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)[source]¶
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 byfield_2d_generate()
, which simulates the random field.For full information please refer to the NAG Library document for g05zr
https://www.nag.com/numeric/nl/nagdoc_29/flhtml/g05/g05zrf.html
- Parameters
- nsint, array-like, shape
The number of sample points to use in each direction, with sample points in the -direction, and sample points in the -direction, . The total number of sample points on the grid is, therefore, .
- xminfloat
The lower bound for the -coordinate, for the region in which the random field is to be simulated.
- xmaxfloat
The upper bound for the -coordinate, for the region in which the random field is to be simulated.
- yminfloat
The lower bound for the -coordinate, for the region in which the random field is to be simulated.
- ymaxfloat
The upper bound for the -coordinate, for the region in which the random field is to be simulated.
- maxmint, array-like, shape
Determines the maximum size of the circulant matrix to use – a maximum of elements in the -direction, and a maximum of elements in the -direction. The maximum size of the circulant matrix is thus .
- varfloat
The multiplicative factor of the variogram .
- icov2int
Determines which of the preset variograms to use. The choices are given below. Note that , where and are correlation lengths in the and directions respectively and are parameters for most of the variograms, and is the variance specified by .
Symmetric stable variogram
where
, ,
, ,
, .
Cauchy variogram
where
, ,
, ,
, .
Differential variogram with compact support
where
, ,
, .
Exponential variogram
where
, ,
, .
Gaussian variogram
where
, ,
, .
Nugget variogram
No parameters need be set for this value of .
Spherical variogram
where
, ,
, .
Bessel variogram
where
is the Bessel function of the first kind,
, ,
, ,
, .
Hole effect variogram
where
, ,
, .
Whittle-Matérn variogram
where
is the modified Bessel function of the second kind,
, ,
, ,
, .
Continuously parameterised variogram with compact support
where
,
is the modified Bessel function of the second kind,
, ,
, ,
, ,
, ,
, .
Generalized hyperbolic distribution variogram
where
is the modified Bessel function of the second kind,
, ,
, ,
, no constraint on ,
, ,
, .
- paramsfloat, array-like, shape
The parameters for the variogram as detailed in the description of .
- normint, optional
Determines which norm to use when calculating the variogram.
The 1-norm is used, i.e., .
The 2-norm (Euclidean norm) is used, i.e., .
- 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.
The embedding matrix is padded with zeros.
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 -coordinates at which values of the random field will be output.
- yyfloat, ndarray, shape
The points of the -coordinates at which values of the random field will be output.
- mint, ndarray, shape
contains , the size of the circulant blocks and contains , the number of blocks, resulting in a final square matrix of size .
- approxint
Indicates whether approximation was used.
No approximation was used.
Approximation was used.
- rhofloat
Indicates the scaling of the covariance matrix. unless approximation was used with or .
- icountint
Indicates the number of negative eigenvalues in the embedding matrix which have had to be set to zero.
- eigfloat, ndarray, shape
Indicates information about the negative eigenvalues in the embedding matrix which have had to be set to zero. contains the smallest eigenvalue, contains the sum of the squares of the negative eigenvalues, and contains the sum of the absolute values of the negative eigenvalues.
- Raises
- NagValueError
- (errno )
On entry, .
Constraint: , .
- (errno )
On entry, and .
Constraint: .
- (errno )
On entry, and .
Constraint: .
- (errno )
On entry, .
Constraint: the minimum calculated value for are .
- (errno )
On entry, .
Constraint: .
- (errno )
On entry, .
Constraint: and .
- (errno )
On entry, .
Constraint: or .
- (errno )
On entry, .
Constraint: for , .
- (errno )
On entry, .
Constraint: dependent on , see documentation.
- (errno )
On entry, .
Constraint: or .
- (errno )
On entry, .
Constraint: , or .
- Notes
A two-dimensional random field in is a function which is random at every point , so is a random variable for each . The random field has a mean function and a symmetric positive semidefinite covariance function . is a Gaussian random field if for any choice of and , the random vector follows a multivariate Normal distribution, which would have a mean vector with entries and a covariance matrix with entries . A Gaussian random field is stationary if is constant for all and for all and hence we can express the covariance function as a function of one variable: . is known as a variogram (or more correctly, a semivariogram) and includes the multiplicative factor representing the variance such that .
The functions
field_2d_predef_setup
andfield_2d_generate()
are used to simulate a two-dimensional stationary Gaussian random field, with mean function zero and variogram , over a domain , using an equally spaced set of points; points in the -direction and points in the -direction. The problem reduces to sampling a Gaussian random vector of size , with mean vector zero and a symmetric covariance matrix , which is an block Toeplitz matrix with Toeplitz blocks of size . Since is in general expensive to factorize, a technique known as the circulant embedding method is used. is embedded into a larger, symmetric matrix , which is an block circulant matrix with circulant blocks of size , where and . can now be factorized as , where is the two-dimensional Fourier matrix ( is the complex conjugate of ), is the diagonal matrix containing the eigenvalues of and . is known as the embedding matrix. The eigenvalues can be calculated by performing a discrete Fourier transform of the first row (or column) of and multiplying by , and so only the first row (or column) of is needed – the whole matrix does not need to be formed.As long as all of the values of are non-negative (i.e., is positive semidefinite), is a covariance matrix for a random vector which has blocks of size . Two samples of can now be simulated from the real and imaginary parts of , where and have elements from the standard Normal distribution. Since , this calculation can be done using a discrete Fourier transform of the vector . Two samples of the random vector can now be recovered by taking the first elements of the first blocks of each sample of – because the original covariance matrix is embedded in , will have the correct distribution.
If is not positive semidefinite, larger embedding matrices can be tried; however if the size of the matrix would have to be larger than , an approximation procedure is used. We write , where and contain the non-negative and negative eigenvalues of respectively. Then is replaced by where and is a scaling factor. The error in approximating the distribution of the random field is given by
Three choices for are available, and are determined by the input argument :
setting sets
setting sets
setting sets .
field_2d_predef_setup
finds a suitable positive semidefinite embedding matrix and outputs its sizes in the vector and the square roots of its eigenvalues in . If approximation is used, information regarding the accuracy of the approximation is output. Note that only the first row (or column) of 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