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Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_rand_field_2d_predef_setup (g05zr)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_rand_field_2d_predef_setup (g05zr) 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 nag_rand_field_2d_generate (g05zs), which simulates the random field.

Syntax

[lam, xx, yy, m, approx, rho, icount, eig, ifail] = g05zr(ns, xmin, xmax, ymin, ymax, maxm, var, icov2, params, 'norm_p', norm_p, 'np', np, 'pad', pad, 'icorr', icorr)
[lam, xx, yy, m, approx, rho, icount, eig, ifail] = nag_rand_field_2d_predef_setup(ns, xmin, xmax, ymin, ymax, maxm, var, icov2, params, 'norm_p', norm_p, 'np', np, 'pad', pad, 'icorr', icorr)

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 functions nag_rand_field_2d_predef_setup (g05zr) and nag_rand_field_2d_generate (g05zs) 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:
nag_rand_field_2d_predef_setup (g05zr) 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.

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

Parameters

Compulsory Input Parameters

1:     ns2 int64int32nag_int array
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 – double scalar
The lower bound for the x-coordinate, for the region in which the random field is to be simulated.
Constraint: xmin<xmax.
3:     xmax – double scalar
The upper bound for the x-coordinate, for the region in which the random field is to be simulated.
Constraint: xmin<xmax.
4:     ymin – double scalar
The lower bound for the y-coordinate, for the region in which the random field is to be simulated.
Constraint: ymin<ymax.
5:     ymax – double scalar
The upper bound for the y-coordinate, for the region in which the random field is to be simulated.
Constraint: ymin<ymax.
6:     maxm2 int64int32nag_int array
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 – double scalar
The multiplicative factor σ2 of the variogram γx.
Constraint: var0.0.
8:     icov2 int64int32nag_int scalar
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:     paramsnp – double array
The parameters for the variogram as detailed in the description of icov2.
Constraint: see icov2 for a description of the individual parameter constraints.

Optional Input Parameters

1:     norm_p int64int32nag_int scalar
Default: norm_p=2
Determines which norm to use when calculating the variogram.
norm_p=1
The 1-norm is used, i.e., x,y=x+y.
norm_p=2
The 2-norm (Euclidean norm) is used, i.e., x,y= x2+y2.
Constraint: norm_p=1 or 2.
2:     np int64int32nag_int scalar
Default: the dimension of the array params.
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.
3:     pad int64int32nag_int scalar
Default: pad=1
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.
Constraint: pad=0 or 1.
4:     icorr int64int32nag_int scalar
Default: icorr=0
Determines which approximation to implement if required, as described in Description.
Constraint: icorr=0, 1 or 2.

Output Parameters

1:     lammaxm1×maxm2 – double array
Contains the square roots of the eigenvalues of the embedding matrix.
2:     xxns1 – double array
The points of the x-coordinates at which values of the random field will be output.
3:     yyns2 – double array
The points of the y-coordinates at which values of the random field will be output.
4:     m2 int64int32nag_int array
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.
5:     approx int64int32nag_int scalar
Indicates whether approximation was used.
approx=0
No approximation was used.
approx=1
Approximation was used.
6:     rho – double scalar
Indicates the scaling of the covariance matrix. rho=1.0 unless approximation was used with icorr=0 or 1.
7:     icount int64int32nag_int scalar
Indicates the number of negative eigenvalues in the embedding matrix which have had to be set to zero.
8:     eig3 – double array
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.
9:     ifail int64int32nag_int scalar
ifail=0 unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:
   ifail=1
Constraint: ns11, ns21.
   ifail=2
Constraint: xmin<xmax.
   ifail=4
Constraint: ymin<ymax.
   ifail=6
Constraint: the minimum calculated value for maxm are _,_.
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
Constraint: var0.0.
   ifail=8
Constraint: icov21 and icov212.
   ifail=9
Constraint: norm_p=1 or 2.
   ifail=10
Constraint: for icov2=_, np=_.
   ifail=11
Constraint: dependent on icov2, see documentation.
   ifail=12
Constraint: pad=0 or 1.
   ifail=13
Constraint: icorr=0, 1 or 2.
   ifail=-99
An unexpected error has been triggered by this routine. Please contact NAG.
   ifail=-399
Your licence key may have expired or may not have been installed correctly.
   ifail=-999
Dynamic memory allocation failed.

Accuracy

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

Further Comments

None.

Example

This example calls nag_rand_field_2d_predef_setup (g05zr) 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).
function g05zr_example


fprintf('g05zr example results\n\n');

% Use symmetric stable variogram
icov2 = int64(1);
params = [0.1; 0.15; 1.2];

% Random Field variance
var = 0.5;
% Domain endpoints
xmin = -1;
xmax =  1;
ymin = -0.5;
ymax =  0.5;
% Number of sample points in x and y
ns = [int64(5), 5];
% Maximum dimensions of circulant matrix
maxm = [int64(64), 64];
% Scaling factor, rhoo = 1.
icorr = int64(2);

% Get square roots of the eigenvalues of the embedding matrix
[lam, xx, yy, m, approx, rho, icount, eig, ifail] = ...
  g05zr( ...
         ns, xmin, xmax, ymin, ymax, maxm, var, ...
         icov2, params, 'icorr', icorr);

fprintf('Size of embedding matrix = %d\n\n', m(1)*m(2));

% Display approximation information if approximation used
if approx == 1
  fprintf('Approximation required\n\n');
  fprintf('rho = %10.5f\n', rho);
  fprintf('eig = %10.5f%10.5f%10.5f\n', eig(1:3));
  fprintf('icount = %d\n', icount);
else
  fprintf('Approximation not required\n\n');
end

% Display square roots of the eigenvalues of the embedding matrix
fprintf('Square roots of eigenvalues of embedding matrix:\n');
mlam = reshape(lam(1:m(1)*m(2)), m(1), m(2));
for i = 1:m(1)
  fprintf('%8.4f',mlam(i,:));
  fprintf('\n');
end

g05zr_plot;



function g05zr_plot

  icov2 = int64(4);
  params = [0.1; 0.1];
  var = 1;
  % Domain endpoints
  xmin = 0;
  xmax = 1;
  ymin = 0;
  ymax = 1;
  % Number of sample points in x and y
  ns = [int64(100), 100];
  % Maximum dimensions of circulant matrix
  maxm = [int64(4096), 4096];
  icorr = int64(0);

  % Get square roots of the eigenvalues of the embedding matrix
  [lam, xx, yy, m, approx, rho, icount, eig, ifail] = ...
  g05zr( ...
         ns, xmin, xmax, ymin, ymax, maxm, var, ...
         icov2, params);

  % Initialize state array
  genid = int64(1);
  subid = int64(1);
  seed  = [int64(14965)];
  [state, ifail] = g05kf( ...
                          genid, subid, seed);

  % Compute 2 random field realisations
  s = int64(2);
  [state, z, ifail] = g05zs( ...
                             ns, s, m, lam, rho, state);

  fig1 = figure;
  zz = reshape(z(:,1),[100,100]);
  contourf(xx,yy,zz);
  axis equal;
  title({'First realization of Random Field', ...
	 'exponential variogram, corr. length = 0.1'});

  fig2 = figure;
  zz = reshape(z(:,2),[100,100]);
  contourf(xx,yy,zz);
  axis equal;
  title({'Second realization of Random Field', ...
	 'exponential variogram, corr. length = 0.1'});

g05zr example results

Size of embedding matrix = 64

Approximation not required

Square roots of eigenvalues of embedding matrix:
  0.8966  0.8234  0.6810  0.5757  0.5391  0.5757  0.6810  0.8234
  0.8940  0.8217  0.6804  0.5756  0.5391  0.5756  0.6804  0.8217
  0.8877  0.8175  0.6792  0.5754  0.5391  0.5754  0.6792  0.8175
  0.8813  0.8133  0.6780  0.5751  0.5390  0.5751  0.6780  0.8133
  0.8787  0.8116  0.6774  0.5750  0.5390  0.5750  0.6774  0.8116
  0.8813  0.8133  0.6780  0.5751  0.5390  0.5751  0.6780  0.8133
  0.8877  0.8175  0.6792  0.5754  0.5391  0.5754  0.6792  0.8175
  0.8940  0.8217  0.6804  0.5756  0.5391  0.5756  0.6804  0.8217
g05zr_fig1.png
g05zr_fig2.png
The two plots shown below illustrate the random fields that can be generated by nag_rand_field_2d_generate (g05zs) using the eigenvalues calculated by nag_rand_field_2d_predef_setup (g05zr). 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.

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