G05 Chapter Contents
G05 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentG05ZPF

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

G05ZPF produces realisations 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

 SUBROUTINE G05ZPF ( NS, S, M, LAM, RHO, STATE, Z, IFAIL)
 INTEGER NS, S, M, STATE(*), IFAIL REAL (KIND=nag_wp) LAM(M), RHO, Z(NS,S)

## 3  Description

A one-dimensional random field $Z\left(x\right)$ in $ℝ$ is a function which is random at every point $x\in ℝ$, so $Z\left(x\right)$ is a random variable for each $x$. The random field has a mean function $\mu \left(x\right)=𝔼\left[Z\left(x\right)\right]$ and a symmetric non-negative definite covariance function $C\left(x,y\right)=𝔼\left[\left(Z\left(x\right)-\mu \left(x\right)\right)\left(Z\left(y\right)-\mu \left(y\right)\right)\right]$. $Z\left(x\right)$ is a Gaussian random field if for any choice of $n\in ℕ$ and ${x}_{1},\dots ,{x}_{n}\in ℝ$, the random vector ${\left[Z\left({x}_{1}\right),\dots ,Z\left({x}_{n}\right)\right]}^{\mathrm{T}}$ follows a multivariate Gaussian 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 $x\in ℝ$ and $C\left(x,y\right)=C\left(x+a,y+a\right)$ for all $x,y,a\in ℝ$ and hence we can express the covariance function $C\left(x,y\right)$ as a function $\gamma$ of one variable: $C\left(x,y\right)=\gamma \left(x-y\right)$. $\gamma$ 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 one-dimensional 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 $N$ gridpoints. The problem reduces to sampling a Gaussian 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 $M\ge 2\left(N-1\right)$, 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 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 ${R}^{*}\left(\mathbf{U}+i\mathbf{V}\right)$, 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 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 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 $S$ realisations 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.

## 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  Parameters

1:     NS – INTEGERInput
On entry: the number of sample points (grid points) to be generated in realisations 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:     S – INTEGERInput
On entry: the number of realisations of the random field to simulate.
Constraint: ${\mathbf{S}}\ge 1$.
3:     M – INTEGERInput
On entry: the size of the embedding matrix, as returned by G05ZMF and G05ZNF.
Constraint: ${\mathbf{M}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,2\left({\mathbf{NS}}-1\right)\right)$.
4:     LAM(M) – REAL (KIND=nag_wp) arrayInput
On entry: must contain the square roots of the eigenvalues of the embedding matrix, as returned by G05ZMF and G05ZNF.
Constraint: ${\mathbf{LAM}}\left(i\right)\ge 0,i=1,2,\dots ,{\mathbf{M}}$.
5:     RHO – REAL (KIND=nag_wp)Input
On entry: indicates the scaling of the covariance matrix, as returned by G05ZMF and G05ZNF.
Constraint: $0.0<{\mathbf{RHO}}\le 1.0$.
6:     STATE($*$) – 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:     Z(NS,S) – REAL (KIND=nag_wp) arrayOutput
On exit: contains the realisations of the random field. Each column of $Z$ contains one realisation of the random field, with ${\mathbf{Z}}\left(i,j\right)$, for $j=1,2,\dots ,{\mathbf{S}}$, corresponding to the gridpoint ${\mathbf{XX}}\left(i\right)$ as returned by G05ZMF or G05ZNF.
8:     IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction 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 parameter, 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 ${\mathbf{IFAIL}}={\mathbf{0}}$ or $-{\mathbf{1}}$, 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, ${\mathbf{NS}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{NS}}\ge 1$.
${\mathbf{IFAIL}}=2$
On entry, ${\mathbf{S}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{S}}\ge 1$.
${\mathbf{IFAIL}}=3$
On entry, ${\mathbf{M}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{NS}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{M}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,2×\left({\mathbf{NS}}-1\right)\right)$.
${\mathbf{IFAIL}}=4$
On entry, at least one element of LAM was negative.
Constraint: all elements of LAM must be non-negative.
${\mathbf{IFAIL}}=5$
On entry, ${\mathbf{RHO}}=⟨\mathit{\text{value}}⟩$.
Constraint: $0.0\le {\mathbf{RHO}}\le 1.0$.
${\mathbf{IFAIL}}=6$
On entry, STATE vector has been corrupted or not initialized.

## 7  Accuracy

Not applicable.

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.

## 9  Example

This example calls G05ZPF to generate $5$ realisations of a random field on $8$ sample points using eigenvalues calculated by G05ZNF for a symmetric stable variogram.

### 9.1  Program Text

Program Text (g05zpfe.f90)

### 9.2  Program Data

Program Data (g05zpfe.d)

### 9.3  Program Results

Program Results (g05zpfe.r)