g05zrc 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 g05zsc, which simulates the random field.

2 Specification

#include <nag.h>

void

g05zrc (const Integer ns[], double xmin, double xmax, double ymin, double ymax, const Integer maxm[], double var, Nag_Variogram cov, Nag_NormType norm, Integer np, const double params[], Nag_EmbedPad pad, Nag_EmbedScale corr, double lam[], double xx[], double yy[], Integer m[], Integer *approx, double *rho, Integer *icount, double eig[], NagError *fail)

The function may be called by the names: g05zrc or nag_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 functions g05zrc and g05zsc 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 corr:

setting $corr = Nag_EmbedScaleTraces$ sets

$ρ = \frac{trace Λ}{trace Λ_{+}},$
setting $corr = Nag_EmbedScaleSqrtTraces$ sets

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

g05zrc 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: $ns [2]$ – const Integer Input

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

ns [0]

sample points in the

x

-direction,

N_{1}

and

ns [1]

sample points in the

y

-direction,

N_{2}

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

ns [0] \times ns [1]

Constraints:

$ns [0] \geq 1$ ;
$ns [1] \geq 1$ .

2: $xmin$ – double 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: $xmax$ – double 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: $ymin$ – double 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: $ymax$ – double 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: $maxm [2]$ – const Integer Input

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

maxm [0]

elements in the

x

-direction, and a maximum of

maxm [1]

elements in the

y

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

maxm [0]

\times

maxm [1]

Constraint:

maxm [i] \geq 2^{k}

, where

k

is the smallest integer satisfying

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

, for

i = 0, 1

7: $var$ – double Input

On entry: the multiplicative factor

σ^{2}

of the variogram

γ (x)

Constraint:

var \geq 0.0

8: $cov$ – Nag_Variogram 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.

$cov = Nag_VgmSymmStab$

Symmetric stable variogram

γ (x) = σ^{2} \exp (- {(x^{'})}^{ν}),

where

$ℓ_{1} = params [0]$ , $ℓ_{1} > 0$ ,
$ℓ_{2} = params [1]$ , $ℓ_{2} > 0$ ,
$ν = params [2]$ , $0 < ν \leq 2$ .

$cov = Nag_VgmCauchy$

Cauchy variogram

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

where

$ℓ_{1} = params [0]$ , $ℓ_{1} > 0$ ,
$ℓ_{2} = params [1]$ , $ℓ_{2} > 0$ ,
$ν = params [2]$ , $ν > 0$ .

$cov = Nag_VgmDifferential$

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

$ℓ_{1} = params [0]$ , $ℓ_{1} > 0$ ,
$ℓ_{2} = params [1]$ , $ℓ_{2} > 0$ .

$cov = Nag_VgmExponential$

Exponential variogram

γ (x) = σ^{2} \exp (- x^{'}),

where

$ℓ_{1} = params [0]$ , $ℓ_{1} > 0$ ,
$ℓ_{2} = params [1]$ , $ℓ_{2} > 0$ .

$cov = Nag_VgmGauss$

Gaussian variogram

γ (x) = σ^{2} \exp ({- (x^{'})}^{2}),

where

$ℓ_{1} = params [0]$ , $ℓ_{1} > 0$ ,
$ℓ_{2} = params [1]$ , $ℓ_{2} > 0$ .

$cov = Nag_VgmNugget$

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 cov.

$cov = Nag_VgmSpherical$

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

$ℓ_{1} = params [0]$ , $ℓ_{1} > 0$ ,
$ℓ_{2} = params [1]$ , $ℓ_{2} > 0$ .

$cov = Nag_VgmBessel$

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,
$ℓ_{1} = params [0]$ , $ℓ_{1} > 0$ ,
$ℓ_{2} = params [1]$ , $ℓ_{2} > 0$ ,
$ν = params [2]$ , $ν \geq 0$ .

$cov = Nag_VgmHole$

Hole effect variogram

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

where

$ℓ_{1} = params [0]$ , $ℓ_{1} > 0$ ,
$ℓ_{2} = params [1]$ , $ℓ_{2} > 0$ .

$cov = Nag_VgmWhittleMatern$

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,
$ℓ_{1} = params [0]$ , $ℓ_{1} > 0$ ,
$ℓ_{2} = params [1]$ , $ℓ_{2} > 0$ ,
$ν = params [2]$ , $ν > 0$ .

$cov = Nag_VgmContParam$

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,
$ℓ_{1} = params [0]$ , $ℓ_{1} > 0$ ,
$ℓ_{2} = params [1]$ , $ℓ_{2} > 0$ ,
$s_{1} = params [2]$ , $s_{1} > 0$ ,
$s_{2} = params [3]$ , $s_{2} > 0$ ,
$ν = params [4]$ , $ν > 0$ .

$cov = Nag_VgmGenHyp$

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,
$ℓ_{1} = params [0]$ , $ℓ_{1} > 0$ ,
$ℓ_{2} = params [1]$ , $ℓ_{2} > 0$ ,
$λ = params [2]$ , no constraint on $λ$ ,
$δ = params [3]$ , $δ > 0$ ,
$κ = params [4]$ , $κ > 0$ .

Constraint:

cov = Nag_VgmSymmStab

Nag_VgmCauchy

Nag_VgmDifferential

Nag_VgmExponential

Nag_VgmGauss

Nag_VgmNugget

Nag_VgmSpherical

Nag_VgmBessel

Nag_VgmHole

Nag_VgmWhittleMatern

Nag_VgmContParam

Nag_VgmGenHyp

9: $norm$ – Nag_NormType Input

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

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

Suggested value:

norm = Nag_TwoNorm

Constraint:

norm = Nag_OneNorm

Nag_TwoNorm

10: $np$ – Integer Input

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

$cov = Nag_VgmNugget$: np must be set to $0$ .
$cov = Nag_VgmDifferential$ , $Nag_VgmExponential$ , $Nag_VgmGauss$ , $Nag_VgmSpherical$ or $Nag_VgmHole$: np must be set to $2$ .
$cov = Nag_VgmSymmStab$ , $Nag_VgmCauchy$ , $Nag_VgmBessel$ or $Nag_VgmWhittleMatern$: np must be set to $3$ .
$cov = Nag_VgmContParam$ or $Nag_VgmGenHyp$: np must be set to $5$ .

11: $params [np]$ – const double Input

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

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

12: $pad$ – Nag_EmbedPad 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.

$pad = Nag_EmbedPadZeros$: The embedding matrix is padded with zeros.
$pad = Nag_EmbedPadValues$: The embedding matrix is padded with values of the variogram.

Suggested value:

pad = Nag_EmbedPadValues

Constraint:

pad = Nag_EmbedPadZeros

Nag_EmbedPadValues

13: $corr$ – Nag_EmbedScale Input

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

Suggested value:

corr = Nag_EmbedScaleTraces

Constraint:

corr = Nag_EmbedScaleTraces

Nag_EmbedScaleSqrtTraces

Nag_EmbedScaleOne

14: $lam [maxm [0] \times maxm [1]]$ – double Output

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

15: $xx [ns [0]]$ – double Output

On exit: the points of the

x

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

16: $yy [ns [1]]$ – double Output

On exit: the points of the

y

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

17: $m [2]$ – Integer Output

On exit:

m [0]

contains

M_{1}

, the size of the circulant blocks and

m [1]

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: $rho$ – double * Output

On exit: indicates the scaling of the covariance matrix.

rho = 1.0

unless approximation was used with

corr = Nag_EmbedScaleTraces

Nag_EmbedScaleSqrtTraces

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: $eig [3]$ – double Output

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

eig [0]

contains the smallest eigenvalue,

eig [1]

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

eig [2]

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

22: $fail$ – 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 $⟨ value ⟩$ had an illegal value.
NE_ENUM_INT: On entry, $np = ⟨ value ⟩$ .
Constraint: for $cov = ⟨ value ⟩$ , $np = ⟨ value ⟩$ .
NE_ENUM_REAL_1: On entry, $params [⟨ value ⟩] = ⟨ value ⟩$ .
Constraint: dependent on cov, see documentation.
NE_INT_ARRAY: On entry, $maxm = [⟨ value ⟩, ⟨ value ⟩]$ .
Constraint: the minimum calculated value for maxm are $[⟨ value ⟩, ⟨ value ⟩]$ .
Where the minima of $maxm [i - 1]$ is given by $2^{k}$ , where $k$ is the smallest integer satisfying $2^{k} \geq 2 (ns [i - 1] - 1)$ , for $i = 1, 2$ .

On entry, $ns = [⟨ value ⟩, ⟨ value ⟩]$ .
Constraint: $ns [0] \geq 1$ , $ns [1] \geq 1$ .
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_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, $var = ⟨ value ⟩$ .
Constraint: $var \geq 0.0$ .
NE_REAL_2: On entry, $xmin = ⟨ value ⟩$ and $xmax = ⟨ value ⟩$ .
Constraint: $xmin < xmax$ .

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

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

g05zrc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

g05zrc 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.

9 Further Comments

None.

10 Example

This example calls g05zrc 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 (

cov = Nag_VgmSymmStab

The two plots shown below illustrate the random fields that can be generated by g05zsc using the eigenvalues calculated by g05zrc. These are for two realizations of a two-dimensional random field, based on eigenvalues of the embedding matrix for points on a