g05zq:: Random Number Generators (NAG Toolbox)

nag_rand_field_2d_user_setup (g05zq) performs the setup required in order to simulate stationary Gaussian random fields in two dimensions, 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_2d_generate (g05zs), which simulates the random field.

Syntax

Description

A two-dimensional random field

Z (x)

ℝ^{2}

is a function which is random at every point

x \in ℝ^{2}

, 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^{2}

, 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^{2}

and

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

for all

x, y, a \in ℝ^{2}

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_2d_user_setup (g05zq) and nag_rand_field_2d_generate (g05zs) are used to simulate a two-dimensional stationary Gaussian random field, with mean function zero and variogram

γ (x)

, over a domain

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

, using an equally spaced set of

N_{1} \times N_{2}

points;

N_{1}

points in the

x

-direction and

N_{2}

points in the

y

-direction. The problem reduces to sampling a Normal random vector

X

of size

N_{1} \times N_{2}

, with mean vector zero and a symmetric covariance matrix

A

, which is an

N_{2}

N_{2}

block Toeplitz matrix with Toeplitz blocks of size

N_{1}

N_{1}

. 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 matrix

B

, which is an

M_{2}

M_{2}

block circulant matrix with circulant blocks of size

M_{1}

M_{1}

, where

M_{1} \geq 2 (N_{1} - 1)

and

M_{2} \geq 2 (N_{2} - 1)

B

can now be factorized as

B = W Λ W^{*} = R^{*} R

, where

W

is the two-dimensional 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_{1} \times M_{2}

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

B

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

The symmetry of

A

as a block matrix, and the symmetry of each block of

A

, depends on whether the variogram

γ

is even or not.

γ

is even in its first coordinate if

γ ({[{- x}_{1}, x_{2}]}^{T}) = γ ({[x_{1}, x_{2}]}^{T})

, even in its second coordinate if

γ ({[x_{1}, {- x}_{2}]}^{T}) = γ ({[x_{1}, x_{2}]}^{T})

, and even if it is even in both coordinates (in two dimensions it is impossible for

γ

to be even in one coordinate and uneven in the other). If

γ

is even then

A

is a symmetric block matrix and has symmetric blocks; if

γ

is uneven then

A

is not a symmetric block matrix and has non-symmetric blocks. In the uneven case,

M_{1}

and

M_{2}

are set to be odd in order to guarantee symmetry in

B

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

which has

M_{2}

blocks of size

M_{1}

. Two samples of

Y

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_{1}

elements of the first

N_{2}

blocks 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}} .

nag_rand_field_2d_user_setup (g05zq) finds a suitable positive semidefinite embedding matrix

B

and outputs its sizes in the vector 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_2d_user_setup (g05zq) to calculate the eigenvalues of the embedding matrix for

25

sample points on a

5

5

grid of a two-dimensional random field characterized by the symmetric stable variogram:

γ (x) = σ^{2} \exp (- {(x^{'})}^{ν}),

where

x^{'} = |\frac{x}{ℓ_{1}} + \frac{y}{ℓ_{2}}|

, and

ℓ_{1}

ℓ_{2}

and

ν

are parameters.

function g05zq_example


fprintf('g05zq example results\n\n');

% Random Field variance
var = 0.5;
% Domain endpoints
xmin = -1;
xmax =  1;
ymin = -0.5;
ymax =  0.5;
% Number of sample points in x and y
ns = [int64(5), 5];
% Maximum dimensions of circulant matrix
maxm = [int64(81), 81];
% Scaling factor rho=1.
icorr = int64(2);

% Put covariance parameters (for cov2) in user
norm_p = int64(2);
l1 = 0.1;
l2 = 0.15;
nu = 1.2;
user = {norm_p; l1; l2; nu};
% cov2 is even
even = int64(1);

% Get square roots of the eigenvalues of the embedding matrix
[lam, xx, yy, m, approx, rho, icount, eig, user, ifail] = ...
  g05zq( ...
         ns, xmin, xmax, ymin, ymax, maxm, var, ...
         @cov2, even, 'icorr', icorr, 'user', user);

fprintf('Size of embedding matrix = %d\n\n', m(1)*m(2));

% 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');
mm = m(1)*m(2);
mlam = reshape(lam(1:mm), m(1), m(2));
for i = 1:m(1)
  fprintf('%8.4f',mlam(i,:));
  fprintf('\n');
end



function [gam, user] = cov2(x, y, user)
  norm_p = user{1};
  l1     = user{2};
  l2     = user{3};
  nu     = user{4};

  tl1 = abs(x)/l1;
  tl2 = abs(y)/l2;

  if norm_p == 1
    rnorm = tl1 +  tl2;
  else
    rnorm = sqrt(tl1^2+tl2^2);
  end

  gam = exp(-(rnorm^nu));

g05zq example results

Size of embedding matrix = 64

Approximation not required

Square roots of eigenvalues of embedding matrix:
  0.8966  0.8234  0.6810  0.5757  0.5391  0.5757  0.6810  0.8234
  0.8940  0.8217  0.6804  0.5756  0.5391  0.5756  0.6804  0.8217
  0.8877  0.8175  0.6792  0.5754  0.5391  0.5754  0.6792  0.8175
  0.8813  0.8133  0.6780  0.5751  0.5390  0.5751  0.6780  0.8133
  0.8787  0.8116  0.6774  0.5750  0.5390  0.5750  0.6774  0.8116
  0.8813  0.8133  0.6780  0.5751  0.5390  0.5751  0.6780  0.8133
  0.8877  0.8175  0.6792  0.5754  0.5391  0.5754  0.6792  0.8175
  0.8940  0.8217  0.6804  0.5756  0.5391  0.5756  0.6804  0.8217

NAG Toolbox: nag_rand_field_2d_user_setup (g05zq)

▸▿ Contents

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

Optional Input Parameters

Output Parameters

Error Indicators and Warnings

Accuracy

Further Comments

Example