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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_rand_field_1d_predef_setup (g05zn)

Purpose

nag_rand_field_1d_predef_setup (g05zn) performs the setup required in order to simulate stationary Gaussian random fields in one dimension, 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_1d_generate (g05zp), which simulates the random field.

Syntax

[lam, xx, m, approx, rho, icount, eig, ifail] = g05zn(ns, xmin, xmax, var, icov1, params, 'maxm', maxm, 'np', np, 'pad', pad, 'icorr', icorr)
[lam, xx, m, approx, rho, icount, eig, ifail] = nag_rand_field_1d_predef_setup(ns, xmin, xmax, var, icov1, params, 'maxm', maxm, 'np', np, 'pad', pad, 'icorr', icorr)

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 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,,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_predef_setup (g05zn) and 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 points. 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 positive semidefinite), 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 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_1d_predef_setup (g05zn) finds a suitable positive semidefinite embedding matrix BB and outputs its size, 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 (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:     ns – int64int32nag_int scalar
The number of sample points (points) to be generated in realisations of the random field.
Constraint: ns1ns1.
2:     xmin – double scalar
The lower bound for the interval over which the random field is to be simulated. Note that if icov1 is set to 1414 (for simulating fractional Brownian motion), xmin is not referenced and the lower bound for the interval is set to zero.
Constraint: if icov114icov114, xmin < xmaxxmin<xmax.
3:     xmax – double scalar
The upper bound for the interval over which the random field is to be simulated. Note that if icov1 is set to 1414 (for simulating fractional Brownian motion), the lower bound for the interval is set to zero and so xmax is required to be greater than zero.
Constraints:
4:     var – double scalar
The multiplicative factor σ2σ2 of the variogram γ(x)γ(x).
Constraint: var0.0var0.0.
5:     icov1 – int64int32nag_int scalar
Determines which of the preset variograms to use. The choices are given below. Note that x = (|x|)/x=|x|, where  is the correlation length and is a parameter for most of the variograms, and σ2σ2 is the variance specified by var.
icov1 = 1icov1=1
Symmetric stable variogram
γ(x) = σ2 exp((x)ν) ,
γ(x) = σ2 exp( - (x) ν ) ,
where
  • = params(1)=params1, > 0>0,
  • ν = params(2)ν=params2, 0ν20ν2.
icov1 = 2icov1=2
Cauchy variogram
γ(x) = σ2 (1 + (x)2)ν ,
γ(x) = σ2 ( 1+ (x) 2 ) -ν ,
where
  • = params(1)=params1, > 0>0,
  • ν = params(2)ν=params2, ν > 0ν>0.
icov1 = 3icov1=3
Differential variogram with compact support
γ(x) =
{ σ2(1 + 8x + 25(x)2 + 32(x)3)(1 − x)8, x < 1, 0, x ≥ 1,
γ(x) = { σ2(1+8x+25(x)2+32(x)3)(1-x)8, x<1, 0, x1,
where
  • = params(1)=params1, > 0>0.
icov1 = 4icov1=4
Exponential variogram
γ(x) = σ2exp(x),
γ(x)=σ2exp(-x),
where
  • = params(1)=params1, > 0>0.
icov1 = 5icov1=5
Gaussian variogram
γ(x) = σ2exp((x)2),
γ(x)=σ2exp(-(x)2),
where
  • = params(1)=params1, > 0>0.
icov1 = 6icov1=6
Nugget variogram
γ(x) =
{ σ2, x = 0, 0, x ≠ 0.
γ(x)={ σ2, x=0, 0, x0.
No parameters need be set for this value of icov1.
icov1 = 7icov1=7
Spherical variogram
γ(x) =
{ σ2(1 − 1.5x + 0.5(x)3), x < 1, 0, x ≥ 1,
γ(x)={ σ2(1-1.5x+0.5(x)3), x<1, 0, x1,
where
  • = params(1)=params1, > 0>0.
icov1 = 8icov1=8
Bessel variogram
γ(x) = σ2(2νΓ(ν + 1)Jν(x))/((x)ν),
γ(x)=σ22νΓ(ν+1)Jν(x)(x)ν,
where
  • Jν( · )Jν(·) is the Bessel function of the first kind,
  • = params(1)=params1, > 0>0,
  • ν = params(2)ν=params2, ν0.5ν-0.5.
icov1 = 9icov1=9
Hole effect variogram
γ(x) = σ2(sin(x))/(x),
γ(x)=σ2sin(x)x,
where
  • = params(1)=params1, > 0>0.
icov1 = 10icov1=10
Whittle-Matérn variogram
γ(x) = σ2(21ν(x)νKν(x))/(Γ(ν)),
γ(x)=σ221-ν(x)νKν(x)Γ(ν),
where
  • Kν( · )Kν(·) is the modified Bessel function of the second kind,
  • = params(1)=params1, > 0>0,
  • ν = params(2)ν=params2, ν > 0ν>0.
icov1 = 11icov1=11
Continuously parameterised variogram with compact support
γ(x) =
{ σ2(21 − ν(x)νKν(x))/(Γ(ν))(1 + 8x′′ + 25(x′′)2 + 32(x′′)3)(1 − x′′)8, x′′ < 1, 0, x′′ ≥ 1,
γ(x)={ σ221-ν(x)νKν(x)Γ(ν)(1+8x+25(x)2+32(x)3)(1-x)8, x<1, 0, x1,
where
  • x = (x)/s x = xs ,
  • Kν( · )Kν(·) is the modified Bessel function of the second kind,
  • = params(1)=params1, > 0>0,
  • s = params(2)s=params2, s > 0s>0 (second correlation length),
  • ν = params(3)ν=params3, ν > 0ν>0.
icov1 = 12icov1=12
Generalized hyperbolic distribution variogram
γ(x) = σ2((δ2 + (x)2)λ/2)/(δλKλ(κδ))Kλ(κ(δ2 + (x)2)(1/2)),
γ(x)=σ2(δ2+(x)2)λ2δλKλ(κδ)Kλ(κ(δ2+(x)2)12),
where
  • Kλ( · )Kλ(·) is the modified Bessel function of the second kind,
  • = params(1)=params1, > 0>0,
  • λ = params(2)λ=params2, no constraint on λλ
  • δ = params(3)δ=params3, δ > 0δ>0,
  • κ = params(4)κ=params4, κ > 0κ>0.
icov1 = 13icov1=13
Cosine variogram
γ(x) = σ2cos(x),
γ(x)=σ2cos(x),
where
  • = params(1)=params1, > 0>0.
icov1 = 14icov1=14
Used for simulating fractional Brownian motion BH(t)BH(t). Fractional Brownian motion itself is not a stationary Gaussian random field, but its increments (i) = BH(ti)BH(ti1)X~(i)=BH(ti)-BH(ti-1) can be simulated in the same way as a stationary random field. The variogram for the so-called ‘increment process’ is
C((ti),(tj)) = γ̃(x) = (δ2H)/2(|x/δ1|2H + |x/δ + 1|2H2|x/δ|2H),
C(X~(ti),X~(tj))=γ~(x)=δ2H2(|xδ-1|2H+|xδ+1|2H-2|xδ|2H),
where
  • x = tjtix=tj-ti,
  • H = params(1)H=params1, 0 < H < 10<H<1, HH is the Hurst parameter,
  • δ = params(2)δ=params2, δ > 0δ>0, normally δ = titi1δ=ti-ti-1 is the (fixed) stepsize.
We scale the increments to set γ(0) = 1γ(0)=1; let X(i) = ((i))/(δH)X(i)=X~(i)δ-H, then
C(X(ti),X(tj)) = γ(x) = (1/2) (|x/δ1|2H + |x/δ + 1|2H2|x/δ|2H) .
C(X(ti),X(tj)) = γ(x) = 12 ( | xδ - 1 | 2H + | xδ + 1 | 2H - 2 | xδ | 2H ) .
The increments X(i)X(i) can then be simulated using nag_rand_field_1d_generate (g05zp), then multiplied by δHδH to obtain the original increments (i)X~(i) for the fractional Brownian motion.
6:     params(np) – double array
The parameters set for the variogram.
Constraint: see icov1 for a description of the individual parameter constraints.

Optional Input Parameters

1:     maxm – int64int32nag_int scalar
The maximum size of the circulant matrix to use. For example, if the embedding matrix is to be allowed to double in size three times before the approximation procedure is used, then choose maxm = 2k + 2 maxm = 2k+2  where k = 1 + log2(ns1) k = 1+ log2(ns-1) .
Default: 23 + ceilinglog2(ns1)23+ceilinglog2(ns-1) 
Constraint: maxm 2k maxm 2 k , where kk is the smallest integer satisfying 2k 2 (ns1) 2 k 2 (ns-1) .
2:     np – int64int32nag_int scalar
Default: The dimension of the array params.
The number of parameters to be set. Different variograms need a different number of parameters.
icov1 = 6icov1=6
np must be set to 00.
icov1 = 3icov1=3, 44, 55, 77, 99 or 1313
np must be set to 11.
icov1 = 1icov1=1, 22, 88, 1010 or 1414
np must be set to 22.
icov1 = 11icov1=11
np must be set to 33.
icov1 = 12icov1=12
np must be set to 44.
3:     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.
4:     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.

Input Parameters Omitted from the MATLAB Interface

None.

Output Parameters

1:     lam(maxm) – double array
Contains the square roots of the eigenvalues of the embedding matrix.
2:     xx(ns) – double array
The points at which values of the random field will be output.
3:     m – int64int32nag_int scalar
The size of the embedding matrix.
4:     approx – int64int32nag_int scalar
Indicates whether approximation was used.
approx = 0approx=0
No approximation was used.
approx = 1approx=1
Approximation was used.
5:     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.
6:     icount – int64int32nag_int scalar
Indicates the number of negative eigenvalues in the embedding matrix which have had to be set to zero.
7:     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.
8:     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: xmin < xmaxxmin<xmax.
  ifail = 3ifail=3
Constraint: xmax > 0.0xmax>0.0.
  ifail = 4ifail=4
Constraint: the calculated minimum value for maxm is __.
Where the minimum calculated value is given by 2k 2 k , where kk is the smallest integer satisfying 2k 2 (ns1) 2 k 2 (ns-1) .
  ifail = 5ifail=5
Constraint: var0.0var0.0.
  ifail = 6ifail=6
Constraint: icov11icov11 and icov114icov114.
  ifail = 7ifail=7
Constraint: for icov1 = _icov1=_.
On entry, np is not the correct number of parameters for the specified variogram.
  ifail = 8ifail=8
Constraint: dependent on icov1, see documentation.
  ifail = 9ifail=9
Constraint: pad = 0pad=0 or 11.
  ifail = 10ifail=10
Constraint: icorr = 0icorr=0, 11 or 22.

Accuracy

Not applicable.

Further Comments

None.

Example

function nag_rand_field_1d_predef_setup_example
icov1 = int64(1);
params = [0.1; 1.2];
var = 0.5;
xmin = -1;
xmax = 1;
ns = int64(8);
icorr = int64(2);
% 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

% Display square roots of the eigenvalues of the embedding matrix
fprintf('Square roots of eigenvalues of embedding matrix:\n');
disp(lam(1:m));
 

Size of embedding matrix = 16

Approximation not required

Square roots of eigenvalues of embedding matrix:
    0.7421
    0.7393
    0.7315
    0.7199
    0.7064
    0.6930
    0.6818
    0.6744
    0.6718
    0.6744
    0.6818
    0.6930
    0.7064
    0.7199
    0.7315
    0.7393


function g05zn_example
icov1 = int64(1);
params = [0.1; 1.2];
var = 0.5;
xmin = -1;
xmax = 1;
ns = int64(8);
icorr = int64(2);
% 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

% Display square roots of the eigenvalues of the embedding matrix
fprintf('Square roots of eigenvalues of embedding matrix:\n');
disp(lam(1:m));
 

Size of embedding matrix = 16

Approximation not required

Square roots of eigenvalues of embedding matrix:
    0.7421
    0.7393
    0.7315
    0.7199
    0.7064
    0.6930
    0.6818
    0.6744
    0.6718
    0.6744
    0.6818
    0.6930
    0.7064
    0.7199
    0.7315
    0.7393



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Chapter Contents
Chapter Introduction
NAG Toolbox

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