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API

Covariance Structures

Isotropic Covariance Structures

GaussianRandomFields.Linear — Type
Linear(λ, [σ = 1], [p = 2])

Linear covariance structure with correlation length λ, (optional) marginal standard deviation σ and (optional) p-norm, defined as

$C(x, y) = \begin{cases} σ \left(1 - \displaystyle\frac{ρ}{λ}\right) & \text{if }ρ ≤ λ\\ 0 & \text{if }ρ>λ\end{cases}$

with $ρ = ||x - y||_p$.

Examples

julia> Linear(0.1)
linear (λ=0.1, σ=1.0, p=2.0)

julia> Linear(1.0, σ=2)
linear (λ=1.0, σ=2.0, p=2.0)

See also: Exponential, Spherical, Whittle, Gaussian, SquaredExponential, Matern

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GaussianRandomFields.Spherical — Type
Spherical(λ, [σ = 1], [p = 2])

Spherical covariance structure with correlation length λ, (optional) marginal standard deviation σ and (optional) p-norm, defined as

$C(x, y) = \begin{cases} σ \left(1 - \displaystyle\frac{3}{2}\frac{ρ}{λ} + \frac{1}{2}\left(\frac{ρ}{λ}\right)^3\right) & \text{for }ρ≤λ\\0 & \text{for }ρ>λ\end{cases}$

with $ρ = ||x - y||_p$.

Examples

julia> Spherical(0.1)
spherical (λ=0.1, σ=1.0, p=2.0)

julia> Spherical(1.0, σ=2)
spherical (λ=1.0, σ=2.0, p=2.0)

See also: Exponential, Linear, Whittle, Gaussian, SquaredExponential, Matern

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GaussianRandomFields.Exponential — Type
Exponential(λ, [σ = 1], [p = 2])

Exponential covariance structure with correlation length λ, (optional) marginal standard deviation σ and (optional) p-norm defined as

$C(x, y) = σ \exp\left(-\displaystyle\frac{ρ}{λ}\right)$

with $ρ = ||x - y||_p$.

Examples

julia> Exponential(0.1)
exponential (λ=0.1, σ=1.0, p=2.0)

julia> Exponential(1.0, σ=2)
exponential (λ=1.0, σ=2.0, p=2.0)

See also: Linear, Spherical, Whittle, Gaussian, SquaredExponential, Matern

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GaussianRandomFields.Whittle — Type
Whittle(λ, [σ = 1], [p = 2])

Whittle covariance structure with correlation length λ, (optional) marginal standard deviation σ and (optional) p-norm, defined as

$C(x, y) = σ \displaystyle\frac{ρ}{λ} K₁\left(\frac{ρ}{λ}\right)$

with $ρ = ||x-y||_p$.

Examples

julia> Whittle(0.1)
Whittle (λ=0.1, σ=1.0, p=2.0)

julia> Whittle(1.0, σ=2)
Whittle (λ=1.0, σ=2.0, p=2.0)

See also: Exponential, Linear, Spherical, Gaussian, SquaredExponential, Matern

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GaussianRandomFields.SquaredExponential — Type
SquaredExponential(λ, [σ = 1], [p = 2])

Squared exponential (Gaussian) covariance structure with correlation length λ, (optional) marginal standard deviation σ and (optional) p-norm, defined as

$C(x, y) = σ \exp\left(-\left(\displaystyle\frac{ρ}{λ}\right)^2\right)$

with $ρ = ||x - y||_p$.

Examples

julia> SquaredExponential(0.1)
Gaussian (λ=0.1, σ=1.0, p=2.0)

julia> SquaredExponential(1, σ=2.)
Gaussian (λ=1.0, σ=2.0, p=2.0)

See also: Exponential, Linear, Spherical, Whittle, Gaussian, Matern

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GaussianRandomFields.Gaussian — Type
Gaussian(λ, [σ = 1], [p = 2])

Gaussian (squared exponential) covariance structure with correlation length λ, (optional) marginal standard deviation σ and (optional) p-norm, defined as

$C(x, y) = σ \exp\left(-\left(\displaystyle\frac{ρ}{λ}\right)^2\right)$

with $ρ = ||x - y||_p$.

Examples

julia> Gaussian(0.1)
Gaussian (λ=0.1, σ=1.0, p=2.0)

julia> Gaussian(1, σ=2.)
Gaussian (λ=1.0, σ=2.0, p=2.0)

See also: Exponential, Linear, Spherical, Whittle, SquaredExponential, Matern

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GaussianRandomFields.Matern — Type
Matern(λ, ν, [σ = 1], [p = 2])

Matérn covariance structure with correlation length λ, smoothness ν, (optional) marginal standard deviation σ and (optional) p-norm, defined as

$C(x, y) = σ \displaystyle\frac{2^{1 - ν}}{Γ(ν)} \left(\frac{ρ}{λ}\right)^ν K_ν\left(\frac{ρ}{λ}\right)$

with $ρ = ||x - y||_p$.

Examples

julia> Matern(0.1, 1.0)
Matérn (λ=0.1, ν=1.0, σ=1.0, p=2.0)

julia> Matern(1, 1, σ=2.0)
Matérn (λ=1.0, ν=1.0, σ=2.0, p=2.0)

See also: Exponential, Linear, Spherical, Whittle, Gaussian, SquaredExponential

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Anisotropic Covariance Structures

GaussianRandomFields.AnisotropicExponential — Type
AnisotropicExponential(A, [σ = 1])

Anisotropic exponential covariance structure with anisotropy matrix A and (optional) marginal standard deviation σ, defined as

$C(x, y) = \exp(-ρᵀ A ρ)$

where $ρ = x - y$.

Examples

julia> A = [1 0.5; 0.5 1];

julia> AnisotropicExponential(A)
anisotropic exponential (A=[1.0 0.5; 0.5 1.0], σ=1.0)
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Covariance Functions

GaussianRandomFields.CovarianceFunction — Type
CovarianceFunction(d, cov)

Covariance function in d dimensions with covariance structure cov.

Examples

julia> CovarianceFunction(1, Exponential(0.1))
1d exponential covariance function (λ=0.1, σ=1.0, p=2.0)

julia> CovarianceFunction(2, Matern(0.1, 1.0))
2d Matérn covariance function (λ=0.1, ν=1.0, σ=1.0, p=2.0)
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GaussianRandomFields.SeparableCovarianceFunction — Type
SeparableCovarianceFunction(cov...)

Separable covariance function in length(cov) dimensions for the covariance structures cov. Usefull for defining anisotropic covariance functions, or if the non-seperable KarhunenLoeve expansion is too expensive.

Examples

julia> SeparableCovarianceFunction(Exponential(0.1), Matern(0.01, 1.0))
2d separable covariance function [ exponential (λ=0.1, σ=1.0, p=2.0), Matérn (λ=0.01, ν=1.0, σ=1.0, p=2.0) ]

See also: CovarianceFunction, KarhunenLoeve

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GaussianRandomFields.apply — Function
apply(cov, pts...)

Returns the covariance matrix, i.e., the covariance function cov evaluated in the points x.

Examples

julia> exponential_covariance = CovarianceFunction(1, Exponential(1))
1d exponential covariance function (λ=1.0, σ=1.0, p=2.0)

julia> pts = range(0, stop=1, length=11)
0.0:0.1:1.0

julia> C = apply(exponential_covariance, pts);

julia> heatmap(C);

julia> whittle_covariance = CovarianceFunction(2, Whittle(1))
2d Whittle covariance function (λ=1.0, σ=1.0, p=2.0)

julia> C = apply(whittle_covariance, pts, pts);

julia> heatmap(C);

See also: CovarianceFunction, Exponential, Whittle

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Gaussian Random Field Generators

GaussianRandomFields.GaussianRandomFieldGenerator — Type

Abstract type GaussianRandomFieldGenerator

The following Gaussian random field generators are implemented:

  • Cholesky: Cholesky factorization of the covariance matrix, exact but expensive for random fields in dimension d > 1
  • Spectral: spectral (eigenvalue) decomposition of the covariance matrix, exact but expensive for random fields in dimension d > 1
  • KarhunenLoeve: Karhunen-Loève expansion, inexact but very efficient for "smooth" random fields when used with a low truncation dimension
  • CirculantEmbedding: circulant embedding method, exact and efficient, but can only be used for random fields on structured grids

See also: Cholesky, Spectral, KarhunenLoeve, CirculantEmbedding

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Cholesky Factorization

GaussianRandomFields.Cholesky — Type
Cholesky()

Retuns a GaussianRandomFieldGenerator based on a Cholesky factorization of the covariance matrix.

Examples

julia> cov = CovarianceFunction(2, Matern(.3, 1))
2d Matérn covariance function (λ=0.3, ν=1.0, σ=1.0, p=2.0)

julia> pts = range(0, stop=1, length=51) 
0.0:0.02:1.0

julia> grf = GaussianRandomField(cov, Cholesky(), pts, pts)
Gaussian random field with 2d Matérn covariance function (λ=0.3, ν=1.0, σ=1.0, p=2.0) on a 51×51 structured grid, using a Cholesky decomposition

julia> heatmap(grf);
Warning

The Cholesky factorization requires the covariance matrix to be symmetric and positive definite. If the covariance matrix is not positive definite, an error will be thrown. Try using the Spectral method in that case.

See also: Spectral, KarhunenLoeve, CirculantEmbedding

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Spectral Decomposition

GaussianRandomFields.Spectral — Type
Spectral()

Returns a GaussianRandomFieldGenerator based on a spectral (eigenvalue) decomposition of the covariance matrix.

Optional Arguments for GaussianRandomField

  • n::Integer: the number of eigenvalues to compute. By default, we compute all eigenvalues.
  • eigensolver::EigenSolver: which method to use for the eigenvalue decomposition (see AbstractEigenSolver). The default is EigenSolver().

Examples

julia> cov = CovarianceFunction(2, Matern(.3, 1))
2d Matérn covariance function (λ=0.3, ν=1.0, σ=1.0, p=2.0)

julia> pts = range(0, stop=1, length=51)
0.0:0.02:1.0

julia> grf = GaussianRandomField(cov, Spectral(), pts, pts)
Gaussian random field with 2d Matérn covariance function (λ=0.3, ν=1.0, σ=1.0, p=2.0) on a 51×51 structured grid, using a spectral decomposition

julia> heatmap(grf);
Tip

Try using the Karhunen-Loève expansion if evaluating the covariance matrix is too expensive.

This is also useful when computing Gaussian random fields on a Finite Element mesh using a truncated KL expansion. Here's an example that computes the first 10 eigenfunctions on an L-shaped domain.

julia> cov = CovarianceFunction(2, Matern(.3, 1))
2d Matérn covariance function (λ=0.3, ν=1.0, σ=1.0, p=2.0)

julia> p, t = Lshape();

julia> grf = GaussianRandomField(cov, Spectral(), p, t, n=10)
Gaussian random field with 2d Matérn covariance function (λ=0.3, ν=1.0, σ=1.0, p=2.0) on a mesh with 998 points and 1861 elements, using a spectral decomposition

See also: Cholesky, KarhunenLoeve, CirculantEmbedding

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Karhunen Loève Decomposition

GaussianRandomFields.KarhunenLoeve — Type
KarhunenLoeve(n)

Returns a GaussianRandomFieldGenerator using a Karhunen-Loève (KL) expansion with n terms.

Optional Arguments for GaussianRandomField

  • quad::QuadratureRule: quadrature rule used for the integral equation (see QuadratureRule), default is EOLE().
  • nq::Integer: number of quadrature points in each dimension, where we require nq^d > n. Default is nq = ceil(n^(1/d)).
  • eigensolver::EigenSolver: which method to use for the eigenvalue decomposition (see AbstractEigenSolver). The default is EigenSolver().

Examples

julia> cov = CovarianceFunction(2, Matern(.3, 1))
2d Matérn covariance function (λ=0.3, ν=1.0, σ=1.0, p=2.0)

julia> pts = range(0, stop=1, length=51)
0.0:0.02:1.0

julia> grf = GaussianRandomField(cov, KarhunenLoeve(300), pts, pts)
Gaussian random field with 2d Matérn covariance function (λ=0.3, ν=1.0, σ=1.0, p=2.0) on a 51×51 structured grid, using a KL expansion with 300 terms

julia> plot_eigenvalues(grf); # plot the eigenvalue decay

julia> plot_eigenfunction(grf, 4); # plots the fourth eigenfunction

If more terms n are used in the expansion, the approximation becomes better.

julia> for n in [1, 2, 5, 10, 20, 50, 100, 200, 500, 1000]
           grf = GaussianRandomField(cov, KarhunenLoeve(n), pts, pts)
           @printf "%.8f\n" rel_error(grf)
       end
0.69838285
0.45494187
0.23231278
0.10079295
0.02647020
0.00978427
0.00356549
0.00107191
0.00019810
0.00004085
Note

Techniqually, the KL expansion is computed using the Nystrom method. For nonstructured grids, we use a bounding box approach. Try using the Spectral method if this is not what you want.

Warning

To avoid an end effect in the eigenvalue decay, choose nq^d ≥ 5n.

julia> grf = GaussianRandomField(cov, KarhunenLoeve(300), pts, pts, quad=GaussLegendre())
Gaussian random field with 2d Matérn covariance function (λ=0.3, ν=1.0, σ=1.0, p=2.0) on a 51×51 structured grid, using a KL expansion with 300 terms

julia> grf = GaussianRandomField(cov, KarhunenLoeve(300), pts, pts, nq=40)
Gaussian random field with 2d Matérn covariance function (λ=0.3, ν=1.0, σ=1.0, p=2.0) on a 51×51 structured grid, using a KL expansion with 300 terms

See also: Cholesky, Spectral, CirculantEmbedding

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GaussianRandomFields.rel_error — Function
rel_error(grf)

Returns the relative error in the Karhunen-Loève approximation of the random field, computed as

$1 - \displaystyle\frac{\sum \theta_j^2}{\sigma^2 \int_D \mathrm{d}x}$.

Only useful for fields defined on a rectangular domain.

Examples

julia> cov = CovarianceFunction(2, Matern(.3, 1))
2d Matérn covariance function (λ=0.3, ν=1.0, σ=1.0, p=2.0)

julia> pts = range(0, stop=1, length=51)
0.0:0.02:1.0

julia> grf = GaussianRandomField(cov, KarhunenLoeve(300), pts, pts)
Gaussian random field with 2d Matérn covariance function (λ=0.3, ν=1.0, σ=1.0, p=2.0) on a 51×51 structured grid, using a KL expansion with 300 terms

julia> @printf "%.8f" rel_error(grf)
0.00046071
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Circulant Embedding

GaussianRandomFields.CirculantEmbedding — Type
CirculantEmbedding()

Returns a GaussianRandomFieldGenerator that uses FFTs to compute samples of the Gaussian random field.

Warning

Circulant embedding can only be applied if the points are specified on a structured grid.

Optional Arguments for GaussianRandomField

  • minnpadding::Integer: minimum amount of padding.
  • measure::Bool: optimize the FFT to increase the efficiency of the sample method. Default is true.
  • primes::Bool: the size of the minimum circulant embedding of the covariance matrix can be written as a product of small primes (2, 3, 5 and 7). Default is false.

Examples

julia> cov = CovarianceFunction(2, Matern(.1, 1))
2d Matérn covariance function (λ=0.1, ν=1.0, σ=1.0, p=2.0)

julia> pts = range(0, stop=1, length=51)
0.0:0.02:1.0

julia> grf = GaussianRandomField(cov, CirculantEmbedding(), pts, pts)
Gaussian random field with 2d Matérn covariance function (λ=0.1, ν=1.0, σ=1.0, p=2.0) on a 51×51 structured grid, using a circulant embedding

julia> contourf(grf);

julia> plot_eigenvalues(grf);
Note

With appropriate ordering, the covariance matrix of a Gaussian random field is a (nested block) Toeplitz matrix. This matrix can be embedded into a larger (nested block) circulant matrix, whose eigenvalues can be rapidly computed using FFT. A difficulty here is that although the covariance matrix is positive semi-definite, its circulant extension in general is not. As a remedy, one can add so-called ghost points outside the domain of interest using the optional flag minpadding.

julia> cov = CovarianceFunction(2, Matern(.3, 1))
2d Matérn covariance function (λ=0.3, ν=1.0, σ=1.0, p=2.0)

julia> pts = range(0, stop=1, length=51)
0.0:0.02:1.0

julia> grf = GaussianRandomField(cov, CirculantEmbedding(), pts, pts);
┌ Warning: 318 negative eigenvalues ≥ -0.5828339433508111 detected, Gaussian random field will be 
│         approximated (ignoring all negative eigenvalues); increase padding if possible
└ @ GaussianRandomFields ~/.julia/dev/GaussianRandomFields/src/generators/circulant_embedding.jl:94
Gaussian random field with 2d Matérn covariance function (λ=0.3, ν=1.0, σ=1.0, p=2.0) on a 51×51 structured grid, using a circulant embedding

julia> grf = GaussianRandomField(cov, CirculantEmbedding(), pts, pts, minpadding=79)
Gaussian random field with 2d Matérn covariance function (λ=0.3, ν=1.0, σ=1.0, p=2.0) on a 51×51 structured grid, using a circulant embedding

See also: Cholesky, Spectral, KarhunenLoeve

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Gaussian Random Fields

GaussianRandomFields.GaussianRandomField — Type
GaussianRandomField([mean,] cov, generator, pts...)
GaussianRandomField([mean,] cov, generator, nodes, elements)

Compute a Gaussian random field with mean mean and covariance structure cov defined in the points pts, and computed using the Gaussian random field generator generator.

Examples

julia> cov = CovarianceFunction(2, Matern(.3, 1))
2d Matérn covariance function (λ=0.3, ν=1.0, σ=1.0, p=2.0)

julia> pts = pts = range(0, stop=1, length=51)
0.0:0.02:1.0

julia> mean = fill(π, (51, 51));

julia> grf = GaussianRandomField(mean, cov, Cholesky(), pts, pts)
Gaussian random field with 2d Matérn covariance function (λ=0.3, ν=1.0, σ=1.0, p=2.0) on a 51×51 structured grid, using a Cholesky decomposition

If no mean is specified, a zero-mean Gaussian random field is assumed.

julia> grf = GaussianRandomField(cov, Cholesky(), pts, pts)
Gaussian random field with 2d Matérn covariance function (λ=0.3, ν=1.0, σ=1.0, p=2.0) on a 51×51 structured grid, using a Cholesky decomposition

The Gaussian random field generator generator can be Cholesky(), Spectral(), KarhunenLoeve(n) (where n is the number of terms in the expansion), or CirculantEmbedding(). The points pts can be specified as arguments of type AbstractVector, in which case a tensor (Kronecker) product is assumed, or as a Finite Element mesh with node table nodes and element table elements.

julia> grf = GaussianRandomField(cov, KarhunenLoeve(500), pts, pts)
Gaussian random field with 2d Matérn covariance function (λ=0.3, ν=1.0, σ=1.0, p=2.0) on a 51×51 structured grid, using a KL expansion with 500 terms

julia> exponential_cov = CovarianceFunction(2, Exponential(.1))
2d exponential covariance function (λ=0.1, σ=1.0, p=2.0)

julia> grf = GaussianRandomField(exponential_cov, CirculantEmbedding(), pts, pts)
Gaussian random field with 2d exponential covariance function (λ=0.1, σ=1.0, p=2.0) on a 51×51 structured grid, using a circulant embedding

Separable Gaussian random fields can be defined using SeparableCovarianceFunction.

julia> separable_cov = SeparableCovarianceFunction(Exponential(.1), Exponential(.1))
2d separable covariance function [ exponential (λ=0.1, σ=1.0, p=2.0), exponential (λ=0.1, σ=1.0, p=2.0) ]

julia> grf = GaussianRandomField(separable_cov, KarhunenLoeve(500), pts, pts)
Gaussian random field with 2d separable covariance function [ exponential (λ=0.1, σ=1.0, p=2.0), exponential (λ=0.1, σ=1.0, p=2.0) ] on a 51×51 structured grid, using a KL expansion with 500 terms

julia> plot_eigenfunction(grf, 3);

We also offer support for anisotropic random fields.

julia> anisotropic_cov = CovarianceFunction(2, AnisotropicExponential([500 400; 400 500]))
2d anisotropic exponential covariance function (A=[500 400; 400 500], σ=1.0)

julia> grf = GaussianRandomField(anisotropic_cov, CirculantEmbedding() , pts, pts)
Gaussian random field with 2d anisotropic exponential covariance function (A=[500 400; 400 500], σ=1.0) on a 51×51 structured grid, using a circulant embedding

julia> heatmap(grf);

For irregular domains, specify the points as matrices containing the nodes and elements of a finite element mesh. To compute the value of the random field at the element centers, use the optional keyword mode="center".

julia> nodes, elements = Lshape();

julia> grf = GaussianRandomField(cov, Spectral(), nodes, elements)
Gaussian random field with 2d Matérn covariance function (λ=0.3, ν=1.0, σ=1.0, p=2.0) on a mesh with 998 points and 1861 elements, using a spectral decomposition

Alternativly, simply pass a Matrix{T} of size N by d to compute the random field at un unstructured grid defined by the given set of points.

Samples from the random field can be computed using the sample function.

julia> sample(grf);

See also: Cholesky, Spectral, KarhunenLoeve, CirculantEmbedding, sample

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StatsBase.sample — Function
sample(grf)
    sample(grf[, xi])

Take a sample from the Gaussian random field grf using the (optional) normally distributed random numbers xi. The vectorxi must have appropriate length.

Examples

julia> cov = CovarianceFunction(2, Whittle(.1))
2d Whittle covariance function (λ=0.1, σ=1.0, p=2.0)

julia> pts = pts = range(0, stop=1, length=51)
0.0:0.02:1.0

julia> grf = GaussianRandomField(cov, CirculantEmbedding(), pts, pts)
Gaussian random field with 2d Whittle covariance function (λ=0.1, σ=1.0, p=2.0) on a 51×51 structured grid, using a circulant embedding

julia> sample(grf);

julia> sample(grf, xi=randn(randdim(grf)));

See also: GaussianRandomField, Matern, CovarianceFunction, [CirculantEmbedding]

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GaussianRandomFields.randdim — Function
randdim(grf)

Returns the number of random numbers used to sample from the Gaussian random field grf.

Examples

julia> cov = CovarianceFunction(2, Whittle(.1))
2d Whittle covariance function (λ=0.1, σ=1.0, p=2.0)

julia> pts = pts = range(0, stop=1, length=51)
0.0:0.02:1.0

julia> grf = GaussianRandomField(cov, KarhunenLoeve(200), pts, pts)
Gaussian random field with 2d Whittle covariance function (λ=0.1, σ=1.0, p=2.0) on a 51×51 structured grid, using a KL expansion with 200 terms

julia> randdim(grf)
200
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Plotting

Standard plotting functions such as plot and plot! (for one-dimensional Gaussian random fields), and heatmap, surface, contour and contourf (for two-dimensional Gaussian random fields) are implemented. There are also some convenience plotting functions defined:

GaussianRandomFields.plot_eigenvalues — Function
plot_eigenvalues(grf)

Log-log plot of the eigenvalues of the Gaussian random field. Only available for Gaussian random field generators of type Spectral, KarhunenLoeve and CirculantEmbedding.

Examples

julia> cov = CovarianceFunction(2, Gaussian(.1))
2d Gaussian covariance function (λ=0.1, σ=1.0, p=2.0)

julia> pts = range(0, stop=1, length=51)
0.0:0.02:1.0

julia> grf = GaussianRandomField(cov, KarhunenLoeve(100), pts, pts)
Gaussian random field with 2d Gaussian covariance function (λ=0.1, σ=1.0, p=2.0) on a 51×51 structured grid, using a KL expansion with 100 terms

julia> plot_eigenvalues(grf);

See also: plot_eigenfunction

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GaussianRandomFields.plot_eigenfunction — Function
plot_eigenfunction(grf, n)

Contour plot of the nth eigenfunction of the Gaussian random field. Only available for Gaussian random field generators of type Spectral and KarhunenLoeve.

Examples

julia> cov = CovarianceFunction(2, Gaussian(.1))
2d Gaussian covariance function (λ=0.1, σ=1.0, p=2.0)

julia> pts = range(0, stop=1, length=51)
0.0:0.02:1.0

julia> grf = GaussianRandomField(cov, KarhunenLoeve(100), pts, pts)
Gaussian random field with 2d Gaussian covariance function (λ=0.1, σ=1.0, p=2.0) on a 51×51 structured grid, using a KL expansion with 100 terms

julia> plot_eigenfunction(grf, 6); # 6th eigenfunction

See also: plot_eigenfunction

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GaussianRandomFields.plot_covariance_matrix — Function
plot_covariance_matrix(grf[, pts...])

Evaluate the covariance function of the Gaussian random field `grf` in the (optional) points `pts` and plot the result.

Examples

julia> cov = CovarianceFunction(2, Matern(.3, 1))
2d Matérn covariance function (λ=0.3, ν=1.0, σ=1.0, p=2.0)

julia> pts = range(0, stop=1, length=11)
0.0:0.1:1.0

julia> grf = GaussianRandomField(cov, CirculantEmbedding(), pts, pts, minpadding=7)
Gaussian random field with 2d Matérn covariance function (λ=0.3, ν=1.0, σ=1.0, p=2.0) on a 11×11 structured grid, using a circulant embedding

julia> plot_covariance_matrix(grf);

julia> pts = range(0, stop=1, length=21)
0.0:0.05:1.0

julia> plot_covariance_matrix(grf, pts, pts);
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Unstructured Meshes

Utilities

GaussianRandomFields.QuadratureRule — Type

Abstract type QuadratureRule

The following quadrature rules are implemented:

  • Midpoint::QuadratureRule: the midpoint rule
  • Trapezoidal::QuadratureRule: the trapezoidal rule
  • Simpson::QuadratureRule: Simpson's rule
  • GaussLegendre::QuadratureRule: Gauss-Legendre quadrature rule
  • EOLE::QuadratureRule: expansion-optimal linear estimation

See also: Midpoint, Trapezoidal, Simpson, GaussLegendre, EOLE

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