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# NAG Toolbox: nag_rand_field_2d_generate (g05zs)

## Purpose

nag_rand_field_2d_generate (g05zs) produces realizations of a stationary Gaussian random field in two dimensions, 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_2d_user_setup (g05zq) or nag_rand_field_2d_predef_setup (g05zr).

## Syntax

[state, z, ifail] = g05zs(ns, s, m, lam, rho, state)
[state, z, ifail] = nag_rand_field_2d_generate(ns, s, m, lam, rho, state)

## Description

A two-dimensional random field $Z\left(\mathbf{x}\right)$ in ${ℝ}^{2}$ is a function which is random at every point $\mathbf{x}\in {ℝ}^{2}$, so $Z\left(\mathbf{x}\right)$ is a random variable for each $\mathbf{x}$. The random field has a mean function $\mu \left(\mathbf{x}\right)=𝔼\left[Z\left(\mathbf{x}\right)\right]$ and a symmetric positive semidefinite covariance function $C\left(\mathbf{x},\mathbf{y}\right)=𝔼\left[\left(Z\left(\mathbf{x}\right)-\mu \left(\mathbf{x}\right)\right)\left(Z\left(\mathbf{y}\right)-\mu \left(\mathbf{y}\right)\right)\right]$. $Z\left(\mathbf{x}\right)$ is a Gaussian random field if for any choice of $n\in ℕ$ and ${\mathbf{x}}_{1},\dots ,{\mathbf{x}}_{n}\in {ℝ}^{2}$, the random vector ${\left[Z\left({\mathbf{x}}_{1}\right),\dots ,Z\left({\mathbf{x}}_{n}\right)\right]}^{\mathrm{T}}$ follows a multivariate Normal distribution, which would have a mean vector $\stackrel{~}{\mathbf{\mu }}$ with entries ${\stackrel{~}{\mu }}_{i}=\mu \left({\mathbf{x}}_{i}\right)$ and a covariance matrix $\stackrel{~}{C}$ with entries ${\stackrel{~}{C}}_{ij}=C\left({\mathbf{x}}_{i},{\mathbf{x}}_{j}\right)$. A Gaussian random field $Z\left(\mathbf{x}\right)$ is stationary if $\mu \left(\mathbf{x}\right)$ is constant for all $\mathbf{x}\in {ℝ}^{2}$ and $C\left(\mathbf{x},\mathbf{y}\right)=C\left(\mathbf{x}+\mathbf{a},\mathbf{y}+\mathbf{a}\right)$ for all $\mathbf{x},\mathbf{y},\mathbf{a}\in {ℝ}^{2}$ and hence we can express the covariance function $C\left(\mathbf{x},\mathbf{y}\right)$ as a function $\gamma$ of one variable: $C\left(\mathbf{x},\mathbf{y}\right)=\gamma \left(\mathbf{x}-\mathbf{y}\right)$. $\gamma$ is known as a variogram (or more correctly, a semivariogram) and includes the multiplicative factor ${\sigma }^{2}$ representing the variance such that $\gamma \left(0\right)={\sigma }^{2}$.
The functions nag_rand_field_2d_user_setup (g05zq) or nag_rand_field_2d_predef_setup (g05zr) along with nag_rand_field_2d_generate (g05zs) are used to simulate a two-dimensional stationary Gaussian random field, with mean function zero and variogram $\gamma \left(\mathbf{x}\right)$, over a domain $\left[{x}_{\mathrm{min}},{x}_{\mathrm{max}}\right]×\left[{y}_{\mathrm{min}},{y}_{\mathrm{max}}\right]$, using an equally spaced set of ${N}_{1}×{N}_{2}$ points; ${N}_{1}$ points in the $x$-direction and ${N}_{2}$ points in the $y$-direction. The problem reduces to sampling a Gaussian random vector $\mathbf{X}$ of size ${N}_{1}×{N}_{2}$, with mean vector zero and a symmetric covariance matrix $A$, which is an ${N}_{2}$ by ${N}_{2}$ block Toeplitz matrix with Toeplitz blocks of size ${N}_{1}$ by ${N}_{1}$. 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 ${M}_{2}$ by ${M}_{2}$ block circulant matrix with circulant bocks of size ${M}_{1}$ by ${M}_{1}$, where ${M}_{1}\ge 2\left({N}_{1}-1\right)$ and ${M}_{2}\ge 2\left({N}_{2}-1\right)$. $B$ can now be factorized as $B=W\Lambda {W}^{*}={R}^{*}R$, where $W$ is the two-dimensional Fourier matrix (${W}^{*}$ is the complex conjugate of $W$), $\Lambda$ is the diagonal matrix containing the eigenvalues of $B$ and $R={\Lambda }^{\frac{1}{2}}{W}^{*}$. $B$ 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 ${M}_{1}×{M}_{2}$, 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 covariance function $\gamma$ is even or not. $\gamma$ is even if $\gamma \left(\mathbf{x}\right)=\gamma \left(-\mathbf{x}\right)$ for all $\mathbf{x}\in {ℝ}^{2}$, and uneven otherwise (in higher dimensions, $\gamma$ can be even in some coordinates and uneven in others, but in two dimensions $\gamma$ is either even in both coordinates or uneven in both coordinates). If $\gamma$ is even then $A$ is a symmetric block matrix and has symmetric blocks; if $\gamma$ 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 $\Lambda$ are non-negative (i.e., $B$ is positive semidefinite), $B$ is a covariance matrix for a random vector $\mathbf{Y}$ which has ${M}_{2}$ ‘blocks’ of size ${M}_{1}$. Two samples of $\mathbf{Y}$ can now be simulated from the real and imaginary parts of ${R}^{*}\left(\mathbf{U}+i\mathbf{V}\right)$, where $\mathbf{U}$ and $\mathbf{V}$ have elements from the standard Normal distribution. Since ${R}^{*}\left(\mathbf{U}+i\mathbf{V}\right)=W{\Lambda }^{\frac{1}{2}}\left(\mathbf{U}+i\mathbf{V}\right)$, this calculation can be done using a discrete Fourier transform of the vector ${\Lambda }^{\frac{1}{2}}\left(\mathbf{U}+i\mathbf{V}\right)$. Two samples of the random vector $\mathbf{X}$ 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 $B$, $\mathbf{X}$ will have the correct distribution.
If $B$ 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. See the documentation of nag_rand_field_2d_user_setup (g05zq) or nag_rand_field_2d_predef_setup (g05zr) for details of the approximation procedure.
nag_rand_field_2d_generate (g05zs) takes the square roots of the eigenvalues of the embedding matrix $B$, and its size vector $M$, as input and outputs $S$ realizations of the random field in $Z$.
One of the initialization functions nag_rand_init_repeat (g05kf) (for a repeatable sequence if computed sequentially) or nag_rand_init_nonrepeat (g05kg) (for a non-repeatable sequence) must be called prior to the first call to nag_rand_field_2d_generate (g05zs).

## References

Dietrich C R and Newsam G N (1997) Fast and exact simulation of stationary Gaussian processes through circulant embedding of the covariance matrix SIAM J. Sci. Comput. 18 1088–1107
Schlather M (1999) Introduction to positive definite functions and to unconditional simulation of random fields Technical Report ST 99–10 Lancaster University
Wood A T A and Chan G (1994) Simulation of stationary Gaussian processes in ${\left[0,1\right]}^{d}$ Journal of Computational and Graphical Statistics 3(4) 409–432

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{ns}\left(2\right)$int64int32nag_int array
The number of sample points to use in each direction, with ${\mathbf{ns}}\left(1\right)$ sample points in the $x$-direction and ${\mathbf{ns}}\left(2\right)$ sample points in the $y$-direction. The total number of sample points on the grid is therefore ${\mathbf{ns}}\left(1\right)×{\mathbf{ns}}\left(2\right)$. This must be the same value as supplied to nag_rand_field_2d_user_setup (g05zq) or nag_rand_field_2d_predef_setup (g05zr) when calculating the eigenvalues of the embedding matrix.
Constraints:
• ${\mathbf{ns}}\left(1\right)\ge 1$;
• ${\mathbf{ns}}\left(2\right)\ge 1$.
2:     $\mathrm{s}$int64int32nag_int scalar
$S$, the number of realizations of the random field to simulate.
Constraint: ${\mathbf{s}}\ge 1$.
3:     $\mathrm{m}\left(2\right)$int64int32nag_int array
Indicates the size, $M$, of the embedding matrix as returned by nag_rand_field_2d_user_setup (g05zq) or nag_rand_field_2d_predef_setup (g05zr). The embedding matrix is a block circulant matrix with circulant blocks. ${\mathbf{m}}\left(1\right)$ is the size of each block, and ${\mathbf{m}}\left(2\right)$ is the number of blocks.
Constraints:
• ${\mathbf{m}}\left(1\right)\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,2\left({\mathbf{ns}}\left(1\right)-1\right)\right)$;
• ${\mathbf{m}}\left(2\right)\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,2\left({\mathbf{ns}}\left(2\right)-1\right)\right)$.
4:     $\mathrm{lam}\left({\mathbf{m}}\left(1\right)×{\mathbf{m}}\left(2\right)\right)$ – double array
Contains the square roots of the eigenvalues of the embedding matrix, as returned by nag_rand_field_2d_user_setup (g05zq) or nag_rand_field_2d_predef_setup (g05zr).
Constraint: ${\mathbf{lam}}\left(i\right)\ge 0$, $i=1,2,\dots ,{\mathbf{m}}\left(1\right)×{\mathbf{m}}\left(2\right)$.
5:     $\mathrm{rho}$ – double scalar
Indicates the scaling of the covariance matrix, as returned by nag_rand_field_2d_user_setup (g05zq) or nag_rand_field_2d_predef_setup (g05zr).
Constraint: $0.0<{\mathbf{rho}}\le 1.0$.
6:     $\mathrm{state}\left(:\right)$int64int32nag_int array
Note: the actual argument supplied must be the array state supplied to the initialization routines nag_rand_init_repeat (g05kf) or nag_rand_init_nonrepeat (g05kg).
Contains information on the selected base generator and its current state.

None.

### Output Parameters

1:     $\mathrm{state}\left(:\right)$int64int32nag_int array
Contains updated information on the state of the generator.
2:     $\mathrm{z}\left({\mathbf{ns}}\left(1\right)×{\mathbf{ns}}\left(2\right),{\mathbf{s}}\right)$ – double array
Contains the realizations of the random field. The $k$th realization (where $k=1,2,\dots ,{\mathbf{s}}$) of the random field on the two-dimensional grid $\left({x}_{i},{y}_{j}\right)$ is stored in ${\mathbf{z}}\left(\left(j-1\right)×{\mathbf{ns}}\left(1\right)+i,k\right)$, for $i=1,2,\dots ,{\mathbf{ns}}\left(1\right)$ and for $j=1,2,\dots ,{\mathbf{ns}}\left(2\right)$. The points are returned in xx and yy by nag_rand_field_2d_user_setup (g05zq) or nag_rand_field_2d_predef_setup (g05zr).
3:     $\mathrm{ifail}$int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ unless the function detects an error (see Error Indicators and Warnings).

## Error Indicators and Warnings

Errors or warnings detected by the function:
${\mathbf{ifail}}=1$
Constraint: ${\mathbf{ns}}\left(1\right)\ge 1$, ${\mathbf{ns}}\left(2\right)\ge 1$.
${\mathbf{ifail}}=2$
Constraint: ${\mathbf{s}}\ge 1$.
${\mathbf{ifail}}=3$
Constraints: ${\mathbf{m}}\left(i\right)\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,2\left({\mathbf{ns}}\left(i\right)\right)-1\right)$, for $i=1,2$.
${\mathbf{ifail}}=4$
On entry, at least one element of lam was negative.
Constraint: all elements of lam must be non-negative.
${\mathbf{ifail}}=5$
Constraint: $0.0<{\mathbf{rho}}\le 1.0$.
${\mathbf{ifail}}=6$
On entry, state vector has been corrupted or not initialized.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please contact NAG.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

Not applicable.

## Further Comments

Because samples are generated in pairs, calling this routine $k$ times, with ${\mathbf{s}}=s$, say, will generate a different sequence of numbers than calling the routine once with ${\mathbf{s}}=ks$, unless $s$ is even.

## Example

This example calls nag_rand_field_2d_generate (g05zs) to generate $5$ realizations of a two-dimensional random field on a $5$ by $5$ grid. This uses eigenvalues of the embedding covariance matrix for a symmetric stable variogram as calculated by nag_rand_field_2d_predef_setup (g05zr) with ${\mathbf{icov2}}=1$.
```function g05zs_example

fprintf('g05zs example results\n\n');

% Use symmetric stable variogram
icov2 = int64(1);
params = [0.1; 0.15; 1.2];

% 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 for circulant matrix
maxm = [int64(64), 64];
% Scaling factor, rho = 1.
icorr = int64(2);

% Get square roots of the eigenvalues of the embedding matrix
[lam, xx, yy, m, approx, rho, icount, eig, ifail] = ...
g05zr( ...
ns, xmin, xmax, ymin, ymax, maxm, var, ...
icov2, params, 'icorr', icorr);

fprintf('\nSize 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

% Initialize state array
genid = int64(1);
subid = int64(1);
seed  = [int64(14965)];
[state, ifail] = g05kf( ...
genid, subid, seed);

% Compute s random field realisations
s = int64(5);
[state, z, ifail] = g05zs( ...
ns, s, m, lam, rho, state);

% Display realisations

% Set row labels to grid points (column label is realisation number).
rlabs = cell(ns(1)*ns(2), 1);
for j=1:ns(2)
for i=1:ns(1)
if i == 1
rlabs{(j-1)*ns(1)+i} = sprintf('%6.1f%6.1f', xx(i), yy(j));
else
rlabs{(j-1)*ns(1)+i} = sprintf('%6.1f     .', xx(i));
end
end
end

% Matrix printing parameters
mtitle = 'Random field realisations (x,y coordinates first):';
matrix = 'General';
diag   = 'Non-unit';
fmt    = 'f10.5';
rlabel = 'Character';
clabel = 'Integer';
clabs  = {' '};
ncols  = int64(80);
indent = int64(0);

[ifail] = x04cb( ...
matrix, diag, z, fmt, mtitle, rlabel, rlabs, clabel, ...
clabs, ncols, indent);

```
```g05zs example results

Size of embedding matrix = 64

Approximation not required

Random field realisations (x,y coordinates first):
1         2         3         4         5
-0.8  -0.4   -0.61951  -0.93149  -0.32975  -0.51201   1.38877
-0.4     .    0.74779   1.33518  -0.51237   0.26595   0.30051
0.0     .   -0.30579   0.51819   0.50961   0.10379   0.36815
0.4     .    0.53797  -0.53992  -0.86589  -0.37098   0.21571
0.8     .   -0.61221  -1.04262   0.00007  -1.22614  -0.06650
-0.8  -0.2    0.01853   0.64126  -0.42978  -0.79178  -0.55728
-0.4     .   -0.77912   0.81079  -0.60613   0.07280   1.61511
0.0     .   -0.23198   1.48744  -0.78145   0.10347   0.07053
0.4     .    0.32356   0.58676   0.05846   0.34828   1.40522
0.8     .   -1.24085  -0.92512   0.27247  -0.66965   0.67073
-0.8   0.0   -1.18183  -0.99775   0.03888   0.01789  -0.65746
-0.4     .    0.26155  -0.01734  -0.14924   0.28886   0.25940
0.0     .    1.14960   0.48850  -0.59023   0.22795  -0.60773
0.4     .   -0.32684  -0.09616  -0.63497  -1.06753  -0.64594
0.8     .    0.10064   1.06148   0.15020  -0.53168  -0.29251
-0.8   0.2   -1.30595  -0.03899  -0.35549  -0.20589  -0.35956
-0.4     .   -0.01776   0.84501   0.20406   0.89039  -0.58338
0.0     .    0.41898   0.93435  -1.10725   0.76913  -0.74579
0.4     .   -1.37738   1.72404  -0.20558  -1.41877   1.21816
0.8     .    0.77866   0.84922  -0.65055   0.83518  -0.26425
-0.8   0.4   -0.65163   0.50492  -0.52463  -1.12816   1.12817
-0.4     .    0.15437   0.20739  -0.12675   1.27782  -0.26157
0.0     .    0.20324   0.54670  -1.73909   0.61580   0.17551
0.4     .   -1.09470   0.83967   0.70226  -0.34259   0.29368
0.8     .    1.08452   1.23097  -0.36003   1.06884   0.23594
```

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