hide long namesshow long names
hide short namesshow short names
Integer type:  int32  int64  nag_int  show int32  show int32  show int64  show int64  show nag_int  show nag_int

PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_rand_field_1d_generate (g05zp)

Purpose

nag_rand_field_1d_generate (g05zp) 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 nag_rand_field_1d_user_setup (g05zm) or nag_rand_field_1d_predef_setup (g05zn).

Syntax

[state, z, ifail] = g05zp(ns, s, lam, rho, state, 'm', m)
[state, z, ifail] = nag_rand_field_1d_generate(ns, s, lam, rho, state, 'm', m)

Description

A one-dimensional random field Z(x)Z(x) in  is a function which is random at every point xx, 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 non-negative definite 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,,xnx1,,xn, 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 xx and C(x,y) = C(x + a,y + a)C(x,y)=C(x+a,y+a) for all x,y,ax,y,a 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_1d_user_setup (g05zm) or nag_rand_field_1d_predef_setup (g05zn), along with nag_rand_field_1d_generate (g05zp), are used to simulate a one-dimensional stationary Gaussian random field, with mean function zero and variogram γ(x)γ(x), over an interval [xmin,xmax][xmin,xmax], using an equally spaced set of NN gridpoints. The problem reduces to sampling a Gaussian random vector XX of size NN, with mean vector zero and a symmetric Toeplitz covariance matrix AA. 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 circulant matrix BB of size M2(N1)M2(N-1), which can now be factorized as B = WΛW* = R*RB=WΛW*=R*R, where WW is the 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 MM, and so only the first row (or column) of BB is needed – the whole matrix does not need to be formed.
As long as all of the values of ΛΛ are non-negative (i.e., BB is non-negative definite), BB is a covariance matrix for a random vector YY, two samples of which 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 NN elements 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 non-negative definite, 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. See the documentation of nag_rand_field_1d_user_setup (g05zm) or nag_rand_field_1d_predef_setup (g05zn) for details of the approximation procedure.
nag_rand_field_1d_generate (g05zp) takes the square roots of the eigenvalues of the embedding matrix BB, and its size MM, as input and outputs SS realisations of the random field in ZZ.
One of the initialization functions nag_rand_init_repeat (g05kf) (for a repeatable sequence if computed sequentially) or nag_rand_init_nonrepeat (g05kg) (for a non-repeatable sequence) must be called prior to the first call to nag_rand_field_1d_generate (g05zp).

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 – int64int32nag_int scalar
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 nag_rand_field_1d_user_setup (g05zm) or nag_rand_field_1d_predef_setup (g05zn) when calculating the eigenvalues of the embedding matrix.
Constraint: ns1ns1.
2:     s – int64int32nag_int scalar
The number of realisations of the random field to simulate.
Constraint: s1s1.
3:     lam(m) – double array
m, the dimension of the array, must satisfy the constraint mmax (1,2(ns1))mmax(1,2(ns-1)).
Must contain the square roots of the eigenvalues of the embedding matrix, as returned by nag_rand_field_1d_user_setup (g05zm) and nag_rand_field_1d_predef_setup (g05zn).
Constraint: lam(i)0,i = 1,2,,mlami0,i=1,2,,m.
4:     rho – double scalar
Indicates the scaling of the covariance matrix, as returned by nag_rand_field_1d_user_setup (g05zm) or nag_rand_field_1d_predef_setup (g05zn).
Constraint: 0.0 < rho1.00.0<rho1.0.
5:     state( : :) – int64int32nag_int array
Note: the actual argument supplied must be the array state supplied to the initialization routines nag_rand_init_repeat (g05kf) or nag_rand_init_nonrepeat (g05kg).
Contains information on the selected base generator and its current state.

Optional Input Parameters

1:     m – int64int32nag_int scalar
Default: The dimension of the array lam.
The size of the embedding matrix, as returned by nag_rand_field_1d_user_setup (g05zm) and nag_rand_field_1d_predef_setup (g05zn).
Constraint: mmax (1,2(ns1))mmax(1,2(ns-1)).

Input Parameters Omitted from the MATLAB Interface

None.

Output Parameters

1:     state( : :) – int64int32nag_int array
Note: the actual argument supplied must be the array state supplied to the initialization routines nag_rand_init_repeat (g05kf) or nag_rand_init_nonrepeat (g05kg).
Contains updated information on the state of the generator.
2:     z(ns,s) – double array
Contains the realisations of the random field. Each column of ZZ contains one realisation of the random field, with z(i,j)zij, for j = 1,2,,sj=1,2,,s, corresponding to the gridpoint xx(i)xxi as returned by nag_rand_field_1d_user_setup (g05zm) or nag_rand_field_1d_predef_setup (g05zn).
3:     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: ns1ns1.
  ifail = 2ifail=2
Constraint: s1s1.
  ifail = 3ifail=3
Constraint: mmax (1,2 × (ns1))mmax(1,2×(ns-1)).
  ifail = 4ifail=4
On entry, at least one element of lam was negative.
Constraint: all elements of lam must be non-negative.
  ifail = 5ifail=5
Constraint: 0.0rho1.00.0rho1.0.
  ifail = 6ifail=6
On entry, state vector has been corrupted or not initialized.

Accuracy

Not applicable.

Further Comments

Because samples are generated in pairs, calling this function kk times, with s = ss=s, say, will generate a different sequence of numbers than calling the function once with s = kss=ks, unless ss is even.

Example

function nag_rand_field_1d_generate_example
icov1 = int64(1);
params = [0.1; 1.2];
var = 0.5;
xmin = -1;
xmax = 1;
ns = int64(8);
icorr = int64(2);
s = int64(5);
% Get square roots of the eigenvalues of the embedding matrix
[lam, xx, m, approx, rho, icount, eig, ifail] = ...
    nag_rand_field_1d_predef_setup(ns, xmin, xmax, var, icov1, params, ...
                                   'icorr', icorr);

fprintf('\nSize of embedding matrix = %d\n\n', m);

% 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

state = initialize_state();

% Compute s random field realisations.
[state, z, ifail] = nag_rand_field_1d_generate(ns, s, lam(1:m), rho, state);

% Display realizations
fprintf('Random field realisations:\n');
fprintf('              1         2         3         4         5\n');
disp([xx, z]);


function state = initialize_state()
  genid = int64(1);
  subid = int64(1);
  seed  = [int64(14965)];
  [state, ifail] = nag_rand_init_repeat(genid, subid, seed)
 

Size of embedding matrix = 16

Approximation not required


state =

                   17
                 1234
                    1
                    0
                18046
                32309
                24932
                23785
                17917
                13895
                19930
                    8
                    0
                 1234
                    1
                    1
                 1234


ifail =

                    0

Random field realisations:
              1         2         3         4         5
   -0.8750   -0.4166   -0.8185   -0.9769    0.6741   -0.6762
   -0.6250    0.0146    1.4538    0.0248    0.5218    1.9466
   -0.3750   -0.5556    0.2913   -0.0853    0.4214   -0.1389
   -0.1250   -0.5568    0.3199   -0.6094    0.2019    0.9085
    0.1250   -0.0423    0.0486    1.4590    0.3608   -0.5288
    0.3750   -0.2806   -0.7969    0.2330    0.1335    0.4012
    0.6250    0.9298   -0.3956   -0.8455   -0.2749    0.5270
    0.8750    0.3222    1.5227   -2.1645    0.1794    1.1937


function g05zp_example
icov1 = int64(1);
params = [0.1; 1.2];
var = 0.5;
xmin = -1;
xmax = 1;
ns = int64(8);
icorr = int64(2);
s = int64(5);
% Get square roots of the eigenvalues of the embedding matrix
[lam, xx, m, approx, rho, icount, eig, ifail] = ...
    g05zn(ns, xmin, xmax, var, icov1, params, 'icorr', icorr);

fprintf('\nSize of embedding matrix = %d\n\n', m);

% 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

state = initialize_state();

% Compute s random field realisations.
[state, z, ifail] = g05zp(ns, s, lam(1:m), rho, state);

% Display realizations
fprintf('Random field realisations:\n');
fprintf('              1         2         3         4         5\n');
disp([xx, z]);


function state = initialize_state()
  genid = int64(1);
  subid = int64(1);
  seed  = [int64(14965)];
  [state, ifail] = g05kf(genid, subid, seed)
 

Size of embedding matrix = 16

Approximation not required


state =

                   17
                 1234
                    1
                    0
                18046
                32309
                24932
                23785
                17917
                13895
                19930
                    8
                    0
                 1234
                    1
                    1
                 1234


ifail =

                    0

Random field realisations:
              1         2         3         4         5
   -0.8750   -0.4166   -0.8185   -0.9769    0.6741   -0.6762
   -0.6250    0.0146    1.4538    0.0248    0.5218    1.9466
   -0.3750   -0.5556    0.2913   -0.0853    0.4214   -0.1389
   -0.1250   -0.5568    0.3199   -0.6094    0.2019    0.9085
    0.1250   -0.0423    0.0486    1.4590    0.3608   -0.5288
    0.3750   -0.2806   -0.7969    0.2330    0.1335    0.4012
    0.6250    0.9298   -0.3956   -0.8455   -0.2749    0.5270
    0.8750    0.3222    1.5227   -2.1645    0.1794    1.1937



PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

© The Numerical Algorithms Group Ltd, Oxford, UK. 2009–2013