NAG CL Interface
g05zpc (field_1d_generate)
1
Purpose
g05zpc produces realizations of a stationary Gaussian random field in one dimension, using the circulant embedding method. The square roots of the eigenvalues of the extended covariance matrix (or embedding matrix) need to be input, and can be calculated using
g05zmc or
g05znc.
2
Specification
void |
g05zpc (Integer ns,
Integer s,
Integer m,
const double lam[],
double rho,
Integer state[],
double z[],
NagError *fail) |
|
The function may be called by the names: g05zpc or nag_rand_field_1d_generate.
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 non-negative definite 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
g05zmc or
g05znc, along with
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. 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 non-negative definite), 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 non-negative definite, 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. See the documentation of
g05zmc or
g05znc for details of the approximation procedure.
g05zpc takes the square roots of the eigenvalues of the embedding matrix , and its size , as input and outputs realizations of the random field in .
One of the initialization functions
g05kfc (for a repeatable sequence if computed sequentially) or
g05kgc (for a non-repeatable sequence) must be called prior to the first call to
g05zpc.
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:
– Integer
Input
-
On entry: the number of sample points to be generated in realizations of the random field. This must be the same value as supplied to
g05zmc or
g05znc when calculating the eigenvalues of the embedding matrix.
Constraint:
.
-
2:
– Integer
Input
-
On entry: , the number of realizations of the random field to simulate.
Constraint:
.
-
3:
– Integer
Input
-
On entry:
, the size of the embedding matrix, as returned by
g05zmc or
g05znc.
Constraint:
.
-
4:
– const double
Input
-
On entry: must contain the square roots of the eigenvalues of the embedding matrix, as returned by
g05zmc or
g05znc.
Constraint:
.
-
5:
– double
Input
-
On entry: indicates the scaling of the covariance matrix, as returned by
g05zmc or
g05znc.
Constraint:
.
-
6:
– Integer
Communication Array
Note: the dimension,
, of this array is dictated by the requirements of associated functions that must have been previously called. This array MUST be the same array passed as argument
state in the previous call to
nag_rand_init_repeatable (g05kfc) or
nag_rand_init_nonrepeatable (g05kgc).
On entry: contains information on the selected base generator and its current state.
On exit: contains updated information on the state of the generator.
-
7:
– double
Output
-
On exit: contains the realizations of the random field. The
th realization, for the
ns sample points, is stored in
, for
. The sample points are as returned in
by
g05zmc or
g05znc.
-
8:
– NagError *
Input/Output
-
The NAG error argument (see
Section 7 in the Introduction to the NAG Library CL Interface).
6
Error Indicators and Warnings
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
See
Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
- NE_BAD_PARAM
-
On entry, argument had an illegal value.
- NE_INT
-
On entry, .
Constraint: .
On entry, .
Constraint: .
- NE_INT_2
-
On entry, and .
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 7.5 in the Introduction to the NAG Library CL Interface for further information.
- NE_INVALID_STATE
-
On entry,
state vector has been corrupted or not initialized.
- NE_NEG_ELEMENT
-
On entry, at least one element of
lam was negative.
Constraint: all elements of
lam must be non-negative.
- NE_NO_LICENCE
-
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.
- NE_REAL
-
On entry, .
Constraint: .
7
Accuracy
Not applicable.
8
Parallelism and Performance
g05zpc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
g05zpc 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.
Because samples are generated in pairs, calling this function times, with , say, will generate a different sequence of numbers than calling the function once with , unless is even.
10
Example
This example calls
g05zpc to generate
realizations of a random field on
sample points using eigenvalues calculated by
g05znc for a symmetric stable variogram.
10.1
Program Text
10.2
Program Data
10.3
Program Results