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 g05zmforg05znf.
The routine may be called by the names g05zpf or nagf_rand_field_1d_generate.
3Description
A one-dimensional 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 non-negative definite covariance function $C(x,y)=\mathbb{E}\left[(Z\left(x\right)-\mu \left(x\right))(Z\left(y\right)-\mu \left(y\right))\right]$. $Z\left(x\right)$ 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 ${[Z\left({x}_{1}\right),\dots ,Z\left({x}_{n}\right)]}^{\mathrm{T}}$ follows a multivariate Normal distribution, which would have a mean vector $\stackrel{~}{\mathbf{\mu}}$ with entries ${\stackrel{~}{\mu}}_{i}=\mu \left({x}_{i}\right)$ and a covariance matrix $\stackrel{~}{C}$ with entries ${\stackrel{~}{C}}_{ij}=C({x}_{i},{x}_{j})$. A Gaussian random field $Z\left(x\right)$ is stationary if $\mu \left(x\right)$ is constant for all $x\in \mathbb{R}$ and $C(x,y)=C(x+a,y+a)$ for all $x,y,a\in \mathbb{R}$ and hence we can express the covariance function $C(x,y)$ as a function $\gamma $ of one variable: $C(x,y)=\gamma (x-y)$. $\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 routines g05zmforg05znf, along with g05zpf, are used to simulate a one-dimensional stationary Gaussian random field, with mean function zero and variogram $\gamma \left(x\right)$, over an interval $[{x}_{\mathrm{min}},{x}_{\mathrm{max}}]$, 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 $M\ge 2(N-1)$, 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 non-negative (i.e., $B$ is non-negative 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 ${R}^{*}(\mathbf{U}+i\mathbf{V})$, 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 $N$ 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 non-negative 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 g05zmforg05znf 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 non-repeatable sequence) must be called prior to the first call to g05zpf.
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
5Arguments
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 g05zmforg05znf 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 g05zmforg05znf.
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: $\mathbf{z}({\mathbf{ns}},{\mathbf{s}})$ – 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 ${\mathbf{z}}(i,j)$, for $i=1,2,\dots ,{\mathbf{ns}}$. The sample points are as returned in ${\mathbf{xx}}$ by g05zmforg05znf.
8: $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $\mathrm{-1}$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $\mathrm{-1}$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $\mathrm{-1}$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ 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).
6Error Indicators and Warnings
If on entry ${\mathbf{ifail}}=0$ or $\mathrm{-1}$, 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, ${\mathbf{ns}}=\u27e8\mathit{\text{value}}\u27e9$. Constraint: ${\mathbf{ns}}\ge 1$.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{s}}=\u27e8\mathit{\text{value}}\u27e9$. Constraint: ${\mathbf{s}}\ge 1$.
${\mathbf{ifail}}=3$
On entry, ${\mathbf{m}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{ns}}=\u27e8\mathit{\text{value}}\u27e9$. Constraint: ${\mathbf{m}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}(1,2\times ({\mathbf{ns}}-1))$.
${\mathbf{ifail}}=4$
On entry, at least one element of lam was negative. Constraint: all elements of lam must be non-negative.
${\mathbf{ifail}}=5$
On entry, ${\mathbf{rho}}=\u27e8\mathit{\text{value}}\u27e9$. 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 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
7Accuracy
Not applicable.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
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 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 g05zpf to generate $5$ realizations of a random field on $8$ sample points using eigenvalues calculated by g05znf for a symmetric stable variogram.