naginterfaces.library.rand.field_​2d_​generate

naginterfaces.library.rand.field_2d_generate(ns, s, m, lam, rho, statecomm)

field_2d_generate produces realizations of a stationary Gaussian random field in two dimensions, 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 field_2d_user_setup() or field_2d_predef_setup().

For full information please refer to the NAG Library document for g05zs

https://support.nag.com/numeric/nl/nagdoc_30.1/flhtml/g05/g05zsf.html

Parameters

nsint, array-like, shape $\left(2\right)$

The number of sample points to use in each direction, with $ns\left[0\right]$ sample points in the $x$-direction and $ns\left[1\right]$ sample points in the $y$-direction. The total number of sample points on the grid is, therefore, $ns\left[0\right]\times ns\left[1\right]$. This must be the same value as supplied to field_2d_user_setup() or field_2d_predef_setup() when calculating the eigenvalues of the embedding matrix.

sint

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

mint, array-like, shape $\left(2\right)$

Indicates the size, $M$, of the embedding matrix as returned by field_2d_user_setup() or field_2d_predef_setup(). The embedding matrix is a block circulant matrix with circulant blocks. $m\left[0\right]$ is the size of each block, and $m\left[1\right]$ is the number of blocks.

lamfloat, array-like, shape $\left(m\right[0]\times m[1\left]\right)$

Contains the square roots of the eigenvalues of the embedding matrix, as returned by field_2d_user_setup() or field_2d_predef_setup().

rhofloat

Indicates the scaling of the covariance matrix, as returned by field_2d_user_setup() or field_2d_predef_setup().

statecommdict, RNG communication object, modified in place

RNG communication structure.

This argument must have been initialized by a prior call to init_repeat() or init_nonrepeat().

Returns

zfloat, ndarray, shape $(:,s)$

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 $z[(j-1)\times ns[0]+i-1,k-1]$, for $i=1,2,\dots ,ns\left[0\right]$ and for $j=1,2,\dots ,ns\left[1\right]$. The points are returned in $\text{xx}$ and $\text{yy}$ by field_2d_user_setup() or field_2d_predef_setup().

Raises

NagValueError

(errno $1$)

On entry, $ns=[⟨value⟩,⟨value⟩]$.

Constraint: $ns\left[0\right]\ge 1$, $ns\left[1\right]\ge 1$.

(errno $2$)

On entry, $s=⟨value⟩$.

Constraint: $s\ge 1$.

(errno $3$)

On entry, $m=[⟨value⟩,⟨value⟩]$, and $ns=[⟨value⟩,⟨value⟩]$.

Constraints: $m[i-1]\ge max(1,2\left(ns\right[i-1\left]\right)-1)$, for $i=1,2$.

(errno $4$)

On entry, at least one element of $lam$ was negative.

Constraint: all elements of $lam$ must be non-negative.

(errno $5$)

On entry, $rho=⟨value⟩$.

Constraint: $0.0<rho\le 1.0$.

(errno $6$)

On entry, $statecomm$[‘state’] vector has been corrupted or not initialized.

Notes

A two-dimensional random field $Z\left(x\right)$ in $R^2$ is a function which is random at every point $x\in R^2$, so $Z\left(x\right)$ is a random variable for each $x$. The random field has a mean function $\mu \left(x\right)=E\left[Z\left(x\right)\right]$ and a symmetric positive semidefinite covariance function $C(x,y)=E\left[(Z\left(x\right)-\mu \left(x\right))(Z\left(y\right)-\mu \left(y\right))\right]$. $Z\left(x\right)$ 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 ${[Z\left(x_1\right),\dots ,Z\left(x_n\right)]}^T$ follows a multivariate Normal distribution, which would have a mean vector $\tilde{\mu }$ with entries ${\tilde{\mu }}_i=\mu \left(x_i\right)$ and a covariance matrix $\tilde{C}$ with entries ${\tilde{C}}_{ij}=C(x_i,x_j)$. A Gaussian random field $Z\left(x\right)$ is stationary if $\mu \left(x\right)$ 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 $\gamma$ of one variable: $C(x,y)=\gamma (x-y)$. $\gamma$ is known as a variogram (or more correctly, a semivariogram) and includes the multiplicative factor ${\sigma }^2$ representing the variance such that $\gamma \left(0\right)={\sigma }^2$.

The functions field_2d_user_setup() or field_2d_predef_setup() along with field_2d_generate are used to simulate a two-dimensional stationary Gaussian random field, with mean function zero and variogram $\gamma \left(x\right)$, over a domain $[x_{\text{min}},x_{\text{max}}]\times [y_{\text{min}},y_{\text{max}}]$, 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 $N_2\times N_2$ block Toeplitz matrix with Toeplitz blocks of size $N_1\times N_1$. 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 $M_2\times M_2$ block circulant matrix with circulant bocks of size $M_1\times M_1$, where $M_1\ge 2(N_1-1)$ and $M_2\ge 2(N_2-1)$. $B$ can now be factorized as $B=W\Lambda W^{\ast }=R^{\ast }R$, where $W$ is the two-dimensional Fourier matrix ($W^{\ast }$ is the complex conjugate of $W$), $\Lambda$ is the diagonal matrix containing the eigenvalues of $B$ and $R={\Lambda }^{\frac{1}{2}}{W}^{\ast }$. $B$ 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 $\gamma$ is even or not. $\gamma$ is even if $\gamma \left(x\right)=\gamma (-x)$ for all $x\in R^2$, and uneven otherwise (in higher dimensions, $\gamma$ can be even in some coordinates and uneven in others, but in two dimensions $\gamma$ is either even in both coordinates or uneven in both coordinates). If $\gamma$ is even then $A$ is a symmetric block matrix and has symmetric blocks; if $\gamma$ 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 $\Lambda$ 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 $R^{\ast }(U+iV)$, where $U$ and $V$ have elements from the standard Normal distribution. Since $R^{\ast }(U+iV)=W{\Lambda }^{\frac{1}{2}}(U+iV)$, this calculation can be done using a discrete Fourier transform of the vector ${\Lambda }^{\frac{1}{2}}(U+iV)$. 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.

If $B$ is not positive semidefinite, larger embedding matrices $B$ can be tried; however if the size of the matrix would have to be larger than $\text{maxm}$, an approximation procedure is used. See the documentation of field_2d_user_setup() or field_2d_predef_setup() for details of the approximation procedure.

field_2d_generate 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 init_repeat() (for a repeatable sequence if computed sequentially) or init_nonrepeat() (for a non-repeatable sequence) must be called prior to the first call to field_2d_generate.

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