nag_rand_field_2d_predef_setup (g05zrc) (PDF version)
g05 Chapter Contents
g05 Chapter Introduction
NAG Library Manual

NAG Library Function Document

nag_rand_field_2d_predef_setup (g05zrc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_rand_field_2d_predef_setup (g05zrc) 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 (g05zsc), which simulates the random field.

2  Specification

#include <nag.h>
#include <nagg05.h>
void  nag_rand_field_2d_predef_setup (const Integer ns[], double xmin, double xmax, double ymin, double ymax, const Integer maxm[], double var, Nag_Variogram cov, Nag_NormType norm, Integer np, const double params[], Nag_EmbedPad pad, Nag_EmbedScale corr, double lam[], double xx[], double yy[], Integer m[], Integer *approx, double *rho, Integer *icount, double eig[], NagError *fail)

3  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 (g05zrc) and nag_rand_field_2d_generate (g05zsc) 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 corr:
nag_rand_field_2d_predef_setup (g05zrc) 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.

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[2]const IntegerInput
On entry: the number of sample points to use in each direction, with ns[0] sample points in the x-direction, N1 and ns[1] sample points in the y-direction, N2. The total number of sample points on the grid is therefore ns[0] × ns[1] .
Constraints:
  • ns[0]1;
  • ns[1]1.
2:     xmindoubleInput
On entry: the lower bound for the x-coordinate, for the region in which the random field is to be simulated.
Constraint: xmin<xmax.
3:     xmaxdoubleInput
On entry: the upper bound for the x-coordinate, for the region in which the random field is to be simulated.
Constraint: xmin<xmax.
4:     ymindoubleInput
On entry: the lower bound for the y-coordinate, for the region in which the random field is to be simulated.
Constraint: ymin<ymax.
5:     ymaxdoubleInput
On entry: the upper bound for the y-coordinate, for the region in which the random field is to be simulated.
Constraint: ymin<ymax.
6:     maxm[2]const IntegerInput
On entry: determines the maximum size of the circulant matrix to use – a maximum of maxm[0] elements in the x-direction, and a maximum of maxm[1] elements in the y-direction. The maximum size of the circulant matrix is thus maxm[0]×maxm[1].
Constraint: maxm[i] 2 k , where k is the smallest integer satisfying 2 k 2 ns[i]-1 , for i=0,1.
7:     vardoubleInput
On entry: the multiplicative factor σ2 of the variogram γx.
Constraint: var0.0.
8:     covNag_VariogramInput
On entry: 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.
cov=Nag_VgmSymmStab
Symmetric stable variogram
γx = σ2 exp - x ν ,
where
  • 1=params[0], 1>0,
  • 2=params[1], 2>0,
  • ν=params[2], 0<ν2.
cov=Nag_VgmCauchy
Cauchy variogram
γx = σ2 1 + x 2 -ν ,
where
  • 1=params[0], 1>0,
  • 2=params[1], 2>0,
  • ν=params[2], ν>0.
cov=Nag_VgmDifferential
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=params[0], 1>0,
  • 2=params[1], 2>0.
cov=Nag_VgmExponential
Exponential variogram
γx = σ2 exp-x ,
where
  • 1=params[0], 1>0,
  • 2=params[1], 2>0.
cov=Nag_VgmGauss
Gaussian variogram
γx = σ2 exp -x 2 ,
where
  • 1=params[0], 1>0,
  • 2=params[1], 2>0.
cov=Nag_VgmNugget
Nugget variogram
γx = σ2, x=0, 0, x0.
No parameters need be set for this value of cov.
cov=Nag_VgmSpherical
Spherical variogram
γx = σ2 1 - 1.5x + 0.5 x 3 , x < 1 , 0, x 1 ,
where
  • 1=params[0], 1>0,
  • 2=params[1], 2>0.
cov=Nag_VgmBessel
Bessel variogram
γx = σ2 2ν Γ ν+1 Jν x x ν ,
where
  • Jν(·) is the Bessel function of the first kind,
  • 1=params[0], 1>0,
  • 2=params[1], 2>0,
  • ν=params[2], ν0.
cov=Nag_VgmHole
Hole effect variogram
γx = σ2 sinx x ,
where
  • 1=params[0], 1>0,
  • 2=params[1], 2>0.
cov=Nag_VgmWhittleMatern
Whittle-Matérn variogram
γx = σ2 21-ν x ν Kν x Γν ,
where
  • Kν(·) is the modified Bessel function of the second kind,
  • 1=params[0], 1>0,
  • 2=params[1], 2>0,
  • ν=params[2], ν>0.
cov=Nag_VgmContParam
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=params[0], 1>0,
  • 2=params[1], 2>0,
  • s1=params[2], s1>0,
  • s2=params[3], s2>0,
  • ν=params[4], ν>0.
cov=Nag_VgmGenHyp
Generalized hyperbolic distribution variogram
γx=σ2δ2+x2λ2δλKλκδKλκδ2+x212,
where
  • Kλ(·) is the modified Bessel function of the second kind,
  • 1=params[0], 1>0,
  • 2=params[1], 2>0,
  • λ=params[2], no constraint on λ,
  • δ=params[3], δ>0,
  • κ=params[4], κ>0.
Constraint: cov=Nag_VgmSymmStab, Nag_VgmCauchy, Nag_VgmDifferential, Nag_VgmExponential, Nag_VgmGauss, Nag_VgmNugget, Nag_VgmSpherical, Nag_VgmBessel, Nag_VgmHole, Nag_VgmWhittleMatern, Nag_VgmContParam or Nag_VgmGenHyp.
9:     normNag_NormTypeInput
On entry: determines which norm to use when calculating the variogram.
norm=Nag_OneNorm
The 1-norm is used, i.e., x,y=x+y.
norm=Nag_TwoNorm
The 2-norm (Euclidean norm) is used, i.e., x,y= x2+y2.
Suggested value: norm=Nag_TwoNorm.
Constraint: norm=Nag_OneNorm or Nag_TwoNorm.
10:   npIntegerInput
On entry: the number of parameters to be set. Different covariance functions need a different number of parameters.
cov=Nag_VgmNugget
np must be set to 0.
cov=Nag_VgmDifferential, Nag_VgmExponential, Nag_VgmGauss, Nag_VgmSpherical or Nag_VgmHole
np must be set to 2.
cov=Nag_VgmSymmStab, Nag_VgmCauchy, Nag_VgmBessel or Nag_VgmWhittleMatern
np must be set to 3.
cov=Nag_VgmContParam or Nag_VgmGenHyp
np must be set to 5.
11:   params[np]const doubleInput
On entry: the parameters for the variogram as detailed in the description of cov.
Constraint: see cov for a description of the individual parameter constraints.
12:   padNag_EmbedPadInput
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.
pad=Nag_EmbedPadZeros
The embedding matrix is padded with zeros.
pad=Nag_EmbedPadValues
The embedding matrix is padded with values of the variogram.
Suggested value: pad=Nag_EmbedPadValues.
Constraint: pad=Nag_EmbedPadZeros or Nag_EmbedPadValues.
13:   corrNag_EmbedScaleInput
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.
14:   lam[maxm[0]×maxm[1]]doubleOutput
On exit: contains the square roots of the eigenvalues of the embedding matrix.
15:   xx[ns[0]]doubleOutput
On exit: the points of the x-coordinates at which values of the random field will be output.
16:   yy[ns[1]]doubleOutput
On exit: the points of the y-coordinates at which values of the random field will be output.
17:   m[2]IntegerOutput
On exit: m[0] contains M1, the size of the circulant blocks and m[1] contains M2, the number of blocks, resulting in a final square matrix of size M1×M2.
18:   approxInteger *Output
On exit: indicates whether approximation was used.
approx=0
No approximation was used.
approx=1
Approximation was used.
19:   rhodouble *Output
On exit: indicates the scaling of the covariance matrix. rho=1.0 unless approximation was used with corr=Nag_EmbedScaleTraces or Nag_EmbedScaleSqrtTraces.
20:   icountInteger *Output
On exit: indicates the number of negative eigenvalues in the embedding matrix which have had to be set to zero.
21:   eig[3]doubleOutput
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.
22:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_ENUM_INT
On entry, np=value.
Constraint: for cov=value, np=value.
NE_ENUM_REAL_1
On entry, params[value]=value.
Constraint: dependent on cov, see documentation.
NE_INT_ARRAY
On entry, maxm=value,value.
Constraint: the minimum calculated value for maxm are value,value.
Where the minima of maxm[i-1] is given by 2 k , where k is the smallest integer satisfying 2 k 2 ns[i-1]-1 , for i=1,2.
On entry, ns=value,value.
Constraint: ns[0]1, ns[1]1.
NE_INTERNAL_ERROR
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.
NE_REAL
On entry, var=value.
Constraint: var0.0.
NE_REAL_2
On entry, xmin=value and xmax=value.
Constraint: xmin<xmax.
On entry, ymin=value and ymax=value.
Constraint: ymin<ymax.

7  Accuracy

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

8  Parallelism and Performance

nag_rand_field_2d_predef_setup (g05zrc) is not threaded by NAG in any implementation.
nag_rand_field_2d_predef_setup (g05zrc) 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 Users' Note for your implementation for any additional implementation-specific information.

9  Further Comments

None.

10  Example

This example calls nag_rand_field_2d_predef_setup (g05zrc) 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 (cov=Nag_VgmSymmStab).

10.1  Program Text

Program Text (g05zrce.c)

10.2  Program Data

Program Data (g05zrce.d)

10.3  Program Results

Program Results (g05zrce.r)

The two plots shown below illustrate the random fields that can be generated by nag_rand_field_2d_generate (g05zsc) using the eigenvalues calculated by nag_rand_field_2d_predef_setup (g05zrc). 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 (cov=Nag_VgmExponential) with correlation lengths both equal to 0.1.
Produced by GNUPLOT 4.4 patchlevel 0 Example Program 1 First realization of two-dimensional Random Field exponential variogram, correlation lengths = 0.1