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 g05zqcorg05zrc.
The function may be called by the names: g05zsc or nag_rand_field_2d_generate.
3Description
A two-dimensional 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 $C(\mathbf{x},\mathbf{y})=\mathbb{E}\left[(Z\left(\mathbf{x}\right)-\mu \left(\mathbf{x}\right))(Z\left(\mathbf{y}\right)-\mu \left(\mathbf{y}\right))\right]$. $Z\left(\mathbf{x}\right)$ 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 ${[Z\left({\mathbf{x}}_{1}\right),\dots ,Z\left({\mathbf{x}}_{n}\right)]}^{\mathrm{T}}$ 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 ${\stackrel{~}{C}}_{ij}=C({\mathbf{x}}_{i},{\mathbf{x}}_{j})$. 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 $C(\mathbf{x},\mathbf{y})=C(\mathbf{x}+\mathbf{a},\mathbf{y}+\mathbf{a})$ for all $\mathbf{x},\mathbf{y},\mathbf{a}\in {\mathbb{R}}^{2}$ and hence we can express the covariance function $C(\mathbf{x},\mathbf{y})$ as a function $\gamma $ of one variable: $C(\mathbf{x},\mathbf{y})=\gamma (\mathbf{x}-\mathbf{y})$. $\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 g05zqcorg05zrc along with g05zsc 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 $[{x}_{\mathrm{min}},{x}_{\mathrm{max}}]\times [{y}_{\mathrm{min}},{y}_{\mathrm{max}}]$, 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 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 ${N}_{2}\times {N}_{2}$ block Toeplitz matrix with Toeplitz blocks of size ${N}_{1}\times {N}_{1}$. 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 ${M}_{2}\times {M}_{2}$ block circulant matrix with circulant bocks of size ${M}_{1}\times {M}_{1}$, where ${M}_{1}\ge 2({N}_{1}-1)$ and ${M}_{2}\ge 2({N}_{2}-1)$. $B$ 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 ${M}_{1}\times {M}_{2}$, 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 $\gamma \left(\mathbf{x}\right)=\gamma (-\mathbf{x})$ 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 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 ${R}^{*}(\mathbf{U}+i\mathbf{V})$, where $\mathbf{U}$ and $\mathbf{V}$ have elements from the standard Normal distribution. Since ${R}^{*}(\mathbf{U}+i\mathbf{V})=W{\Lambda}^{\frac{1}{2}}(\mathbf{U}+i\mathbf{V})$, this calculation can be done using a discrete Fourier transform of the vector ${\Lambda}^{\frac{1}{2}}(\mathbf{U}+i\mathbf{V})$. 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 g05zqcorg05zrc 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 non-repeatable sequence) must be called prior to the first call to g05zsc.
4References
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 ${[0,1]}^{d}$Journal of Computational and Graphical Statistics3(4) 409–432
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 g05zqcorg05zrc when calculating the eigenvalues of the embedding matrix.
Constraints:
${\mathbf{ns}}\left[0\right]\ge 1$;
${\mathbf{ns}}\left[1\right]\ge 1$.
2: $\mathbf{s}$ – IntegerInput
On entry: $S$, the number of realizations of the random field to simulate.
On entry: indicates the size, $M$, of the embedding matrix as returned by g05zqcorg05zrc. 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.
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.
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 two-dimensional grid $({x}_{i},{y}_{j})$ is stored in ${\mathbf{z}}\left[\left(k-1\right)\times {\mathbf{ns}}\left[0\right]\times {\mathbf{ns}}\left[1\right]+(j-1)\times {\mathbf{ns}}\left[0\right]+i-1\right]$, 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 g05zqcorg05zrc.
8: $\mathbf{fail}$ – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error 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 $\u27e8\mathit{\text{value}}\u27e9$ had an illegal value.
NE_INT
On entry, ${\mathbf{s}}=\u27e8\mathit{\text{value}}\u27e9$. Constraint: ${\mathbf{s}}\ge 1$.
NE_INT_ARRAY
On entry, ${\mathbf{ns}}=[\u27e8\mathit{\text{value}}\u27e9,\u27e8\mathit{\text{value}}\u27e9]$. Constraint: ${\mathbf{ns}}\left[0\right]\ge 1$, ${\mathbf{ns}}\left[1\right]\ge 1$.
NE_INT_ARRAY_2
On entry, ${\mathbf{m}}=[\u27e8\mathit{\text{value}}\u27e9,\u27e8\mathit{\text{value}}\u27e9]$, and ${\mathbf{ns}}=[\u27e8\mathit{\text{value}}\u27e9,\u27e8\mathit{\text{value}}\u27e9]$. Constraints: ${\mathbf{m}}\left[i-1\right]\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}(1,2\left({\mathbf{ns}}\left[i-1\right]\right)-1)$, 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 non-negative.
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, ${\mathbf{rho}}=\u27e8\mathit{\text{value}}\u27e9$. Constraint: $0.0<{\mathbf{rho}}\le 1.0$.
7Accuracy
Not applicable.
8Parallelism 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 implementation-specific information.
9Further 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.
10Example
This example calls g05zsc to generate $5$ realizations of a two-dimensional random field on a $5\times 5$ grid. This uses eigenvalues of the embedding covariance matrix for a symmetric stable variogram as calculated by g05zrc with ${\mathbf{cov}}=\mathrm{Nag\_VgmSymmStab}$.