g05zpc 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 g05zmcorg05znc.
The function may be called by the names: g05zpc or nag_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 functions g05zmcorg05znc, along with g05zpc, 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 g05zmcorg05znc for details of the approximation procedure.
g05zpc 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 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 g05zpc.
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 g05zmcorg05znc 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 g05zmcorg05znc.
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.
On exit: contains the realizations of the random field. The $j$th realization, for the ns sample points, is stored in ${\mathbf{z}}\left[\left(j-1\right)\times {\mathbf{ns}}+i-1\right]$, for $i=1,2,\dots ,{\mathbf{ns}}$. The sample points are as returned in ${\mathbf{xx}}$ by g05zmcorg05znc.
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, ${\mathbf{ns}}=\u27e8\mathit{\text{value}}\u27e9$. Constraint: ${\mathbf{ns}}\ge 1$.
On entry, ${\mathbf{s}}=\u27e8\mathit{\text{value}}\u27e9$. Constraint: ${\mathbf{s}}\ge 1$.
NE_INT_2
On entry, ${\mathbf{m}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{ns}}=\u27e8\mathit{\text{value}}\u27e9$. Constraint: ${\mathbf{m}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}(1,2\times ({\mathbf{ns}}-1))$.
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: $0.0\le {\mathbf{rho}}\le 1.0$.
7Accuracy
Not applicable.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
g05zpc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
g05zpc 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 function $k$ times, with ${\mathbf{s}}=s$, say, will generate a different sequence of numbers than calling the function once with ${\mathbf{s}}=ks$, unless $s$ is even.
10Example
This example calls g05zpc to generate $5$ realizations of a random field on $8$ sample points using eigenvalues calculated by g05znc for a symmetric stable variogram.