NAG Library Function Document
nag_rand_field_1d_user_setup (g05zmc)
1 Purpose
nag_rand_field_1d_user_setup (g05zmc) performs the setup required in order to simulate stationary Gaussian random fields in one dimension, 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_1d_generate (g05zpc), which simulates the random field.
2 Specification
#include <nag.h> |
#include <nagg05.h> |
void |
nag_rand_field_1d_user_setup (Integer ns,
double xmin,
double xmax,
Integer maxm,
double var,
Nag_EmbedPad pad,
Nag_EmbedScale corr,
double lam[],
double xx[],
Integer *m,
Integer *approx,
double *rho,
Integer *icount,
double eig[],
Nag_Comm *comm,
NagError *fail) |
|
3 Description
A one-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_1d_user_setup (g05zmc) and
nag_rand_field_1d_generate (g05zpc) are used to simulate a one-dimensional stationary Gaussian random field, with mean function zero and variogram
, over an interval
, using an equally spaced set of
points on the interval. The problem reduces to sampling a Normal random vector
of size
, with mean vector zero and a symmetric Toeplitz covariance matrix
. Since
is in general expensive to factorize, a technique known as the
circulant embedding method is used.
is embedded into a larger, symmetric circulant matrix
of size
, which can now be factorized as
, where
is the 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 , two samples of which 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 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_1d_user_setup (g05zmc) finds a suitable positive semidefinite embedding matrix
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
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:
ns – IntegerInput
On entry: the number of sample points to be generated in realizations of the random field.
Constraint:
.
- 2:
xmin – doubleInput
On entry: the lower bound for the interval over which the random field is to be simulated.
Constraint:
.
- 3:
xmax – doubleInput
On entry: the upper bound for the interval over which the random field is to be simulated.
Constraint:
.
- 4:
maxm – IntegerInput
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 where .
Suggested value:
Constraint:
, where is the smallest integer satisfying .
- 5:
var – doubleInput
On entry: the multiplicative factor of the variogram .
Constraint:
.
- 6:
cov1 – function, supplied by the userExternal Function
cov1 must evaluate the variogram
, without the multiplicative factor
, for all
. The value returned in
gamma is multiplied internally by
var.
The specification of
cov1 is:
void |
cov1 (double x,
double *gamma,
Nag_Comm *comm)
|
|
- 1:
x – doubleInput
On entry: the value at which the variogram is to be evaluated.
- 2:
gamma – double *Output
On exit: the value of the variogram .
- 3:
comm – Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to
cov1.
- user – double *
- iuser – Integer *
- p – Pointer
The type Pointer will be
void *. Before calling nag_rand_field_1d_user_setup (g05zmc) you may allocate memory and initialize these pointers with various quantities for use by
cov1 when called from nag_rand_field_1d_user_setup (g05zmc) (see
Section 3.2.1.1 in the Essential Introduction).
- 7:
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 .
- 8:
corr – Nag_EmbedScaleInput
On entry: determines which approximation to implement if required, as described in
Section 3.
Suggested value:
.
Constraint:
, or .
- 9:
lam[maxm] – doubleOutput
On exit: contains the square roots of the eigenvalues of the embedding matrix.
- 10:
xx[ns] – doubleOutput
On exit: the points at which values of the random field will be output.
- 11:
m – Integer *Output
On exit: the size of the embedding matrix.
- 12:
approx – Integer *Output
On exit: indicates whether approximation was used.
- No approximation was used.
- Approximation was used.
- 13:
rho – double *Output
On exit: indicates the scaling of the covariance matrix. unless approximation was used with or .
- 14:
icount – Integer *Output
On exit: indicates the number of negative eigenvalues in the embedding matrix which have had to be set to zero.
- 15:
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.
- 16:
comm – Nag_Comm *Communication Structure
-
The NAG communication argument (see
Section 3.2.1.1 in the Essential Introduction).
- 17:
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_INT
-
On entry,
.
Constraint: the minimum calculated value for
maxm is
.
Where the minimum calculated value is given by
, where
is the smallest integer satisfying
.
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: .
7 Accuracy
If on exit
, 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
Not applicable.
None.
10 Example
This example calls nag_rand_field_1d_user_setup (g05zmc) to calculate the eigenvalues of the embedding matrix for
sample points of a 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_1d_predef_setup (g05znc). It is used here purely for illustrative purposes.
10.1 Program Text
Program Text (g05zmce.c)
10.2 Program Data
Program Data (g05zmce.d)
10.3 Program Results
Program Results (g05zmce.r)