NAG Library Function Document
nag_rand_field_2d_predef_setup (g05zrc)
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 in is a function which is random at every point , so is a random variable for each . The random field has a mean function and a symmetric positive semidefinite covariance function . is a Gaussian random field if for any choice of and , the random vector follows a multivariate Normal distribution, which would have a mean vector with entries and a covariance matrix with entries . A Gaussian random field is stationary if is constant for all and for all and hence we can express the covariance function as a function of one variable: . is known as a variogram (or more correctly, a semivariogram) and includes the multiplicative factor representing the variance such that .
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
, over a domain
, using an equally spaced set of
points;
points in the
-direction and
points in the
-direction. The problem reduces to sampling a Gaussian random vector
of size
, with mean vector zero and a symmetric covariance matrix
, which is an
by
block Toeplitz matrix with Toeplitz blocks of size
by
. Since
is in general expensive to factorize, a technique known as the
circulant embedding method is used.
is embedded into a larger, symmetric matrix
, which is an
by
block circulant matrix with circulant blocks of size
by
, where
and
.
can now be factorized as
, where
is the two-dimensional Fourier matrix (
is the complex conjugate of
),
is the diagonal matrix containing the eigenvalues of
and
.
is known as the embedding matrix. The eigenvalues can be calculated by performing a discrete Fourier transform of the first row (or column) of
and multiplying by
, and so only the first row (or column) of
is needed – the whole matrix does not need to be formed.
As long as all of the values of are non-negative (i.e., is positive semidefinite), is a covariance matrix for a random vector which has blocks of size . Two samples of can now be simulated from the real and imaginary parts of , where and have elements from the standard Normal distribution. Since , this calculation can be done using a discrete Fourier transform of the vector . Two samples of the random vector can now be recovered by taking the first elements of the first blocks of each sample of – because the original covariance matrix is embedded in , will have the correct distribution.
If
is not positive semidefinite, larger embedding matrices
can be tried; however if the size of the matrix would have to be larger than
maxm, an approximation procedure is used. We write
, where
and
contain the non-negative and negative eigenvalues of
respectively. Then
is replaced by
where
and
is a scaling factor. The error
in approximating the distribution of the random field is given by
Three choices for
are available, and are determined by the input argument
corr:
- setting sets
- setting sets
- setting sets .
nag_rand_field_2d_predef_setup (g05zrc) finds a suitable positive semidefinite embedding matrix
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
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[] – const IntegerInput
-
On entry: the number of sample points to use in each direction, with sample points in the -direction, and sample points in the -direction, . The total number of sample points on the grid is therefore .
- 2:
xmin – doubleInput
On entry: the lower bound for the -coordinate, for the region in which the random field is to be simulated.
Constraint:
.
- 3:
xmax – doubleInput
On entry: the upper bound for the -coordinate, for the region in which the random field is to be simulated.
Constraint:
.
- 4:
ymin – doubleInput
On entry: the lower bound for the -coordinate, for the region in which the random field is to be simulated.
Constraint:
.
- 5:
ymax – doubleInput
On entry: the upper bound for the -coordinate, for the region in which the random field is to be simulated.
Constraint:
.
- 6:
maxm[] – const IntegerInput
On entry: determines the maximum size of the circulant matrix to use – a maximum of elements in the -direction, and a maximum of elements in the -direction. The maximum size of the circulant matrix is thus .
Constraint:
, where is the smallest integer satisfying , for .
- 7:
var – doubleInput
On entry: the multiplicative factor of the variogram .
Constraint:
.
- 8:
cov – Nag_VariogramInput
-
On entry: determines which of the preset variograms to use. The choices are given below. Note that
, where
and
are correlation lengths in the
and
directions respectively and are parameters for most of the variograms, and
is the variance specified by
var.
- Symmetric stable variogram
where
- , ,
- , ,
- , .
- Cauchy variogram
where
- , ,
- , ,
- , .
- Differential variogram with compact support
where
- , ,
- , .
- Exponential variogram
where
- , ,
- , .
- Gaussian variogram
where
- , ,
- , .
- Nugget variogram
No parameters need be set for this value of cov.
- Spherical variogram
where
- , ,
- , .
- Bessel variogram
where
- is the Bessel function of the first kind,
- , ,
- , ,
- , .
- Hole effect variogram
where
- , ,
- , .
- Whittle-Matérn variogram
where
- is the modified Bessel function of the second kind,
- , ,
- , ,
- , .
- Continuously parameterised variogram with compact support
where
- ,
- is the modified Bessel function of the second kind,
- , ,
- , ,
- , ,
- , ,
- , .
- Generalized hyperbolic distribution variogram
where
- is the modified Bessel function of the second kind,
- , ,
- , ,
- , no constraint on ,
- , ,
- , .
Constraint:
, , , , , , , , , , or .
- 9:
norm – Nag_NormTypeInput
On entry: determines which norm to use when calculating the variogram.
- The 1-norm is used, i.e., .
- The 2-norm (Euclidean norm) is used, i.e., .
Suggested value:
.
Constraint:
or .
- 10:
np – IntegerInput
On entry: the number of parameters to be set. Different covariance functions need a different number of parameters.
- np must be set to .
- , , , or
- np must be set to .
- , , or
- np must be set to .
- or
- np must be set to .
- 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:
pad – Nag_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.
- The embedding matrix is padded with zeros.
- The embedding matrix is padded with values of the variogram.
Suggested value:
.
Constraint:
or .
- 13:
corr – Nag_EmbedScaleInput
On entry: determines which approximation to implement if required, as described in
Section 3.
Suggested value:
.
Constraint:
, or .
- 14:
lam[] – doubleOutput
On exit: contains the square roots of the eigenvalues of the embedding matrix.
- 15:
xx[] – doubleOutput
On exit: the points of the -coordinates at which values of the random field will be output.
- 16:
yy[] – doubleOutput
On exit: the points of the -coordinates at which values of the random field will be output.
- 17:
m[] – IntegerOutput
On exit: contains , the size of the circulant blocks and contains , the number of blocks, resulting in a final square matrix of size .
- 18:
approx – Integer *Output
On exit: indicates whether approximation was used.
- No approximation was used.
- Approximation was used.
- 19:
rho – double *Output
On exit: indicates the scaling of the covariance matrix. unless approximation was used with or .
- 20:
icount – Integer *Output
On exit: indicates the number of negative eigenvalues in the embedding matrix which have had to be set to zero.
- 21:
eig[] – doubleOutput
On exit: indicates information about the negative eigenvalues in the embedding matrix which have had to be set to zero. contains the smallest eigenvalue, contains the sum of the squares of the negative eigenvalues, and contains the sum of the absolute values of the negative eigenvalues.
- 22:
fail – NagError *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 had an illegal value.
- NE_ENUM_INT
-
On entry, .
Constraint: for , .
- NE_ENUM_REAL_1
-
On entry,
.
Constraint: dependent on
cov, see documentation.
- NE_INT_ARRAY
-
On entry,
.
Constraint: the minimum calculated value for
maxm are
.
Where the minima of
is given by
, where
is the smallest integer satisfying
, for
.
On entry, .
Constraint: , .
- 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, .
Constraint: .
- NE_REAL_2
-
On entry, and .
Constraint: .
On entry, and .
Constraint: .
7 Accuracy
If on exit
, 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.
None.
10 Example
This example calls nag_rand_field_2d_predef_setup (g05zrc) to calculate the eigenvalues of the embedding matrix for sample points on a by grid of a two-dimensional random field characterized by the symmetric stable variogram ().
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
by
grid. The random field is characterized by the
exponential variogram (
) with correlation lengths both equal to
.