NAG Library Routine Document

g05zrf (field_2d_predef_setup)

 Contents

    1  Purpose
    7  Accuracy

1
Purpose

g05zrf performs the setup required in order to simulate stationary Gaussian random fields in two dimensions, for a preset 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

Fortran Interface
Subroutine g05zrf ( ns, xmin, xmax, ymin, ymax, maxm, var, icov2, norm, np, params, pad, icorr, lam, xx, yy, m, approx, rho, icount, eig, ifail)
Integer, Intent (In):: ns(2), maxm(2), icov2, norm, np, pad, icorr
Integer, Intent (Inout):: ifail
Integer, Intent (Out):: m(2), approx, icount
Real (Kind=nag_wp), Intent (In):: xmin, xmax, ymin, ymax, var, params(np)
Real (Kind=nag_wp), Intent (Out):: lam(MAXM(1)*MAXM(2)), xx(NS(1)), yy(NS(2)), rho, eig(3)
C Header Interface
#include nagmk26.h
void  g05zrf_ (const Integer ns[], const double *xmin, const double *xmax, const double *ymin, const double *ymax, const Integer maxm[], const double *var, const Integer *icov2, const Integer *norm, const Integer *np, const double params[], const Integer *pad, const Integer *icorr, double lam[], double xx[], double yy[], Integer m[], Integer *approx, double *rho, Integer *icount, double eig[], Integer *ifail)

3
Description

A two-dimensional random field Zx in 2 is a function which is random at every point x2, so Zx is a random variable for each x. The random field has a mean function μx=𝔼Zx and a symmetric positive semidefinite covariance function Cx,y=𝔼Zx-μxZy-μy. Zx is a Gaussian random field if for any choice of n and x1,,xn2, the random vector Zx1,,ZxnT follows a multivariate Normal distribution, which would have a mean vector μ~ with entries μ~i=μxi and a covariance matrix C~ with entries C~ij=Cxi,xj. A Gaussian random field Zx is stationary if μx is constant for all x2 and Cx,y=Cx+a,y+a for all x,y,a2 and hence we can express the covariance function Cx,y as a function γ of one variable: Cx,y=γx-y. γ is known as a variogram (or more correctly, a semivariogram) and includes the multiplicative factor σ2 representing the variance such that γ0=σ2.
The routines g05zrf and g05zsf are used to simulate a two-dimensional stationary Gaussian random field, with mean function zero and variogram γx, over a domain xmin,xmax×ymin,ymax, using an equally spaced set of N1×N2 points; N1 points in the x-direction and N2 points in the y-direction. The problem reduces to sampling a Gaussian random vector X of size N1×N2, with mean vector zero and a symmetric covariance matrix A, which is an N2 by N2 block Toeplitz matrix with Toeplitz blocks of size N1 by N1. 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 M2 by M2 block circulant matrix with circulant blocks of size M1 by M1, where M12N1-1 and M22N2-1. B can now be factorized as B=WΛW*=R*R, where W is the two-dimensional Fourier matrix (W* is the complex conjugate of W), Λ is the diagonal matrix containing the eigenvalues of B and R=Λ12W*. 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 M1×M2, 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 Λ are non-negative (i.e., B is positive semidefinite), B is a covariance matrix for a random vector Y which has M2 blocks of size M1. Two samples of Y can now be simulated from the real and imaginary parts of R*U+iV, where U and V have elements from the standard Normal distribution. Since R*U+iV=WΛ12U+iV, this calculation can be done using a discrete Fourier transform of the vector Λ12U+iV. Two samples of the random vector X can now be recovered by taking the first N1 elements of the first N2 blocks of each sample of Y – because the original covariance matrix A is embedded in B, 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 Λ=Λ++Λ-, where Λ+ and Λ- contain the non-negative and negative eigenvalues of B respectively. Then B is replaced by ρB+ where B+=WΛ+W* and ρ0,1 is a scaling factor. The error ε in approximating the distribution of the random field is given by
ε= 1-ρ 2 traceΛ + ρ2 traceΛ- M .  
Three choices for ρ are available, and are determined by the input argument icorr:
g05zrf 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.

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 (1997) Algorithm AS 312: An Algorithm for Simulating Stationary Gaussian Random Fields Journal of the Royal Statistical Society, Series C (Applied Statistics) (Volume 46) 1 171–181

5
Arguments

1:     ns2 – Integer arrayInput
On entry: the number of sample points to use in each direction, with ns1 sample points in the x-direction, N1 and ns2 sample points in the y-direction, N2. The total number of sample points on the grid is therefore ns1×ns2.
Constraints:
  • ns11;
  • ns21.
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: xmin<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: xmin<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: ymin<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: ymin<ymax.
6:     maxm2 – Integer arrayInput
On entry: determines the maximum size of the circulant matrix to use – a maximum of maxm1 elements in the x-direction, and a maximum of maxm2 elements in the y-direction. The maximum size of the circulant matrix is thus maxm1×maxm2.
Constraint: maxmi 2 k , where k is the smallest integer satisfying 2 k 2 nsi-1 , for i=1,2.
7:     var – Real (Kind=nag_wp)Input
On entry: the multiplicative factor σ2 of the variogram γx.
Constraint: var0.0.
8:     icov2 – IntegerInput
On entry: determines which of the preset variograms to use. The choices are given below. Note that x = x1 , y2 , where 1 and 2 are correlation lengths in the x and y directions respectively and are parameters for most of the variograms, and σ2 is the variance specified by var.
icov2=1
Symmetric stable variogram
γx = σ2 exp - x ν ,  
where
  • 1=params1, 1>0,
  • 2=params2, 2>0,
  • ν=params3, 0<ν2.
icov2=2
Cauchy variogram
γx = σ2 1 + x 2 -ν ,  
where
  • 1=params1, 1>0,
  • 2=params2, 2>0,
  • ν=params3, ν>0.
icov2=3
Differential variogram with compact support
γx = σ2 1 + 8 x + 25 x 2 + 32 x 3 1 - x 8 , x<1 , 0 , x 1 ,  
where
  • 1=params1, 1>0,
  • 2=params2, 2>0.
icov2=4
Exponential variogram
γx = σ2 exp-x ,  
where
  • 1=params1, 1>0,
  • 2=params2, 2>0.
icov2=5
Gaussian variogram
γx = σ2 exp -x 2 ,  
where
  • 1=params1, 1>0,
  • 2=params2, 2>0.
icov2=6
Nugget variogram
γx = σ2, x=0, 0, x0.  
No parameters need be set for this value of icov2.
icov2=7
Spherical variogram
γx = σ2 1 - 1.5x + 0.5 x 3 , x < 1 , 0, x 1 ,  
where
  • 1=params1, 1>0,
  • 2=params2, 2>0.
icov2=8
Bessel variogram
γx = σ2 2ν Γ ν+1 Jν x x ν ,  
where
  • Jν(·) is the Bessel function of the first kind,
  • 1=params1, 1>0,
  • 2=params2, 2>0,
  • ν=params3, ν0.
icov2=9
Hole effect variogram
γx = σ2 sinx x ,  
where
  • 1=params1, 1>0,
  • 2=params2, 2>0.
icov2=10
Whittle-Matérn variogram
γx = σ2 21-ν x ν Kν x Γν ,  
where
  • Kν(·) is the modified Bessel function of the second kind,
  • 1=params1, 1>0,
  • 2=params2, 2>0,
  • ν=params3, ν>0.
icov2=11
Continuously parameterised variogram with compact support
γx = σ2 21-ν xν Kν x Γν 1+8x+25x2+32x31-x8, x<1, 0, x1,  
where
  • x′′ = x 1s1 , y 2s2 ,
  • Kν(·) is the modified Bessel function of the second kind,
  • 1=params1, 1>0,
  • 2=params2, 2>0,
  • s1=params3, s1>0,
  • s2=params4, s2>0,
  • ν=params5, ν>0.
icov2=12
Generalized hyperbolic distribution variogram
γx=σ2δ2+x2λ2δλKλκδKλκδ2+x212,  
where
  • Kλ(·) is the modified Bessel function of the second kind,
  • 1=params1, 1>0,
  • 2=params2, 2>0,
  • λ=params3, no constraint on λ,
  • δ=params4, δ>0,
  • κ=params5, κ>0.
Constraint: icov2=1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 or 12.
9:     norm – IntegerInput
On entry: determines which norm to use when calculating the variogram.
norm=1
The 1-norm is used, i.e., x,y=x+y.
norm=2
The 2-norm (Euclidean norm) is used, i.e., x,y= x2+y2.
Suggested value: norm=2.
Constraint: norm=1 or 2.
10:   np – IntegerInput
On entry: the number of parameters to be set. Different covariance functions need a different number of parameters.
icov2=6
np must be set to 0.
icov2=3, 4, 5, 7 or 9
np must be set to 2.
icov2=1, 2, 8 or 10
np must be set to 3.
icov2=11 or 12
np must be set to 5.
11:   paramsnp – Real (Kind=nag_wp) arrayInput
On entry: the parameters for the variogram as detailed in the description of icov2.
Constraint: see icov2 for a description of the individual parameter constraints.
12:   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.
pad=0
The embedding matrix is padded with zeros.
pad=1
The embedding matrix is padded with values of the variogram.
Suggested value: pad=1.
Constraint: pad=0 or 1.
13:   icorr – IntegerInput
On entry: determines which approximation to implement if required, as described in Section 3.
Suggested value: icorr=0.
Constraint: icorr=0, 1 or 2.
14:   lammaxm1×maxm2 – Real (Kind=nag_wp) arrayOutput
On exit: contains the square roots of the eigenvalues of the embedding matrix.
15:   xxns1 – Real (Kind=nag_wp) arrayOutput
On exit: the points of the x-coordinates at which values of the random field will be output.
16:   yyns2 – Real (Kind=nag_wp) arrayOutput
On exit: the points of the y-coordinates at which values of the random field will be output.
17:   m2 – Integer arrayOutput
On exit: m1 contains M1, the size of the circulant blocks and m2 contains M2, the number of blocks, resulting in a final square matrix of size M1×M2.
18:   approx – IntegerOutput
On exit: indicates whether approximation was used.
approx=0
No approximation was used.
approx=1
Approximation was used.
19:   rho – Real (Kind=nag_wp)Output
On exit: indicates the scaling of the covariance matrix. rho=1.0 unless approximation was used with icorr=0 or 1.
20:   icount – IntegerOutput
On exit: indicates the number of negative eigenvalues in the embedding matrix which have had to be set to zero.
21:   eig3 – 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. eig1 contains the smallest eigenvalue, eig2 contains the sum of the squares of the negative eigenvalues, and eig3 contains the sum of the absolute values of the negative eigenvalues.
22:   ifail – IntegerInput/Output
On entry: ifail must be set to 0, -1​ 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​ 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 -1​ or ​1 is used it is essential to test the value of ifail on exit.
On exit: ifail=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6
Error Indicators and Warnings

If on entry 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:
ifail=1
On entry, ns=value,value.
Constraint: ns11, ns21.
ifail=2
On entry, xmin=value and xmax=value.
Constraint: xmin<xmax.
ifail=4
On entry, ymin=value and ymax=value.
Constraint: ymin<ymax.
ifail=6
On entry, maxm=value,value.
Constraint: the minimum calculated value for maxm are value,value.
Where the minima of maxmi is given by 2 k , where k is the smallest integer satisfying 2 k 2 nsi-1 , for i=1,2.
ifail=7
On entry, var=value.
Constraint: var0.0.
ifail=8
On entry, icov2=value.
Constraint: icov21 and icov212.
ifail=9
On entry, norm=value.
Constraint: norm=1 or 2.
ifail=10
On entry, np=value.
Constraint: for icov2=value, np=value.
ifail=11
On entry, paramsvalue=value.
Constraint: dependent on icov2, see documentation.
ifail=12
On entry, pad=value.
Constraint: pad=0 or 1.
ifail=13
On entry, icorr=value.
Constraint: icorr=0, 1 or 2.
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.
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.
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

If on exit approx=1, see the comments in Section 3 regarding the quality of approximation; increase the values in maxm to attempt to avoid approximation.

8
Parallelism and Performance

g05zrf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
g05zrf 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.

9
Further Comments

None.

10
Example

This example calls g05zrf 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 (icov2=1).

10.1
Program Text

Program Text (g05zrfe.f90)

10.2
Program Data

Program Data (g05zrfe.d)

10.3
Program Results

Program Results (g05zrfe.r)

The two plots shown below illustrate the random fields that can be generated by g05zsf using the eigenvalues calculated by g05zrf. These are for two realizations of a two-dimensional random field, based on eigenvalues of the embedding matrix for points on a 100 by 100 grid. The random field is characterized by the exponential variogram (icov2=4) with correlation lengths both equal to 0.1.
GnuplotProduced by GNUPLOT 4.6 patchlevel 3 Example Program 1 First realization of two-dimensional Random Field exponential variogram, correlation lengths = 0.1 gnuplot_plot_1