G05 Chapter Contents
G05 Chapter Introduction
NAG Library Manual

NAG Library Routine DocumentG05ZQF

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

G05ZQF performs the setup required in order to simulate stationary Gaussian random fields in two dimensions, for a user-defined 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 G05ZSF, which simulates the random field.

2  Specification

 SUBROUTINE G05ZQF ( NS, XMIN, XMAX, YMIN, YMAX, MAXM, VAR, COV2, EVEN, PAD, ICORR, LAM, XX, YY, M, APPROX, RHO, ICOUNT, EIG, IUSER, RUSER, IFAIL)
 INTEGER NS(2), MAXM(2), EVEN, PAD, ICORR, M(2), APPROX, ICOUNT, IUSER(*), IFAIL REAL (KIND=nag_wp) XMIN, XMAX, YMIN, YMAX, VAR, LAM(MAXM(1)*MAXM(2)), XX(NS(1)), YY(NS(2)), RHO, EIG(3), RUSER(*) EXTERNAL COV2

3  Description

A two-dimensional random field $Z\left(\mathbf{x}\right)$ in ${ℝ}^{2}$ is a function which is random at every point $\mathbf{x}\in {ℝ}^{2}$, so $Z\left(\mathbf{x}\right)$ is a random variable for each $\mathbf{x}$. The random field has a mean function $\mu \left(\mathbf{x}\right)=𝔼\left[Z\left(\mathbf{x}\right)\right]$ and a symmetric positive semidefinite covariance function $C\left(\mathbf{x},\mathbf{y}\right)=𝔼\left[\left(Z\left(\mathbf{x}\right)-\mu \left(\mathbf{x}\right)\right)\left(Z\left(\mathbf{y}\right)-\mu \left(\mathbf{y}\right)\right)\right]$. $Z\left(\mathbf{x}\right)$ is a Gaussian random field if for any choice of $n\in ℕ$ and ${\mathbf{x}}_{1},\dots ,{\mathbf{x}}_{n}\in {ℝ}^{2}$, the random vector ${\left[Z\left({\mathbf{x}}_{1}\right),\dots ,Z\left({\mathbf{x}}_{n}\right)\right]}^{\mathrm{T}}$ follows a multivariate Gaussian distribution, which would have a mean vector $\stackrel{~}{\mathbf{\mu }}$ with entries ${\stackrel{~}{\mu }}_{i}=\mu \left({\mathbf{x}}_{i}\right)$ and a covariance matrix $\stackrel{~}{C}$ with entries ${\stackrel{~}{C}}_{ij}=C\left({\mathbf{x}}_{i},{\mathbf{x}}_{j}\right)$. A Gaussian random field $Z\left(\mathbf{x}\right)$ is stationary if $\mu \left(\mathbf{x}\right)$ is constant for all $\mathbf{x}\in {ℝ}^{2}$ and $C\left(\mathbf{x},\mathbf{y}\right)=C\left(\mathbf{x}+\mathbf{a},\mathbf{y}+\mathbf{a}\right)$ for all $\mathbf{x},\mathbf{y},\mathbf{a}\in {ℝ}^{2}$ and hence we can express the covariance function $C\left(\mathbf{x},\mathbf{y}\right)$ as a function $\gamma$ of one variable: $C\left(\mathbf{x},\mathbf{y}\right)=\gamma \left(\mathbf{x}-\mathbf{y}\right)$. $\gamma$ 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 G05ZQF and G05ZSF are used to simulate a two-dimensional stationary Gaussian random field, with mean function zero and variogram $\gamma \left(\mathbf{x}\right)$, over a domain $\left[{x}_{\mathrm{min}},{x}_{\mathrm{max}}\right]×\left[{y}_{\mathrm{min}},{y}_{\mathrm{max}}\right]$, using an equally spaced set of ${N}_{1}×{N}_{2}$ gridpoints; ${N}_{1}$ gridpoints in the $x$-direction and ${N}_{2}$ gridpoints in the $y$-direction. The problem reduces to sampling a Gaussian random vector $\mathbf{X}$ of size ${N}_{1}×{N}_{2}$, with mean vector zero and a symmetric covariance matrix $A$, which is an ${N}_{2}$ by ${N}_{2}$ block Toeplitz matrix with Toeplitz blocks of size ${N}_{1}$ by ${N}_{1}$. 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 matrix $B$, which is an ${M}_{2}$ by ${M}_{2}$ block circulant matrix with circulant blocks of size ${M}_{1}$ by ${M}_{1}$, where ${M}_{1}\ge 2\left({N}_{1}-1\right)$ and ${M}_{2}\ge 2\left({N}_{2}-1\right)$. $B$ can now be factorized as $B=W\Lambda {W}^{*}={R}^{*}R$, where $W$ is the two-dimensional 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 ${M}_{1}×{M}_{2}$, and so only the first row (or column) of $B$ is needed – the whole matrix does not need to be formed.
The symmetry of $A$ as a block matrix, and the symmetry of each block of $A$, depends on whether the variogram $\gamma$ is even or not. $\gamma$ is even in its first coordinate if $\gamma \left({\left[{-x}_{1},{x}_{2}\right]}^{\mathrm{T}}\right)=\gamma \left({\left[{x}_{1},{x}_{2}\right]}^{\mathrm{T}}\right)$, even in its second coordinate if $\gamma \left({\left[{x}_{1},{-x}_{2}\right]}^{\mathrm{T}}\right)=\gamma \left({\left[{x}_{1},{x}_{2}\right]}^{\mathrm{T}}\right)$, and even if it is even in both coordinates (in two dimensions it is impossible for $\gamma$ to be even in one coordinate and uneven in the other). If $\gamma$ is even then $A$ is a symmetric block matrix and has symmetric blocks; if $\gamma$ is uneven then $A$ is not a symmetric block matrix and has non-symmetric blocks. In the uneven case, ${M}_{1}$ and ${M}_{2}$ are set to be odd in order to guarantee symmetry in $B$.
As long as all of the values of $\Lambda$ are non-negative (i.e., $B$ is positive semidefinite), $B$ is a covariance matrix for a random vector $\mathbf{Y}$ which has ${M}_{2}$ blocks of size ${M}_{1}$. Two samples of $\mathbf{Y}$ 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 ${N}_{1}$ elements of the first ${N}_{2}$ blocks 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 non-negative 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
 $ε= 1-ρ 2 trace⁡Λ + ρ2 trace⁡Λ- M .$
Three choices for $\rho$ are available, and are determined by the input parameter ICORR:
• setting ${\mathbf{ICORR}}=0$ sets
 $ρ= trace⁡Λ trace⁡Λ+ ,$
• setting ${\mathbf{ICORR}}=1$ sets
 $ρ= trace⁡Λ trace⁡Λ+ ,$
• setting ${\mathbf{ICORR}}=2$ sets $\rho =1$.
G05ZQF finds a suitable positive semidefinite embedding matrix $B$ and outputs its sizes in the vector 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.
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($2$) – INTEGER arrayInput
On entry: the number of sample points (gridpoints) to use in each direction, with ${\mathbf{NS}}\left(1\right)$ sample points in the $x$-direction, ${N}_{1}$ and ${\mathbf{NS}}\left(2\right)$ sample points in the $y$-direction, ${N}_{2}$. The total number of sample points on the grid is therefore ${\mathbf{NS}}\left(1\right)×{\mathbf{NS}}\left(2\right)$.
Constraints:
• ${\mathbf{NS}}\left(1\right)\ge 1$;
• ${\mathbf{NS}}\left(2\right)\ge 1$.
2:     XMIN – REAL (KIND=nag_wp)Input
On entry: the lower bound for the $x$-coordinate, for the region in which the random field is to be simulated.
Constraint: ${\mathbf{XMIN}}<{\mathbf{XMAX}}$.
3:     XMAX – REAL (KIND=nag_wp)Input
On entry: the upper bound for the $x$-coordinate, for the region in which the random field is to be simulated.
Constraint: ${\mathbf{XMIN}}<{\mathbf{XMAX}}$.
4:     YMIN – REAL (KIND=nag_wp)Input
On entry: the lower bound for the $y$-coordinate, for the region in which the random field is to be simulated.
Constraint: ${\mathbf{YMIN}}<{\mathbf{YMAX}}$.
5:     YMAX – REAL (KIND=nag_wp)Input
On entry: the upper bound for the $y$-coordinate, for the region in which the random field is to be simulated.
Constraint: ${\mathbf{YMIN}}<{\mathbf{YMAX}}$.
6:     MAXM($2$) – INTEGER arrayInput
On entry: determines the maximum size of the circulant matrix to use – a maximum of ${\mathbf{MAXM}}\left(1\right)$ elements in the $x$-direction, and a maximum of ${\mathbf{MAXM}}\left(2\right)$ elements in the $y$-direction. The maximum size of the circulant matrix is thus ${\mathbf{MAXM}}\left(1\right)$$×$${\mathbf{MAXM}}\left(2\right)$.
Constraints:
• if ${\mathbf{EVEN}}=1$, ${\mathbf{MAXM}}\left(i\right)\ge {2}^{k}$, where $k$ is the smallest integer satisfying ${2}^{k}\ge 2\left({\mathbf{NS}}\left(i\right)-1\right)$, for $i=1,2$ ;
• if ${\mathbf{EVEN}}=0$, ${\mathbf{MAXM}}\left(i\right)\ge {3}^{k}$, where $k$ is the smallest integer satisfying ${3}^{k}\ge 2\left({\mathbf{NS}}\left(i\right)-1\right)$, for $i=1,2$ .
7:     VAR – REAL (KIND=nag_wp)Input
On entry: the multiplicative factor ${\sigma }^{2}$ of the variogram $\gamma \left(\mathbf{x}\right)$.
Constraint: ${\mathbf{VAR}}\ge 0.0$.
8:     COV2 – SUBROUTINE, supplied by the user.External Procedure
COV2 must evaluate the variogram $\gamma \left(\mathbf{x}\right)$ for all $\mathbf{x}$ if ${\mathbf{EVEN}}=0$, and for all $\mathbf{x}$ with non-negative entries if ${\mathbf{EVEN}}=1$. The value returned in GAMMA is multiplied internally by VAR.
The specification of COV2 is:
 SUBROUTINE COV2 ( X, Y, GAMMA, IUSER, RUSER)
 INTEGER IUSER(*) REAL (KIND=nag_wp) X, Y, GAMMA, RUSER(*)
1:     X – REAL (KIND=nag_wp)Input
On entry: the coordinate $x$ at which the variogram $\gamma \left(\mathbf{x}\right)$ is to be evaluated.
2:     Y – REAL (KIND=nag_wp)Input
On entry: the coordinate $y$ at which the variogram $\gamma \left(\mathbf{x}\right)$ is to be evaluated.
3:     GAMMA – REAL (KIND=nag_wp)Output
On exit: the value of the variogram $\gamma \left(\mathbf{x}\right)$.
4:     IUSER($*$) – INTEGER arrayUser Workspace
5:     RUSER($*$) – REAL (KIND=nag_wp) arrayUser Workspace
COV2 is called with the parameters IUSER and RUSER as supplied to G05ZQF. You are free to use the arrays IUSER and RUSER to supply information to COV2 as an alternative to using COMMON global variables.
COV2 must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which G05ZQF is called. Parameters denoted as Input must not be changed by this procedure.
9:     EVEN – INTEGERInput
On entry: indicates whether the covariance function supplied is even or uneven.
${\mathbf{EVEN}}=0$
The covariance function is uneven.
${\mathbf{EVEN}}=1$
The covariance function is even.
Constraint: ${\mathbf{EVEN}}=0$ or $1$.
10:   PAD – INTEGERInput
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.
${\mathbf{PAD}}=0$
The embedding matrix is padded with zeros.
${\mathbf{PAD}}=1$
The embedding matrix is padded with values of the variogram.
Suggested value: ${\mathbf{PAD}}=1\text{.}$
Constraint: ${\mathbf{PAD}}=0$ or $1$.
11:   ICORR – INTEGERInput
On entry: determines which approximation to implement if required, as described in Section 3.
Suggested value: ${\mathbf{ICORR}}=0$.
Constraint: ${\mathbf{ICORR}}=0$, $1$ or $2$.
12:   LAM(${\mathbf{MAXM}}\left(1\right)×{\mathbf{MAXM}}\left(2\right)$) – REAL (KIND=nag_wp) arrayOutput
On exit: contains the square roots of the eigenvalues of the embedding matrix.
13:   XX(${\mathbf{NS}}\left(1\right)$) – REAL (KIND=nag_wp) arrayOutput
On exit: the gridpoints of the $x$-coordinates at which values of the random field will be output.
14:   YY(${\mathbf{NS}}\left(2\right)$) – REAL (KIND=nag_wp) arrayOutput
On exit: the gridpoints of the $y$-coordinates at which values of the random field will be output.
15:   M($2$) – INTEGER arrayOutput
On exit: ${\mathbf{M}}\left(1\right)$ contains ${M}_{1}$, the size of the circulant blocks and ${\mathbf{M}}\left(2\right)$ contains ${M}_{2}$, the number of blocks, resulting in a final square matrix of size ${M}_{1}×{M}_{2}$.
16:   APPROX – INTEGEROutput
On exit: indicates whether approximation was used.
${\mathbf{APPROX}}=0$
No approximation was used.
${\mathbf{APPROX}}=1$
Approximation was used.
17:   RHO – REAL (KIND=nag_wp)Output
On exit: indicates the scaling of the covariance matrix. ${\mathbf{RHO}}=1$ unless approximation was used with ${\mathbf{ICORR}}=0$ or $1$.
18:   ICOUNT – INTEGEROutput
On exit: indicates the number of negative eigenvalues in the embedding matrix which have had to be set to zero.
19:   EIG($3$) – REAL (KIND=nag_wp) arrayOutput
On exit: indicates information about the negative eigenvalues in the embedding matrix which have had to be set to zero. ${\mathbf{EIG}}\left(1\right)$ contains the smallest eigenvalue, ${\mathbf{EIG}}\left(2\right)$ contains the sum of the squares of the negative eigenvalues, and ${\mathbf{EIG}}\left(3\right)$ contains the sum of the absolute values of the negative eigenvalues.
20:   IUSER($*$) – INTEGER arrayUser Workspace
21:   RUSER($*$) – REAL (KIND=nag_wp) arrayUser Workspace
IUSER and RUSER are not used by G05ZQF, but are passed directly to COV2 and may be used to pass information to this routine as an alternative to using COMMON global variables.
22:   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, ${\mathbf{NS}}=\left[⟨\mathit{\text{value}}⟩,⟨\mathit{\text{value}}⟩\right]$.
Constraint: ${\mathbf{NS}}\left(1\right)\ge 1$, ${\mathbf{NS}}\left(2\right)\ge 1$.
${\mathbf{IFAIL}}=2$
On entry, ${\mathbf{XMIN}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{XMAX}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{XMIN}}<{\mathbf{XMAX}}$.
${\mathbf{IFAIL}}=4$
On entry, ${\mathbf{YMIN}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{YMAX}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{YMIN}}<{\mathbf{YMAX}}$.
${\mathbf{IFAIL}}=6$
On entry, ${\mathbf{MAXM}}=\left[⟨\mathit{\text{value}}⟩,⟨\mathit{\text{value}}⟩\right]$.
Constraint: the calculated minimum value for MAXM are $\left[⟨\mathit{\text{value}}⟩,⟨\mathit{\text{value}}⟩\right]$.
Where, if ${\mathbf{EVEN}}=1$, the minimum calculated value of ${\mathbf{MAXM}}\left(i\right)$ is given by ${2}^{k}$, where $k$ is the smallest integer satisfying ${2}^{k}\ge 2\left({\mathbf{NS}}\left(i\right)-1\right)$, and if ${\mathbf{EVEN}}=0$, the minimum calculated value of ${\mathbf{MAXM}}\left(i\right)$ is given by ${3}^{k}$, where $k$ is the smallest integer satisfying ${3}^{k}\ge 2{\mathbf{NS}}\left(i\right)-1$, for $i=1,2$.
${\mathbf{IFAIL}}=7$
On entry, ${\mathbf{VAR}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{VAR}}\ge 0.0$.
${\mathbf{IFAIL}}=9$
On entry, ${\mathbf{EVEN}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{EVEN}}=0$ or $1$.
${\mathbf{IFAIL}}=10$
On entry, ${\mathbf{PAD}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{PAD}}=0$ or $1$.
${\mathbf{IFAIL}}=11$
On entry, ${\mathbf{ICORR}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{ICORR}}=0$, $1$ or $2$.

Not applicable.

None.

9  Example

This example calls G05ZQF to calculate the eigenvalues of the embedding matrix for $25$ sample points on a $5$ by $5$ grid of a two-dimensional random field characterized by the symmetric stable variogram:
 $γx = σ2 exp - x′ ν ,$
where ${x}^{\prime }=\left|\frac{x}{{\ell }_{1}}+\frac{y}{{\ell }_{2}}\right|$, and ${\ell }_{1}$, ${\ell }_{2}$ and $\nu$ are parameters.
It should be noted that the symmetric stable variogram is one of the pre-defined variograms available in G05ZRF. It is used here purely for illustrative purposes.

9.1  Program Text

Program Text (g05zqfe.f90)

9.2  Program Data

Program Data (g05zqfe.d)

9.3  Program Results

Program Results (g05zqfe.r)