G05ZRF : NAG Library, Mark 24

Note: before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

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.

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 Gaussian 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}

gridpoints;

N_{1}

gridpoints in the

x

-direction and

N_{2}

gridpoints 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}

N_{2}

block Toeplitz matrix with Toeplitz blocks of size

N_{1}

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}

M_{2}

block circulant matrix with circulant blocks of size

M_{1}

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 parameter 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.

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

On entry: the number of sample points (gridpoints) 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:

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

On entry: the lower bound for the

x

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

Constraint:

XMIN < XMAX

On entry: the upper bound for the

x

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

Constraint:

XMIN < XMAX

On entry: the lower bound for the

y

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

Constraint:

YMIN < YMAX

On entry: the upper bound for the

y

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

Constraint:

YMIN < YMAX

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

On entry: the multiplicative factor

σ^{2}

of the variogram

γ (x)

Constraint:

VAR \geq 0.0

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 $λ$ ,
$δ = PARAMS (4)$ , $δ > 0$ ,
$κ = PARAMS (5)$ , $κ > 0$ .

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

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 $5$ .

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.

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

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

Suggested value:

ICORR = 0

Constraint:

ICORR = 0

1

2

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

On exit: the gridpoints of the

x

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

On exit: the gridpoints of the

y

-coordinates at which values of the random field will be 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}

On exit: indicates whether approximation was used.

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

On exit: indicates the scaling of the covariance matrix.

RHO = 1

unless approximation was used with

ICORR = 0

1

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

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.

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).

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:

On entry,

NS = [⟨value⟩, ⟨value⟩]

.
Constraint:

NS (1) \geq 1

NS (2) \geq 1

On entry,

XMIN = ⟨value⟩

and

XMAX = ⟨value⟩

.
Constraint:

XMIN < XMAX

On entry,

YMIN = ⟨value⟩

and

YMAX = ⟨value⟩

.
Constraint:

YMIN < YMAX

On entry,

MAXM = [⟨value⟩, ⟨value⟩]

.
Constraint: the calculated minimum value for MAXM are

[⟨value⟩, ⟨value⟩]

Where the minimum calculated value of

MAXM (i)

is given by

2^{k}

, where

k

is the smallest integer satisfying

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

On entry,

VAR = ⟨value⟩

.
Constraint:

VAR \geq 0.0

On entry,

ICOV2 = ⟨value⟩

.
Constraint:

ICOV2 \geq 1

and

ICOV2 \leq 12

On entry,

NORM = ⟨value⟩

.
Constraint:

NORM = 1

2

On entry,

NP = ⟨value⟩

.
Constraint: for

ICOV2 = ⟨value⟩

NP = ⟨value⟩

On entry,

PARAMS (⟨value⟩) = ⟨value⟩

.
Constraint: dependent on ICOV2, see documentation.

On entry,

PAD = ⟨value⟩

.
Constraint:

PAD = 0

1

On entry,

ICORR = ⟨value⟩

.
Constraint:

ICORR = 0

1

2

This example calls G05ZRF to calculate the eigenvalues of the embedding matrix for

25

sample points on a

5

5

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

ICOV2 = 1

NAG Library Routine Document

G05ZRF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine DocumentG05ZRF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

G05ZRF