nag_rngs_corr_matrix (g05qbc) (PDF version)
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NAG C Library Manual

NAG Library Function Document

nag_rngs_corr_matrix (g05qbc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_rngs_corr_matrix (g05qbc) generates a random correlation matrix with given eigenvalues.

2  Specification

#include <nag.h>
#include <nagg05.h>
void  nag_rngs_corr_matrix (Nag_OrderType order, Integer n, const double d[], double c[], Integer pdc, double eps, Integer igen, Integer iseed[], NagError *fail)

3  Description

Given n eigenvalues, λ1,λ2,,λn, such that
i=1nλi=n
and
λi 0,   i= 1,2,,n,
nag_rngs_corr_matrix (g05qbc) 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_rngs_orthog_matrix (g05qac) 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_rngs_init_repeatable (g05kbc) (for a repeatable sequence if computed sequentially) or nag_rngs_init_nonrepeatable (g05kcc) (for a non-repeatable sequence) must be called prior to the first call to nag_rngs_corr_matrix (g05qbc).

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:     orderNag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by order=Nag_RowMajor. See Section 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     nIntegerInput
On entry: n, the dimension of the correlation matrix to be generated.
Constraint: n1.
3:     d[n]const doubleInput
On entry: the n eigenvalues, λi, for i=1,2,,n.
Constraints:
  • d[i-1]0.0, for i=1,2,,n;
  • i=1nd[i-1]=n to within eps.
4:     c[dim]doubleOutput
Note: the dimension, dim, of the array c must be at least pdc×n.
The i,jth element of the matrix C is stored in
  • c[j-1×pdc+i-1] when order=Nag_ColMajor;
  • c[i-1×pdc+j-1] when order=Nag_RowMajor.
On exit: a random correlation matrix, C, of dimension n.
5:     pdcIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array c.
Constraint: pdcn.
6:     epsdoubleInput
On entry: εthe maximum acceptable error in the diagonal elements.
Suggested value: eps=0.00001.
Constraint: epsn×machine precision (see Chapter x02).
7:     igenIntegerInput
On entry: must contain the identification number for the generator to be used to return a pseudorandom number and should remain unchanged following initialization by a prior call to nag_rngs_init_repeatable (g05kbc) or nag_rngs_init_nonrepeatable (g05kcc).
8:     iseed[4]IntegerCommunication Array
On entry: contains values which define the current state of the selected generator.
On exit: contains updated values defining the new state of the selected generator.
9:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_DIAG_ELEMENTS
The error in a diagonal element is greater than eps. The value of eps should be increased. Otherwise the program could be rerun with a different value used for the seed of the random number generator, see nag_rngs_init_repeatable (g05kbc) or nag_rngs_init_nonrepeatable (g05kcc).
NE_EIGVAL_SUM
On entry, the eigenvalues do not sum to n.
NE_INT
On entry, n=value.
Constraint: n1.
On entry, pdc=value.
Constraint: pdc>0.
NE_INT_2
On entry, pdc=value and n=value.
Constraint: pdcn.
NE_INTERNAL_ERROR
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.
NE_NEGATIVE_EIGVAL
On entry, an eigenvalue is negative.
NE_REAL
On entry, eps=value.
Constraint: epsn×machine precision.

7  Accuracy

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

8  Further Comments

The time taken by nag_rngs_corr_matrix (g05qbc) is approximately proportional to n2.

9  Example

Following initialization of the pseudorandom number generator by a call to nag_rngs_init_repeatable (g05kbc), a 3 by 3 correlation matrix with eigenvalues of 0.7, 0.9 and 1.4 is generated and printed.

9.1  Program Text

Program Text (g05qbce.c)

9.2  Program Data

Program Data (g05qbce.d)

9.3  Program Results

Program Results (g05qbce.r)


nag_rngs_corr_matrix (g05qbc) (PDF version)
g05 Chapter Contents
g05 Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012