NAG Library Routine Document

G05ZMF

+− 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

G05ZMF performs the setup required in order to simulate stationary Gaussian random fields in one dimension, 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 G05ZPF, which simulates the random field.

2 Specification

SUBROUTINE G05ZMF (

NS, XMIN, XMAX, MAXM, VAR, COV1, PAD, ICORR, LAM, XX, M, APPROX, RHO, ICOUNT, EIG, IUSER, RUSER, IFAIL)

INTEGER	NS, MAXM, PAD, ICORR, M, APPROX, ICOUNT, IUSER(*), IFAIL
REAL (KIND=nag_wp)	XMIN, XMAX, VAR, LAM(MAXM), XX(NS), RHO, EIG(3), RUSER(*)
EXTERNAL	COV1

3 Description

A one-dimensional random field

Z (x)

ℝ

is a function which is random at every point

x \in ℝ

, 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

, 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

and

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

for all

x, y, a \in ℝ

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 and G05ZPF are used to simulate a one-dimensional stationary Gaussian random field, with mean function zero and variogram

γ (x)

, over an interval

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

, using an equally spaced set of

N

gridpoints. The problem reduces to sampling a Gaussian random vector

X

of size

N

, with mean vector zero and a symmetric Toeplitz covariance matrix

A

. 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 circulant matrix

B

of size

M \geq 2 (N - 1)

, which can now be factorized as

B = W Λ W^{*} = R^{*} R

, where

W

is the 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

, 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

, two samples of which 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

elements 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 $ρ = 1$ .

G05ZMF finds a suitable positive semidefinite embedding matrix

B

and outputs its size, 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 – INTEGERInput

On entry: the number of sample points (points) to be generated in realisations of the random field.

Constraint:

NS \geq 1

2: XMIN – REAL (KIND=nag_wp)Input

On entry: the lower bound for the interval over 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 interval over which the random field is to be simulated.

Constraint:

XMIN < XMAX

4: MAXM – INTEGERInput

On entry: the maximum size of the circulant matrix to use. For example, if the embedding matrix is to be allowed to double in size three times before the approximation procedure is used, then choose

MAXM = 2^{k + 2}

where

k = 1 + ⌈\log_{2} (NS - 1)⌉

Constraint:

MAXM \geq 2^{k}

, where

k

is the smallest integer satisfying

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

5: VAR – REAL (KIND=nag_wp)Input

On entry: the multiplicative factor

σ^{2}

of the variogram

γ (x)

Constraint:

VAR \geq 0.0

6: COV1 – SUBROUTINE, supplied by the user.External Procedure

COV1 must evaluate the variogram

γ (x)

, without the multiplicative factor

σ^{2}

, for all

x \geq 0

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

The specification of COV1 is:

SUBROUTINE COV1 (

X, GAMMA, IUSER, RUSER)

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

1: X – REAL (KIND=nag_wp)Input: On entry: the value $x$ at which the variogram $γ (x)$ is to be evaluated.
2: GAMMA – REAL (KIND=nag_wp)Output: On exit: the value of the variogram $\frac{γ (x)}{σ^{2}}$ .
3: IUSER( $*$ ) – INTEGER arrayUser Workspace
4: RUSER( $*$ ) – REAL (KIND=nag_wp) arrayUser Workspace: COV1 is called with the parameters IUSER and RUSER as supplied to G05ZMF. You are free to use the arrays IUSER and RUSER to supply information to COV1 as an alternative to using COMMON global variables.

COV1 must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which G05ZMF is called. Parameters denoted as Input must not be changed by this procedure.

7: 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

8: 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

9: LAM(MAXM) – REAL (KIND=nag_wp) arrayOutput

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

10: XX(NS) – REAL (KIND=nag_wp) arrayOutput

On exit: the points at which values of the random field will be output.

11: M – INTEGEROutput

On exit: the size of the embedding matrix.

12: APPROX – INTEGEROutput

On exit: indicates whether approximation was used.

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

13: 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

14: ICOUNT – INTEGEROutput

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

15: 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.

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

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

IUSER and RUSER are not used by G05ZMF, but are passed directly to COV1 and may be used to pass information to this routine as an alternative to using COMMON global variables.

18: 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, $NS = ⟨value⟩$ .
Constraint: $NS \geq 1$ .

$IFAIL = 2$: On entry, $XMIN = ⟨value⟩$ and $XMAX = ⟨value⟩$ .
Constraint: $XMIN < XMAX$ .

$IFAIL = 4$: On entry, $MAXM = ⟨value⟩$ .
Constraint: the calculated minimum value for MAXM is $⟨value⟩$ .
Where the minimum calculated value is given by $2^{k}$ , where $k$ is the smallest integer satisfying $2^{k} \geq 2 (NS - 1)$ .

$IFAIL = 5$: On entry, $VAR = ⟨value⟩$ .
Constraint: $VAR \geq 0.0$ .

$IFAIL = 7$: On entry, $PAD = ⟨value⟩$ .
Constraint: $PAD = 0$ or $1$ .

$IFAIL = 8$: On entry, $ICORR = ⟨value⟩$ .
Constraint: $ICORR = 0$ , $1$ or $2$ .

7 Accuracy

Not applicable.

8 Further Comments

None.

9 Example

This example calls G05ZMF to calculate the eigenvalues of the embedding matrix for

8

sample points of a random field characterized by the symmetric stable variogram:

γ (x) = σ^{2} \exp (- {(x^{'})}^{ν}),

where

x^{'} = \frac{x}{ℓ}

, and

ℓ

and

ν

are parameters.

It should be noted that the symmetric stable variogram is one of the pre-defined variograms available in G05ZNF. It is used here purely for illustrative purposes.

NAG Library Routine DocumentG05ZMF

+− 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

G05ZMF