inference

The Debiased Whittle Likelihood estimator is based on the periodogram of the data and the expected periodogram corresponding to the combination of a sampling grid and covariance model.

periodogram

This module provides tools to define a periodogram and to obtain its expectation, the expected periodogram.

The periodogram is defined according to, $$ I(\mathbf{k}) = \frac{1}{|n|} \left| \sum_{\mathbf{s}} { g_{\mathbf{s}} X_{\mathbf{s}} \exp(-i\mathbf{k}\cdot\mathbf{s}) } \right|^2 $$ where the summation is over grid points, and where \(g_{\mathbf{s}}\) is obtained from both the grid mask and the periodogram taper function. In practice, this is implemented using the Fast Fourier Transform.

Periodogram(taper=None)

Provides the capability to compute the periodogram of the data.

Attributes:

Name	Type	Description
taper	function handle	tapering function
fold	boolean	Whether to fold the periodogram.

fold property writable

Whether to compute a folded version of the periodogram.

__call__(sample: Union[xp.ndarray, SampleOnRectangularGrid])

Computes the periodogram of the data.

Parameters:

Name	Type	Description	Default
sample	Union[ndarray, SampleOnRectangularGrid]	Sampled data on the grid. Can either be an ndarray, or an instance of SampleOnRectangularGrid. In the latter case, repeated calls to this method will access cached values of the periodogram rather than carrying out the same computation again.	required

Returns:

Name	Type	Description
periodogram	ndarray	Periodogram of the data - shape (2 * n1 + 1, ..., 2 * nk + 1) if the fold attribute is False - shape (n1, ..., nk) if the fold attribute is True

__setattr__(key, value)

Sets attribute and update version of the object, which will update its hash, so that stored periodogram values are not used if properties of the periodogram are changed.

Parameters:

Name	Type	Description	Default
key		name of the attribute	required
value		value of the attribute	required

ExpectedPeriodogram(grid: RectangularGrid, periodogram: Periodogram)

Provides the capability to compute the expected periodogram on a fixed grid for any covariance model.

Attributes:

Name	Type	Description
grid	RectangularGrid	sampling grid
periodogram	Periodogram	periodogram for which we require the expectation. This is necessary to account for tapering for instance.

Notes

In dimension 1, the formula for the expected periodogram can be written as,

\[ \overline{I}(\omega_k) = \sum_{\tau=-n + 1}^{n - 1} c_g(\tau) c_X(\tau) e^{i \omega_k \tau}, \]

or equivalently,

\[ \overline{I}(\omega_k) = \sum_{\tau=0}^{2n - 1} \left[ c_g(\tau) c_X(\tau) + c_g(- n + \tau) c_X(- n + \tau) \right] e^{i \omega_k \tau}. \]

The latter can naturally be implemented via FFT and generalizes to higher-dimension domains.

grid: RectangularGrid property writable

Sampling grid

periodogram property writable

periodogram for which we require the expectation.

__call__(model: CovarianceModel) -> xp.ndarray

Compute the expected periodogram for this covariance model.

Parameters:

Name	Type	Description	Default
model	CovarianceModel	Covariance model under which we compute the expectation of the periodogram	required

Returns:

Name	Type	Description
ep	ndarray	The shape depends n univariate versus multivariate (\(p\)) and on unique versus vectorized model (\(m\)), as per the table below. Shape of ep \(p=1\) \(p>1\) Unique model (n1, ..., nd) (n1, ..., nd, p, p) Vectorized model (n1, ..., nd, m) (n1, ..., nd, m, p, p) If the fold attribute of the periodogram is False, the shape of the returned array is instead (2 * n1 + 1, ..., 2 * nd + 1).

compute_ep(acv: xp.ndarray, fold: bool = True, d: Tuple[int, int] = (0, 0), apply_cg: bool = True)

Computes the expected periodogram, and more generally any diagonal of the covariance matrix of the Discrete Fourier Transform identitied by the two-dimensional offset d. The standard expected periodogram corresponds to the default d = (0, 0).

Parameters:

Name	Type	Description	Default
acv	ndarray	Autocovariance evaluated on the grid's lags. For a grid with shape (n1, ..., nd), the first d dimensions of acv should have sizes (2 * n1 - 1, ..., 2 * nd - 1). The standard way to obtain acv is through the call of the autocov method of a rectangular grid. acv may have extra dimensions. The following other cases are standard (see the documentation of the call and gradient methods of models.ModelInterface): Univariate data, vectorized models. acv will have shape (2 * n1 - 1, ..., 2 * nd - 1, m) where m is the number of model parameter vectors. Univariate data, gradient. acv will have shape (2 * n1 - 1, ..., 2 * nd - 1, k) where k is the number of parameters with respect to which we request the gradient. Multivariate data, unique model parameter vector. acv will have shape (2 * n1 - 1, ..., 2 * nd - 1, p, p) where p is the number of variates Multivariate data, vectorized models. acv will have shape (2 * n1 - 1, ..., 2 * nd - 1, m, p, p) Multivariate data, gradient. acv will have shape (2 * n1 - 1, ..., 2 * nd - 1, k, p, p) where k is the number of parameters with respect to which we take the gradient. The case of gradient computation combined with vectorized models (not shown above) might work, but has not been tested as of now.	required
fold	bool	Whether to apply folding of the expected periodogram	True
d	Tuple[int, int]	Offset that identifies a hyper-diagonal of the covariance matrix of the DFT.	(0, 0)

Returns:

Name	Type	Description
ep	ndarray	Expectation of the periodogram. Same shape as acv when fold is True.

Notes

For standard use cases, this should not be called directly. Instead, one should directly call the call method.

gradient(model: CovarianceModel, params: list[ModelParameter]) -> ndarray

Provides the gradient of the expected periodogram with respect to the parameters of the model at all frequencies of the Fourier grid. The last dimension of the returned array indexes the parameters.

Parameters:

Name	Type	Description	Default
model	CovarianceModel	Covariance model. It should implement the gradient method.	required
params	list[ModelParameter]	Parameters with which to take the gradient.	required

Returns:

Name	Type	Description
gradient	ndarray	Array providing the gradient of the expected periodogram at all Fourier frequencies with respect to the requested parameters. The last dimension of the returned array indexes the parameters.

Notes

This requires that the model's _gradient method be implemented.

cov_dft_matrix(model: CovarianceModel)

Provides the complex-valued covariance matrix of the Discrete Fourier Transform. Implemented via FFT, but still requires the full covariance matrix of the data, hence this is not viable for large grids.

Parameters:

Name	Type	Description	Default
model	CovarianceModel	Covariance model for which we request the covariance matrix of the Discrete Fourier Transform	required

Returns:

Name	Type	Description
cov_dft	ndarray(n1, n2, ..., nd)	Covariance matrix of the Discrete Fourier Transform of the data.

Notes

The result of this method corresponds to,

\[ E[\mathbf{J} \mathbf{J}^*] = E[U^* \mathbf{X} \mathbf{X}^* U]=U^* E[\mathbf{X} \mathbf{X}^T] U= U^* C_X U \]

where \(\mathbf{J}\) is the Discrete Fourier Transform (DFT) of the data \(\mathbf{X}\), whose covariance matrix is denoted by \(C_X\), and \(U\) is the DFT matrix.

rel_dft_matrix(model: CovarianceModel)

Provides the relation matrix of the Discrete Fourier Transform. Requires storing the full covariance matrix, hence not viable for large grids. Useful however to check other methods.

Parameters:

Name	Type	Description	Default
model	CovarianceModel	Covariance model for which we request the covariance matrix of the Discrete Fourier Transform	required

Returns:

Name	Type	Description
rel_dft	ndarray(n1, n2, ..., nk)	Relation matrix of the Discrete Fourier Transform of the data

cov_dft_diagonals(model: CovarianceModel, m: Tuple[int, int])

Returns the covariance of the DFT over a given diagonal.

Parameters:

Name	Type	Description	Default
model	CovarianceModel	Covariance model. The covariance matrix of the DFT will depend on both the model and the sampling.	required
m	Tuple[int, int]	Offset in Fourier frequency indices. More precisely, m = (m1, m2) is the offset between two frequencies. In dimension 1 this would correspond to i2 - i1 = m1 in terms of the indices of Fourier frequencies.	required

Returns:

Name	Type	Description
cov_dft	ndarray	The covariance of the DFT between Fourier frequencies separated by the offset.

Notes

When m is zero everywhere, this is just the expected periodogram.

cov_diagonals(model: CovarianceModel, m: Tuple[int, int])

Returns the covariance of the periodogram (valid only in 2d).

Parameters:

Name	Type	Description	Default
model	CovarianceModel	True covariance model	required
m	Tuple[int, int]	frequency offset	required

Returns:

Name	Type	Description
cov	ndarray	Covariance of the periodogram between frequencies offset by m

cov_dft_antidiagonals(model: CovarianceModel, m: Tuple[int, int])

Returns the covariance of the DFT over a given diagonal. More precisely, m = (m1, m2) is the offset between two frequencies. In dimension 1 this would correspond to i2 + i1 = m1 in terms of the indices of Fourier frequencies

cov_antidiagonals(model: CovarianceModel, m: Tuple[int, int])

Parameters:

Name	Type	Description	Default
model	CovarianceModel		required
m	Tuple[int, int]		required

SeparableExpectedPeriodogram(grid: RectangularGrid, periodogram: Periodogram)

Bases: ExpectedPeriodogram

Class to obtain the expected periodogram on a rectangular grid for a separable covariance model, in which case separability offers computational gains since the full expected periodogram can be computed as the outer product of the expected periodograms in the lower dimensions.

gradient(model)

Provides the derivatives of the expected periodogram with respect to the parameters of the model at all frequencies of the Fourier grid. The last dimension is used for different parameters.

compute_ep(acv: xp.ndarray, fold: bool = True, d: Tuple[int, int] = (0, 0)) -> xp.ndarray

Computes the expected periodogram for the passed finite autocovariance function, in the case where...

Parameters:

Name	Type	Description	Default
acv	ndarray		required
fold	bool		True
d	Tuple[int, int]		(0, 0)

autocov(cov_func, shape)

Compute the covariance function on a grid of lags determined by the passed shape. In d=1 the lags would be -(n-1)...n-1, but then a iffshit is applied so that the lags are 0 ... n-1 -n+1 ... -1. This may look weird but it makes it easier to do the folding operation when computing the expecting periodogram

compute_ep(acf: ndarray, spatial_kernel: ndarray, grid: ndarray = None, fold: bool = True) -> ndarray

Computes the expected periodogram, for the passed covariance function and grid. The grid is an array, in the simplest case just an array of ones :param cov_func: covariance function :param grid: array :param fold: True if we compute the expected periodogram on the natural Fourier grid, False if we compute it on a frequency grid with twice higher resolution :return:

compute_ep_old(cov_func, grid, fold=True)

Computes the expected periodogram, for the passed covariance function and grid. The grid is an array, in the simplest case just an array of ones :param cov_func: covariance function :param grid: array :param fold: True if we compute the expected periodogram on the natural Fourier grid, False if we compute it on a frequency grid with twice higher resolution :return:

multivariate_periodogram

This module provides a class for the definition of a multi-variate periodogram.

Periodogram()

This class defines a periodogram for a multivariate random field.

__call__(z: List[xp.ndarray], return_fft: bool = False) -> xp.ndarray

Compute the multivariate periodogram. The data z is expected to be a list of p arrays with the same shape, where p is the number of variates.

Parameters:

Name	Type	Description	Default
z	List[ndarray]	Data, list of arrays corresponding to the distinct variates	required
return_fft	bool	If true, returns the Discrete Fourier Transform rather than the periodogram	False

Returns:

Type	Description
periodogram	Shape (n1, n2, ..., nd, p, p) if the data is p-variate and over d spatial dimensions.

likelihood

This module provides tools to define the Debiased Whittle Likelihood and the associated estimator. More specifically, the Debiased Whittle Likelihood is defined by, $$ l(\boldsymbol{\theta}) = \sum_{\mathbf{k}\in\Omega} \left[ \log\overline{I}(\mathbf{k}; \boldsymbol{\theta}) + \frac { I(\mathbf{k}) } { \overline{I}(\mathbf{k}; \boldsymbol{\theta}) } \right] $$ where \(I(\mathbf{k})\) is the periodogram at space-time frequency \(\mathbf{k}\), \(\overline{I}(\mathbf{k}; \boldsymbol{\theta})\) is the expected periodogram and \(\Omega\) is the set of Fourier frequency (one can choose to select a subset of frequencies though).

The Debiased Whittle Likelihood Estimator then uses numerical optimization to approximately solve,

$$ \widehat{\boldsymbol{\theta}} = \argmin_{\boldsymbol{\theta}\in\Theta} l(\boldsymbol{\theta}), $$ where \(\Theta\) is the parameter space, defined within the specified model.

MultivariateDebiasedWhittle(periodogram: MultPeriodogram, expected_periodogram: ExpectedPeriodogram)

Implements the Debiased Whittle Likelihood for multivariate data. This requires the use of a multivariate periodogram. Currently, only implemented for bi-variate.

Attributes:

Name	Type	Description
periodogram	Periodogram	Multivariate periodogram applied to the multivariate random field
expected_periodogram	ExpectedPeriodogram	Object used to compute the expectation of the periodogram

__call__(z: xp.ndarray, model: CovarianceModel, params_for_gradient: list[ModelParameter] = None)

Computes the likelihood for these data

fisher(model: CovarianceModel, params_for_gradient: list[ModelParameter])

Provides the expectation of the hessian matrix

jmatrix_sample(model: CovarianceModel, params_for_gradient: list[ModelParameter], n_sims: int = 400, block_size: int = 100) -> xp.ndarray

Computes the sample covariance matrix of the gradient of the debiased Whittle likelihood from simulated realisations.

Parameters:

Name	Type	Description	Default
model	CovarianceModel	Covariance model to sample from	required
params_for_gradient	list[ModelParameter]	Parameters with respect to which we take the gradient	required
n_sims	int	Number of samples used for the estimate covariance matrix	400
block_size	int	Number of samples per simulations. A higher number should improve computational efficiency, but for large grids this may cause Out Of Memory issues.	100

Returns:

Type	Description
ndarray	Sample covariance matrix of the gradient of the likelihood

DebiasedWhittle(periodogram: Periodogram, expected_periodogram: ExpectedPeriodogram)

Implements the Debiased Whittle likelihood for univariate data.

Attributes:

Name	Type	Description
periodogram	Periodogram	Periodogram applied to the data
expected_periodogram	ExpectedPeriodogram	Object used to compute the expectation of the periodogram
frequency_mask	ndarray	mask of zero and ones to select frequencies over which the summation is carried out in the computation of the Whittle.

Examples:

>>> import numpy as np
>>> np.random.seed(1712)
>>> from debiased_spatial_whittle.grids.base import RectangularGrid
>>> from debiased_spatial_whittle.models.univariate import SquaredExponentialModel
>>> from debiased_spatial_whittle.sampling.simulation import SamplerOnRectangularGrid
>>> from debiased_spatial_whittle.inference.periodogram import Periodogram, ExpectedPeriodogram
>>> grid = RectangularGrid(shape=(256, 256))
>>> model1 = SquaredExponentialModel()
>>> model1.rho = 12
>>> model1.sigma = 1
>>> model2 = SquaredExponentialModel()
>>> model2.rho = 4
>>> model2.sigma = 1
>>> sampler = SamplerOnRectangularGrid(model1, grid)
>>> per = Periodogram()
>>> ep = ExpectedPeriodogram(grid, per)
>>> dbw = DebiasedWhittle(per, ep)
>>> sample = sampler()
>>> dbw(sample, model1), dbw(sample, model2)
(-8.37515633567113, -7.315812857735173)

whittle(periodogram: xp.ndarray, expected_periodogram: xp.ndarray)

Compute the Whittle distance between periodogram values and expectation.

Parameters:

Name	Type	Description	Default
periodogram	ndarray	periodogram of the data on Fourier grid	required
expected_periodogram	ndarray	expected periodogram or spectral density values on same Fourier grid	required

Returns:

Name	Type	Description
whittle_value	float	whittle distance between periodogram and expected periodogram

Notes

In standard use cases, this method should not be called directly. Instead, one should use the call method.

__call__(sample: xp.ndarray, model: CovarianceModel, params_for_gradient: list[ModelParameter] = None) -> xp.float64

Computes the Debiased Whittle likelihood for these data

Parameters:

Name	Type	Description	Default
sample	ndarray	sample data	required
model	CovarianceModel	covariance model used to compute the likelihood of the data	required
params_for_gradient	list[ModelParameter]	parameters with respect to which we require the derivative of the likelihood. Default, None	None

Returns:

Name	Type	Description
likelihood	float	likelihood value
gradient	ndarray	gradient with respect to the parameters provided in params_for_gradient. If the latter is None, this second output is not returned.

expected(true_model: CovarianceModel, eval_model: CovarianceModel)

Evaluate the expectation of the Debiased Whittle likelihood estimator for a given parameter.

Parameters:

Name	Type	Description	Default
true_model	CovarianceModel	Covariance model of the process	required
eval_model	CovarianceModel	Covariance model for which we evaluate the likelihood	required

Returns:

Name	Type	Description
expected	float	Expectation of the Debiased Whittle likelihood under true_model, evaluated at eval_model

fisher(model: CovarianceModel, params_for_gradient: list[ModelParameter])

Provides the Fisher Information Matrix.

Parameters:

Name	Type	Description	Default
model	CovarianceModel	True covariance model	required
params_for_gradient	list[ModelParameter]	Parameters with respect to which the Fisher is obtained	required

Returns:

Name	Type	Description
fisher	ndarray	Fisher covariance matrix

Examples:

>>> from debiased_spatial_whittle.grids.base import RectangularGrid
>>> from debiased_spatial_whittle.models.univariate import ExponentialModel
>>> model = ExponentialModel(rho=30, sigma=1.41)
>>> periodogram = Periodogram()
>>> grid = RectangularGrid((67, 192))
>>> ep = ExpectedPeriodogram(grid, periodogram)
>>> dbw = DebiasedWhittle(periodogram, ep)
>>> dbw.fisher(model, [model.param.rho, model.param.sigma])
array([[ 1.03736229e-03, -4.49238561e-02],
       [-4.49238561e-02,  2.01197123e+00]])

jmatrix(model: CovarianceModel, params_for_gradient: list[ModelParameter], mcmc_mode: bool = False)

Provides the variance matrix of the score (gradient of likelihood) under the specified model.

Parameters:

Name	Type	Description	Default
model	CovarianceModel	Covariance model	required
params_for_gradient	list[ModelParameter]	Parameters with respect to which we take the derivative	required
mcmc_mode	bool	Whether we use mcmc approximation	False

Returns:

Type	Description
ndarray	The predicted covariance matrix of the score, with parameters ordered according to params_for_gradient

jmatrix_sample(model: CovarianceModel, params_for_gradient: list[ModelParameter], n_sims: int = 1000, block_size: int = 100) -> xp.ndarray

Computes the sample covariance matrix of the gradient of the debiased Whittle likelihood from simulated realisations. Specifically, this simulates n_sims samples from model, computes the gradient for each sample using the call method, and computes the sample covariance of those gradients.

Parameters:

Name	Type	Description	Default
model	CovarianceModel	Covariance model to sample from	required
params_for_gradient	list[ModelParameter]	Parameters with respect to which we take the gradient	required
n_sims	int	Number of samples used for the estimate covariance matrix	1000
block_size	int	Number of samples per simulations. A higher number should improve computational efficiency, but for large grids this may cause Out Of Memory issues.	100

Returns:

Type	Description
ndarray	Sample covariance matrix of the gradient of the likelihood

Examples:

>>> import numpy.random as nrrandom
>>> nrrandom.seed(1712)
>>> from debiased_spatial_whittle.grids.base import RectangularGrid
>>> from debiased_spatial_whittle.models.univariate import ExponentialModel
>>> model = ExponentialModel(rho=12, sigma=1.41)
>>> periodogram = Periodogram()
>>> grid = RectangularGrid((67, 192))
>>> ep = ExpectedPeriodogram(grid, periodogram)
>>> dbw = DebiasedWhittle(periodogram, ep)
>>> dbw.jmatrix_sample(model, [model.param.rho, model.param.sigma], n_sims=20)
array([[ 1.79844275e-06, -3.36165062e-05],
       [-3.36165062e-05,  8.20809861e-04]])

variance_of_estimates(model: CovarianceModel, params: list[ModelParameter], jmat: xp.ndarray = None)

Compute the covariance matrix of the estimated parameters specified by params under the specified covariance model.

Parameters:

Name	Type	Description	Default
model	CovarianceModel	Covariance model	required
params	list[ModelParameter]	Estimated parameters. The method returns the covariance matrix of estimates of those parameters	required
jmat	ndarray	The variance of the score, if it has already been pre-computed. If not provided, it is computed exactly which can be computationally expensive.	None

Returns:

Type	Description
cov_mat	Covariance matrix of the parameter estimates.

Examples:

>>> import numpy.random as nrrandom
>>> nrrandom.seed(1712)
>>> from debiased_spatial_whittle.grids.base import RectangularGrid
>>> from debiased_spatial_whittle.models.univariate import ExponentialModel
>>> model = ExponentialModel(rho=12., sigma=4.)
>>> periodogram = Periodogram()
>>> grid = RectangularGrid((67, 192))
>>> ep = ExpectedPeriodogram(grid, periodogram)
>>> dbw = DebiasedWhittle(periodogram, ep)
>>> jmat = dbw.jmatrix_sample(model, [model.param.rho, model.param.sigma], n_sims=20)
>>> dbw.variance_of_estimates(model, [model.param.rho, model.param.sigma], jmat)
array([[8.27761908, 1.34780351],
       [1.34780351, 0.22064392]])

Estimator(likelihood: DebiasedWhittle, use_gradients: bool = False, max_iter: int = 100, optim_options: dict = dict(), method: str = 'L-BFGS-B')

Class to define an estimator that uses a likelihood.

Attributes:

Name	Type	Description
likelihood	DebiasedWhittle	Debiased Whittle likelihood used for fitting.
use_gradients	bool	Whether to use gradients in the optimization procedure. Not working at the moment.
max_iter	int	Maximum number of iterations of the optimization procedure
optim_options	dict	Additional options passed to the optimizer.
method	string	Optimization procedure. Should be one of the methods available in scipy's local or global optimizers.

Parameters:

Name	Type	Description	Default
likelihood	DebiasedWhittle	Debiased Whittle likelihood used for fitting.	required
use_gradients	bool	Whether to use gradients in the optimization procedure	False
max_iter	int	Maximum number of iterations of the optimization procedure	100
optim_options	dict	Additional options passed to the optimizer.	dict()
method	str	Optimization procedure	'L-BFGS-B'

__call__(model: CovarianceModel, sample: Union[xp.ndarray, SampleOnRectangularGrid], opt_callback: Callable = None)

Fits the passed covariance model to the passed data.

Parameters:

Name	Type	Description	Default
model	CovarianceModel	Covariance model to be fitted to the data. Only free parameters are estimated, that is parameters of the covariance model set to None.	required
sample	Union[ndarray, SampleOnRectangularGrid]	Sampled random field	required
opt_callback	Callable	Callback function called by the optimizer	None

Returns:

Name	Type	Description
model	CovarianceModel	The fitted covariance model

Notes

This directly updates the parameters of the passed covariance model.

covmat(model: CovarianceModel, params: list[ModelParameter] = None)

Compute an approximate covariance matrix of the parameter estimates under the specified covariance model.

Parameters:

Name	Type	Description	Default
model	CovarianceModel	True covariance model	required
params	list[ModelParameter]	estimated parameters	None

Returns:

Name	Type	Description
covmat	ndarray	Covariance matrix.

least_squares

This module provides a method for a least-squares fit of the expected periodogram to the periodogram. This can be used for instance to obtain an initial guess before using Debiased Whittle estimation.

This module provides a method for a least-squares fit of the expected periodogram to the periodogram. This can be used for instance to obtain an initial guess before using Debiased Whittle estimation.

This module provides a method for a least-squares fit of the expected periodogram to the periodogram. This can be used for instance to obtain an initial guess before using Debiased Whittle estimation.

LeastSquareEstimator(periodogram: Periodogram, expected_periodogram: ExpectedPeriodogram, verbose: int = 0)

Implements Least-Square estimation using scipy.optimize's non-linear least-square optimization algorithm. Can be used for instanced to obtain initial guesses for the optimization of the Debiased Spatial Whittle.

Attributes:

Name	Type	Description
periodogram		Periodogram applied to the data
expected_periodogram		Expected periodogram used for the fit

Parameters:

Name	Type	Description	Default
periodogram	Periodogram	Periodogram object that will be applied to the data	required
expected_periodogram	ExpectedPeriodogram	Expected periodogram object	required
verbose	int	Verbosity level passed on to the least_squares function of scipy.optimize	0

__call__(data: xp.array, model: CovarianceModel) -> CovarianceModel

Carries out the Least-Square estimation.

Parameters:

Name	Type	Description	Default
data	array	Sampled random field	required
model	CovarianceModel	Covariance model	required

Returns:

Type	Description
model	Fitted covariance model

diagnostics

This module provides some diagnostic tools to assess how a covariance model fits a sampled random field.

GoodnessOfFit(model: CovarianceModel, grid: RectangularGrid, sample, n_bins: int = 10)

Class to perform a goodness of fit analysis between a model and a sampled random field.

Parameters:

Name	Type	Description	Default
model	CovarianceModel	Covariance model for the data	required
grid	RectangularGrid	Sampling grid	required
sample		sampled random field data	required
n_bins	int	Number of bins used in the goodness-of-fit analysis	10