NAG Library Routine Document
G05ZNF
1 Purpose
G05ZNF performs the setup required in order to simulate stationary Gaussian random fields in one dimension, 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
G05ZPF, which simulates the random field.
2 Specification
SUBROUTINE G05ZNF ( |
NS, XMIN, XMAX, MAXM, VAR, ICOV1, NP, PARAMS, PAD, ICORR, LAM, XX, M, APPROX, RHO, ICOUNT, EIG, IFAIL) |
INTEGER |
NS, MAXM, ICOV1, NP, PAD, ICORR, M, APPROX, ICOUNT, IFAIL |
REAL (KIND=nag_wp) |
XMIN, XMAX, VAR, PARAMS(NP), LAM(MAXM), XX(NS), RHO, EIG(3) |
|
3 Description
A one-dimensional random field Zx in ℝ is a function which is random at every point x∈ℝ, so Zx is a random variable for each x. The random field has a mean function μx=𝔼Zx and a symmetric positive semidefinite covariance function Cx,y=𝔼Zx-μxZy-μy. Zx is a Gaussian random field if for any choice of n∈ℕ and x1,…,xn∈ℝ, the random vector Zx1,…,ZxnT follows a multivariate Gaussian distribution, which would have a mean vector μ~ with entries μ~i=μxi and a covariance matrix C~ with entries C~ij=Cxi,xj. A Gaussian random field Zx is stationary if μx is constant for all x∈ℝ and Cx,y=Cx+a,y+a for all x,y,a∈ℝ and hence we can express the covariance function Cx,y as a function γ of one variable: Cx,y=γx-y. γ is known as a variogram (or more correctly, a semivariogram) and includes the multiplicative factor σ2 representing the variance such that γ0=σ2.
The routines G05ZNF and
G05ZPF are used to simulate a one-dimensional stationary Gaussian random field, with mean function zero and variogram
γx, over an interval
xmin,xmax, using an equally spaced set of
N points. The problem reduces to sampling a Gaussian random vector
X of size
N, with mean vector zero and a symmetric Toeplitz covariance matrix
A. Since
A is in general expensive to factorize, a technique known as the
circulant embedding method is used.
A is embedded into a larger, symmetric circulant matrix
B of size
M≥2N-1, which can now be factorized as
B=WΛW*=R*R, where
W is the Fourier matrix (
W* is the complex conjugate of
W),
Λ is the diagonal matrix containing the eigenvalues of
B and
R=Λ12W*.
B is known as the embedding matrix. The eigenvalues can be calculated by performing a discrete Fourier transform of the first row (or column) of
B and multiplying by
M, and so only the first row (or column) of
B is needed – the whole matrix does not need to be formed.
As long as all of the values of Λ are non-negative (i.e., B is positive semidefinite), B is a covariance matrix for a random vector Y, two samples of which can now be simulated from the real and imaginary parts of R*U+iV, where U and V have elements from the standard Normal distribution. Since R*U+iV=WΛ12U+iV, this calculation can be done using a discrete Fourier transform of the vector Λ12U+iV. Two samples of the random vector X can now be recovered by taking the first N elements of each sample of Y – because the original covariance matrix A is embedded in B, X will have the correct distribution.
If
B is not positive semidefinite, larger embedding matrices
B can be tried; however if the size of the matrix would have to be larger than
MAXM, an approximation procedure is used. We write
Λ=Λ++Λ-, where
Λ+ and
Λ- contain the non-negative and negative eigenvalues of
B respectively. Then
B is replaced by
ρB+ where
B+=WΛ+W* and
ρ∈0,1 is a scaling factor. The error
ε in approximating the distribution of the random field is given by
Three choices for
ρ are available, and are determined by the input parameter
ICORR:
- setting ICORR=0 sets
- setting ICORR=1 sets
- setting ICORR=2 sets ρ=1.
G05ZNF finds a suitable positive semidefinite embedding matrix
B 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
B 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 Parameters
- 1: NS – INTEGERInput
On entry: the number of sample points (points) to be generated in realisations of the random field.
Constraint:
NS≥1.
- 2: XMIN – REAL (KIND=nag_wp)Input
On entry: the lower bound for the interval over which the random field is to be simulated. Note that if
ICOV1 is set to
14 (for simulating fractional Brownian motion),
XMIN is not referenced and the lower bound for the interval is set to zero.
Constraint:
if ICOV1≠14, XMIN<XMAX.
- 3: XMAX – REAL (KIND=nag_wp)Input
-
On entry: the upper bound for the interval over which the random field is to be simulated. Note that if
ICOV1 is set to
14 (for simulating fractional Brownian motion), the lower bound for the interval is set to zero and so
XMAX is required to be greater than zero.
Constraints:
- if ICOV1≠14, XMIN<XMAX;
- if ICOV1=14, XMAX>0.0.
- 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
MAXM
=
2k+2
where
k
=
1+
log2
NS-1
.
Constraint:
MAXM ≥ 2 k , where k is the smallest integer satisfying 2 k ≥ 2 NS-1 .
- 5: VAR – REAL (KIND=nag_wp)Input
On entry: the multiplicative factor σ2 of the variogram γx.
Constraint:
VAR≥0.0.
- 6: ICOV1 – INTEGERInput
-
On entry: determines which of the preset variograms to use. The choices are given below. Note that
x′=xℓ, where
ℓ is the correlation length and is a parameter for most of the variograms, and
σ2 is the variance specified by
VAR.
- ICOV1=1
- Symmetric stable variogram
where
- ℓ=PARAMS1, ℓ>0,
- ν=PARAMS2, 0≤ν≤2.
- ICOV1=2
- Cauchy variogram
where
- ℓ=PARAMS1, ℓ>0,
- ν=PARAMS2, ν>0.
- ICOV1=3
- Differential variogram with compact support
where
- ICOV1=4
- Exponential variogram
where
- ICOV1=5
- Gaussian variogram
where
- ICOV1=6
- Nugget variogram
No parameters need be set for this value of ICOV1.
- ICOV1=7
- Spherical variogram
where
- ICOV1=8
- Bessel variogram
where
- Jν(·) is the Bessel function of the first kind,
- ℓ=PARAMS1, ℓ>0,
- ν=PARAMS2, ν≥-0.5.
- ICOV1=9
- Hole effect variogram
where
- ICOV1=10
- Whittle-Matérn variogram
where
- Kν(·) is the modified Bessel function of the second kind,
- ℓ=PARAMS1, ℓ>0,
- ν=PARAMS2, ν>0.
- ICOV1=11
- Continuously parameterised variogram with compact support
where
-
x′′
=
x′s
,
- Kν(·) is the modified Bessel function of the second kind,
- ℓ=PARAMS1, ℓ>0,
- s=PARAMS2, s>0 (second correlation length),
- ν=PARAMS3, ν>0.
- ICOV1=12
- Generalized hyperbolic distribution variogram
where
- Kλ(·) is the modified Bessel function of the second kind,
- ℓ=PARAMS1, ℓ>0,
- λ=PARAMS2, no constraint on λ
- δ=PARAMS3, δ>0,
- κ=PARAMS4, κ>0.
- ICOV1=13
- Cosine variogram
where
- ICOV1=14
- Used for simulating fractional Brownian motion BHt. Fractional Brownian motion itself is not a stationary Gaussian random field, but its increments X~i=BHti-BHti-1 can be simulated in the same way as a stationary random field. The variogram for the so-called ‘increment process’ is
where
- x=tj-ti,
- H=PARAMS1, 0<H<1, H is the Hurst parameter,
- δ=PARAMS2, δ>0, normally δ=ti-ti-1 is the (fixed) stepsize.
We scale the increments to set
γ0=1; let
Xi=X~iδ-H, then
The increments
Xi can then be simulated using
G05ZPF, then multiplied by
δH to obtain the original increments
X~i for the fractional Brownian motion.
- 7: NP – INTEGERInput
On entry: the number of parameters to be set. Different variograms need a different number of parameters.
- ICOV1=6
- NP must be set to 0.
- ICOV1=3, 4, 5, 7, 9 or 13
- NP must be set to 1.
- ICOV1=1, 2, 8, 10 or 14
- NP must be set to 2.
- ICOV1=11
- NP must be set to 3.
- ICOV1=12
- NP must be set to 4.
- 8: PARAMS(NP) – REAL (KIND=nag_wp) arrayInput
On entry: the parameters set for the variogram.
Constraint:
see
ICOV1 for a description of the individual parameter constraints.
- 9: PAD – INTEGERInput
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.
- PAD=0
- The embedding matrix is padded with zeros.
- PAD=1
- The embedding matrix is padded with values of the variogram.
Suggested value:
PAD=1.
Constraint:
PAD=0 or 1.
- 10: ICORR – INTEGERInput
On entry: determines which approximation to implement if required, as described in
Section 3.
Suggested value:
ICORR=0.
Constraint:
ICORR=0, 1 or 2.
- 11: LAM(MAXM) – REAL (KIND=nag_wp) arrayOutput
On exit: contains the square roots of the eigenvalues of the embedding matrix.
- 12: XX(NS) – REAL (KIND=nag_wp) arrayOutput
On exit: the points at which values of the random field will be output.
- 13: M – INTEGEROutput
On exit: the size of the embedding matrix.
- 14: APPROX – INTEGEROutput
On exit: indicates whether approximation was used.
- APPROX=0
- No approximation was used.
- APPROX=1
- Approximation was used.
- 15: RHO – REAL (KIND=nag_wp)Output
On exit: indicates the scaling of the covariance matrix. RHO=1 unless approximation was used with ICORR=0 or 1.
- 16: ICOUNT – INTEGEROutput
On exit: indicates the number of negative eigenvalues in the embedding matrix which have had to be set to zero.
- 17: EIG(3) – 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. EIG1 contains the smallest eigenvalue, EIG2 contains the sum of the squares of the negative eigenvalues, and EIG3 contains the sum of the absolute values of the negative eigenvalues.
- 18: IFAIL – INTEGERInput/Output
-
On entry:
IFAIL must be set to
0,
-1 or 1. 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
-1 or 1 is recommended. If the output of error messages is undesirable, then the value
1 is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is
0.
When the value -1 or 1 is used it is essential to test the value of IFAIL on exit.
On exit:
IFAIL=0 unless the routine detects an error or a warning has been flagged (see
Section 6).
6 Error Indicators and Warnings
If on entry
IFAIL=0 or
-1, explanatory error messages are output on the current error message unit (as defined by
X04AAF).
Errors or warnings detected by the routine:
- IFAIL=1
-
On entry, NS=value.
Constraint: NS≥1.
- IFAIL=2
-
On entry, ICOV1≠14, XMIN=value and
XMAX=value.
Constraint: XMIN<XMAX.
- IFAIL=3
-
On entry, ICOV1=14 and XMAX=value.
Constraint: XMAX>0.0.
- IFAIL=4
-
On entry,
MAXM=value.
Constraint: the calculated minimum value for
MAXM is
value.
Where the minimum calculated value is given by 2 k , where k is the smallest integer satisfying 2 k ≥ 2 NS-1 .
- IFAIL=5
-
On entry, VAR=value.
Constraint: VAR≥0.0.
- IFAIL=6
-
On entry, ICOV1=value.
Constraint: ICOV1≥1 and ICOV1≤14.
- IFAIL=7
-
On entry, NP=value.
Constraint: for ICOV1=value, NP=value.
- IFAIL=8
-
On entry,
PARAMSvalue=value.
Constraint: dependent on
ICOV1, see documentation.
- IFAIL=9
-
On entry, PAD=value.
Constraint: PAD=0 or 1.
- IFAIL=10
-
On entry, ICORR=value.
Constraint: ICORR=0, 1 or 2.
7 Accuracy
Not applicable.
8 Further Comments
None.
9 Example
This example calls G05ZNF to calculate the eigenvalues of the embedding matrix for 8 sample points of a random field characterized by the symmetric stable variogram (ICOV1=1).
9.1 Program Text
Program Text (g05znfe.f90)
9.2 Program Data
Program Data (g05znfe.d)
9.3 Program Results
Program Results (g05znfe.r)