g05zm:: Random Number Generators (NAG Toolbox)

nag_rand_field_1d_user_setup (g05zm) performs the setup required in order to simulate stationary Gaussian random fields in one dimension, for a user-defined variogram, using the circulant embedding method. Specifically, the eigenvalues of the extended covariance matrix (or embedding matrix) are calculated, and their square roots output, for use by nag_rand_field_1d_generate (g05zp), which simulates the random field.

Syntax

Description

A one-dimensional random field

Z (x)

ℝ

is a function which is random at every point

x \in ℝ

, so

Z (x)

is a random variable for each

x

. The random field has a mean function

μ (x) = 𝔼 [Z (x)]

and a symmetric positive semidefinite covariance function

C (x, y) = 𝔼 [(Z (x) - μ (x)) (Z (y) - μ (y))]

Z (x)

is a Gaussian random field if for any choice of

n \in ℕ

and

x_{1}, \dots, x_{n} \in ℝ

, the random vector

{[Z (x_{1}), \dots, Z (x_{n})]}^{T}

follows a multivariate Normal distribution, which would have a mean vector

\tilde{μ}

with entries

{\tilde{μ}}_{i} = μ (x_{i})

and a covariance matrix

\tilde{C}

with entries

{\tilde{C}}_{i j} = C (x_{i}, x_{j})

. A Gaussian random field

Z (x)

is stationary if

μ (x)

is constant for all

x \in ℝ

and

C (x, y) = C (x + a, y + a)

for all

x, y, a \in ℝ

and hence we can express the covariance function

C (x, y)

as a function

γ

of one variable:

C (x, y) = γ (x - y)

γ

is known as a variogram (or more correctly, a semivariogram) and includes the multiplicative factor

σ^{2}

representing the variance such that

γ (0) = σ^{2}

The functions nag_rand_field_1d_user_setup (g05zm) and nag_rand_field_1d_generate (g05zp) are used to simulate a one-dimensional stationary Gaussian random field, with mean function zero and variogram

γ (x)

, over an interval

[x_{\min}, x_{\max}]

, using an equally spaced set of

N

points on the interval. The problem reduces to sampling a Normal random vector

X

of size

N

, with mean vector zero and a symmetric Toeplitz covariance matrix

A

. Since

A

is in general expensive to factorize, a technique known as the circulant embedding method is used.

A

is embedded into a larger, symmetric circulant matrix

B

of size

M \geq 2 (N - 1)

, which can now be factorized as

B = W Λ W^{*} = R^{*} R

, where

W

is the Fourier matrix (

W^{*}

is the complex conjugate of

W

Λ

is the diagonal matrix containing the eigenvalues of

B

and

R = Λ^{\frac{1}{2}} W^{*}

B

is known as the embedding matrix. The eigenvalues can be calculated by performing a discrete Fourier transform of the first row (or column) of

B

and multiplying by

M

, and so only the first row (or column) of

B

is needed – the whole matrix does not need to be formed.

As long as all of the values of

Λ

are non-negative (i.e.,

B

is positive semidefinite),

B

is a covariance matrix for a random vector

Y

, two samples of which can now be simulated from the real and imaginary parts of

R^{*} (U + i V)

, where

U

and

V

have elements from the standard Normal distribution. Since

R^{*} (U + i V) = W Λ^{\frac{1}{2}} (U + i V)

, this calculation can be done using a discrete Fourier transform of the vector

Λ^{\frac{1}{2}} (U + i V)

. Two samples of the random vector

X

can now be recovered by taking the first

N

elements of each sample of

Y

– because the original covariance matrix

A

is embedded in

B

X

will have the correct distribution.

B

is not positive semidefinite, larger embedding matrices

B

can be tried; however if the size of the matrix would have to be larger than maxm, an approximation procedure is used. We write

Λ = Λ_{+} + Λ_{-}

, where

Λ_{+}

and

Λ_{-}

contain the non-negative and negative eigenvalues of

B

respectively. Then

B

is replaced by

ρ B_{+}

where

B_{+} = W Λ_{+} W^{*}

and

ρ \in (0, 1]

is a scaling factor. The error

ε

in approximating the distribution of the random field is given by

ε = \sqrt{\frac{{(1 - ρ)}^{2} trace Λ + ρ^{2} trace Λ_{-}}{M}} .

Three choices for

ρ

are available, and are determined by the input argument icorr:

setting $icorr = 0$ sets
$ρ = \frac{trace Λ}{trace Λ_{+}},$
setting $icorr = 1$ sets
$ρ = \sqrt{\frac{trace Λ}{trace Λ_{+}}},$
setting $icorr = 2$ sets $ρ = 1$ .

nag_rand_field_1d_user_setup (g05zm) finds a suitable positive semidefinite embedding matrix

B

and outputs its size, m, and the square roots of its eigenvalues in lam. If approximation is used, information regarding the accuracy of the approximation is output. Note that only the first row (or column) of

B

is actually formed and stored.

References

Parameters

Compulsory Input Parameters

Optional Input Parameters

Output Parameters

Error Indicators and Warnings

Accuracy

Further Comments

Example

This example calls nag_rand_field_1d_user_setup (g05zm) to calculate the eigenvalues of the embedding matrix for

8

sample points of a random field characterized by the symmetric stable variogram:

γ (x) = σ^{2} \exp (- {(x^{'})}^{ν}),

where

x^{'} = \frac{x}{ℓ}

, and

ℓ

and

ν

are parameters.

function g05zm_example


fprintf('g05zm example results\n\n');

% Random field variance
var = 0.5;
% Domain endpoints
xmin = -1;
xmax = 1;
% Scaling factor rho = 1
icorr = int64(2);
% Number of sample points
ns = int64(8);

% Put covariance parameters in communication array
l    = 0.1;
nu   = 1.2;
user = [l, nu];

% Get square roots of the eigenvalues of the embedding matrix
[lam, xx, m, approx, rho, icount, eig, user, ifail] = ...
g05zm(...
       ns, xmin, xmax, var, @cov1, 'icorr', icorr, 'user', user);

fprintf('\nSize of embedding matrix = %d\n\n', m);

% Display approximation information if approximation used
if approx == 1
  fprintf('Approximation required\n\n');
  fprintf('rho = %10.5f\n', rho);
  fprintf('eig = %10.5f%10.5f%10.5f\n', eig(1:3));
  fprintf('icount = %d\n', icount);
else
  fprintf('Approximation not required\n\n');
end

% Display square roots of the eigenvalues of the embedding matrix
fprintf('Square roots of eigenvalues of embedding matrix:\n');
fprintf('%9.5f%9.5f%9.5f%9.5f\n',lam(1:m));



function [gam, user] = cov1(x, user)
  if x == 0
    gam = 1;
  else
    l  = user(1);
    nu = user(2);
    gam = exp(-(abs(x)/l)^nu);
  end

g05zm example results


Size of embedding matrix = 16

Approximation not required

Square roots of eigenvalues of embedding matrix:
  0.74207  0.73932  0.73150  0.71991
  0.70639  0.69304  0.68184  0.67442
  0.67182  0.67442  0.68184  0.69304
  0.70639  0.71991  0.73150  0.73932

NAG Toolbox: nag_rand_field_1d_user_setup (g05zm)

▸▿ Contents

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

Optional Input Parameters

Output Parameters

Error Indicators and Warnings

Accuracy

Further Comments

Example