NAG Library Routine Document

G05ZSF

+− 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

1 Purpose

G05ZSF produces realisations of a stationary Gaussian random field in two dimensions, using the circulant embedding method. The square roots of the eigenvalues of the extended covariance matrix (or embedding matrix) need to be input, and can be calculated using G05ZQF or G05ZRF.

2 Specification

SUBROUTINE G05ZSF (

NS, S, M, LAM, RHO, STATE, Z, IFAIL)

INTEGER	NS(2), S, M(2), STATE(*), IFAIL
REAL (KIND=nag_wp)	LAM(M(1)M(2)), RHO, Z(NS(1)NS(2),S)

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 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 G05ZMF or G05ZNF along with G05ZPF 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 bocks 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.

The symmetry of

A

as a block matrix, and the symmetry of each block of

A

, depends on whether the covariance function

γ

is even or not.

γ

is even if

γ (x) = γ (- x)

for all

x \in ℝ^{2}

, and uneven otherwise (in higher dimensions,

γ

can be even in some coordinates and uneven in others, but in two dimensions

γ

is either even in both coordinates or uneven in both coordinates). If

γ

is even then

A

is a symmetric block matrix and has symmetric blocks; if

γ

is uneven then

A

is not a symmetric block matrix and has non-symmetric blocks. In the uneven case,

M_{1}

and

M_{2}

are set to be odd in order to guarantee symmetry in

B

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. See the documentation of G05ZQF or G05ZRF for details of the approximation procedure.

G05ZSF takes the square roots of the eigenvalues of the embedding matrix

B

, and its size vector

M

, as input and outputs

S

realisations of the random field in

Z

One of the initialization routines G05KFF (for a repeatable sequence if computed sequentially) or G05KGF (for a non-repeatable sequence) must be called prior to the first call to G05ZSF.

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 (1994) Simulation of stationary Gaussian processes in

{[0, 1]}^{d}

Journal of Computational and Graphical Statistics 3(4) 409–432

5 Parameters

1: NS( $2$ ) – INTEGER arrayInput

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

NS (1)

sample points in the

x

-direction and

NS (2)

sample points in the

y

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

NS (1) \times NS (1)

. This must be the same value as supplied to G05ZQF or G05ZRF when calculating the eigenvalues of the embedding matrix.

Constraints:

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

2: S – INTEGERInput

On entry: the number of realisations of the random field to simulate.

Constraint:

S \geq 1

3: M( $2$ ) – INTEGER arrayInput

On entry: indicates the size of the embedding matrix as returned by G05ZQF or G05ZRF. The embedding matrix is a block circulant matrix with circulant blocks.

M (1)

is the size of each block, and

M (2)

is the number of blocks.

Constraints:

$M (1) \geq \max (1, 2 (NS (1) - 1))$ ;
$M (2) \geq \max (1, 2 (NS (2) - 1))$ .

4: LAM( $M (1) \times M (2)$ ) – REAL (KIND=nag_wp) arrayInput

On entry: contains the square roots of the eigenvalues of the embedding matrix, as returned by G05ZQF or G05ZRF.

Constraint:

LAM (i) = 0

i = 1, 2, \dots, M (1) \times M (2)

5: RHO – REAL (KIND=nag_wp)Input

On entry: indicates the scaling of the covariance matrix, as returned by G05ZQF or G05ZRF.

Constraint:

0.0 < RHO \leq 1.0

6: STATE( $*$ ) – INTEGER arrayCommunication Array

Note: the actual argument supplied must be the array STATE supplied to the initialization routines G05KFF or G05KGF.

On entry: contains information on the selected base generator and its current state.

On exit: contains updated information on the state of the generator.

7: Z( $NS (1) \times NS (2)$ ,S) – REAL (KIND=nag_wp) arrayOutput

On exit: contains the realisations of the random field. Each column of

Z

contains one realisation of the random field, with

Z (i + (j - 1) NS (1), k)

, for

k = 1, 2, \dots, S

, corresponding to the gridpoint

XX (i)

and

YY (j)

as returned by G05ZQF or G05ZRF.

8: 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

- 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, $S = ⟨value⟩$ .
Constraint: $S \geq 1$ .

$IFAIL = 3$: On entry, $M = [⟨value⟩, ⟨value⟩]$ , and $NS = [⟨value⟩, ⟨value⟩]$ .
Constraints: $M (i) \geq \max (1, 2 (NS (i)) - 1)$ , for $i = 1, 2$ .

$IFAIL = 4$: On entry, at least one element of LAM was negative.
Constraint: all elements of LAM must be non-negative.

$IFAIL = 5$: On entry, $RHO = ⟨value⟩$ .
Constraint: $0.0 < RHO \leq 1.0$ .

$IFAIL = 6$: On entry, STATE vector has been corrupted or not initialized.

7 Accuracy

Not applicable.

8 Further Comments

Because samples are generated in pairs, calling this routine

k

times, with

S = s

, say, will generate a different sequence of numbers than calling the routine once with

S = k s

, unless

s

is even.

9 Example

This example calls G05ZSF to generate

5

realisations of a two-dimensional random field on a

5

5

grid. This uses eigenvalues of the embedding covariance matrix for a symmetric stable variogram as calculated by G05ZRF with

ICOV2 = 1

NAG Library Routine DocumentG05ZSF

+− 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

G05ZSF