corr_matrix (g05pyc)


    1  Purpose
    7  Accuracy

1  Purpose

nag_rand_corr_matrix (g05pyc) generates a random correlation matrix with given eigenvalues.

2  Specification

#include <nag.h>
#include <nagg05.h>
void  nag_rand_corr_matrix (Integer n, const double d[], double eps, Integer state[], double c[], Integer pdc, NagError *fail)

3  Description

Given n eigenvalues, λ1,λ2,,λn, such that
λi 0,   i= 1,2,,n,  
nag_rand_corr_matrix (g05pyc) will generate a random correlation matrix, C, of dimension n, with eigenvalues λ1,λ2,,λn.
The method used is based on that described by Lin and Bendel (1985). Let D be the diagonal matrix with values λ1,λ2,,λn and let A be a random orthogonal matrix generated by nag_rand_orthog_matrix (g05pxc) then the matrix C0=A D AT is a random covariance matrix with eigenvalues λ1,λ2,,λn. The matrix C0 is transformed into a correlation matrix by means of n-1 elementary rotation matrices Pi such that C = Pn-1 Pn-2 P1 C0 P1T Pn-2T Pn-1T . The restriction on the sum of eigenvalues implies that for any diagonal element of C0>1, there is another diagonal element <1. The Pi are constructed from such pairs, chosen at random, to produce a unit diagonal element corresponding to the first element. This is repeated until all diagonal elements are 1 to within a given tolerance ε.
The randomness of C should be interpreted only to the extent that A is a random orthogonal matrix and C is computed from A using the Pi which are chosen as arbitrarily as possible.
One of the initialization functions nag_rand_init_repeatable (g05kfc) (for a repeatable sequence if computed sequentially) or nag_rand_init_nonrepeatable (g05kgc) (for a non-repeatable sequence) must be called prior to the first call to nag_rand_corr_matrix (g05pyc).

4  References

Lin S P and Bendel R B (1985) Algorithm AS 213: Generation of population correlation on matrices with specified eigenvalues Appl. Statist. 34 193–198

5  Arguments

1:     n IntegerInput
On entry: n, the dimension of the correlation matrix to be generated.
Constraint: n1.
2:     d[n] const doubleInput
On entry: the n eigenvalues, λi, for i=1,2,,n.
  • d[i-1]0.0, for i=1,2,,n;
  • i=1nd[i-1]=n to within eps.
3:     eps doubleInput
On entry: the maximum acceptable error in the diagonal elements.
Suggested value: eps=0.00001.
Constraint: epsn×machine precision (see Chapter x02).
4:     state[dim] IntegerCommunication Array
Note: the dimension, dim, of this array is dictated by the requirements of associated functions that must have been previously called. This array MUST be the same array passed as argument state in the previous call to nag_rand_init_repeatable (g05kfc) or nag_rand_init_nonrepeatable (g05kgc).
On entry: contains information on the selected base generator and its current state.
On exit: contains updated information on the state of the generator.
5:     c[n×pdc] doubleOutput
On exit: a random correlation matrix, C, of dimension n.
6:     pdc IntegerInput
On entry: the stride separating row elements of the matrix C in the array c.
Constraint: pdcn.
7:     fail NagError *Input/Output
The NAG error argument (see Section 2.7 in How to Use the NAG Library and its Documentation).

6  Error Indicators and Warnings

Dynamic memory allocation failed.
See Section in How to Use the NAG Library and its Documentation for further information.
On entry, argument value had an illegal value.
The diagonals of the returned matrix are not unity, try increasing the value of eps, or rerun the code using a different seed.
On entry, the eigenvalues do not sum to n.
On entry, n=value.
Constraint: n1.
On entry, pdc=value.
Constraint: pdc>0.
On entry, pdc=value and n=value.
Constraint: pdcn.
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
An unexpected error has been triggered by this function. Please contact NAG.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
On entry, state vector has been corrupted or not initialized.
On entry, an eigenvalue is negative.
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
On entry, eps=value.
Constraint: epsn×machine precision.

7  Accuracy

The maximum error in a diagonal element is given by eps.

8  Parallelism and Performance

nag_rand_corr_matrix (g05pyc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_rand_corr_matrix (g05pyc) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the x06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9  Further Comments

The time taken by nag_rand_corr_matrix (g05pyc) is approximately proportional to n2.

10  Example

Following initialization of the pseudorandom number generator by a call to nag_rand_init_repeatable (g05kfc), a 3 by 3 correlation matrix with eigenvalues of 0.7, 0.9 and 1.4 is generated and printed.

10.1  Program Text

Program Text (g05pyce.c)

10.2  Program Data

Program Data (g05pyce.d)

10.3  Program Results

Program Results (g05pyce.r)

nag_rand_corr_matrix (g05pyc) (PDF version)
g05 Chapter Contents
g05 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2016