NAG Library Routine Document
G05ZMF 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 G05ZPF
, which simulates the random field.
|SUBROUTINE G05ZMF (
||NS, XMIN, XMAX, MAXM, VAR, COV1, PAD, ICORR, LAM, XX, M, APPROX, RHO, ICOUNT, EIG, IUSER, RUSER, IFAIL)
||NS, MAXM, PAD, ICORR, M, APPROX, ICOUNT, IUSER(*), IFAIL
||XMIN, XMAX, VAR, LAM(MAXM), XX(NS), RHO, EIG(3), RUSER(*)
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 Gaussian 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 routines G05ZMF and G05ZPF
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
gridpoints. The problem reduces to sampling a Gaussian random vector
, with mean vector zero and a symmetric Toeplitz covariance matrix
is in general expensive to factorize, a technique known as the circulant embedding method
is embedded into a larger, symmetric circulant matrix
, which can now be factorized as
is the Fourier matrix (
is the complex conjugate of
is the diagonal matrix containing the eigenvalues of
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.
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
contain the non-negative and negative eigenvalues of
is replaced by
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 parameter ICORR
- setting sets
- setting sets
- setting sets .
G05ZMF 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.
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
- 1: NS – INTEGERInput
On entry: the number of sample points (points) to be generated in realisations of the random field.
- 2: XMIN – REAL (KIND=nag_wp)Input
On entry: the lower bound for the interval over which the random field is to be simulated.
- 3: XMAX – REAL (KIND=nag_wp)Input
On entry: the upper bound for the interval over which the random field is to be simulated.
- 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 .
, where is the smallest integer satisfying .
- 5: VAR – REAL (KIND=nag_wp)Input
On entry: the multiplicative factor of the variogram .
- 6: COV1 – SUBROUTINE, supplied by the user.External Procedure
must evaluate the variogram
, without the multiplicative factor
, for all
. The value returned in GAMMA
is multiplied internally by VAR
The specification of COV1
||X, GAMMA, RUSER(*)
- 1: X – REAL (KIND=nag_wp)Input
On entry: the value at which the variogram is to be evaluated.
- 2: GAMMA – REAL (KIND=nag_wp)Output
On exit: the value of the variogram .
- 3: IUSER() – INTEGER arrayUser Workspace
- 4: RUSER() – REAL (KIND=nag_wp) arrayUser Workspace
is called with the parameters IUSER
as supplied to G05ZMF. You are free to use the arrays IUSER
to supply information to COV1
as an alternative to using COMMON global variables.
must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which G05ZMF is called. Parameters denoted as Input
be changed by this procedure.
- 7: PAD – INTEGERInput
: 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.
- 8: ICORR – INTEGERInput
: determines which approximation to implement if required, as described in Section 3
, or .
- 9: LAM(MAXM) – REAL (KIND=nag_wp) arrayOutput
On exit: contains the square roots of the eigenvalues of the embedding matrix.
- 10: XX(NS) – REAL (KIND=nag_wp) arrayOutput
On exit: the points at which values of the random field will be output.
- 11: M – INTEGEROutput
On exit: the size of the embedding matrix.
- 12: APPROX – INTEGEROutput
: indicates whether approximation was used.
- No approximation was used.
- Approximation was used.
- 13: RHO – REAL (KIND=nag_wp)Output
On exit: indicates the scaling of the covariance matrix. unless approximation was used with or .
- 14: ICOUNT – INTEGEROutput
On exit: indicates the number of negative eigenvalues in the embedding matrix which have had to be set to zero.
- 15: EIG() – REAL (KIND=nag_wp) arrayOutput
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: IUSER() – INTEGER arrayUser Workspace
- 17: RUSER() – REAL (KIND=nag_wp) arrayUser Workspace
are not used by G05ZMF, but are passed directly to COV1
and may be used to pass information to this routine as an alternative to using COMMON global variables.
- 18: IFAIL – INTEGERInput/Output
must be set to
. If you are unfamiliar with this parameter you should refer to Section 3.3
in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value
is recommended. If the output of error messages is undesirable, then the value
is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is
. When the value is used it is essential to test the value of IFAIL on exit.
unless the routine detects an error or a warning has been flagged (see Section 6
6 Error Indicators and Warnings
If on entry
, explanatory error messages are output on the current error message unit (as defined by X04AAF
Errors or warnings detected by the routine:
On entry, .
On entry, and .
Constraint: the calculated minimum value for MAXM
Where the minimum calculated value is given by , where is the smallest integer satisfying .
On entry, .
On entry, .
Constraint: or .
On entry, .
Constraint: , or .
This example calls G05ZMF to calculate the eigenvalues of the embedding matrix for
sample points of a random field characterized by the symmetric stable variogram:
It should be noted that the symmetric stable variogram is one of the pre-defined variograms available in G05ZNF
. It is used here purely for illustrative purposes.
9.1 Program Text
Program Text (g05zmfe.f90)
9.2 Program Data
Program Data (g05zmfe.d)
9.3 Program Results
Program Results (g05zmfe.r)