g05zp:: Random Number Generators (NAG Toolbox)

nag_rand_field_1d_generate (g05zp) produces realizations of a stationary Gaussian random field in one dimension, using the circulant embedding method. The square roots of the eigenvalues of the extended covariance matrix (or embedding matrix) need to be input, and can be calculated using nag_rand_field_1d_user_setup (g05zm) or nag_rand_field_1d_predef_setup (g05zn).

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 non-negative definite 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) or nag_rand_field_1d_predef_setup (g05zn), along with 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. 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 non-negative definite),

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 non-negative definite, 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. See the documentation of nag_rand_field_1d_user_setup (g05zm) or nag_rand_field_1d_predef_setup (g05zn) for details of the approximation procedure.

References

Parameters

Compulsory Input Parameters

Optional Input Parameters

Output Parameters

Error Indicators and Warnings

Accuracy

Further Comments

Example

function g05zp_example


fprintf('g05zp example results\n\n');

% Choose the Symmetric stable variogram
icov1 = int64(1);
params = [0.1; 1.2];

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

% Get square roots of the eigenvalues of the embedding matrix
[lam, xx, m, approx, rho, icount, eig, ifail] = ...
  g05zn( ...
         ns, xmin, xmax, var, icov1, params, 'icorr', icorr);

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

state = initialize_state();

% Compute s random field realisations.
s = int64(5);
[state, z, ifail] = g05zp( ...
                           ns, s, lam(1:m), rho, state);

% Display realizations
fprintf('Random field realisations:\n     ');
fprintf('%10d',[1:5]);
fprintf('\n');
disp([xx, z]);



function state = initialize_state()
  genid = int64(1);
  subid = int64(1);
  seed  = [int64(14965)];
  [state, ifail] = g05kf( ...
                          genid, subid, seed);

g05zp example results


Size of embedding matrix = 16

Approximation not required

Random field realisations:
              1         2         3         4         5
   -0.8750   -0.4166   -0.8185   -0.9769    0.6741   -0.6762
   -0.6250    0.0146    1.4538    0.0248    0.5218    1.9466
   -0.3750   -0.5556    0.2913   -0.0853    0.4214   -0.1389
   -0.1250   -0.5568    0.3199   -0.6094    0.2019    0.9085
    0.1250   -0.0423    0.0486    1.4590    0.3608   -0.5288
    0.3750   -0.2806   -0.7969    0.2330    0.1335    0.4012
    0.6250    0.9298   -0.3956   -0.8455   -0.2749    0.5270
    0.8750    0.3222    1.5227   -2.1645    0.1794    1.1937

NAG Toolbox: nag_rand_field_1d_generate (g05zp)

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Purpose