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Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_rand_field_2d_user_setup (g05zq)

Purpose

nag_rand_field_2d_user_setup (g05zq) 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 nag_rand_field_2d_generate (g05zs), which simulates the random field.

Syntax

[lam, xx, yy, m, approx, rho, icount, eig, user, ifail] = g05zq(ns, xmin, xmax, ymin, ymax, maxm, var, cov2, even, 'pad', pad, 'icorr', icorr, 'user', user)
[lam, xx, yy, m, approx, rho, icount, eig, user, ifail] = nag_rand_field_2d_user_setup(ns, xmin, xmax, ymin, ymax, maxm, var, cov2, even, 'pad', pad, 'icorr', icorr, 'user', user)

Description

A two-dimensional random field Z(x)Z(x) in 22 is a function which is random at every point x2x2, so Z(x)Z(x) is a random variable for each xx. The random field has a mean function μ(x) = 𝔼[Z(x)]μ(x)=𝔼[Z(x)] and a symmetric positive semidefinite covariance function C(x,y) = 𝔼[(Z(x)μ(x))(Z(y)μ(y))]C(x,y)=𝔼[(Z(x)-μ(x))(Z(y)-μ(y))]. Z(x)Z(x) is a Gaussian random field if for any choice of nn and x1,,xn2x1,,xn2, the random vector [Z(x1),,Z(xn)]T[Z(x1),,Z(xn)]T follows a multivariate Gaussian distribution, which would have a mean vector μ̃μ~ with entries μ̃i = μ(xi)μ~i=μ(xi) and a covariance matrix C~ with entries ij = C(xi,xj)C~ij=C(xi,xj). A Gaussian random field Z(x)Z(x) is stationary if μ(x)μ(x) is constant for all x2x2 and C(x,y) = C(x + a,y + a)C(x,y)=C(x+a,y+a) for all x,y,a2x,y,a2 and hence we can express the covariance function C(x,y)C(x,y) as a function γγ of one variable: C(x,y) = γ(xy)C(x,y)=γ(x-y). γγ is known as a variogram (or more correctly, a semivariogram) and includes the multiplicative factor σ2σ2 representing the variance such that γ(0) = σ2γ(0)=σ2.
The functions nag_rand_field_2d_user_setup (g05zq) and nag_rand_field_2d_generate (g05zs) are used to simulate a two-dimensional stationary Gaussian random field, with mean function zero and variogram γ(x)γ(x), over a domain [xmin,xmax] × [ymin,ymax][xmin,xmax]×[ymin,ymax], using an equally spaced set of N1 × N2N1×N2 gridpoints; N1N1 gridpoints in the xx-direction and N2N2 gridpoints in the yy-direction. The problem reduces to sampling a Gaussian random vector XX of size N1 × N2N1×N2, with mean vector zero and a symmetric covariance matrix AA, which is an N2N2 by N2N2 block Toeplitz matrix with Toeplitz blocks of size N1N1 by N1N1. Since AA is in general expensive to factorize, a technique known as the circulant embedding method is used. AA is embedded into a larger, symmetric matrix BB, which is an M2M2 by M2M2 block circulant matrix with circulant blocks of size M1M1 by M1M1, where M12(N11)M12(N1-1) and M22(N21)M22(N2-1). BB can now be factorized as B = WΛW* = R*RB=WΛW*=R*R, where WW is the two-dimensional Fourier matrix (W*W* is the complex conjugate of WW), ΛΛ is the diagonal matrix containing the eigenvalues of BB and R = Λ(1/2)W*R=Λ12W*. BB is known as the embedding matrix. The eigenvalues can be calculated by performing a discrete Fourier transform of the first row (or column) of BB and multiplying by M1 × M2M1×M2, and so only the first row (or column) of BB is needed – the whole matrix does not need to be formed.
The symmetry of AA as a block matrix, and the symmetry of each block of AA, depends on whether the variogram γγ is even or not. γγ is even in its first coordinate if γ([x1,x2]T) = γ([x1,x2]T)γ([-x1,x2]T)=γ([x1,x2]T), even in its second coordinate if γ([x1,x2]T) = γ([x1,x2]T)γ([x1,-x2]T)=γ([x1,x2]T), and even if it is even in both coordinates (in two dimensions it is impossible for γγ to be even in one coordinate and uneven in the other). If γγ is even then AA is a symmetric block matrix and has symmetric blocks; if γγ is uneven then AA is not a symmetric block matrix and has non-symmetric blocks. In the uneven case, M1M1 and M2M2 are set to be odd in order to guarantee symmetry in BB.
As long as all of the values of ΛΛ are non-negative (i.e., BB is positive semidefinite), BB is a covariance matrix for a random vector YY which has M2M2 blocks of size M1M1. Two samples of YY can now be simulated from the real and imaginary parts of R*(U + iV)R*(U+iV), where UU and VV have elements from the standard Normal distribution. Since R*(U + iV) = WΛ(1/2)(U + iV)R*(U+iV)=WΛ12(U+iV), this calculation can be done using a discrete Fourier transform of the vector Λ(1/2)(U + iV)Λ12(U+iV). Two samples of the random vector XX can now be recovered by taking the first N1N1 elements of the first N2N2 blocks of each sample of YY – because the original covariance matrix AA is embedded in BB, XX will have the correct distribution.
If BB is not positive semidefinite, larger embedding matrices BB 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+ + ΛΛ=Λ++Λ-, where Λ+Λ+ and ΛΛ- contain the non-negative and negative eigenvalues of BB respectively. Then BB is replaced by ρB+ρB+ where B+ = WΛ+W*B+=WΛ+W* and ρ(0,1]ρ(0,1] is a scaling factor. The error εε in approximating the distribution of the random field is given by
ε = sqrt( ( (1ρ)2 traceΛ + ρ2 traceΛ )/M ) .
ε= (1-ρ) 2 traceΛ + ρ2 traceΛ- M .
Three choices for ρρ are available, and are determined by the input parameter icorr:
nag_rand_field_2d_user_setup (g05zq) finds a suitable positive semidefinite embedding matrix BB 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 BB 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 (1994) Simulation of stationary Gaussian processes in [0,1]d[0,1]d Journal of Computational and Graphical Statistics 3(4) 409–432

Parameters

Compulsory Input Parameters

1:     ns(22) – int64int32nag_int array
The number of sample points (gridpoints) to use in each direction, with ns(1)ns1 sample points in the xx-direction, N1N1 and ns(2)ns2 sample points in the yy-direction, N2N2. The total number of sample points on the grid is therefore ns(1) × ns(2)ns1×ns2.
Constraints:
  • ns(1)1ns11;
  • ns(2)1ns21.
2:     xmin – double scalar
The lower bound for the xx-coordinate, for the region in which the random field is to be simulated.
Constraint: xmin < xmaxxmin<xmax.
3:     xmax – double scalar
The upper bound for the xx-coordinate, for the region in which the random field is to be simulated.
Constraint: xmin < xmaxxmin<xmax.
4:     ymin – double scalar
The lower bound for the yy-coordinate, for the region in which the random field is to be simulated.
Constraint: ymin < ymaxymin<ymax.
5:     ymax – double scalar
The upper bound for the yy-coordinate, for the region in which the random field is to be simulated.
Constraint: ymin < ymaxymin<ymax.
6:     maxm(22) – int64int32nag_int array
Determines the maximum size of the circulant matrix to use – a maximum of maxm(1)maxm1 elements in the xx-direction, and a maximum of maxm(2)maxm2 elements in the yy-direction. The maximum size of the circulant matrix is thus maxm(1)maxm1 × ×maxm(2)maxm2.
Constraints:
  • if even = 1even=1, maxm(i) 2k maxmi 2 k , where kk is the smallest integer satisfying 2k 2 (ns(i)1) 2 k 2 (nsi-1) , for i = 1,2i=1,2 ;
  • if even = 0even=0, maxm(i) 3k maxmi 3 k , where kk is the smallest integer satisfying 3k 2 (ns(i)1) 3 k 2 ( nsi-1 ) , for i = 1,2i=1,2 .
7:     var – double scalar
The multiplicative factor σ2σ2 of the variogram γ(x)γ(x).
Constraint: var0.0var0.0.
8:     cov2 – function handle or string containing name of m-file
cov2 must evaluate the variogram γ(x)γ(x) for all xx if even = 0even=0, and for all xx with non-negative entries if even = 1even=1. The value returned in gamma is multiplied internally by var.
[gamma, user] = cov2(x, y, user)

Input Parameters

1:     x – double scalar
The coordinate xx at which the variogram γ(x)γ(x) is to be evaluated.
2:     y – double scalar
The coordinate yy at which the variogram γ(x)γ(x) is to be evaluated.
3:     user – Any MATLAB object
cov2 is called from nag_rand_field_2d_user_setup (g05zq) with the object supplied to nag_rand_field_2d_user_setup (g05zq).

Output Parameters

1:     gamma – double scalar
The value of the variogram γ(x)γ(x).
2:     user – Any MATLAB object
9:     even – int64int32nag_int scalar
Indicates whether the covariance function supplied is even or uneven.
even = 0even=0
The covariance function is uneven.
even = 1even=1
The covariance function is even.
Constraint: even = 0even=0 or 11.

Optional Input Parameters

1:     pad – int64int32nag_int scalar
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 = 0pad=0
The embedding matrix is padded with zeros.
pad = 1pad=1
The embedding matrix is padded with values of the variogram.
Default: pad = 1pad=1
Constraint: pad = 0pad=0 or 11.
2:     icorr – int64int32nag_int scalar
Determines which approximation to implement if required, as described in Section [Description].
Default: icorr = 0icorr=0
Constraint: icorr = 0icorr=0, 11 or 22.
3:     user – Any MATLAB object
user is not used by nag_rand_field_2d_user_setup (g05zq), but is passed to cov2. Note that for large objects it may be more efficient to use a global variable which is accessible from the m-files than to use user.

Input Parameters Omitted from the MATLAB Interface

iuser ruser

Output Parameters

1:     lam(maxm(1) × maxm(2)maxm1×maxm2) – double array
Contains the square roots of the eigenvalues of the embedding matrix.
2:     xx(ns(1)ns1) – double array
The gridpoints of the xx-coordinates at which values of the random field will be output.
3:     yy(ns(2)ns2) – double array
The gridpoints of the yy-coordinates at which values of the random field will be output.
4:     m(22) – int64int32nag_int array
m(1)m1 contains M1M1, the size of the circulant blocks and m(2)m2 contains M2M2, the number of blocks, resulting in a final square matrix of size M1 × M2M1×M2.
5:     approx – int64int32nag_int scalar
Indicates whether approximation was used.
approx = 0approx=0
No approximation was used.
approx = 1approx=1
Approximation was used.
6:     rho – double scalar
Indicates the scaling of the covariance matrix. rho = 1.0rho=1.0 unless approximation was used with icorr = 0icorr=0 or 11.
7:     icount – int64int32nag_int scalar
Indicates the number of negative eigenvalues in the embedding matrix which have had to be set to zero.
8:     eig(33) – double array
Indicates information about the negative eigenvalues in the embedding matrix which have had to be set to zero. eig(1)eig1 contains the smallest eigenvalue, eig(2)eig2 contains the sum of the squares of the negative eigenvalues, and eig(3)eig3 contains the sum of the absolute values of the negative eigenvalues.
9:     user – Any MATLAB object
10:   ifail – int64int32nag_int scalar
ifail = 0ifail=0 unless the function detects an error (see [Error Indicators and Warnings]).

Error Indicators and Warnings

Errors or warnings detected by the function:
  ifail = 1ifail=1
Constraint: ns(1)1ns11, ns(2)1ns21.
  ifail = 2ifail=2
Constraint: xmin < xmaxxmin<xmax.
  ifail = 4ifail=4
Constraint: ymin < ymaxymin<ymax.
  ifail = 6ifail=6
Constraint: the calculated minimum value for maxm are [_,_][_,_].
Where, if even = 1even=1, the minimum calculated value of maxm(i)maxmi is given by 2k 2 k , where kk is the smallest integer satisfying 2k 2 (ns(i)1) 2 k 2 (nsi-1) , and if even = 0even=0, the minimum calculated value of maxm(i)maxmi is given by 3k 3 k , where kk is the smallest integer satisfying 3k 2 ns(i)1 3 k 2 nsi-1 , for i = 1,2i=1,2.
  ifail = 7ifail=7
Constraint: var0.0var0.0.
  ifail = 9ifail=9
Constraint: even = 0even=0 or 11.
  ifail = 10ifail=10
Constraint: pad = 0pad=0 or 11.
  ifail = 11ifail=11
Constraint: icorr = 0icorr=0, 11 or 22.

Accuracy

Not applicable.

Further Comments

None.

Example

function nag_rand_field_2d_user_setup_example
norm_p = int64(2);
l1 = 0.1;
l2 = 0.15;
nu = 1.2;
var = 0.5;
xmin = -1;
xmax = 1;
ymin = -0.5;
ymax = 0.5;
even = int64(1);
ns = [int64(5), 5];
maxm = [int64(81), 81];
icorr = int64(2);
% Put covariance parameters in user
user = {norm_p; l1; l2; nu};

% Get square roots of the eigenvalues of the embedding matrix
[lam, xx, yy, m, approx, rho, icount, eig, user, ifail] = ...
      nag_rand_field_2d_user_setup(ns, xmin, xmax, ymin, ymax, maxm, var, ...
            @cov2, even, 'icorr', icorr, 'user', user);

fprintf('\nSize 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');
reshape(lam(1:64), m(1), m(2))


function [gam, user] = cov2(x, y, user)
  norm_p = user{1};
  l1     = user{2};
  l2     = user{3};
  nu     = user{4};

  tl1 = abs(x)/l1;
  tl2 = abs(y)/l2;

  if norm_p == 1
    rnorm = tl1 +  tl2;
  else
    rnorm = sqrt(tl1^2+tl2^2);
  end

  gam = exp(-(rnorm^nu));
 

Size of embedding matrix = 64

Approximation not required

Square roots of eigenvalues of embedding matrix:

ans =

    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


function g05zq_example
norm_p = int64(2);
l1 = 0.1;
l2 = 0.15;
nu = 1.2;
var = 0.5;
xmin = -1;
xmax = 1;
ymin = -0.5;
ymax = 0.5;
even = int64(1);
ns = [int64(5), 5];
maxm = [int64(81), 81];
icorr = int64(2);
% Put covariance parameters in user
user = {norm_p; l1; l2; nu};

% Get square roots of the eigenvalues of the embedding matrix
[lam, xx, yy, m, approx, rho, icount, eig, user, ifail] = ...
      g05zq(ns, xmin, xmax, ymin, ymax, maxm, var, ...
            @cov2, even, 'icorr', icorr, 'user', user);

fprintf('\nSize 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');
reshape(lam(1:64), m(1), m(2))


function [gam, user] = cov2(x, y, user)
  norm_p = user{1};
  l1     = user{2};
  l2     = user{3};
  nu     = user{4};

  tl1 = abs(x)/l1;
  tl2 = abs(y)/l2;

  if norm_p == 1
    rnorm = tl1 +  tl2;
  else
    rnorm = sqrt(tl1^2+tl2^2);
  end

  gam = exp(-(rnorm^nu));
 

Size of embedding matrix = 64

Approximation not required

Square roots of eigenvalues of embedding matrix:

ans =

    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



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