NAG Library Routine Document
g05zpf (field_1d_generate)
1
Purpose
g05zpf produces realizations of a stationary Gaussian random field in one dimension, 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
g05zmf or
g05znf.
2
Specification
Fortran Interface
Integer, Intent (In)  ::  ns, s, m  Integer, Intent (Inout)  ::  state(*), ifail  Real (Kind=nag_wp), Intent (In)  ::  lam(m), rho  Real (Kind=nag_wp), Intent (Out)  ::  z(ns,s) 

C Header Interface
#include nagmk26.h
void 
g05zpf_ (const Integer *ns, const Integer *s, const Integer *m, const double lam[], const double *rho, Integer state[], double z[], Integer *ifail) 

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 nonnegative definite 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 routines
g05zmf or
g05znf, along with
g05zpf, 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. 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 nonnegative definite), $B$ 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 nonnegative definite, 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
g05zmf or
g05znf for details of the approximation procedure.
g05zpf takes the square roots of the eigenvalues of the embedding matrix $B$, and its size $M$, as input and outputs $S$ realizations of the random field in $Z$.
One of the initialization routines
g05kff (for a repeatable sequence if computed sequentially) or
g05kgf (for a nonrepeatable sequence) must be called prior to the first call to
g05zpf.
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: $\mathbf{ns}$ – IntegerInput

On entry: the number of sample points to be generated in realizations of the random field. This must be the same value as supplied to
g05zmf or
g05znf when calculating the eigenvalues of the embedding matrix.
Constraint:
${\mathbf{ns}}\ge 1$.
 2: $\mathbf{s}$ – IntegerInput

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

On entry:
$M$, the size of the embedding matrix, as returned by
g05zmf or
g05znf.
Constraint:
${\mathbf{m}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,2\left({\mathbf{ns}}1\right)\right)$.
 4: $\mathbf{lam}\left({\mathbf{m}}\right)$ – Real (Kind=nag_wp) arrayInput

On entry: must contain the square roots of the eigenvalues of the embedding matrix, as returned by
g05zmf or
g05znf.
Constraint:
${\mathbf{lam}}\left(i\right)\ge 0,i=1,2,\dots ,{\mathbf{m}}$.
 5: $\mathbf{rho}$ – Real (Kind=nag_wp)Input

On entry: indicates the scaling of the covariance matrix, as returned by
g05zmf or
g05znf.
Constraint:
$0.0<{\mathbf{rho}}\le 1.0$.
 6: $\mathbf{state}\left(*\right)$ – Integer arrayCommunication Array

Note: the actual argument supplied
must be the array
state supplied to the initialization routines
g05kff or
g05kgf.
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: $\mathbf{z}\left({\mathbf{ns}},{\mathbf{s}}\right)$ – Real (Kind=nag_wp) arrayOutput

On exit: contains the realizations of the random field. The
$j$th realization, for the
ns sample points, is stored in
${\mathbf{z}}\left(i,j\right)$, for
$i=1,2,\dots ,{\mathbf{ns}}$. The sample points are as returned in
${\mathbf{xx}}$ by
g05zmf or
g05znf.
 8: $\mathbf{ifail}$ – IntegerInput/Output

On entry:
ifail must be set to
$0$,
$1\text{ or}1$. If you are unfamiliar with this argument you should refer to
Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value
$1\text{ or}1$ is recommended. If the output of error messages is undesirable, then the value
$1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is
$0$.
When the value $\mathbf{1}\text{ or}\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit:
${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see
Section 6).
6
Error Indicators and Warnings
If on entry
${\mathbf{ifail}}=0$ or
$1$, explanatory error messages are output on the current error message unit (as defined by
x04aaf).
Errors or warnings detected by the routine:
 ${\mathbf{ifail}}=1$

On entry, ${\mathbf{ns}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{ns}}\ge 1$.
 ${\mathbf{ifail}}=2$

On entry, ${\mathbf{s}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{s}}\ge 1$.
 ${\mathbf{ifail}}=3$

On entry, ${\mathbf{m}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{ns}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{m}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,2\times \left({\mathbf{ns}}1\right)\right)$.
 ${\mathbf{ifail}}=4$

On entry, at least one element of
lam was negative.
Constraint: all elements of
lam must be nonnegative.
 ${\mathbf{ifail}}=5$

On entry, ${\mathbf{rho}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: $0.0\le {\mathbf{rho}}\le 1.0$.
 ${\mathbf{ifail}}=6$

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.
See
Section 3.9 in How to Use the NAG Library and its Documentation for further information.
 ${\mathbf{ifail}}=399$
Your licence key may have expired or may not have been installed correctly.
See
Section 3.8 in How to Use the NAG Library and its Documentation for further information.
 ${\mathbf{ifail}}=999$
Dynamic memory allocation failed.
See
Section 3.7 in How to Use the NAG Library and its Documentation for further information.
7
Accuracy
Not applicable.
8
Parallelism and Performance
g05zpf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
g05zpf 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 routine. 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
g05zpf to generate
$5$ realizations of a random field on
$8$ sample points using eigenvalues calculated by
g05znf for a symmetric stable variogram.
10.1
Program Text
Program Text (g05zpfe.f90)
10.2
Program Data
Program Data (g05zpfe.d)
10.3
Program Results
Program Results (g05zpfe.r)