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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_rand_field_2d_generate (g05zs)

Purpose

nag_rand_field_2d_generate (g05zs) produces realisations 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(x)$Z\left(\mathbf{x}\right)$ in 2${ℝ}^{2}$ is a function which is random at every point x2$\mathbf{x}\in {ℝ}^{2}$, so Z(x)$Z\left(\mathbf{x}\right)$ is a random variable for each x$\mathbf{x}$. The random field has a mean function μ(x) = 𝔼[Z(x)]$\mu \left(\mathbf{x}\right)=𝔼\left[Z\left(\mathbf{x}\right)\right]$ and a symmetric positive semidefinite covariance function C(x,y) = 𝔼[(Z(x)μ(x))(Z(y)μ(y))]$C\left(\mathbf{x},\mathbf{y}\right)=𝔼\left[\left(Z\left(\mathbf{x}\right)-\mu \left(\mathbf{x}\right)\right)\left(Z\left(\mathbf{y}\right)-\mu \left(\mathbf{y}\right)\right)\right]$. Z(x)$Z\left(\mathbf{x}\right)$ is a Gaussian random field if for any choice of n$n\in ℕ$ and x1,,xn2${\mathbf{x}}_{1},\dots ,{\mathbf{x}}_{n}\in {ℝ}^{2}$, the random vector [Z(x1),,Z(xn)]T${\left[Z\left({\mathbf{x}}_{1}\right),\dots ,Z\left({\mathbf{x}}_{n}\right)\right]}^{\mathrm{T}}$ follows a multivariate Gaussian distribution, which would have a mean vector μ̃$\stackrel{~}{\mathbf{\mu }}$ with entries μ̃i = μ(xi)${\stackrel{~}{\mu }}_{i}=\mu \left({\mathbf{x}}_{i}\right)$ and a covariance matrix $\stackrel{~}{C}$ with entries ij = C(xi,xj)${\stackrel{~}{C}}_{ij}=C\left({\mathbf{x}}_{i},{\mathbf{x}}_{j}\right)$. A Gaussian random field Z(x)$Z\left(\mathbf{x}\right)$ is stationary if μ(x)$\mu \left(\mathbf{x}\right)$ is constant for all x2$\mathbf{x}\in {ℝ}^{2}$ and C(x,y) = C(x + a,y + a)$C\left(\mathbf{x},\mathbf{y}\right)=C\left(\mathbf{x}+\mathbf{a},\mathbf{y}+\mathbf{a}\right)$ for all x,y,a2$\mathbf{x},\mathbf{y},\mathbf{a}\in {ℝ}^{2}$ and hence we can express the covariance function C(x,y)$C\left(\mathbf{x},\mathbf{y}\right)$ as a function γ$\gamma$ of one variable: C(x,y) = γ(xy)$C\left(\mathbf{x},\mathbf{y}\right)=\gamma \left(\mathbf{x}-\mathbf{y}\right)$. γ$\gamma$ is known as a variogram (or more correctly, a semivariogram) and includes the multiplicative factor σ2${\sigma }^{2}$ representing the variance such that γ(0) = σ2$\gamma \left(0\right)={\sigma }^{2}$.
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 two-dimensional stationary Gaussian random field, with mean function zero and variogram γ(x)$\gamma \left(\mathbf{x}\right)$, over a domain [xmin,xmax] × [ymin,ymax]$\left[{x}_{\mathrm{min}},{x}_{\mathrm{max}}\right]×\left[{y}_{\mathrm{min}},{y}_{\mathrm{max}}\right]$, using an equally spaced set of N1 × N2${N}_{1}×{N}_{2}$ gridpoints; N1${N}_{1}$ gridpoints in the x$x$-direction and N2${N}_{2}$ gridpoints in the y$y$-direction. The problem reduces to sampling a Gaussian random vector X$\mathbf{X}$ of size N1 × N2${N}_{1}×{N}_{2}$, with mean vector zero and a symmetric covariance matrix A$A$, which is an N2${N}_{2}$ by N2${N}_{2}$ block Toeplitz matrix with Toeplitz blocks of size N1${N}_{1}$ by N1${N}_{1}$. Since A$A$ is in general expensive to factorize, a technique known as the circulant embedding method is used. A$A$ is embedded into a larger, symmetric matrix B$B$, which is an M2${M}_{2}$ by M2${M}_{2}$ block circulant matrix with circulant bocks of size M1${M}_{1}$ by M1${M}_{1}$, where M12(N11)${M}_{1}\ge 2\left({N}_{1}-1\right)$ and M22(N21)${M}_{2}\ge 2\left({N}_{2}-1\right)$. B$B$ can now be factorized as B = WΛW* = R*R$B=W\Lambda {W}^{*}={R}^{*}R$, where W$W$ is the two-dimensional Fourier matrix (W*${W}^{*}$ is the complex conjugate of W$W$), Λ$\Lambda$ is the diagonal matrix containing the eigenvalues of B$B$ and R = Λ(1/2)W*$R={\Lambda }^{\frac{1}{2}}{W}^{*}$. B$B$ 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$B$ and multiplying by M1 × M2${M}_{1}×{M}_{2}$, and so only the first row (or column) of B$B$ is needed – the whole matrix does not need to be formed.
The symmetry of A$A$ as a block matrix, and the symmetry of each block of A$A$, depends on whether the covariance function γ$\gamma$ is even or not. γ$\gamma$ is even if γ(x) = γ(x)$\gamma \left(\mathbf{x}\right)=\gamma \left(-\mathbf{x}\right)$ for all x2$\mathbf{x}\in {ℝ}^{2}$, and uneven otherwise (in higher dimensions, γ$\gamma$ can be even in some coordinates and uneven in others, but in two dimensions γ$\gamma$ is either even in both coordinates or uneven in both coordinates). If γ$\gamma$ is even then A$A$ is a symmetric block matrix and has symmetric blocks; if γ$\gamma$ is uneven then A$A$ is not a symmetric block matrix and has non-symmetric blocks. In the uneven case, M1${M}_{1}$ and M2${M}_{2}$ are set to be odd in order to guarantee symmetry in B$B$.
As long as all of the values of Λ$\Lambda$ are non-negative (i.e., B$B$ is positive semidefinite), B$B$ is a covariance matrix for a random vector Y$\mathbf{Y}$ which has M2${M}_{2}$ ‘blocks’ of size M1${M}_{1}$. Two samples of Y$\mathbf{Y}$ can now be simulated from the real and imaginary parts of R*(U + iV)${R}^{*}\left(\mathbf{U}+i\mathbf{V}\right)$, where U$\mathbf{U}$ and V$\mathbf{V}$ have elements from the standard Normal distribution. Since R*(U + iV) = WΛ(1/2)(U + iV)${R}^{*}\left(\mathbf{U}+i\mathbf{V}\right)=W{\Lambda }^{\frac{1}{2}}\left(\mathbf{U}+i\mathbf{V}\right)$, this calculation can be done using a discrete Fourier transform of the vector Λ(1/2)(U + iV)${\Lambda }^{\frac{1}{2}}\left(\mathbf{U}+i\mathbf{V}\right)$. Two samples of the random vector X$\mathbf{X}$ can now be recovered by taking the first N1${N}_{1}$ elements of the first N2${N}_{2}$ blocks of each sample of Y$Y$ – because the original covariance matrix A$A$ is embedded in B$B$, X$\mathbf{X}$ will have the correct distribution.
If B$B$ is not positive semidefinite, larger embedding matrices B$B$ 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$B$, and its size vector M$M$, as input and outputs S$S$ realisations of the random field in Z$Z$.
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 [0,1]d${\left[0,1\right]}^{d}$ Journal of Computational and Graphical Statistics 3(4) 409–432

Parameters

Compulsory Input Parameters

1:     ns(2$2$) – int64int32nag_int array
The number of sample points (gridpoints) to use in each direction, with ns(1)${\mathbf{ns}}\left(1\right)$ sample points in the x$x$-direction and ns(2)${\mathbf{ns}}\left(2\right)$ sample points in the y$y$-direction. The total number of sample points on the grid is therefore ns(1) × ns(1) ${\mathbf{ns}}\left(1\right)×{\mathbf{ns}}\left(1\right)$. 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:
• ns(1)1${\mathbf{ns}}\left(1\right)\ge 1$;
• ns(2)1${\mathbf{ns}}\left(2\right)\ge 1$.
2:     s – int64int32nag_int scalar
The number of realisations of the random field to simulate.
Constraint: s1${\mathbf{s}}\ge 1$.
3:     m(2$2$) – int64int32nag_int array
Indicates the size 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. m(1)${\mathbf{m}}\left(1\right)$ is the size of each block, and m(2)${\mathbf{m}}\left(2\right)$ is the number of blocks.
Constraints:
• m(1)max (1,2(ns(1)1))${\mathbf{m}}\left(1\right)\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,2\left({\mathbf{ns}}\left(1\right)-1\right)\right)$;
• m(2)max (1,2(ns(2)1))${\mathbf{m}}\left(2\right)\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,2\left({\mathbf{ns}}\left(2\right)-1\right)\right)$.
4:     lam(m(1) × m(2)${\mathbf{m}}\left(1\right)×{\mathbf{m}}\left(2\right)$) – 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: lam(i) = 0${\mathbf{lam}}\left(i\right)=0$, i = 1,2,,m(1) × m(2)$i=1,2,\dots ,{\mathbf{m}}\left(1\right)×{\mathbf{m}}\left(2\right)$.
5:     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 < rho1.0$0.0<{\mathbf{rho}}\le 1.0$.
6:     state( : $:$) – 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.

None.

Output Parameters

1:     state( : $:$) – 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 updated information on the state of the generator.
2:     z(ns(1) × ns(2)${\mathbf{ns}}\left(1\right)×{\mathbf{ns}}\left(2\right)$,s) – double array
Contains the realisations of the random field. Each column of Z$Z$ contains one realisation of the random field, with z(i + (j1)ns(1),k)${\mathbf{z}}\left(i+\left(j-1\right){\mathbf{ns}}\left(1\right),k\right)$, for k = 1,2,,s$k=1,2,\dots ,{\mathbf{s}}$, corresponding to the gridpoint xx(i)${\mathbf{xx}}\left(i\right)$ and yy(j)${\mathbf{yy}}\left(j\right)$ as returned by nag_rand_field_2d_user_setup (g05zq) or nag_rand_field_2d_predef_setup (g05zr) .
3:     ifail – int64int32nag_int scalar
${\mathrm{ifail}}={\mathbf{0}}$ unless the function detects an error (see [Error Indicators and Warnings]).

Error Indicators and Warnings

Errors or warnings detected by the function:
ifail = 1${\mathbf{ifail}}=1$
Constraint: ns(1)1${\mathbf{ns}}\left(1\right)\ge 1$, ns(2)1${\mathbf{ns}}\left(2\right)\ge 1$.
ifail = 2${\mathbf{ifail}}=2$
Constraint: s1${\mathbf{s}}\ge 1$.
ifail = 3${\mathbf{ifail}}=3$
Constraints: m(i)max (1,2(ns(i))1)${\mathbf{m}}\left(i\right)\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,2\left({\mathbf{ns}}\left(i\right)\right)-1\right)$, for i = 1,2$i=1,2$.
ifail = 4${\mathbf{ifail}}=4$
On entry, at least one element of lam was negative.
Constraint: all elements of lam must be non-negative.
ifail = 5${\mathbf{ifail}}=5$
Constraint: 0.0 < rho1.0$0.0<{\mathbf{rho}}\le 1.0$.
ifail = 6${\mathbf{ifail}}=6$
On entry, state vector has been corrupted or not initialized.

Accuracy

Not applicable.

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

Example

```function nag_rand_field_2d_generate_example
icov2 = int64(1); % symmetric stable
params = [0.1; 0.15; 1.2];
var = 0.5;
xmin = -1;
xmax = 1;
ymin = -0.5;
ymax = 0.5;
ns = [int64(5), 5];
maxm = [int64(64), 64];
icorr = int64(2);
s = int64(5);

% Get square roots of the eigenvalues of the embedding matrix
[lam, xx, yy, m, approx, rho, icount, eig, ifail] = ...
nag_rand_field_2d_predef_setup(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] = nag_rand_init_repeat(genid, subid, seed);

% Compute s random field realisations
[state, z, ifail] = nag_rand_field_2d_generate(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

title = 'Random field realisations (x,y coordinates first):';
[ifail] = ...
nag_file_print_matrix_real_gen_comp('g', 'n', z, 'f10.5', title, 'c', rlabs, 'i', {''}, int64(80), int64(0))
```
```

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

ifail =

0

```
```function g05zs_example
icov2 = int64(1); % symmetric stable
params = [0.1; 0.15; 1.2];
var = 0.5;
xmin = -1;
xmax = 1;
ymin = -0.5;
ymax = 0.5;
ns = [int64(5), 5];
maxm = [int64(64), 64];
icorr = int64(2);
s = int64(5);

% 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
[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

title = 'Random field realisations (x,y coordinates first):';
[ifail] = ...
x04cb('g', 'n', z, 'f10.5', title, 'c', rlabs, 'i', {''}, int64(80), int64(0))
```
```

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

ifail =

0

```