naginterfaces.library.rand.field_​1d_​predef_​setup

naginterfaces.library.rand.field_1d_predef_setup(ns, xmin, xmax, var, icov1, params, maxm=None, pad=1, icorr=0)

field_1d_predef_setup performs the setup required in order to simulate stationary Gaussian random fields in one dimension, 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_1d_generate(), which simulates the random field.

For full information please refer to the NAG Library document for g05zn

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/g05/g05znf.html

Parameters

nsint

The number of sample points to be generated in realizations of the random field.

xminfloat

The lower bound for the interval over which the random field is to be simulated. Note that if $icov1=14$ (for simulating fractional Brownian motion), $xmin$ is not referenced and the lower bound for the interval is set to zero.

xmaxfloat

The upper bound for the interval over which the random field is to be simulated. Note that if $icov1=14$ (for simulating fractional Brownian motion), the lower bound for the interval is set to zero and so $xmax$ is required to be greater than zero.

varfloat

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

icov1int

Determines which of the preset variograms to use. The choices are given below. Note that $x^{\prime }=\frac{\left|x\right|}{\ell }$, where $\ell$ is the correlation length and is a parameter for most of the variograms, and ${\sigma }^2$ is the variance specified by $var$.

$icov1=1$

Symmetric stable variogram

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

where

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

$\nu =params\left[1\right]$, $0\le \nu \le 2$.

$icov1=2$

Cauchy variogram

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

where

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

$\nu =params\left[1\right]$, $\nu >0$.

$icov1=3$

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

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

$icov1=4$

Exponential variogram

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

where

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

$icov1=5$

Gaussian variogram

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

where

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

$icov1=6$

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 $icov1$.

$icov1=7$

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

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

$icov1=8$

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,

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

$\nu =params\left[1\right]$, $\nu \ge -0.5$.

$icov1=9$

Hole effect variogram

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

where

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

$icov1=10$

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,

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

$\nu =params\left[1\right]$, $\nu >0$.

$icov1=11$

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

$x^{\prime \prime }=\frac{x^{\prime }}{s}$,

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

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

$s=params\left[1\right]$, $s>0$ (second correlation length),

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

$icov1=12$

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,

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

$\lambda =params\left[1\right]$, no constraint on $\lambda$

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

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

$icov1=13$

Cosine variogram

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

where

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

$icov1=14$

Used for simulating fractional Brownian motion $B^H\left(t\right)$. Fractional Brownian motion itself is not a stationary Gaussian random field, but its increments $\tilde{X}\left(i\right)=B^H\left(t_i\right)-B^H\left(t_{i-1}\right)$ can be simulated in the same way as a stationary random field. The variogram for the so-called ‘increment process’ is

$$C(\tilde{X}\left(t_i\right),\tilde{X}\left(t_j\right))=\tilde{\gamma }\left(x\right)=\frac{{\delta }^{2H}}{2}({|\frac{x}{\delta }-1|}^{2H}+{|\frac{x}{\delta }+1|}^{2H}-2{\left|\frac{x}{\delta }\right|}^{2H})\text{,}$$

where

$x=t_j-t_i$,

$H=params\left[0\right]$, $0<H<1$, $H$ is the Hurst parameter,

$\delta =params\left[1\right]$, $\delta >0$, normally $\delta =t_i-t_{i-1}$ is the (fixed) step size.

We scale the increments to set $\gamma \left(0\right)=1$; let $X\left(i\right)=\frac{\tilde{X}\left(i\right)}{{\delta }^{-H}}$, then

$$C(X\left(t_i\right),X\left(t_j\right))=\gamma \left(x\right)=\frac{1}{2}({|\frac{x}{\delta }-1|}^{2H}+{|\frac{x}{\delta }+1|}^{2H}-2{\left|\frac{x}{\delta }\right|}^{2H})\text{.}$$

The increments $X\left(i\right)$ can then be simulated using field_1d_generate(), then multiplied by ${\delta }^H$ to obtain the original increments $\tilde{X}\left(i\right)$ for the fractional Brownian motion.

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

The parameters set for the variogram.

maxmNone or int, optional

Note: if this argument is None then a default value will be used, determined as follows: $2^{3+ceiling\left({\log }_2\left((ns-1)\right)\right)}$.

The maximum size of the circulant matrix to use. For example, if the embedding matrix is to be allowed to double in size three times before the approximation procedure is used, then choose $maxm=2^{k+2}$ where $k=1+\lceil log2(ns-1)\rceil$.

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 $\left(maxm\right)$

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

xxfloat, ndarray, shape $\left(ns\right)$

The points at which values of the random field will be output.

mint

The size of the embedding matrix.

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, $ns=⟨value⟩$.

Constraint: $ns\ge 1$.

(errno $2$)

On entry, $icov1\ne 14$, $xmin=⟨value⟩$ and $xmax=⟨value⟩$.

Constraint: $xmin<xmax$.

(errno $3$)

On entry, $icov1=14$ and $xmax=⟨value⟩$.

Constraint: $xmax>0.0$.

(errno $4$)

On entry, $maxm=⟨value⟩$.

Constraint: the minimum calculated value for $maxm$ is $⟨value⟩$.

(errno $5$)

On entry, $var=⟨value⟩$.

Constraint: $var\ge 0.0$.

(errno $6$)

On entry, $icov1=⟨value⟩$.

Constraint: $icov1\ge 1$ and $icov1\le 14$.

(errno $7$)

On entry, $\text{np}=⟨value⟩$.

Constraint: for $icov1=⟨value⟩$, $\text{np}=⟨value⟩$.

(errno $8$)

On entry, $params[⟨value⟩]=⟨value⟩$.

Constraint: dependent on $icov1$.

(errno $9$)

On entry, $pad=⟨value⟩$.

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

(errno $10$)

On entry, $icorr=⟨value⟩$.

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

Notes

A one-dimensional random field $Z\left(x\right)$ in $R$ is a function which is random at every point $x\in R$, 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 $x_1,\dots ,x_n\in R$, 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 $x\in R$ 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 $\gamma$ 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_1d_predef_setup and field_1d_generate() are used to simulate a one-dimensional stationary Gaussian random field, with mean function zero and variogram $\gamma \left(x\right)$, over an interval $[x_{\text{min}},x_{\text{max}}]$, 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 $M\ge 2(N-1)$, which can now be factorized as $B=W\Lambda W^{\ast }=R^{\ast }R$, where $W$ is the 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 $M$, 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 $Y$, two samples of which 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 $N$ elements 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_1d_predef_setup finds a suitable positive semidefinite embedding matrix $B$ and outputs its size, $m$, 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