G05ZNF (PDF version)
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NAG Library Manual

NAG Library Routine DocumentG05ZNF

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

1  Purpose

G05ZNF 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 G05ZPF, which simulates the random field.

2  Specification

 SUBROUTINE G05ZNF ( NS, XMIN, XMAX, MAXM, VAR, ICOV1, NP, PARAMS, PAD, ICORR, LAM, XX, M, APPROX, RHO, ICOUNT, EIG, IFAIL)
 INTEGER NS, MAXM, ICOV1, NP, PAD, ICORR, M, APPROX, ICOUNT, IFAIL REAL (KIND=nag_wp) XMIN, XMAX, VAR, PARAMS(NP), LAM(MAXM), XX(NS), RHO, EIG(3)

3  Description

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 $C\left(x,y\right)=𝔼\left[\left(Z\left(x\right)-\mu \left(x\right)\right)\left(Z\left(y\right)-\mu \left(y\right)\right)\right]$. $Z\left(x\right)$ is a Gaussian random field if for any choice of $n\in ℕ$ and ${x}_{1},\dots ,{x}_{n}\in ℝ$, the random vector ${\left[Z\left({x}_{1}\right),\dots ,Z\left({x}_{n}\right)\right]}^{\mathrm{T}}$ follows a multivariate Gaussian distribution, which would have a mean vector $\stackrel{~}{\mathbf{\mu }}$ with entries ${\stackrel{~}{\mu }}_{i}=\mu \left({x}_{i}\right)$ and a covariance matrix $\stackrel{~}{C}$ with entries ${\stackrel{~}{C}}_{ij}=C\left({x}_{i},{x}_{j}\right)$. A Gaussian random field $Z\left(x\right)$ is stationary if $\mu \left(x\right)$ is constant for all $x\in ℝ$ and $C\left(x,y\right)=C\left(x+a,y+a\right)$ for all $x,y,a\in ℝ$ and hence we can express the covariance function $C\left(x,y\right)$ as a function $\gamma$ of one variable: $C\left(x,y\right)=\gamma \left(x-y\right)$. $\gamma$ 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 routines G05ZNF and G05ZPF are used to simulate a one-dimensional stationary Gaussian random field, with mean function zero and variogram $\gamma \left(x\right)$, over an interval $\left[{x}_{\mathrm{min}},{x}_{\mathrm{max}}\right]$, using an equally spaced set of $N$ points. The problem reduces to sampling a Gaussian random vector $\mathbf{X}$ 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 $M\ge 2\left(N-1\right)$, which can now be factorized as $B=W\Lambda {W}^{*}={R}^{*}R$, where $W$ is the Fourier matrix (${W}^{*}$ is the complex conjugate of $W$), $\Lambda$ is the diagonal matrix containing the eigenvalues of $B$ and $R={\Lambda }^{\frac{1}{2}}{W}^{*}$. $B$ 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 $\Lambda$ are non-negative (i.e., $B$ is positive semidefinite), $B$ is a covariance matrix for a random vector $\mathbf{Y}$, two samples of which can now be simulated from the real and imaginary parts of ${R}^{*}\left(\mathbf{U}+i\mathbf{V}\right)$, where $\mathbf{U}$ and $\mathbf{V}$ have elements from the standard Normal distribution. Since ${R}^{*}\left(\mathbf{U}+i\mathbf{V}\right)=W{\Lambda }^{\frac{1}{2}}\left(\mathbf{U}+i\mathbf{V}\right)$, this calculation can be done using a discrete Fourier transform of the vector ${\Lambda }^{\frac{1}{2}}\left(\mathbf{U}+i\mathbf{V}\right)$. Two samples of the random vector $\mathbf{X}$ can now be recovered by taking the first $N$ elements of each sample of $\mathbf{Y}$ – because the original covariance matrix $A$ is embedded in $B$, $\mathbf{X}$ 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 ${B}_{+}=W{\Lambda }_{+}{W}^{*}$ and $\rho \in \left(0,1\right]$ is a scaling factor. The error $\epsilon$ in approximating the distribution of the random field is given by
 $ε= 1-ρ 2 trace⁡Λ + ρ2 trace⁡Λ- M .$
Three choices for $\rho$ are available, and are determined by the input parameter ICORR:
• setting ${\mathbf{ICORR}}=0$ sets
 $ρ= trace⁡Λ trace⁡Λ+ ,$
• setting ${\mathbf{ICORR}}=1$ sets
 $ρ= trace⁡Λ trace⁡Λ+ ,$
• setting ${\mathbf{ICORR}}=2$ sets $\rho =1$.
G05ZNF 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.

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 (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

5  Parameters

1:     NS – INTEGERInput
On entry: the number of sample points (points) to be generated in realisations of the random field.
Constraint: ${\mathbf{NS}}\ge 1$.
2:     XMIN – REAL (KIND=nag_wp)Input
On entry: the lower bound for the interval over which the random field is to be simulated. Note that if ICOV1 is set to $14$ (for simulating fractional Brownian motion), XMIN is not referenced and the lower bound for the interval is set to zero.
Constraint: if ${\mathbf{ICOV1}}\ne 14$, ${\mathbf{XMIN}}<{\mathbf{XMAX}}$.
3:     XMAX – REAL (KIND=nag_wp)Input
On entry: the upper bound for the interval over which the random field is to be simulated. Note that if ICOV1 is set to $14$ (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.
Constraints:
• if ${\mathbf{ICOV1}}\ne 14$, ${\mathbf{XMIN}}<{\mathbf{XMAX}}$;
• if ${\mathbf{ICOV1}}=14$, ${\mathbf{XMAX}}>0.0$.
4:     MAXM – INTEGERInput
On entry: 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 ${\mathbf{MAXM}}={2}^{k+2}$ where $k=1+⌈{\mathrm{log}}_{2}\left({\mathbf{NS}}-1\right)⌉$.
Constraint: ${\mathbf{MAXM}}\ge {2}^{k}$, where $k$ is the smallest integer satisfying ${2}^{k}\ge 2\left({\mathbf{NS}}-1\right)$.
5:     VAR – REAL (KIND=nag_wp)Input
On entry: the multiplicative factor ${\sigma }^{2}$ of the variogram $\gamma \left(x\right)$.
Constraint: ${\mathbf{VAR}}\ge 0.0$.
6:     ICOV1 – INTEGERInput
On entry: 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.
${\mathbf{ICOV1}}=1$
Symmetric stable variogram
 $γx = σ2 exp - x′ ν ,$
where
• $\ell ={\mathbf{PARAMS}}\left(1\right)$, $\ell >0$,
• $\nu ={\mathbf{PARAMS}}\left(2\right)$, $0\le \nu \le 2$.
${\mathbf{ICOV1}}=2$
Cauchy variogram
 $γx = σ2 1+ x′ 2 -ν ,$
where
• $\ell ={\mathbf{PARAMS}}\left(1\right)$, $\ell >0$,
• $\nu ={\mathbf{PARAMS}}\left(2\right)$, $\nu >0$.
${\mathbf{ICOV1}}=3$
Differential variogram with compact support
 $γx = σ21+8x′+25x′2+32x′31-x′8, x′<1, 0, x′≥1,$
where
• $\ell ={\mathbf{PARAMS}}\left(1\right)$, $\ell >0$.
${\mathbf{ICOV1}}=4$
Exponential variogram
 $γx=σ2exp-x′,$
where
• $\ell ={\mathbf{PARAMS}}\left(1\right)$, $\ell >0$.
${\mathbf{ICOV1}}=5$
Gaussian variogram
 $γx=σ2exp-x′2,$
where
• $\ell ={\mathbf{PARAMS}}\left(1\right)$, $\ell >0$.
${\mathbf{ICOV1}}=6$
Nugget variogram
 $γx= σ2, x=0, 0, x≠0.$
No parameters need be set for this value of ICOV1.
${\mathbf{ICOV1}}=7$
Spherical variogram
 $γx= σ21-1.5x′+0.5x′3, x′<1, 0, x′≥1,$
where
• $\ell ={\mathbf{PARAMS}}\left(1\right)$, $\ell >0$.
${\mathbf{ICOV1}}=8$
Bessel variogram
 $γx=σ22νΓν+1Jνx′x′ν,$
where
• ${J}_{\nu }\left(·\right)$ is the Bessel function of the first kind,
• $\ell ={\mathbf{PARAMS}}\left(1\right)$, $\ell >0$,
• $\nu ={\mathbf{PARAMS}}\left(2\right)$, $\nu \ge -0.5$.
${\mathbf{ICOV1}}=9$
Hole effect variogram
 $γx=σ2sinx′x′,$
where
• $\ell ={\mathbf{PARAMS}}\left(1\right)$, $\ell >0$.
${\mathbf{ICOV1}}=10$
Whittle-Matérn variogram
 $γx=σ221-νx′νKνx′Γν,$
where
• ${K}_{\nu }\left(·\right)$ is the modified Bessel function of the second kind,
• $\ell ={\mathbf{PARAMS}}\left(1\right)$, $\ell >0$,
• $\nu ={\mathbf{PARAMS}}\left(2\right)$, $\nu >0$.
${\mathbf{ICOV1}}=11$
Continuously parameterised variogram with compact support
 $γx= σ221-νx′νKνx′Γν1+8x′′+25x′′2+32x′′31-x′′8, x′′<1, 0, x′′≥1,$
where
• ${x}^{\prime \prime }=\frac{{x}^{\prime }}{s}$,
• ${K}_{\nu }\left(·\right)$ is the modified Bessel function of the second kind,
• $\ell ={\mathbf{PARAMS}}\left(1\right)$, $\ell >0$,
• $s={\mathbf{PARAMS}}\left(2\right)$, $s>0$ (second correlation length),
• $\nu ={\mathbf{PARAMS}}\left(3\right)$, $\nu >0$.
${\mathbf{ICOV1}}=12$
Generalized hyperbolic distribution variogram
 $γx=σ2δ2+x′2λ2δλKλκδKλκδ2+x′212,$
where
• ${K}_{\lambda }\left(·\right)$ is the modified Bessel function of the second kind,
• $\ell ={\mathbf{PARAMS}}\left(1\right)$, $\ell >0$,
• $\lambda ={\mathbf{PARAMS}}\left(2\right)$, no constraint on $\lambda$
• $\delta ={\mathbf{PARAMS}}\left(3\right)$, $\delta >0$,
• $\kappa ={\mathbf{PARAMS}}\left(4\right)$, $\kappa >0$.
${\mathbf{ICOV1}}=13$
Cosine variogram
 $γx=σ2cosx′,$
where
• $\ell ={\mathbf{PARAMS}}\left(1\right)$, $\ell >0$.
${\mathbf{ICOV1}}=14$
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 $\stackrel{~}{X}\left(i\right)={B}^{H}\left({t}_{i}\right)-{B}^{H}\left({t}_{i-1}\right)$ can be simulated in the same way as a stationary random field. The variogram for the so-called ‘increment process’ is
 $CX~ti,X~tj=γ~x=δ2H2xδ-12H+xδ+12H-2xδ2H,$
where
• $x={t}_{j}-{t}_{i}$,
• $H={\mathbf{PARAMS}}\left(1\right)$, $0, $H$ is the Hurst parameter,
• $\delta ={\mathbf{PARAMS}}\left(2\right)$, $\delta >0$, normally $\delta ={t}_{i}-{t}_{i-1}$ is the (fixed) stepsize.
We scale the increments to set $\gamma \left(0\right)=1$; let $X\left(i\right)=\frac{\stackrel{~}{X}\left(i\right)}{{\delta }^{-H}}$, then
 $CXti,Xtj = γx = 12 xδ - 1 2H + xδ + 1 2H - 2 xδ 2H .$
The increments $X\left(i\right)$ can then be simulated using G05ZPF, then multiplied by ${\delta }^{H}$ to obtain the original increments $\stackrel{~}{X}\left(i\right)$ for the fractional Brownian motion.
7:     NP – INTEGERInput
On entry: the number of parameters to be set. Different variograms need a different number of parameters.
${\mathbf{ICOV1}}=6$
NP must be set to $0$.
${\mathbf{ICOV1}}=3$, $4$, $5$, $7$, $9$ or $13$
NP must be set to $1$.
${\mathbf{ICOV1}}=1$, $2$, $8$, $10$ or $14$
NP must be set to $2$.
${\mathbf{ICOV1}}=11$
NP must be set to $3$.
${\mathbf{ICOV1}}=12$
NP must be set to $4$.
8:     PARAMS(NP) – REAL (KIND=nag_wp) arrayInput
On entry: the parameters set for the variogram.
Constraint: see ICOV1 for a description of the individual parameter constraints.
9:     PAD – INTEGERInput
On entry: 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.
${\mathbf{PAD}}=0$
The embedding matrix is padded with zeros.
${\mathbf{PAD}}=1$
The embedding matrix is padded with values of the variogram.
Suggested value: ${\mathbf{PAD}}=1\text{.}$
Constraint: ${\mathbf{PAD}}=0$ or $1$.
10:   ICORR – INTEGERInput
On entry: determines which approximation to implement if required, as described in Section 3.
Suggested value: ${\mathbf{ICORR}}=0$.
Constraint: ${\mathbf{ICORR}}=0$, $1$ or $2$.
11:   LAM(MAXM) – REAL (KIND=nag_wp) arrayOutput
On exit: contains the square roots of the eigenvalues of the embedding matrix.
12:   XX(NS) – REAL (KIND=nag_wp) arrayOutput
On exit: the points at which values of the random field will be output.
13:   M – INTEGEROutput
On exit: the size of the embedding matrix.
14:   APPROX – INTEGEROutput
On exit: indicates whether approximation was used.
${\mathbf{APPROX}}=0$
No approximation was used.
${\mathbf{APPROX}}=1$
Approximation was used.
15:   RHO – REAL (KIND=nag_wp)Output
On exit: indicates the scaling of the covariance matrix. ${\mathbf{RHO}}=1$ unless approximation was used with ${\mathbf{ICORR}}=0$ or $1$.
16:   ICOUNT – INTEGEROutput
On exit: indicates the number of negative eigenvalues in the embedding matrix which have had to be set to zero.
17:   EIG($3$) – REAL (KIND=nag_wp) arrayOutput
On exit: indicates information about the negative eigenvalues in the embedding matrix which have had to be set to zero. ${\mathbf{EIG}}\left(1\right)$ contains the smallest eigenvalue, ${\mathbf{EIG}}\left(2\right)$ contains the sum of the squares of the negative eigenvalues, and ${\mathbf{EIG}}\left(3\right)$ contains the sum of the absolute values of the negative eigenvalues.
18:   IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{​ or ​}1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is $0$. When the value $-\mathbf{1}\text{​ or ​}\mathbf{1}$ is used it is essential to test the value of IFAIL on exit.
On exit: ${\mathbf{IFAIL}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

6  Error Indicators and Warnings

If on entry ${\mathbf{IFAIL}}={\mathbf{0}}$ or $-{\mathbf{1}}$, explanatory error messages are output on the current error message unit (as defined by X04AAF).
Errors or warnings detected by the routine:
${\mathbf{IFAIL}}=1$
On entry, ${\mathbf{NS}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{NS}}\ge 1$.
${\mathbf{IFAIL}}=2$
On entry, ${\mathbf{ICOV1}}\ne 14$, ${\mathbf{XMIN}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{XMAX}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{XMIN}}<{\mathbf{XMAX}}$.
${\mathbf{IFAIL}}=3$
On entry, ${\mathbf{ICOV1}}=14$ and ${\mathbf{XMAX}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{XMAX}}>0.0$.
${\mathbf{IFAIL}}=4$
On entry, ${\mathbf{MAXM}}=⟨\mathit{\text{value}}⟩$.
Constraint: the calculated minimum value for MAXM is $⟨\mathit{\text{value}}⟩$.
Where the minimum calculated value is given by ${2}^{k}$, where $k$ is the smallest integer satisfying ${2}^{k}\ge 2\left({\mathbf{NS}}-1\right)$.
${\mathbf{IFAIL}}=5$
On entry, ${\mathbf{VAR}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{VAR}}\ge 0.0$.
${\mathbf{IFAIL}}=6$
On entry, ${\mathbf{ICOV1}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{ICOV1}}\ge 1$ and ${\mathbf{ICOV1}}\le 14$.
${\mathbf{IFAIL}}=7$
On entry, ${\mathbf{NP}}=⟨\mathit{\text{value}}⟩$.
Constraint: for ${\mathbf{ICOV1}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{NP}}=⟨\mathit{\text{value}}⟩$.
${\mathbf{IFAIL}}=8$
On entry, ${\mathbf{PARAMS}}\left(⟨\mathit{\text{value}}⟩\right)=⟨\mathit{\text{value}}⟩$.
Constraint: dependent on ICOV1, see documentation.
${\mathbf{IFAIL}}=9$
On entry, ${\mathbf{PAD}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{PAD}}=0$ or $1$.
${\mathbf{IFAIL}}=10$
On entry, ${\mathbf{ICORR}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{ICORR}}=0$, $1$ or $2$.

Not applicable.

None.

9  Example

This example calls G05ZNF to calculate the eigenvalues of the embedding matrix for $8$ sample points of a random field characterized by the symmetric stable variogram (${\mathbf{ICOV1}}=1$).

9.1  Program Text

Program Text (g05znfe.f90)

9.2  Program Data

Program Data (g05znfe.d)

9.3  Program Results

Program Results (g05znfe.r)

G05ZNF (PDF version)
G05 Chapter Contents
G05 Chapter Introduction
NAG Library Manual