NAG Library Routine Document

G05ZQF

+− 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

G05ZQF performs the setup required in order to simulate stationary Gaussian random fields in two dimensions, for a user-defined 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

SUBROUTINE G05ZQF (

NS, XMIN, XMAX, YMIN, YMAX, MAXM, VAR, COV2, EVEN, PAD, ICORR, LAM, XX, YY, M, APPROX, RHO, ICOUNT, EIG, IUSER, RUSER, IFAIL)

INTEGER	NS(2), MAXM(2), EVEN, PAD, ICORR, M(2), APPROX, ICOUNT, IUSER(*), IFAIL
REAL (KIND=nag_wp)	XMIN, XMAX, YMIN, YMAX, VAR, LAM(MAXM(1)MAXM(2)), XX(NS(1)), YY(NS(2)), RHO, EIG(3), RUSER()
EXTERNAL	COV2

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 G05ZQF 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.

The symmetry of

A

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

A

, depends on whether the variogram

γ

is even or not.

γ

is even in its first coordinate if

γ ({[{- x}_{1}, x_{2}]}^{T}) = γ ({[x_{1}, x_{2}]}^{T})

, even in its second coordinate if

γ ({[x_{1}, {- x}_{2}]}^{T}) = γ ({[x_{1}, x_{2}]}^{T})

, and even if it is even in both coordinates (in two dimensions it is impossible for

γ

to be even in one coordinate and uneven in the other). 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. 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 $ρ = 1$ .

G05ZQF 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 (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,

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 arrayInput

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)

Constraints:

if $EVEN = 1$ , $MAXM (i) \geq 2^{k}$ , where $k$ is the smallest integer satisfying $2^{k} \geq 2 (NS (i) - 1)$ , for $i = 1, 2$ ;
if $EVEN = 0$ , $MAXM (i) \geq 3^{k}$ , where $k$ is the smallest integer satisfying $3^{k} \geq 2 (NS (i) - 1)$ , 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: COV2 – SUBROUTINE, supplied by the user.External Procedure

COV2 must evaluate the variogram

γ (x)

for all

x

EVEN = 0

, and for all

x

with non-negative entries if

EVEN = 1

. The value returned in GAMMA is multiplied internally by VAR.

The specification of COV2 is:

SUBROUTINE COV2 (

X, Y, GAMMA, IUSER, RUSER)

INTEGER	IUSER(*)
REAL (KIND=nag_wp)	X, Y, GAMMA, RUSER(*)

1: X – REAL (KIND=nag_wp)Input: On entry: the coordinate $x$ at which the variogram $γ (x)$ is to be evaluated.
2: Y – REAL (KIND=nag_wp)Input: On entry: the coordinate $y$ at which the variogram $γ (x)$ is to be evaluated.
3: GAMMA – REAL (KIND=nag_wp)Output: On exit: the value of the variogram $γ (x)$ .
4: IUSER( $*$ ) – INTEGER arrayUser Workspace
5: RUSER( $*$ ) – REAL (KIND=nag_wp) arrayUser Workspace: COV2 is called with the parameters IUSER and RUSER as supplied to G05ZQF. You are free to use the arrays IUSER and RUSER to supply information to COV2 as an alternative to using COMMON global variables.

COV2 must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which G05ZQF is called. Parameters denoted as Input must not be changed by this procedure.

9: EVEN – INTEGERInput

On entry: indicates whether the covariance function supplied is even or uneven.

$EVEN = 0$: The covariance function is uneven.
$EVEN = 1$: The covariance function is even.

Constraint:

EVEN = 0

1

10: 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

1

11: ICORR – INTEGERInput

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

Suggested value:

ICORR = 0

Constraint:

ICORR = 0

1

2

12: LAM( $MAXM (1) \times MAXM (2)$ ) – REAL (KIND=nag_wp) arrayOutput

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

13: XX( $NS (1)$ ) – REAL (KIND=nag_wp) arrayOutput

On exit: the gridpoints of the

x

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

14: YY( $NS (2)$ ) – REAL (KIND=nag_wp) arrayOutput

On exit: the gridpoints of the

y

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

15: M( $2$ ) – INTEGER arrayOutput

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}

16: APPROX – INTEGEROutput

On exit: indicates whether approximation was used.

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

17: RHO – REAL (KIND=nag_wp)Output

On exit: indicates the scaling of the covariance matrix.

RHO = 1

unless approximation was used with

ICORR = 0

1

18: ICOUNT – INTEGEROutput

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

19: 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.

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.

20: IUSER( $*$ ) – INTEGER arrayUser Workspace

21: RUSER( $*$ ) – REAL (KIND=nag_wp) arrayUser Workspace

IUSER and RUSER are not used by G05ZQF, but are passed directly to COV2 and may be used to pass information to this routine as an alternative to using COMMON global variables.

22: 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, $XMIN = ⟨value⟩$ 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 calculated minimum value for MAXM are $[⟨value⟩, ⟨value⟩]$ .
Where, if $EVEN = 1$ , the minimum calculated value of $MAXM (i)$ is given by $2^{k}$ , where $k$ is the smallest integer satisfying $2^{k} \geq 2 (NS (i) - 1)$ , and if $EVEN = 0$ , the minimum calculated value of $MAXM (i)$ is given by $3^{k}$ , where $k$ is the smallest integer satisfying $3^{k} \geq 2 NS (i) - 1$ , for $i = 1, 2$ .

$IFAIL = 7$: On entry, $VAR = ⟨value⟩$ .
Constraint: $VAR \geq 0.0$ .

$IFAIL = 9$: On entry, $EVEN = ⟨value⟩$ .
Constraint: $EVEN = 0$ or $1$ .

$IFAIL = 10$: On entry, $PAD = ⟨value⟩$ .
Constraint: $PAD = 0$ or $1$ .

$IFAIL = 11$: On entry, $ICORR = ⟨value⟩$ .
Constraint: $ICORR = 0$ , $1$ or $2$ .

7 Accuracy

Not applicable.

8 Further Comments

None.

9 Example

This example calls G05ZQF 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:

γ (x) = σ^{2} \exp (- {(x^{'})}^{ν}),

where

x^{'} = |\frac{x}{ℓ_{1}} + \frac{y}{ℓ_{2}}|

, and

ℓ_{1}

ℓ_{2}

and

ν

are parameters.

It should be noted that the symmetric stable variogram is one of the pre-defined variograms available in G05ZRF. It is used here purely for illustrative purposes.

NAG Library Routine DocumentG05ZQF

+− 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

G05ZQF