g05zrf performs the setup required in order to simulate stationary Gaussian random fields in two dimensions, 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 g05zsf, which simulates the random field.

2 Specification

Fortran Interface

Subroutine g05zrf (

ns, xmin, xmax, ymin, ymax, maxm, var, icov2, norm, np, params, pad, icorr, lam, xx, yy, m, approx, rho, icount, eig, ifail)

Integer, Intent (In)	::	ns(2), maxm(2), icov2, norm, np, pad, icorr
Integer, Intent (Inout)	::	ifail
Integer, Intent (Out)	::	m(2), approx, icount
Real (Kind=nag_wp), Intent (In)	::	xmin, xmax, ymin, ymax, var, params(np)
Real (Kind=nag_wp), Intent (Out)	::	lam(MAXM(1)*MAXM(2)), xx(NS(1)), yy(NS(2)), rho, eig(3)

C Header Interface

#include <nag.h>

void

g05zrf_ (const Integer ns[], const double *xmin, const double *xmax, const double *ymin, const double *ymax, const Integer maxm[], const double *var, const Integer *icov2, const Integer *norm, const Integer *np, const double params[], const Integer *pad, const Integer *icorr, double lam[], double xx[], double yy[], Integer m[], Integer *approx, double *rho, Integer *icount, double eig[], Integer *ifail)

The routine may be called by the names g05zrf or nagf_rand_field_2d_predef_setup.

3 Description

A two-dimensional random field

Z (x)

ℝ^{2}

is a function which is random at every point

x \in ℝ^{2}

, so

Z (x)

is a random variable for each

x

. The random field has a mean function

μ (x) = 𝔼 [Z (x)]

and a symmetric positive semidefinite covariance function

C (x, y) = 𝔼 [(Z (x) - μ (x)) (Z (y) - μ (y))]

Z (x)

is a Gaussian random field if for any choice of

n \in ℕ

and

x_{1}, \dots, x_{n} \in ℝ^{2}

, the random vector

{[Z (x_{1}), \dots, Z (x_{n})]}^{T}

follows a multivariate Normal distribution, which would have a mean vector

\tilde{μ}

with entries

{\tilde{μ}}_{i} = μ (x_{i})

and a covariance matrix

\tilde{C}

with entries

{\tilde{C}}_{i j} = C (x_{i}, x_{j})

. A Gaussian random field

Z (x)

is stationary if

μ (x)

is constant for all

x \in ℝ^{2}

and

C (x, y) = C (x + a, y + a)

for all

x, y, a \in ℝ^{2}

and hence we can express the covariance function

C (x, y)

as a function

γ

of one variable:

C (x, y) = γ (x - y)

γ

is known as a variogram (or more correctly, a semivariogram) and includes the multiplicative factor

σ^{2}

representing the variance such that

γ (0) = σ^{2}

The routines g05zrf and g05zsf are used to simulate a two-dimensional stationary Gaussian random field, with mean function zero and variogram

γ (x)

, over a domain

[x_{\min}, x_{\max}] \times [y_{\min}, y_{\max}]

, using an equally spaced set of

N_{1} \times N_{2}

points;

N_{1}

points in the

x

-direction and

N_{2}

points in the

y

-direction. The problem reduces to sampling a Gaussian random vector

X

of size

N_{1} \times N_{2}

, with mean vector zero and a symmetric covariance matrix

A

, which is an

N_{2} \times N_{2}

block Toeplitz matrix with Toeplitz blocks of size

N_{1} \times N_{1}

. 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 matrix

B

, which is an

M_{2} \times M_{2}

block circulant matrix with circulant blocks of size

M_{1} \times M_{1}

, where

M_{1} \geq 2 (N_{1} - 1)

and

M_{2} \geq 2 (N_{2} - 1)

B

can now be factorized as

B = W Λ W^{*} = R^{*} R

, where

W

is the two-dimensional Fourier matrix (

W^{*}

is the complex conjugate of

W

Λ

is the diagonal matrix containing the eigenvalues of

B

and

R = Λ^{\frac{1}{2}} W^{*}

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_{1} \times M_{2}

, 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

which has

M_{2}

blocks of size

M_{1}

. Two samples of

Y

can now be simulated from the real and imaginary parts of

R^{*} (U + i V)

, where

U

and

V

have elements from the standard Normal distribution. Since

R^{*} (U + i V) = W Λ^{\frac{1}{2}} (U + i V)

, this calculation can be done using a discrete Fourier transform of the vector

Λ^{\frac{1}{2}} (U + i V)

. Two samples of the random vector

X

can now be recovered by taking the first

N_{1}

elements of the first

N_{2}

blocks of each sample of

Y

– because the original covariance matrix

A

is embedded in

B

X

will have the correct distribution.

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

ρ B_{+}

where

B_{+} = W Λ_{+} W^{*}

and

ρ \in (0, 1]

is a scaling factor. The error

ε

in approximating the distribution of the random field is given by

ε = \sqrt{\frac{{(1 - ρ)}^{2} trace Λ + ρ^{2} trace Λ_{-}}{M}} .

Three choices for

ρ

are available, and are determined by the input argument icorr:

setting $icorr = 0$ sets

$ρ = \frac{trace Λ}{trace Λ_{+}},$
setting $icorr = 1$ sets

$ρ = \sqrt{\frac{trace Λ}{trace Λ_{+}}},$
setting $icorr = 2$ sets $ρ = 1$ .

g05zrf finds a suitable positive semidefinite embedding matrix

B

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

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 Arguments

1: $ns (2)$ – Integer array Input

On entry: the number of sample points to use in each direction, with

ns (1)

sample points in the

x

-direction,

N_{1}

and

ns (2)

sample points in the

y

-direction,

N_{2}

. The total number of sample points on the grid is, therefore,

ns (1) \times ns (2)

Constraints:

$ns (1) \geq 1$ ;
$ns (2) \geq 1$ .

2: $xmin$ – Real (Kind=nag_wp) Input

On entry: the lower bound for the

x

-coordinate, for the region in which the random field is to be simulated.

Constraint:

xmin < xmax

3: $xmax$ – Real (Kind=nag_wp) Input

On entry: the upper bound for the

x

-coordinate, for the region in which the random field is to be simulated.

Constraint:

xmin < xmax

4: $ymin$ – Real (Kind=nag_wp) Input

On entry: the lower bound for the

y

-coordinate, for the region in which the random field is to be simulated.

Constraint:

ymin < ymax

5: $ymax$ – Real (Kind=nag_wp) Input

On entry: the upper bound for the

y

-coordinate, for the region in which the random field is to be simulated.

Constraint:

ymin < ymax

6: $maxm (2)$ – Integer array Input

On entry: determines the maximum size of the circulant matrix to use – a maximum of

maxm (1)

elements in the

x

-direction, and a maximum of

maxm (2)

elements in the

y

-direction. The maximum size of the circulant matrix is thus

maxm (1)

\times

maxm (2)

Constraint:

maxm (i) \geq 2^{k}

, where

k

is the smallest integer satisfying

2^{k} \geq 2 (ns (i) - 1)

, for

i = 1, 2

7: $var$ – Real (Kind=nag_wp) Input

On entry: the multiplicative factor

σ^{2}

of the variogram

γ (x)

Constraint:

var \geq 0.0

8: $icov2$ – Integer Input

On entry: determines which of the preset variograms to use. The choices are given below. Note that

x^{'} = ‖ \frac{x}{ℓ_{1}}, \frac{y}{ℓ_{2}} ‖

, where

ℓ_{1}

and

ℓ_{2}

are correlation lengths in the

x

and

y

directions respectively and are parameters for most of the variograms, and

σ^{2}

is the variance specified by var.

$icov2 = 1$

Symmetric stable variogram

γ (x) = σ^{2} \exp (- {(x^{'})}^{ν}),

where

$ℓ_{1} = params (1)$ , $ℓ_{1} > 0$ ,
$ℓ_{2} = params (2)$ , $ℓ_{2} > 0$ ,
$ν = params (3)$ , $0 < ν \leq 2$ .

$icov2 = 2$

Cauchy variogram

γ (x) = σ^{2} {(1 + {(x^{'})}^{2})}^{- ν},

where

$ℓ_{1} = params (1)$ , $ℓ_{1} > 0$ ,
$ℓ_{2} = params (2)$ , $ℓ_{2} > 0$ ,
$ν = params (3)$ , $ν > 0$ .

$icov2 = 3$

Differential variogram with compact support

γ (x) = {\begin{matrix} σ^{2} (1 + 8 x^{'} + 25 {(x^{'})}^{2} + 32 {(x^{'})}^{3}) {(1 - x^{'})}^{8}, & x^{'} < 1, \\ 0, & x^{'} \geq 1, \end{matrix}

where

$ℓ_{1} = params (1)$ , $ℓ_{1} > 0$ ,
$ℓ_{2} = params (2)$ , $ℓ_{2} > 0$ .

$icov2 = 4$

Exponential variogram

γ (x) = σ^{2} \exp (- x^{'}),

where

$ℓ_{1} = params (1)$ , $ℓ_{1} > 0$ ,
$ℓ_{2} = params (2)$ , $ℓ_{2} > 0$ .

$icov2 = 5$

Gaussian variogram

γ (x) = σ^{2} \exp ({- (x^{'})}^{2}),

where

$ℓ_{1} = params (1)$ , $ℓ_{1} > 0$ ,
$ℓ_{2} = params (2)$ , $ℓ_{2} > 0$ .

$icov2 = 6$

Nugget variogram

γ (x) = {\begin{matrix} σ^{2}, & x = 0, \\ 0, & x \neq 0 . \end{matrix}

No parameters need be set for this value of icov2.

$icov2 = 7$

Spherical variogram

γ (x) = {\begin{matrix} σ^{2} (1 - 1.5 x^{'} + 0.5 {(x^{'})}^{3}), & x^{'} < 1, \\ 0, & x^{'} \geq 1, \end{matrix}

where

$ℓ_{1} = params (1)$ , $ℓ_{1} > 0$ ,
$ℓ_{2} = params (2)$ , $ℓ_{2} > 0$ .

$icov2 = 8$

Bessel variogram

γ (x) = σ^{2} \frac{2^{ν} Γ (ν + 1) J_{ν} (x^{'})}{{(x^{'})}^{ν}},

where

$J_{ν} (\cdot)$ is the Bessel function of the first kind,
$ℓ_{1} = params (1)$ , $ℓ_{1} > 0$ ,
$ℓ_{2} = params (2)$ , $ℓ_{2} > 0$ ,
$ν = params (3)$ , $ν \geq 0$ .

$icov2 = 9$

Hole effect variogram

γ (x) = σ^{2} \frac{\sin (x^{'})}{x^{'}},

where

$ℓ_{1} = params (1)$ , $ℓ_{1} > 0$ ,
$ℓ_{2} = params (2)$ , $ℓ_{2} > 0$ .

$icov2 = 10$

Whittle-Matérn variogram

γ (x) = σ^{2} \frac{2^{1 - ν} {(x^{'})}^{ν} K_{ν} (x^{'})}{Γ (ν)},

where

$K_{ν} (\cdot)$ is the modified Bessel function of the second kind,
$ℓ_{1} = params (1)$ , $ℓ_{1} > 0$ ,
$ℓ_{2} = params (2)$ , $ℓ_{2} > 0$ ,
$ν = params (3)$ , $ν > 0$ .

$icov2 = 11$

Continuously parameterised variogram with compact support

γ (x) = {\begin{matrix} σ^{2} \frac{2^{1 - ν} {(x^{'})}^{ν} K_{ν} (x^{'})}{Γ (ν)} (1 + 8 x^{''} + 25 {(x^{''})}^{2} + 32 {(x^{''})}^{3}) {(1 - x^{''})}^{8}, & x^{''} < 1, \\ 0, & x^{''} \geq 1, \end{matrix}

where

$x^{′′} = ‖ \frac{x^{'}}{ℓ_{1} s_{1}}, \frac{y^{'}}{ℓ_{2} s_{2}} ‖$ ,
$K_{ν} (\cdot)$ is the modified Bessel function of the second kind,
$ℓ_{1} = params (1)$ , $ℓ_{1} > 0$ ,
$ℓ_{2} = params (2)$ , $ℓ_{2} > 0$ ,
$s_{1} = params (3)$ , $s_{1} > 0$ ,
$s_{2} = params (4)$ , $s_{2} > 0$ ,
$ν = params (5)$ , $ν > 0$ .

$icov2 = 12$

Generalized hyperbolic distribution variogram

γ (x) = σ^{2} \frac{{(δ^{2} + {(x^{'})}^{2})}^{\frac{λ}{2}}}{δ^{λ} K_{λ} (κ δ)} K_{λ} (κ {(δ^{2} + {(x^{'})}^{2})}^{\frac{1}{2}}),

where

$K_{λ} (\cdot)$ is the modified Bessel function of the second kind,
$ℓ_{1} = params (1)$ , $ℓ_{1} > 0$ ,
$ℓ_{2} = params (2)$ , $ℓ_{2} > 0$ ,
$λ = params (3)$ , no constraint on $λ$ ,
$δ = params (4)$ , $δ > 0$ ,
$κ = params (5)$ , $κ > 0$ .

Constraint:

icov2 = 1

2

3

4

5

6

7

8

9

10

11

12

9: $norm$ – Integer Input

On entry: determines which norm to use when calculating the variogram.

$norm = 1$: The 1-norm is used, i.e., $‖ x, y ‖ = | x | + | y |$ .
$norm = 2$: The 2-norm (Euclidean norm) is used, i.e., $‖ x, y ‖ = \sqrt{x^{2} + y^{2}}$ .

Suggested value:

norm = 2

Constraint:

norm = 1

2

10: $np$ – Integer Input

On entry: the number of parameters to be set. Different covariance functions need a different number of parameters.

$icov2 = 6$: np must be set to $0$ .
$icov2 = 3$ , $4$ , $5$ , $7$ or $9$: np must be set to $2$ .
$icov2 = 1$ , $2$ , $8$ or $10$: np must be set to $3$ .
$icov2 = 11$ or $12$: np must be set to $5$ .

11: $params (np)$ – Real (Kind=nag_wp) array Input

On entry: the parameters for the variogram as detailed in the description of icov2.

Constraint: see icov2 for a description of the individual parameter constraints.

12: $pad$ – Integer Input

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

1

13: $icorr$ – Integer Input

On entry: determines which approximation to implement if required, as described in Section 3.

Suggested value:

icorr = 0

Constraint:

icorr = 0

1

2

14: $lam (maxm (1) \times maxm (2))$ – Real (Kind=nag_wp) array Output

On exit: contains the square roots of the eigenvalues of the embedding matrix.

15: $xx (ns (1))$ – Real (Kind=nag_wp) array Output

On exit: the points of the

x

-coordinates at which values of the random field will be output.

16: $yy (ns (2))$ – Real (Kind=nag_wp) array Output

On exit: the points of the

y

-coordinates at which values of the random field will be output.

17: $m (2)$ – Integer array Output

On exit:

m (1)

contains

M_{1}

, the size of the circulant blocks and

m (2)

contains

M_{2}

, the number of blocks, resulting in a final square matrix of size

M_{1} \times M_{2}

18: $approx$ – Integer Output

On exit: indicates whether approximation was used.

$approx = 0$: No approximation was used.
$approx = 1$: Approximation was used.

19: $rho$ – Real (Kind=nag_wp) Output

On exit: indicates the scaling of the covariance matrix.

rho = 1.0

unless approximation was used with

icorr = 0

1

20: $icount$ – Integer Output

On exit: indicates the number of negative eigenvalues in the embedding matrix which have had to be set to zero.

21: $eig (3)$ – Real (Kind=nag_wp) array Output

On exit: indicates information about the negative eigenvalues in the embedding matrix which have had to be set to zero.

eig (1)

contains the smallest eigenvalue,

eig (2)

contains the sum of the squares of the negative eigenvalues, and

eig (3)

contains the sum of the absolute values of the negative eigenvalues.

22: $ifail$ – Integer Input/Output

On entry: ifail must be set to

0

−1

1

to set behaviour on detection of an error; these values have no effect when no error is detected.

A value of

0

causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of

−1

means that an error message is printed while a value of

1

means that it is not.

If halting is not appropriate, the value

−1

1

is recommended. If message printing is undesirable, then the value

1

is recommended. Otherwise, the value

0

is recommended. 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

−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, $ns = [⟨ value ⟩, ⟨ value ⟩]$ .
Constraint: $ns (1) \geq 1$ , $ns (2) \geq 1$ .

$ifail = 2$: On entry, $xmin = ⟨ value ⟩$ and $xmax = ⟨ value ⟩$ .
Constraint: $xmin < xmax$ .

$ifail = 4$: On entry, $ymin = ⟨ value ⟩$ and $ymax = ⟨ value ⟩$ .
Constraint: $ymin < ymax$ .

$ifail = 6$: On entry, $maxm = [⟨ value ⟩, ⟨ value ⟩]$ .
Constraint: the minimum calculated value for maxm are $[⟨ value ⟩, ⟨ value ⟩]$ .
Where the minima of $maxm (i)$ is given by $2^{k}$ , where $k$ is the smallest integer satisfying $2^{k} \geq 2 (ns (i) - 1)$ , for $i = 1, 2$ .

$ifail = 7$: On entry, $var = ⟨ value ⟩$ .
Constraint: $var \geq 0.0$ .

$ifail = 8$: On entry, $icov2 = ⟨ value ⟩$ .
Constraint: $icov2 \geq 1$ and $icov2 \leq 12$ .

$ifail = 9$: On entry, $norm = ⟨ value ⟩$ .
Constraint: $norm = 1$ or $2$ .

$ifail = 10$: On entry, $np = ⟨ value ⟩$ .
Constraint: for $icov2 = ⟨ value ⟩$ , $np = ⟨ value ⟩$ .

$ifail = 11$: On entry, $params (⟨ value ⟩) = ⟨ value ⟩$ .
Constraint: dependent on icov2, see documentation.

$ifail = 12$: On entry, $pad = ⟨ value ⟩$ .
Constraint: $pad = 0$ or $1$ .

$ifail = 13$: On entry, $icorr = ⟨ value ⟩$ .
Constraint: $icorr = 0$ , $1$ or $2$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

If on exit

approx = 1

, 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

g05zrf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

g05zrf 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 routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

None.

10 Example

This example calls g05zrf to calculate the eigenvalues of the embedding matrix for

25

sample points on a

5 \times 5

grid of a two-dimensional random field characterized by the symmetric stable variogram (

icov2 = 1

The two plots shown below illustrate the random fields that can be generated by g05zsf using the eigenvalues calculated by g05zrf. These are for two realizations of a two-dimensional random field, based on eigenvalues of the embedding matrix for points on a