naginterfaces.library.rand.field_​2d_​predef_​setup

naginterfaces.library.rand.field_2d_predef_setup(ns, xmin, xmax, ymin, ymax, maxm, var, icov2, params, norm=2, pad=1, icorr=0)

field_2d_predef_setup performs the setup required in order to simulate stationary Gaussian random fields in two dimensions, for a preset 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 field_2d_generate(), which simulates the random field.

For full information please refer to the NAG Library document for g05zr

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/g05/g05zrf.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, $N_1$ and $ns\left[1\right]$ sample points in the $y$-direction, $N_2$. The total number of sample points on the grid is, therefore, $ns\left[0\right]\times ns\left[1\right]$.

xminfloat

The lower bound for the $x$-coordinate, for the region in which the random field is to be simulated.

xmaxfloat

The upper bound for the $x$-coordinate, for the region in which the random field is to be simulated.

yminfloat

The lower bound for the $y$-coordinate, for the region in which the random field is to be simulated.

ymaxfloat

The upper bound for the $y$-coordinate, for the region in which the random field is to be simulated.

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

Determines the maximum size of the circulant matrix to use – a maximum of $maxm\left[0\right]$ elements in the $x$-direction, and a maximum of $maxm\left[1\right]$ elements in the $y$-direction. The maximum size of the circulant matrix is thus $maxm\left[0\right]$$\times$$maxm\left[1\right]$.

varfloat

The multiplicative factor ${\sigma }^2$ of the variogram $\gamma \left(x\right)$.

icov2int

Determines which of the preset variograms to use. The choices are given below. Note that $x^{\prime }=\Vert \frac{x}{{\ell }_1},\frac{y}{{\ell }_2}\Vert$, where ${\ell }_1$ and ${\ell }_2$ are correlation lengths in the $x$ and $y$ directions respectively and are parameters for most of the variograms, and ${\sigma }^2$ is the variance specified by $var$.

$icov2=1$

Symmetric stable variogram

$$\gamma \left(x\right)={\sigma }^{2}exp(-{\left(x^{\prime }\right)}^{\nu })\text{,}$$

where

${\ell }_1=params\left[0\right]$, ${\ell }_1>0$,

${\ell }_2=params\left[1\right]$, ${\ell }_2>0$,

$\nu =params\left[2\right]$, $0<\nu \le 2$.

$icov2=2$

Cauchy variogram

$$\gamma \left(x\right)={\sigma }^2{(1+{\left(x^{\prime }\right)}^2)}^{-\nu }\text{,}$$

where

${\ell }_1=params\left[0\right]$, ${\ell }_1>0$,

${\ell }_2=params\left[1\right]$, ${\ell }_2>0$,

$\nu =params\left[2\right]$, $\nu >0$.

$icov2=3$

Differential variogram with compact support

$$\begin{array}{c}\gamma \left(x\right)=\{\begin{array}{cc}{\sigma }^2(1+8x^{\prime }+25{\left(x^{\prime }\right)}^2+32{\left(x^{\prime }\right)}^3){(1-x^{\prime })}^8\text{,}& x^{\prime }<1\text{,}\\ 0\text{,}& x^{\prime }\ge 1\text{,}\end{array}\end{array}$$

where

${\ell }_1=params\left[0\right]$, ${\ell }_1>0$,

${\ell }_2=params\left[1\right]$, ${\ell }_2>0$.

$icov2=4$

Exponential variogram

$$\gamma \left(x\right)={\sigma }^{2}exp(-x^{\prime })\text{,}$$

where

${\ell }_1=params\left[0\right]$, ${\ell }_1>0$,

${\ell }_2=params\left[1\right]$, ${\ell }_2>0$.

$icov2=5$

Gaussian variogram

$$\gamma \left(x\right)={\sigma }^{2}exp\left({-\left(x^{\prime }\right)}^2\right)\text{,}$$

where

${\ell }_1=params\left[0\right]$, ${\ell }_1>0$,

${\ell }_2=params\left[1\right]$, ${\ell }_2>0$.

$icov2=6$

Nugget variogram

$$\begin{array}{c}\gamma \left(x\right)=\{\begin{array}{cc}{\sigma }^2\text{,}& x=0\text{,}\\ 0\text{,}& x\ne 0\text{.}\end{array}\end{array}$$

No parameters need be set for this value of $icov2$.

$icov2=7$

Spherical variogram

$$\begin{array}{c}\gamma \left(x\right)=\{\begin{array}{cc}{\sigma }^2(1-1.5x^{\prime }+0.5{\left(x^{\prime }\right)}^3)\text{,}& x^{\prime }<1\text{,}\\ 0\text{,}& x^{\prime }\ge 1\text{,}\end{array}\end{array}$$

where

${\ell }_1=params\left[0\right]$, ${\ell }_1>0$,

${\ell }_2=params\left[1\right]$, ${\ell }_2>0$.

$icov2=8$

Bessel variogram

$$\gamma \left(x\right)={\sigma }^2\frac{2^{\nu }\Gamma (\nu +1)J_{\nu }\left(x^{\prime }\right)}{{\left(x^{\prime }\right)}^{\nu }}\text{,}$$

where

$J_{\nu }(\cdot )$ is the Bessel function of the first kind,

${\ell }_1=params\left[0\right]$, ${\ell }_1>0$,

${\ell }_2=params\left[1\right]$, ${\ell }_2>0$,

$\nu =params\left[2\right]$, $\nu \ge 0$.

$icov2=9$

Hole effect variogram

$$\gamma \left(x\right)={\sigma }^2\frac{\sin \left(x^{\prime }\right)}{x^{\prime }}\text{,}$$

where

${\ell }_1=params\left[0\right]$, ${\ell }_1>0$,

${\ell }_2=params\left[1\right]$, ${\ell }_2>0$.

$icov2=10$

Whittle-Matérn variogram

$$\gamma \left(x\right)={\sigma }^2\frac{2^{1-\nu }{\left(x^{\prime }\right)}^{\nu }{K}_{\nu }\left(x^{\prime }\right)}{\Gamma \left(\nu \right)}\text{,}$$

where

$K_{\nu }(\cdot )$ is the modified Bessel function of the second kind,

${\ell }_1=params\left[0\right]$, ${\ell }_1>0$,

${\ell }_2=params\left[1\right]$, ${\ell }_2>0$,

$\nu =params\left[2\right]$, $\nu >0$.

$icov2=11$

Continuously parameterised variogram with compact support

$$\begin{array}{c}\gamma \left(x\right)=\{\begin{array}{cc}{\sigma }^2\frac{2^{1-\nu }{\left(x^{\prime }\right)}^{\nu }{K}_{\nu }\left(x^{\prime }\right)}{\Gamma \left(\nu \right)}(1+8x^{\prime \prime }+25{\left(x^{\prime \prime }\right)}^2+32{\left(x^{\prime \prime }\right)}^3){(1-x^{\prime \prime })}^8\text{,}& x^{\prime \prime }<1\text{,}\\ 0\text{,}& x^{\prime \prime }\ge 1\text{,}\end{array}\end{array}$$

where

$x^{\text{''}}=\Vert \frac{x^{\prime }}{{\ell }_1{s}_1},\frac{y^{\prime }}{{\ell }_2{s}_2}\Vert$,

$K_{\nu }(\cdot )$ is the modified Bessel function of the second kind,

${\ell }_1=params\left[0\right]$, ${\ell }_1>0$,

${\ell }_2=params\left[1\right]$, ${\ell }_2>0$,

$s_1=params\left[2\right]$, $s_1>0$,

$s_2=params\left[3\right]$, $s_2>0$,

$\nu =params\left[4\right]$, $\nu >0$.

$icov2=12$

Generalized hyperbolic distribution variogram

$$\gamma \left(x\right)={\sigma }^2\frac{{({\delta }^2+{\left(x^{\prime }\right)}^2)}^{\frac{\lambda }{2}}}{{\delta }^{\lambda }{K}_{\lambda }\left(\kappa \delta \right)}{K}_{\lambda }\left(\kappa {({\delta }^2+{\left(x^{\prime }\right)}^2)}^{\frac{1}{2}}\right)\text{,}$$

where

$K_{\lambda }(\cdot )$ is the modified Bessel function of the second kind,

${\ell }_1=params\left[0\right]$, ${\ell }_1>0$,

${\ell }_2=params\left[1\right]$, ${\ell }_2>0$,

$\lambda =params\left[2\right]$, no constraint on $\lambda$,

$\delta =params\left[3\right]$, $\delta >0$,

$\kappa =params\left[4\right]$, $\kappa >0$.

paramsfloat, array-like, shape $\left(\text{np}\right)$

The parameters for the variogram as detailed in the description of $icov2$.

normint, optional

Determines which norm to use when calculating the variogram.

$norm=1$

The 1-norm is used, i.e., $\parallel x,y\parallel =\left|x\right|+\left|y\right|$.

$norm=2$

The 2-norm (Euclidean norm) is used, i.e., $\parallel x,y\parallel =\sqrt{x^2+y^2}$.

padint, optional

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.

icorrint, optional

Determines which approximation to implement if required, as described in Notes.

Returns

lamfloat, ndarray, shape $(:)$

Contains the square roots of the eigenvalues of the embedding matrix.

xxfloat, ndarray, shape $(:)$

The points of the $x$-coordinates at which values of the random field will be output.

yyfloat, ndarray, shape $(:)$

The points of the $y$-coordinates at which values of the random field will be output.

mint, ndarray, shape $\left(2\right)$

$m\left[0\right]$ contains $M_1$, the size of the circulant blocks and $m\left[1\right]$ contains $M_2$, the number of blocks, resulting in a final square matrix of size $M_1\times M_2$.

approxint

Indicates whether approximation was used.

$approx=0$

No approximation was used.

$approx=1$

Approximation was used.

rhofloat

Indicates the scaling of the covariance matrix. $rho=1.0$ unless approximation was used with $icorr=0$ or $1$.

icountint

Indicates the number of negative eigenvalues in the embedding matrix which have had to be set to zero.

eigfloat, ndarray, shape $\left(3\right)$

Indicates information about the negative eigenvalues in the embedding matrix which have had to be set to zero. $eig\left[0\right]$ contains the smallest eigenvalue, $eig\left[1\right]$ contains the sum of the squares of the negative eigenvalues, and $eig\left[2\right]$ contains the sum of the absolute values of the negative eigenvalues.

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, $xmin=⟨value⟩$ and $xmax=⟨value⟩$.

Constraint: $xmin<xmax$.

(errno $4$)

On entry, $ymin=⟨value⟩$ and $ymax=⟨value⟩$.

Constraint: $ymin<ymax$.

(errno $6$)

On entry, $maxm=[⟨value⟩,⟨value⟩]$.

Constraint: the minimum calculated value for $maxm$ are $[⟨value⟩,⟨value⟩]$.

(errno $7$)

On entry, $var=⟨value⟩$.

Constraint: $var\ge 0.0$.

(errno $8$)

On entry, $icov2=⟨value⟩$.

Constraint: $icov2\ge 1$ and $icov2\le 12$.

(errno $9$)

On entry, $norm=⟨value⟩$.

Constraint: $norm=1$ or $2$.

(errno $10$)

On entry, $\text{np}=⟨value⟩$.

Constraint: for $icov2=⟨value⟩$, $\text{np}=⟨value⟩$.

(errno $11$)

On entry, $params[⟨value⟩]=⟨value⟩$.

Constraint: dependent on $icov2$, see documentation.

(errno $12$)

On entry, $pad=⟨value⟩$.

Constraint: $pad=0$ or $1$.

(errno $13$)

On entry, $icorr=⟨value⟩$.

Constraint: $icorr=0$, $1$ or $2$.

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_predef_setup and 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 blocks 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.

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 $maxm$, an approximation procedure is used. We write $\Lambda ={\Lambda }_{+}+{\Lambda }_{-}$, where ${\Lambda }_{+}$ and ${\Lambda }_{-}$ contain the non-negative and negative eigenvalues of $B$ respectively. Then $B$ is replaced by $\rho B_{+}$ where $B_{+}=W{\Lambda }_{+}{W}^{\ast }$ and $\rho \in (0,1]$ is a scaling factor. The error $ϵ$ in approximating the distribution of the random field is given by

$$ϵ=\sqrt{\frac{{(1-\rho )}^{2}trace\left(\Lambda \right)+{\rho }^{2}trace\left({\Lambda }_{-}\right)}{M}}\text{.}$$

Three choices for $\rho$ are available, and are determined by the input argument $icorr$:

setting $icorr=0$ sets

$$\rho =\frac{trace\left(\Lambda \right)}{trace\left({\Lambda }_{+}\right)}\text{,}$$

setting $icorr=1$ sets

$$\rho =\sqrt{\frac{trace\left(\Lambda \right)}{trace\left({\Lambda }_{+}\right)}}\text{,}$$

setting $icorr=2$ sets $\rho =1$.

field_2d_predef_setup finds a suitable positive semidefinite embedding matrix $B$ and outputs its sizes in the vector $m$ 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.

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, 1997, Algorithm AS 312: An Algorithm for Simulating Stationary Gaussian Random Fields, Journal of the Royal Statistical Society, Series C (Applied Statistics) (Volume 46) (1), 171–181