nag_rand_field_1d_generate (g05zpc) : NAG Library, Mark 24

1 Purpose

nag_rand_field_1d_generate (g05zpc) produces realizations 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 nag_rand_field_1d_user_setup (g05zmc) or nag_rand_field_1d_predef_setup (g05znc).

2 Specification

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 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

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 functions nag_rand_field_1d_user_setup (g05zmc) or nag_rand_field_1d_predef_setup (g05znc), along with nag_rand_field_1d_generate (g05zpc), 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

points. The problem reduces to sampling a Normal 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 nag_rand_field_1d_user_setup (g05zmc) or nag_rand_field_1d_predef_setup (g05znc) for details of the approximation procedure.

nag_rand_field_1d_generate (g05zpc) takes the square roots of the eigenvalues of the embedding matrix

B

, and its size

M

, as input and outputs

S

realizations of the random field in

Z

One of the initialization functions nag_rand_init_repeatable (g05kfc) (for a repeatable sequence if computed sequentially) or nag_rand_init_nonrepeatable (g05kgc) (for a non-repeatable sequence) must be called prior to the first call to nag_rand_field_1d_generate (g05zpc).

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

1: ns – IntegerInput

On entry: the number of sample points to be generated in realizations of the random field. This must be the same value as supplied to nag_rand_field_1d_user_setup (g05zmc) or nag_rand_field_1d_predef_setup (g05znc) when calculating the eigenvalues of the embedding matrix.

Constraint:

ns \geq 1

2: s – IntegerInput

On entry:

S

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

Constraint:

s \geq 1

3: m – IntegerInput

On entry:

M

, the size of the embedding matrix, as returned by nag_rand_field_1d_user_setup (g05zmc) or nag_rand_field_1d_predef_setup (g05znc).

Constraint:

m \geq \max (1, 2 (ns - 1))

4: lam[m] – const doubleInput

On entry: must contain the square roots of the eigenvalues of the embedding matrix, as returned by nag_rand_field_1d_user_setup (g05zmc) or nag_rand_field_1d_predef_setup (g05znc).

Constraint:

lam [i - 1] \geq 0, i = 1, 2, \dots, m

5: rho – doubleInput

On entry: indicates the scaling of the covariance matrix, as returned by nag_rand_field_1d_user_setup (g05zmc) or nag_rand_field_1d_predef_setup (g05znc).

Constraint:

0.0 < rho \leq 1.0

6: state[

\dim

] – IntegerCommunication 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.

7: z[

ns \times s

] – doubleOutput

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

j

th realization, for the ns sample points, is stored in

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

, for

i = 1, 2, \dots, ns

. The sample points are as returned in

xx

by nag_rand_field_1d_user_setup (g05zmc) or nag_rand_field_1d_predef_setup (g05znc).

8: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_BAD_PARAM

On entry, argument

⟨value⟩

had an illegal value.

NE_INT

On entry,

ns = ⟨value⟩

.
Constraint:

ns \geq 1

On entry,

s = ⟨value⟩

.
Constraint:

s \geq 1

NE_INT_2

On entry,

m = ⟨value⟩

and

ns = ⟨value⟩

.
Constraint:

m \geq \max (1, 2 \times (ns - 1))

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.

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_REAL

On entry,

rho = ⟨value⟩

.
Constraint:

0.0 \leq rho \leq 1.0

NAG Library Function Document

nag_rand_field_1d_generate (g05zpc)

+− 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 Library Function Documentnag_rand_field_1d_generate (g05zpc)

+− 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 Library Function Document

nag_rand_field_1d_generate (g05zpc)