NAG CL Interface
g05znc (field_​1d_​predef_​setup)

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1 Purpose

g05znc 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 g05zpc, which simulates the random field.

2 Specification

#include <nag.h>
void  g05znc (Integer ns, double xmin, double xmax, Integer maxm, double var, Nag_Variogram cov, Integer np, const double params[], Nag_EmbedPad pad, Nag_EmbedScale corr, double lam[], double xx[], Integer *m, Integer *approx, double *rho, Integer *icount, double eig[], NagError *fail)
The function may be called by the names: g05znc or nag_rand_field_1d_predef_setup.

3 Description

A one-dimensional random field Z(x) in is a function which is random at every point x, so Z(x) is a random variable for each x. The random field has a mean function μ(x)=𝔼[Z(x)] and a symmetric positive semidefinite covariance function C(x,y)=𝔼[(Z(x)-μ(x))(Z(y)-μ(y))]. Z(x) is a Gaussian random field if for any choice of n and x1,,xn, the random vector [Z(x1),,Z(xn)]T follows a multivariate Normal distribution, which would have a mean vector μ~ with entries μ~i=μ(xi) and a covariance matrix C~ with entries C~ij=C(xi,xj). A Gaussian random field Z(x) is stationary if μ(x) is constant for all x and C(x,y)=C(x+a,y+a) for all x,y,a and hence we can express the covariance function C(x,y) as a function γ of one variable: C(x,y)=γ(x-y). γ is known as a variogram (or more correctly, a semivariogram) and includes the multiplicative factor σ2 representing the variance such that γ(0)=σ2.
The functions g05znc and g05zpc are used to simulate a one-dimensional stationary Gaussian random field, with mean function zero and variogram γ(x), over an interval [xmin,xmax], using an equally spaced set of N points. The problem reduces to sampling a Normal random vector X of size N, with mean vector zero and a symmetric Toeplitz covariance matrix A. 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 circulant matrix B of size M2(N-1), which can now be factorized as B=WΛW*=R*R, where W is the 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 M, 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, two samples of which 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)=WΛ12(U+iV), this calculation can be done using a discrete Fourier transform of the vector Λ12(U+iV). Two samples of the random vector X can now be recovered by taking the first N elements 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 ρ(0,1] 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 corr:
g05znc finds a suitable positive semidefinite embedding matrix B 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 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: ns Integer Input
On entry: the number of sample points to be generated in realizations of the random field.
Constraint: ns1.
2: xmin double Input
On entry: the lower bound for the interval over which the random field is to be simulated. Note that if cov=Nag_VgmBrownian (for simulating fractional Brownian motion), xmin is not referenced and the lower bound for the interval is set to zero.
Constraint: if covNag_VgmBrownian, xmin<xmax.
3: xmax double Input
On entry: the upper bound for the interval over which the random field is to be simulated. Note that if cov=Nag_VgmBrownian (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.
  • if covNag_VgmBrownian, xmin<xmax;
  • if cov=Nag_VgmBrownian, xmax>0.0.
4: maxm Integer Input
On entry: 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 where k = 1+ log2(ns-1) .
Suggested value: 2k+2​ where ​ k = 1+ log2(ns-1) .
Constraint: maxm 2 k , where k is the smallest integer satisfying 2 k 2 (ns-1) .
5: var double Input
On entry: the multiplicative factor σ2 of the variogram γ(x).
Constraint: var0.0.
6: cov Nag_Variogram Input
On entry: determines which of the preset variograms to use. The choices are given below. Note that x=|x|, where is the correlation length and is a parameter for most of the variograms, and σ2 is the variance specified by var.
Symmetric stable variogram
γ(x) = σ2 exp(- (x) ν) ,  
  • =params[0], >0,
  • ν=params[1], 0ν2.
Cauchy variogram
γ(x) = σ2 (1+ (x) 2 ) -ν ,  
  • =params[0], >0,
  • ν=params[1], ν>0.
Differential variogram with compact support
γ(x) = { σ2(1+8x+25(x)2+32(x)3)(1-x)8, x<1, 0, x1,  
  • =params[0], >0.
Exponential variogram
  • =params[0], >0.
Gaussian variogram
  • =params[0], >0.
Nugget variogram
γ(x)={ σ2, x=0, 0, x0.  
No parameters need be set for this value of cov.
Spherical variogram
γ(x)={ σ2(1-1.5x+0.5(x)3), x<1, 0, x1,  
  • =params[0], >0.
Bessel variogram
  • Jν(·) is the Bessel function of the first kind,
  • =params[0], >0,
  • ν=params[1], ν-0.5.
Hole effect variogram
  • =params[0], >0.
Whittle-Matérn variogram
  • Kν(·) is the modified Bessel function of the second kind,
  • =params[0], >0,
  • ν=params[1], ν>0.
Continuously parameterised variogram with compact support
γ(x)={ σ221-ν(x)νKν(x)Γ(ν)(1+8x+25(x)2+32(x)3)(1-x)8, x<1, 0, x1,  
  • x = xs ,
  • Kν(·) is the modified Bessel function of the second kind,
  • =params[0], >0,
  • s=params[1], s>0 (second correlation length),
  • ν=params[2], ν>0.
Generalized hyperbolic distribution variogram
  • Kλ(·) is the modified Bessel function of the second kind,
  • =params[0], >0,
  • λ=params[1], no constraint on λ
  • δ=params[2], δ>0,
  • κ=params[3], κ>0.
Cosine variogram
  • =params[0], >0.
Used for simulating fractional Brownian motion BH(t). Fractional Brownian motion itself is not a stationary Gaussian random field, but its increments 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
  • x=tj-ti,
  • H=params[0], 0<H<1, H is the Hurst parameter,
  • δ=params[1], δ>0, normally δ=ti-ti-1 is the (fixed) step size.
We scale the increments to set γ(0)=1; let X(i)=X~(i)δ-H, then
C(X(ti),X(tj)) = γ(x) = 12 ( |xδ-1| 2H + |xδ+1| 2H -2 |xδ| 2H ) .  
The increments X(i) can then be simulated using g05zpc, then multiplied by δH to obtain the original increments X~(i) for the fractional Brownian motion.
Constraint: cov=Nag_VgmSymmStab, Nag_VgmCauchy, Nag_VgmDifferential, Nag_VgmExponential, Nag_VgmGauss, Nag_VgmNugget, Nag_VgmSpherical, Nag_VgmBessel, Nag_VgmHole, Nag_VgmWhittleMatern, Nag_VgmContParam, Nag_VgmGenHyp, Nag_VgmCosine or Nag_VgmBrownian.
7: np Integer Input
On entry: the number of parameters to be set. Different variograms need a different number of parameters.
np must be set to 0.
cov=Nag_VgmDifferential, Nag_VgmExponential, Nag_VgmGauss, Nag_VgmSpherical, Nag_VgmHole or Nag_VgmCosine
np must be set to 1.
cov=Nag_VgmSymmStab, Nag_VgmCauchy, Nag_VgmBessel, Nag_VgmWhittleMatern or Nag_VgmBrownian
np must be set to 2.
np must be set to 3.
np must be set to 4.
8: params[np] const double Input
On entry: the parameters set for the variogram.
Constraint: see cov for a description of the individual parameter constraints.
9: pad Nag_EmbedPad Input
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.
The embedding matrix is padded with zeros.
The embedding matrix is padded with values of the variogram.
Suggested value: pad=Nag_EmbedPadValues.
Constraint: pad=Nag_EmbedPadZeros or Nag_EmbedPadValues.
10: corr Nag_EmbedScale Input
On entry: determines which approximation to implement if required, as described in Section 3.
Suggested value: corr=Nag_EmbedScaleTraces.
Constraint: corr=Nag_EmbedScaleTraces, Nag_EmbedScaleSqrtTraces or Nag_EmbedScaleOne.
11: lam[maxm] double Output
On exit: contains the square roots of the eigenvalues of the embedding matrix.
12: xx[ns] double Output
On exit: the points at which values of the random field will be output.
13: m Integer * Output
On exit: the size of the embedding matrix.
14: approx Integer * Output
On exit: indicates whether approximation was used.
No approximation was used.
Approximation was used.
15: rho double * Output
On exit: indicates the scaling of the covariance matrix. rho=1.0 unless approximation was used with corr=Nag_EmbedScaleTraces or Nag_EmbedScaleSqrtTraces.
16: icount Integer * Output
On exit: indicates the number of negative eigenvalues in the embedding matrix which have had to be set to zero.
17: eig[3] double Output
On exit: indicates information about the negative eigenvalues in the embedding matrix which have had to be set to zero. eig[0] contains the smallest eigenvalue, eig[1] contains the sum of the squares of the negative eigenvalues, and eig[2] contains the sum of the absolute values of the negative eigenvalues.
18: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument value had an illegal value.
On entry, np=value.
Constraint: for cov=value, np=value.
On entry, cov=Nag_VgmBrownian and xmax=value.
Constraint: xmax>0.0.
On entry, params[value]=value.
Constraint: dependent on cov.
On entry, covNag_VgmBrownian, xmin=value and xmax=value.
Constraint: xmin<xmax.
On entry, maxm=value.
Constraint: the minimum calculated value for maxm is value.
Where the minimum calculated value is given by 2 k , where k is the smallest integer satisfying 2 k 2 (ns-1) .
On entry, ns=value.
Constraint: ns1.
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
On entry, var=value.
Constraint: var0.0.

7 Accuracy

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

8 Parallelism and Performance

g05znc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
g05znc 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 function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments


10 Example

This example calls g05znc to calculate the eigenvalues of the embedding matrix for 8 sample points of a random field characterized by the symmetric stable variogram (cov=Nag_VgmSymmStab).

10.1 Program Text

Program Text (g05znce.c)

10.2 Program Data

Program Data (g05znce.d)

10.3 Program Results

Program Results (g05znce.r)