g05zsc (field_2d_generate) : NAG Library CL Interface, Mark 27

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 Normal 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 functions g05zqc or g05zrc along with g05zsc 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}

points;

N_{1}

points in the

x

-direction and

N_{2}

points 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 g05zqc or g05zrc for details of the approximation procedure.

g05zsc takes the square roots of the eigenvalues of the embedding matrix

B

, and its size vector

M

, as input and outputs

S

realizations of the random field in

Z

One of the initialization functions g05kfc (for a repeatable sequence if computed sequentially) or g05kgc (for a non-repeatable sequence) must be called prior to the first call to g05zsc.

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 Arguments

ns [2]

– const Integer Input

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

ns [0]

sample points in the

x

-direction and

ns [1]

sample points in the

y

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

ns [0] \times ns [1]

. This must be the same value as supplied to g05zqc or g05zrc when calculating the eigenvalues of the embedding matrix.

Constraints:

$ns [0] \geq 1$ ;
$ns [1] \geq 1$ .

s

– Integer Input

On entry:

S

, the number of realizations of the random field to simulate.

Constraint:

s \geq 1

m [2]

– const Integer Input

On entry: indicates the size,

M

, of the embedding matrix as returned by g05zqc or g05zrc. The embedding matrix is a block circulant matrix with circulant blocks.

m [0]

is the size of each block, and

m [1]

is the number of blocks.

Constraints:

$m [0] \geq \max (1, 2 (ns [0] - 1))$ ;
$m [1] \geq \max (1, 2 (ns [1] - 1))$ .

lam [m [0] \times m [1]]

– const double Input

On entry: contains the square roots of the eigenvalues of the embedding matrix, as returned by g05zqc or g05zrc.

Constraint:

lam [i - 1] \geq 0

, for

i = 1, 2, \dots, m [0] \times m [1]

rho

– double Input

On entry: indicates the scaling of the covariance matrix, as returned by g05zqc or g05zrc.

Constraint:

0.0 < rho \leq 1.0

state [\dim]

– Integer Communication Array

Note: the dimension,

\dim

, of this array is dictated by the requirements of associated functions that must have been previously called. This array MUST be the same array passed as argument state in the previous call to nag_rand_init_repeatable (g05kfc) or nag_rand_init_nonrepeatable (g05kgc).

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

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

z [\dim]

– double Output

Note: the dimension, dim, of the array z must be at least

s \times ns [0] \times ns [1]

On exit: contains the realizations of the random field. The

k

th realization (where

k = 1, 2, \dots, s

) of the random field on the two-dimensional grid

(x_{i}, y_{j})

is stored in

z [(k - 1) \times ns [0] \times ns [1] + (j - 1) \times ns [0] + i - 1]

, for

i = 1, 2, \dots, ns [0]

and for

j = 1, 2, \dots, ns [1]

. The points are returned in xx and yy by g05zqc or g05zrc.

fail

– NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL

Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.

NE_BAD_PARAM

On entry, argument

〈value〉

had an illegal value.

NE_INT

On entry,

s = 〈value〉

.
Constraint:

s \geq 1

NE_INT_ARRAY

On entry,

ns = [〈value〉, 〈value〉]

.
Constraint:

ns [0] \geq 1

ns [1] \geq 1

NE_INT_ARRAY_2

On entry,

m = [〈value〉, 〈value〉]

, and

ns = [〈value〉, 〈value〉]

.
Constraints:

m [i - 1] \geq \max (1, 2 (ns [i - 1]) - 1)

, for

i = 1, 2

NE_INTERNAL_ERROR

An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.

NE_INVALID_STATE

On entry, state vector has been corrupted or not initialized.

NE_NEG_ELEMENT

On entry, at least one element of lam was negative.
Constraint: all elements of lam must be non-negative.

NE_NO_LICENCE

Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.

NE_REAL

On entry,

rho = 〈value〉

.
Constraint:

0.0 < rho \leq 1.0

8 Parallelism and Performance

g05zsc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

g05zsc makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.

Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

NAG CL Interface
g05zsc (field_2d_generate)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG CL Interfaceg05zsc (field_​2d_​generate)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG CL Interface
g05zsc (field_2d_generate)