g05zmc performs the setup required in order to simulate stationary Gaussian random fields in one dimension, for a user-defined 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 g05zpc, which simulates the random field.
The function may be called by the names: g05zmc or nag_rand_field_1d_user_setup.
3Description
A one-dimensional random field $Z\left(x\right)$ in $\mathbb{R}$ is a function which is random at every point $x\in \mathbb{R}$, so $Z\left(x\right)$ is a random variable for each $x$. The random field has a mean function $\mu \left(x\right)=\mathbb{E}\left[Z\left(x\right)\right]$ and a symmetric positive semidefinite covariance function $C(x,y)=\mathbb{E}\left[(Z\left(x\right)-\mu \left(x\right))(Z\left(y\right)-\mu \left(y\right))\right]$. $Z\left(x\right)$ is a Gaussian random field if for any choice of $n\in \mathbb{N}$ and ${x}_{1},\dots ,{x}_{n}\in \mathbb{R}$, the random vector ${[Z\left({x}_{1}\right),\dots ,Z\left({x}_{n}\right)]}^{\mathrm{T}}$ follows a multivariate Normal 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 ${\stackrel{~}{C}}_{ij}=C({x}_{i},{x}_{j})$. A Gaussian random field $Z\left(x\right)$ is stationary if $\mu \left(x\right)$ is constant for all $x\in \mathbb{R}$ and $C(x,y)=C(x+a,y+a)$ for all $x,y,a\in \mathbb{R}$ 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 g05zmc and g05zpc are used to simulate a one-dimensional stationary Gaussian random field, with mean function zero and variogram $\gamma \left(x\right)$, over an interval $[{x}_{\mathrm{min}},{x}_{\mathrm{max}}]$, using an equally spaced set of $N$ points on the interval. The problem reduces to sampling a Normal 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 $M\ge 2(N-1)$, 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 ${R}^{*}(\mathbf{U}+i\mathbf{V})$, where $\mathbf{U}$ and $\mathbf{V}$ have elements from the standard Normal distribution. Since ${R}^{*}(\mathbf{U}+i\mathbf{V})=W{\Lambda}^{\frac{1}{2}}(\mathbf{U}+i\mathbf{V})$, this calculation can be done using a discrete Fourier transform of the vector ${\Lambda}^{\frac{1}{2}}(\mathbf{U}+i\mathbf{V})$. 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 $\rho \in (0,1]$ is a scaling factor. The error $\epsilon $ in approximating the distribution of the random field is given by
g05zmc 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.
4References
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 Statistics3(4) 409–432
5Arguments
1: $\mathbf{ns}$ – IntegerInput
On entry: the number of sample points to be generated in realizations of the random field.
Constraint:
${\mathbf{ns}}\ge 1$.
2: $\mathbf{xmin}$ – doubleInput
On entry: the lower bound for the interval over which the random field is to be simulated.
Constraint:
${\mathbf{xmin}}<{\mathbf{xmax}}$.
3: $\mathbf{xmax}$ – doubleInput
On entry: the upper bound for the interval over which the random field is to be simulated.
Constraint:
${\mathbf{xmin}}<{\mathbf{xmax}}$.
4: $\mathbf{maxm}$ – 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+\lceil {\mathrm{log}}_{2}({\mathbf{ns}}-1)\rceil $.
Constraint:
${\mathbf{maxm}}\ge {2}^{k}$, where $k$ is the smallest integer satisfying ${2}^{k}\ge 2({\mathbf{ns}}-1)$.
5: $\mathbf{var}$ – doubleInput
On entry: the multiplicative factor ${\sigma}^{2}$ of the variogram $\gamma \left(x\right)$.
Constraint:
${\mathbf{var}}\ge 0.0$.
6: $\mathbf{cov1}$ – function, supplied by the userExternal Function
cov1 must evaluate the variogram $\gamma \left(x\right)$, without the multiplicative factor ${\sigma}^{2}$, for all $x\ge 0$. The value returned in gamma is multiplied internally by var.
On entry: the value $x$ at which the variogram $\gamma \left(x\right)$ is to be evaluated.
2: $\mathbf{gamma}$ – double *Output
On exit: the value of the variogram $\frac{\gamma \left(x\right)}{{\sigma}^{2}}$.
3: $\mathbf{comm}$ – Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to cov1.
user – double *
iuser – Integer *
p – Pointer
The type Pointer will be void *. Before calling g05zmc you may allocate memory and initialize these pointers with various quantities for use by cov1 when called from g05zmc (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).
Note:cov1 should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by g05zmc. If your code inadvertently does return any NaNs or infinities, g05zmc is likely to produce unexpected results.
7: $\mathbf{pad}$ – Nag_EmbedPadInput
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.
${\mathbf{pad}}=\mathrm{Nag\_EmbedPadZeros}$
The embedding matrix is padded with zeros.
${\mathbf{pad}}=\mathrm{Nag\_EmbedPadValues}$
The embedding matrix is padded with values of the variogram.
On exit: the points at which values of the random field will be output.
11: $\mathbf{m}$ – Integer *Output
On exit: the size of the embedding matrix.
12: $\mathbf{approx}$ – Integer *Output
On exit: indicates whether approximation was used.
${\mathbf{approx}}=0$
No approximation was used.
${\mathbf{approx}}=1$
Approximation was used.
13: $\mathbf{rho}$ – double *Output
On exit: indicates the scaling of the covariance matrix. ${\mathbf{rho}}=1.0$ unless approximation was used with ${\mathbf{corr}}=\mathrm{Nag\_EmbedScaleTraces}$ or $\mathrm{Nag\_EmbedScaleSqrtTraces}$.
14: $\mathbf{icount}$ – Integer *Output
On exit: indicates the number of negative eigenvalues in the embedding matrix which have had to be set to zero.
15: $\mathbf{eig}\left[3\right]$ – doubleOutput
On exit: indicates information about the negative eigenvalues in the embedding matrix which have had to be set to zero. ${\mathbf{eig}}\left[0\right]$ contains the smallest eigenvalue, ${\mathbf{eig}}\left[1\right]$ contains the sum of the squares of the negative eigenvalues, and ${\mathbf{eig}}\left[2\right]$ contains the sum of the absolute values of the negative eigenvalues.
16: $\mathbf{comm}$ – Nag_Comm *
The NAG communication argument (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).
17: $\mathbf{fail}$ – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error Indicators and Warnings
NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM
On entry, argument $\u27e8\mathit{\text{value}}\u27e9$ had an illegal value.
NE_INT
On entry, ${\mathbf{maxm}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: the minimum calculated value for maxm is $\u27e8\mathit{\text{value}}\u27e9$.
Where the minimum calculated value is given by ${2}^{k}$, where $k$ is the smallest integer satisfying ${2}^{k}\ge 2({\mathbf{ns}}-1)$.
On entry, ${\mathbf{ns}}=\u27e8\mathit{\text{value}}\u27e9$. Constraint: ${\mathbf{ns}}\ge 1$.
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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_REAL
On entry, ${\mathbf{var}}=\u27e8\mathit{\text{value}}\u27e9$. Constraint: ${\mathbf{var}}\ge 0.0$.
NE_REAL_2
On entry, ${\mathbf{xmin}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{xmax}}=\u27e8\mathit{\text{value}}\u27e9$. Constraint: ${\mathbf{xmin}}<{\mathbf{xmax}}$.
7Accuracy
If on exit ${\mathbf{approx}}=1$, see the comments in Section 3 regarding the quality of approximation; increase the value of maxm to attempt to avoid approximation.
8Parallelism and Performance
g05zmc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
g05zmc makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
9Further Comments
None.
10Example
This example calls g05zmc to calculate the eigenvalues of the embedding matrix for $8$ sample points of a random field characterized by the symmetric stable variogram:
where ${x}^{\prime}=\frac{x}{\ell}$, and $\ell $ and $\nu $ are parameters.
It should be noted that the symmetric stable variogram is one of the pre-defined variograms available in g05znc. It is used here purely for illustrative purposes.