NAG CL Interface
g05zsc (field_2d_generate)
1
Purpose
g05zsc 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
g05zqc or
g05zrc.
2
Specification
void 
g05zsc (const Integer ns[],
Integer s,
const Integer m[],
const double lam[],
double rho,
Integer state[],
double z[],
NagError *fail) 

The function may be called by the names: g05zsc or nag_rand_field_2d_generate.
3
Description
A twodimensional random field $Z\left(\mathbf{x}\right)$ in ${\mathbb{R}}^{2}$ is a function which is random at every point $\mathbf{x}\in {\mathbb{R}}^{2}$, so $Z\left(\mathbf{x}\right)$ is a random variable for each $\mathbf{x}$. The random field has a mean function $\mu \left(\mathbf{x}\right)=\mathbb{E}\left[Z\left(\mathbf{x}\right)\right]$ and a symmetric positive semidefinite covariance function $C\left(\mathbf{x},\mathbf{y}\right)=\mathbb{E}\left[\left(Z\left(\mathbf{x}\right)\mu \left(\mathbf{x}\right)\right)\left(Z\left(\mathbf{y}\right)\mu \left(\mathbf{y}\right)\right)\right]$. $Z\left(\mathbf{x}\right)$ is a Gaussian random field if for any choice of $n\in \mathbb{N}$ and ${\mathbf{x}}_{1},\dots ,{\mathbf{x}}_{n}\in {\mathbb{R}}^{2}$, 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 $\mathbf{x}\in {\mathbb{R}}^{2}$ and $C\left(\mathbf{x},\mathbf{y}\right)=C\left(\mathbf{x}+\mathbf{a},\mathbf{y}+\mathbf{a}\right)$ for all $\mathbf{x},\mathbf{y},\mathbf{a}\in {\mathbb{R}}^{2}$ 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
g05zqc or
g05zrc along with
g05zsc are used to simulate a twodimensional stationary Gaussian random field, with mean function zero and variogram
$\gamma \left(\mathbf{x}\right)$, over a domain
$\left[{x}_{\mathrm{min}},{x}_{\mathrm{max}}\right]\times \left[{y}_{\mathrm{min}},{y}_{\mathrm{max}}\right]$, 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 Gaussian random vector
$\mathbf{X}$ of size
${N}_{1}\times {N}_{2}$, 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 twodimensional 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
${M}_{1}\times {M}_{2}$, 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 covariance function $\gamma $ is even or not. $\gamma $ is even if $\gamma \left(\mathbf{x}\right)=\gamma \left(\mathbf{x}\right)$ for all $\mathbf{x}\in {\mathbb{R}}^{2}$, 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 nonsymmetric 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 nonnegative (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
g05zqc or
g05zrc for details of the approximation procedure.
g05zsc 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
g05kfc (for a repeatable sequence if computed sequentially) or
g05kgc (for a nonrepeatable sequence) must be called prior to the first call to
g05zsc.
4
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
5
Arguments

1:
$\mathbf{ns}\left[2\right]$ – const Integer
Input

On entry: the number of sample points to use in each direction, with
${\mathbf{ns}}\left[0\right]$ sample points in the
$x$direction and
${\mathbf{ns}}\left[1\right]$ sample points in the
$y$direction. The total number of sample points on the grid is therefore
${\mathbf{ns}}\left[0\right]\times {\mathbf{ns}}\left[1\right]$. This must be the same value as supplied to
g05zqc or
g05zrc when calculating the eigenvalues of the embedding matrix.
Constraints:
 ${\mathbf{ns}}\left[0\right]\ge 1$;
 ${\mathbf{ns}}\left[1\right]\ge 1$.

2:
$\mathbf{s}$ – Integer
Input

On entry: $S$, the number of realizations of the random field to simulate.
Constraint:
${\mathbf{s}}\ge 1$.

3:
$\mathbf{m}\left[2\right]$ – const Integer
Input

On entry: indicates the size,
$M$, of the embedding matrix as returned by
g05zqc or
g05zrc. The embedding matrix is a block circulant matrix with circulant blocks.
${\mathbf{m}}\left[0\right]$ is the size of each block, and
${\mathbf{m}}\left[1\right]$ is the number of blocks.
Constraints:
 ${\mathbf{m}}\left[0\right]\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,2\left({\mathbf{ns}}\left[0\right]1\right)\right)$;
 ${\mathbf{m}}\left[1\right]\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,2\left({\mathbf{ns}}\left[1\right]1\right)\right)$.

4:
$\mathbf{lam}\left[{\mathbf{m}}\left[0\right]\times {\mathbf{m}}\left[1\right]\right]$ – const double
Input

On entry: contains the square roots of the eigenvalues of the embedding matrix, as returned by
g05zqc or
g05zrc.
Constraint:
${\mathbf{lam}}\left[\mathit{i}1\right]\ge 0$, for $\mathit{i}=1,2,\dots ,{\mathbf{m}}\left[0\right]\times {\mathbf{m}}\left[1\right]$.

5:
$\mathbf{rho}$ – double
Input

On entry: indicates the scaling of the covariance matrix, as returned by
g05zqc or
g05zrc.
Constraint:
$0.0<{\mathbf{rho}}\le 1.0$.

6:
$\mathbf{state}\left[\mathit{dim}\right]$ – Integer
Communication Array
Note: the dimension,
$\mathit{dim}$, of this array is dictated by the requirements of associated functions that must have been previously called. This array MUST be the same array passed as argument
state in the previous call to
nag_rand_init_repeatable (g05kfc) or
nag_rand_init_nonrepeatable (g05kgc).
On entry: contains information on the selected base generator and its current state.
On exit: contains updated information on the state of the generator.

7:
$\mathbf{z}\left[\mathit{dim}\right]$ – double
Output

Note: the dimension,
dim, of the array
z
must be at least
${\mathbf{s}}\times {\mathbf{ns}}\left[0\right]\times {\mathbf{ns}}\left[1\right]$.
On exit: contains the realizations of the random field.
The
$k$th realization (where
$k=1,2,\dots ,{\mathbf{s}}$) of the random field on the twodimensional grid
$\left({x}_{i},{y}_{j}\right)$ is stored in
${\mathbf{z}}\left[\left(k1\right)\times {\mathbf{ns}}\left[0\right]\times {\mathbf{ns}}\left[1\right]+\left(j1\right)\times {\mathbf{ns}}\left[0\right]+i1\right]$, for
$i=1,2,\dots ,{\mathbf{ns}}\left[0\right]$ and for
$j=1,2,\dots ,{\mathbf{ns}}\left[1\right]$. The points are returned in
xx and
yy by
g05zqc or
g05zrc.

8:
$\mathbf{fail}$ – NagError *
Input/Output

The NAG error argument (see
Section 7 in the Introduction to the NAG Library CL Interface).
6
Error Indicators and Warnings
 NE_ALLOC_FAIL

Dynamic memory allocation failed.
See
Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
 NE_BAD_PARAM

On entry, argument $\u2329\mathit{\text{value}}\u232a$ had an illegal value.
 NE_INT

On entry, ${\mathbf{s}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{s}}\ge 1$.
 NE_INT_ARRAY

On entry, ${\mathbf{ns}}=\left[\u2329\mathit{\text{value}}\u232a,\u2329\mathit{\text{value}}\u232a\right]$.
Constraint: ${\mathbf{ns}}\left[0\right]\ge 1$, ${\mathbf{ns}}\left[1\right]\ge 1$.
 NE_INT_ARRAY_2

On entry, ${\mathbf{m}}=\left[\u2329\mathit{\text{value}}\u232a,\u2329\mathit{\text{value}}\u232a\right]$, and ${\mathbf{ns}}=\left[\u2329\mathit{\text{value}}\u232a,\u2329\mathit{\text{value}}\u232a\right]$.
Constraints: ${\mathbf{m}}\left[i1\right]\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,2\left({\mathbf{ns}}\left[i1\right]\right)1\right)$, for $i=1,2$.
 NE_INTERNAL_ERROR

An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact
NAG for assistance.
See
Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
 NE_INVALID_STATE

On entry,
state vector has been corrupted or not initialized.
 NE_NEG_ELEMENT

On entry, at least one element of
lam was negative.
Constraint: all elements of
lam must be nonnegative.
 NE_NO_LICENCE

Your licence key may have expired or may not have been installed correctly.
See
Section 8 in the Introduction to the NAG Library CL Interface for further information.
 NE_REAL

On entry, ${\mathbf{rho}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: $0.0<{\mathbf{rho}}\le 1.0$.
7
Accuracy
Not applicable.
8
Parallelism and Performance
g05zsc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
g05zsc makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the
X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the
Users' Note for your implementation for any additional implementationspecific information.
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.
10
Example
This example calls
g05zsc to generate
$5$ realizations of a twodimensional random field on a
$5$ by
$5$ grid. This uses eigenvalues of the embedding covariance matrix for a symmetric stable variogram as calculated by
g05zrc with
${\mathbf{cov}}=\mathrm{Nag\_VgmSymmStab}$.
10.1
Program Text
10.2
Program Data
10.3
Program Results