NAG Library Routine Document

G05ZPF

+− 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

G05ZPF produces realisations of a stationary Gaussian random field in one dimension, 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 G05ZMF or G05ZNF.

2 Specification

SUBROUTINE G05ZPF (

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

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

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 non-negative definite 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 or G05ZNF, along with 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 non-negative definite),

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 non-negative definite, 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 G05ZMF or G05ZNF for details of the approximation procedure.

G05ZPF takes the square roots of the eigenvalues of the embedding matrix

B

, and its size

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 G05ZPF.

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 (grid points) to be generated in realisations of the random field. This must be the same value as supplied to G05ZMF or G05ZNF when calculating the eigenvalues of the embedding matrix.
Constraint: $NS \geq 1$ .
2: S – INTEGERInput: On entry: the number of realisations of the random field to simulate.
Constraint: $S \geq 1$ .
3: M – INTEGERInput: On entry: the size of the embedding matrix, as returned by G05ZMF and G05ZNF.
Constraint: $M \geq \max (1, 2 (NS - 1))$ .
4: LAM(M) – REAL (KIND=nag_wp) arrayInput: On entry: must contain the square roots of the eigenvalues of the embedding matrix, as returned by G05ZMF and G05ZNF.
Constraint: $LAM (i) \geq 0, i = 1, 2, \dots, M$ .
5: RHO – REAL (KIND=nag_wp)Input: On entry: indicates the scaling of the covariance matrix, as returned by G05ZMF and G05ZNF.
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,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 $Z (i, j)$ , for $j = 1, 2, \dots, S$ , corresponding to the gridpoint $XX (i)$ as returned by G05ZMF or G05ZNF.
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, $NS = ⟨value⟩$ .
Constraint: $NS \geq 1$ .

$IFAIL = 2$: On entry, $S = ⟨value⟩$ .
Constraint: $S \geq 1$ .

$IFAIL = 3$: On entry, $M = ⟨value⟩$ and $NS = ⟨value⟩$ .
Constraint: $M \geq \max (1, 2 \times (NS - 1))$ .

$IFAIL = 4$: 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: $0.0 \leq RHO \leq 1.0$ .

$IFAIL = 6$: 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 G05ZPF to generate

5

realisations of a random field on

8

sample points using eigenvalues calculated by G05ZNF for a symmetric stable variogram.

NAG Library Routine DocumentG05ZPF