NAG CL Interface
g05zmc (field_1d_user_setup)
1
Purpose
g05zmc performs the setup required in order to simulate stationary Gaussian random fields in one dimension, for a userdefined variogram, using the
circulant embedding method. Specifically, the eigenvalues of the extended covariance matrix (or embedding matrix) are calculated, and their square roots output, for use by
g05zpc, which simulates the random field.
2
Specification
void 
g05zmc (Integer ns,
double xmin,
double xmax,
Integer maxm,
double var,
Nag_EmbedPad pad,
Nag_EmbedScale corr,
double lam[],
double xx[],
Integer *m,
Integer *approx,
double *rho,
Integer *icount,
double eig[],
Nag_Comm *comm,
NagError *fail) 

The function may be called by the names: g05zmc or nag_rand_field_1d_user_setup.
3
Description
A onedimensional random field $Z\left(x\right)$ in $\mathbb{R}$ is a function which is random at every point $x\in \mathbb{R}$, so $Z\left(x\right)$ is a random variable for each $x$. The random field has a mean function $\mu \left(x\right)=\mathbb{E}\left[Z\left(x\right)\right]$ and a symmetric positive semidefinite covariance function $C\left(x,y\right)=\mathbb{E}\left[\left(Z\left(x\right)\mu \left(x\right)\right)\left(Z\left(y\right)\mu \left(y\right)\right)\right]$. $Z\left(x\right)$ is a Gaussian random field if for any choice of $n\in \mathbb{N}$ and ${x}_{1},\dots ,{x}_{n}\in \mathbb{R}$, the random vector ${\left[Z\left({x}_{1}\right),\dots ,Z\left({x}_{n}\right)\right]}^{\mathrm{T}}$ follows a multivariate Normal distribution, which would have a mean vector $\stackrel{~}{\mathbf{\mu}}$ with entries ${\stackrel{~}{\mu}}_{i}=\mu \left({x}_{i}\right)$ and a covariance matrix $\stackrel{~}{C}$ with entries ${\stackrel{~}{C}}_{ij}=C\left({x}_{i},{x}_{j}\right)$. A Gaussian random field $Z\left(x\right)$ is stationary if $\mu \left(x\right)$ is constant for all $x\in \mathbb{R}$ and $C\left(x,y\right)=C\left(x+a,y+a\right)$ for all $x,y,a\in \mathbb{R}$ and hence we can express the covariance function $C\left(x,y\right)$ as a function $\gamma $ of one variable: $C\left(x,y\right)=\gamma \left(xy\right)$. $\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
g05zmc and
g05zpc are used to simulate a onedimensional stationary Gaussian random field, with mean function zero and variogram
$\gamma \left(x\right)$, over an interval
$\left[{x}_{\mathrm{min}},{x}_{\mathrm{max}}\right]$, using an equally spaced set of
$N$ points on the interval. The problem reduces to sampling a Normal random vector
$\mathbf{X}$ 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
$M\ge 2\left(N1\right)$, which can now be factorized as
$B=W\Lambda {W}^{*}={R}^{*}R$, where
$W$ is the 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
$M$, 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 $\Lambda $ are nonnegative (i.e., $B$ is positive semidefinite), $B$ is a covariance matrix for a random vector $\mathbf{Y}$, two samples of which 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 $N$ elements of each sample of $\mathbf{Y}$ – 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. We write
$\Lambda ={\Lambda}_{+}+{\Lambda}_{}$, where
${\Lambda}_{+}$ and
${\Lambda}_{}$ contain the nonnegative and negative eigenvalues of
$B$ respectively. Then
$B$ is replaced by
$\rho {B}_{+}$ where
${B}_{+}=W{\Lambda}_{+}{W}^{*}$ and
$\rho \in \left(0,1\right]$ is a scaling factor. The error
$\epsilon $ in approximating the distribution of the random field is given by
Three choices for
$\rho $ are available, and are determined by the input argument
corr:
 setting ${\mathbf{corr}}=\mathrm{Nag\_EmbedScaleTraces}$ sets
 setting ${\mathbf{corr}}=\mathrm{Nag\_EmbedScaleSqrtTraces}$ sets
 setting ${\mathbf{corr}}=\mathrm{Nag\_EmbedScaleOne}$ sets $\rho =1$.
g05zmc finds a suitable positive semidefinite embedding matrix
$B$ and outputs its size,
m, and the square roots of its eigenvalues in
lam. If approximation is used, information regarding the accuracy of the approximation is output. Note that only the first row (or column) of
$B$ is actually formed and stored.
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:
$\mathbf{ns}$ – Integer
Input

On entry: the number of sample points to be generated in realizations of the random field.
Constraint:
${\mathbf{ns}}\ge 1$.

2:
$\mathbf{xmin}$ – double
Input

On entry: the lower bound for the interval over which the random field is to be simulated.
Constraint:
${\mathbf{xmin}}<{\mathbf{xmax}}$.

3:
$\mathbf{xmax}$ – double
Input

On entry: the upper bound for the interval over which the random field is to be simulated.
Constraint:
${\mathbf{xmin}}<{\mathbf{xmax}}$.

4:
$\mathbf{maxm}$ – Integer
Input

On entry: the maximum size of the circulant matrix to use. For example, if the embedding matrix is to be allowed to double in size three times before the approximation procedure is used, then choose ${\mathbf{maxm}}={2}^{k+2}$ where $k=1+\u2308{\mathrm{log}}_{2}\left({\mathbf{ns}}1\right)\u2309$.
Suggested value:
${2}^{k+2}\text{ where}k=1+\u2308{\mathrm{log}}_{2}\left({\mathbf{ns}}1\right)\u2309$.
Constraint:
${\mathbf{maxm}}\ge {2}^{k}$, where $k$ is the smallest integer satisfying ${2}^{k}\ge 2\left({\mathbf{ns}}1\right)$.

5:
$\mathbf{var}$ – double
Input

On entry: the multiplicative factor ${\sigma}^{2}$ of the variogram $\gamma \left(x\right)$.
Constraint:
${\mathbf{var}}\ge 0.0$.

6:
$\mathbf{cov1}$ – function, supplied by the user
External Function

cov1 must evaluate the variogram
$\gamma \left(x\right)$, without the multiplicative factor
${\sigma}^{2}$, for all
$x\ge 0$. The value returned in
gamma is multiplied internally by
var.
The specification of
cov1 is:
void 
cov1 (double x,
double *gamma,
Nag_Comm *comm)



1:
$\mathbf{x}$ – double
Input

On entry: the value $x$ at which the variogram $\gamma \left(x\right)$ is to be evaluated.

2:
$\mathbf{gamma}$ – double *
Output

On exit: the value of the variogram $\frac{\gamma \left(x\right)}{{\sigma}^{2}}$.

3:
$\mathbf{comm}$ – Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to
cov1.
 user – double *
 iuser – Integer *
 p – Pointer
The type Pointer will be
void *. Before calling
g05zmc you may allocate memory and initialize these pointers with various quantities for use by
cov1 when called from
g05zmc (see
Section 3.1.1 in the Introduction to the NAG Library CL Interface).
Note: cov1 should not return floatingpoint NaN (Not a Number) or infinity values, since these are not handled by
g05zmc. If your code inadvertently
does return any NaNs or infinities,
g05zmc is likely to produce unexpected results.

7:
$\mathbf{pad}$ – Nag_EmbedPad
Input

On entry: determines whether the embedding matrix is padded with zeros, or padded with values of the variogram. The choice of padding may affect how big the embedding matrix must be in order to be positive semidefinite.
 ${\mathbf{pad}}=\mathrm{Nag\_EmbedPadZeros}$
 The embedding matrix is padded with zeros.
 ${\mathbf{pad}}=\mathrm{Nag\_EmbedPadValues}$
 The embedding matrix is padded with values of the variogram.
Suggested value:
${\mathbf{pad}}=\mathrm{Nag\_EmbedPadValues}$.
Constraint:
${\mathbf{pad}}=\mathrm{Nag\_EmbedPadZeros}$ or $\mathrm{Nag\_EmbedPadValues}$.

8:
$\mathbf{corr}$ – Nag_EmbedScale
Input

On entry: determines which approximation to implement if required, as described in
Section 3.
Suggested value:
${\mathbf{corr}}=\mathrm{Nag\_EmbedScaleTraces}$.
Constraint:
${\mathbf{corr}}=\mathrm{Nag\_EmbedScaleTraces}$, $\mathrm{Nag\_EmbedScaleSqrtTraces}$ or $\mathrm{Nag\_EmbedScaleOne}$.

9:
$\mathbf{lam}\left[{\mathbf{maxm}}\right]$ – double
Output

On exit: contains the square roots of the eigenvalues of the embedding matrix.

10:
$\mathbf{xx}\left[{\mathbf{ns}}\right]$ – double
Output

On exit: the points at which values of the random field will be output.

11:
$\mathbf{m}$ – Integer *
Output

On exit: the size of the embedding matrix.

12:
$\mathbf{approx}$ – Integer *
Output

On exit: indicates whether approximation was used.
 ${\mathbf{approx}}=0$
 No approximation was used.
 ${\mathbf{approx}}=1$
 Approximation was used.

13:
$\mathbf{rho}$ – double *
Output

On exit: indicates the scaling of the covariance matrix. ${\mathbf{rho}}=1.0$ unless approximation was used with ${\mathbf{corr}}=\mathrm{Nag\_EmbedScaleTraces}$ or $\mathrm{Nag\_EmbedScaleSqrtTraces}$.

14:
$\mathbf{icount}$ – Integer *
Output

On exit: indicates the number of negative eigenvalues in the embedding matrix which have had to be set to zero.

15:
$\mathbf{eig}\left[3\right]$ – double
Output

On exit: indicates information about the negative eigenvalues in the embedding matrix which have had to be set to zero. ${\mathbf{eig}}\left[0\right]$ contains the smallest eigenvalue, ${\mathbf{eig}}\left[1\right]$ contains the sum of the squares of the negative eigenvalues, and ${\mathbf{eig}}\left[2\right]$ contains the sum of the absolute values of the negative eigenvalues.

16:
$\mathbf{comm}$ – Nag_Comm *

The NAG communication argument (see
Section 3.1.1 in the Introduction to the NAG Library CL Interface).

17:
$\mathbf{fail}$ – 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,
${\mathbf{maxm}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: the minimum calculated value for
maxm is
$\u2329\mathit{\text{value}}\u232a$.
Where the minimum calculated value is given by
${2}^{k}$, where
$k$ is the smallest integer satisfying
${2}^{k}\ge 2\left({\mathbf{ns}}1\right)$.
On entry, ${\mathbf{ns}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{ns}}\ge 1$.
 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_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, ${\mathbf{var}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{var}}\ge 0.0$.
 NE_REAL_2

On entry, ${\mathbf{xmin}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{xmax}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{xmin}}<{\mathbf{xmax}}$.
7
Accuracy
If on exit
${\mathbf{approx}}=1$, see the comments in
Section 3 regarding the quality of approximation; increase the value of
maxm to attempt to avoid approximation.
8
Parallelism and Performance
g05zmc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
g05zmc 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.
None.
10
Example
This example calls
g05zmc to calculate the eigenvalues of the embedding matrix for
$8$ sample points of a random field characterized by the symmetric stable variogram:
where
${x}^{\prime}=\frac{x}{\ell}$, and
$\ell $ and
$\nu $ are parameters.
It should be noted that the symmetric stable variogram is one of the predefined variograms available in
g05znc. It is used here purely for illustrative purposes.
10.1
Program Text
10.2
Program Data
10.3
Program Results