NAG Library Function Document
nag_rand_field_2d_user_setup (g05zqc)
1
Purpose
nag_rand_field_2d_user_setup (g05zqc) performs the setup required in order to simulate stationary Gaussian random fields in two dimensions, for a user-defined 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_user_setup (const Integer ns[],
double xmin,
double xmax,
double ymin,
double ymax,
const Integer maxm[],
double var,
Nag_Parity parity,
Nag_EmbedPad pad,
Nag_EmbedScale corr,
double lam[],
double xx[],
double yy[],
Integer m[],
Integer *approx,
double *rho,
Integer *icount,
double eig[],
Nag_Comm *comm,
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_user_setup (g05zqc) 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 Normal 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.
The symmetry of as a block matrix, and the symmetry of each block of , depends on whether the variogram is even or not. is even in its first coordinate if , even in its second coordinate if , and even if it is even in both coordinates (in two dimensions it is impossible for to be even in one coordinate and uneven in the other). If is even then is a symmetric block matrix and has symmetric blocks; if is uneven then is not a symmetric block matrix and has non-symmetric blocks. In the uneven case, and are set to be odd in order to guarantee symmetry in .
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_user_setup (g05zqc) 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 (1994) Simulation of stationary Gaussian processes in Journal of Computational and Graphical Statistics 3(4) 409–432
5
Arguments
- 1:
– 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:
– doubleInput
-
On entry: the lower bound for the -coordinate, for the region in which the random field is to be simulated.
Constraint:
.
- 3:
– doubleInput
-
On entry: the upper bound for the -coordinate, for the region in which the random field is to be simulated.
Constraint:
.
- 4:
– doubleInput
-
On entry: the lower bound for the -coordinate, for the region in which the random field is to be simulated.
Constraint:
.
- 5:
– doubleInput
-
On entry: the upper bound for the -coordinate, for the region in which the random field is to be simulated.
Constraint:
.
- 6:
– 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 .
Constraints:
- if , , where is the smallest integer satisfying , for ;
- if , , where is the smallest integer satisfying , for .
- 7:
– doubleInput
-
On entry: the multiplicative factor of the variogram .
Constraint:
.
- 8:
– function, supplied by the userExternal Function
-
cov2 must evaluate the variogram
for all
if
, and for all
with non-negative entries if
. The value returned in
gamma is multiplied internally by
var.
The specification of
cov2 is:
void |
cov2 (double x,
double y,
double *gamma,
Nag_Comm *comm)
|
|
- 1:
– doubleInput
-
On entry: the coordinate at which the variogram is to be evaluated.
- 2:
– doubleInput
-
On entry: the coordinate at which the variogram is to be evaluated.
- 3:
– double *Output
-
On exit: the value of the variogram .
- 4:
– Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to
cov2.
- user – double *
- iuser – Integer *
- p – Pointer
The type Pointer will be
void *. Before calling
nag_rand_field_2d_user_setup (g05zqc) you may allocate memory and initialize these pointers with various quantities for use by
cov2 when called from
nag_rand_field_2d_user_setup (g05zqc) (see
Section 3.3.1.1 in How to Use the NAG Library and its Documentation).
Note: cov2 should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by
nag_rand_field_2d_user_setup (g05zqc). If your code inadvertently
does return any NaNs or infinities,
nag_rand_field_2d_user_setup (g05zqc) is likely to produce unexpected results.
- 9:
– Nag_ParityInput
-
On entry: indicates whether the covariance function supplied is even or uneven.
- The covariance function is uneven.
- The covariance function is even.
Constraint:
or .
- 10:
– 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 .
- 11:
– Nag_EmbedScaleInput
-
On entry: determines which approximation to implement if required, as described in
Section 3.
Suggested value:
.
Constraint:
, or .
- 12:
– doubleOutput
-
On exit: contains the square roots of the eigenvalues of the embedding matrix.
- 13:
– doubleOutput
-
On exit: the points of the -coordinates at which values of the random field will be output.
- 14:
– doubleOutput
-
On exit: the points of the -coordinates at which values of the random field will be output.
- 15:
– IntegerOutput
-
On exit: contains , the size of the circulant blocks and contains , the number of blocks, resulting in a final square matrix of size .
- 16:
– Integer *Output
-
On exit: indicates whether approximation was used.
- No approximation was used.
- Approximation was used.
- 17:
– double *Output
-
On exit: indicates the scaling of the covariance matrix. unless approximation was used with or .
- 18:
– Integer *Output
-
On exit: indicates the number of negative eigenvalues in the embedding matrix which have had to be set to zero.
- 19:
– 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.
- 20:
– Nag_Comm *
-
The NAG communication argument (see
Section 3.3.1.1 in How to Use the NAG Library and its Documentation).
- 21:
– NagError *Input/Output
-
The NAG error argument (see
Section 3.7 in How to Use the NAG Library and its Documentation).
6
Error Indicators and Warnings
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
See
Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
- NE_BAD_PARAM
-
On entry, argument had an illegal value.
- NE_INT_ARRAY
-
On entry,
.
Constraint: the minima for
maxm are
.
Where, if
, the minimum calculated value of
is given by
, where
is the smallest integer satisfying
, and if
, the minimum calculated value 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.
See
Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
- NE_NO_LICENCE
-
Your licence key may have expired or may not have been installed correctly.
See
Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
- 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_user_setup (g05zqc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_rand_field_2d_user_setup (g05zqc) 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.
None.
10
Example
This example calls
nag_rand_field_2d_user_setup (g05zqc) 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:
where
, and
,
and
are parameters.
It should be noted that the symmetric stable variogram is one of the pre-defined variograms available in
nag_rand_field_2d_predef_setup (g05zrc). It is used here purely for illustrative purposes.
10.1
Program Text
Program Text (g05zqce.c)
10.2
Program Data
Program Data (g05zqce.d)
10.3
Program Results
Program Results (g05zqce.r)