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spam.excursions package

spam.excursions package#

Submodules#

spam.excursions.elkc module#

Library of SPAM functions for computation Lipschitz-Killing Curvatures. Copyright (C) 2020 SPAM Contributors

This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version.

This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

You should have received a copy of the GNU General Public License along with this program. If not, see <http://www.gnu.org/licenses/>.

spam.excursions.elkc.expectedMesures(kappa, j, n, hittingSet='tail', distributionType='gauss', mu=0.0, std=1.0, correlationType='gauss', lc=1.0, nu=2.0, a=1.0)[source]#

Compute the Lipschitz-Killing Curvatures E(LKC)

Parameters:

  • kappa (float or list) – value of the threshold

  • j (int) – number of the functionnal

  • n (int) – spatial dimension

  • hittingSet (string) – the hitting set

  • distributionType (string) – type of distribution (see gaussianMinkowskiFunctionals)

  • mu (float) – mean value of the distribution

  • std (float) – standard deviation of the distribution

  • correlationType (string) – type of correlation function

  • lc (float) – correlation length

  • nu (float) – parameter used for correlationType=’matern’

  • a (float or list of floats) – size(s) of the object if float, the object is considered as a cube

Returns:

the Gaussian Minkowski functionnal

Return type:

Same type as kappa

spam.excursions.elkc.gaussianMinkowskiFunctionals(kappa, j, hittingSet='tail', distributionType='gauss', mu=0.0, std=1.0)[source]#

Evaluate the Gaussian Minkowski functionals j

Parameters:

  • kappa (float or list of floats) – value(s) of the threshold(s)

  • j (int) – number of the functionnal

  • hittingSet (string) – hitting set

  • distributionType (string) – type of the underlying distribution. Implemented: gauss and log

  • mu (float) – mean value of the distribution

  • std (float) – standard deviation of the distribution

Returns:

mink – the Gaussian Minkowski functionnal

Return type:

same type as kappa

spam.excursions.elkc.secondSpectralMoment(std, lc, correlationType='gauss', nu=2.0)[source]#

Compute the second spectral moment for a given covariance function:

\[{\lambda}_2 = C''(h)|_{h=0}\]
Parameters:

  • std (float) – standard deviation

  • lc (float) – correlation length

  • correlationType (string) – type of correlation function

  • nu (float) – parameter used for correlationType=’matern’

Returns:

the second spectral moment

Return type:

float

spam.excursions.elkc.flag(n, j)[source]#

Compute the flag coefficients [ n, j ] = ( n, j ) w_n / w_{n-j}w_j

Parameters:

  • n (int) – first parameter of the binom

  • j (int) – second parameter of the binom

Returns:

the flag coefficient [n , j]

Return type:

float

spam.excursions.elkc.hermitePolynomial(x, n)[source]#

Evaluate a x the probabilitic Hermite polynomial n

Parameters:

  • x (float) – point of evaluation

  • n (int) – number of Hermite polynomia

Returns:

He_n(x) the value of the polynomial

Return type:

float

spam.excursions.elkc.ballVolume(n, r=1.0)[source]#

Compute the volume of a nth dimensional ball

Parameters:

  • n (int) – spatial dimension

  • r (float) – radius of the ball

Returns:

volume of the ball

Return type:

float

spam.excursions.elkc.lkcCuboid(i, n, a)[source]#

Compute the Lipschitz-Killing Curvatures of a parallelepiped

Parameters:

  • i (int) – number of the LKC. 0 <= i <= n

  • n (int) – spatial dimension

  • a (float or list of float) – size(s) of the cuboid. If float, the object is considered to be a cube.

Returns:

The Lipschitz-Killing Curvature

Return type:

float

spam.excursions.elkc.getThresholdFromVolume(targetVolume, initialThreshold, hittingSet='tail', distributionType='gauss', mu=0.0, std=1.0, correlationType='gauss', nu=2.0, a=1.0)[source]#
spam.excursions.elkc.getCorrelationLengthFormArea(targetArea, initialLength, threshold, hittingSet='tail', distributionType='gauss', mu=0.0, std=1.0, correlationType='gauss', nu=2.0, a=1.0)[source]#

spam.excursions.randomFields module#

This module is a wrapper around the python library gstools made to intuitively generate realisations of correlated Random Fields on a regular grid with a given size, discretisation and characteristic length.

spam.excursions.randomFields.simulateRandomField(lengths=1.0, nNodes=100, covarianceModel='Gaussian', covarianceParameters={}, dim=3, nRea=1, seed=None, vtkFile=None)[source]#

Generates realisations of Correlated Random Field based on the python library gstools.

Parameters:

  • lengths (float | list of floats, default=1.0) – Length of the definition domain in every directions (should be size of dim). If a single float is given it will be used in each directions.

  • nNodes (int | list of int, default=100) – Number of nodes in every directions. If a single float is given it will be used in each directions.

  • covarianceModel (string, default="Gaussian") – The name of the covariance model used by gstools. To be chosen among this list of models.

  • covarianceParameters (dict, default={}) – A dictionnary of the keyword arguments of the covariance model used (see gstools documentation).

  • dim (int default=3) – Spatial dimention

  • nRea (int, default=1) – number of realisations

  • seed (int, default=None) – Defines the seed for the random numbers generator. Setting a seed to a certain integer yields the same realisations.

  • vtkFile (string, default=None) – If not None, the base name of the vtk files saved.

Returns:

The realisations of the random fields.

Return type:

List of nRea numpy arrays if shape nNodes

Example

>>> from spam.excursions import simulateRandomField
>>>
>>> # generation of two realisations of a Gaussian Correlated Random field
>>> realisations = simulateRandomField(
>>>     lengths=25,
>>>     nNodes=100,
>>>     dim=3,
>>>     nRea=2,
>>>     covarianceModel="Gaussian",
>>>     covarianceParameters={"len_scale": 10},
>>>     vtkFile="gaussian_randomfield"
>>> )
>>>
>>> for realisation in realisations:
>>>     print(realisation.shape)
>>>
>>> # generation of one realisation of a Matern Correlated Random field
>>> realisations = simulateRandomField(
>>>     lengths=25,
>>>     nNodes=100,
>>>     dim=3,
>>>     covarianceModel="Matern",
>>>     covarianceParameters={"len_scale": 10, "nu": 0.2},
>>>     vtkFile="matern_randomfield"
>>> )

Warning

If no rescaling factors are given in the covariance parameters it is set to 1 which deviates from gstools default behavior.

spam.excursions.randomFields.parametersLogToGauss(muLog, stdLog, lcLog=1)[source]#

Gives the underlying Gaussian standard deviation and mean value of the log normal distribution of standard deviation and mean value.

Parameters:

  • muLog (float) – Mean value of the log normal distribution.

  • stdLog (float) – Standard deviation of the log normal distribution.

  • lcLog (float, default=1) – Correlation length of the underlying log normal correlated Random Field.

Returns:

  • muGauss (float) – Mean value of the Gaussian distribution.

  • stdGauss (float) – Standard deviation of the Gaussian distribution.

  • lcGauss (float) – Correlation length of the underlying Gaussian correlated Random Field.

spam.excursions.randomFields.fieldGaussToUniform(g, a=0.0, b=1.0)[source]#

Transforms a Gaussian Random Field into a uniform Random Field.

Parameters:

  • g (array) – The values of the Gaussian Random Fields.

  • a (float, default=0.0) – The minimum borne of the uniform distribution.

  • b (float, default=1.0) – The maximum borne of the uniform distribution.

Returns:

The uniform Random Field.

Return type:

array

spam.excursions.randomFields.fieldUniformToGauss(g, a=0.0, b=1.0)[source]#

Transforms a uniform Random Field into a Gaussian Random Field.

Parameters:

  • g (array) – The values of the uniform Random Fields.

  • a (float, default=0.0) – The minimum borne of the uniform distribution.

  • b (float, default=1.0) – The maximum borne of the uniform distribution.

Returns:

The Gaussian Random Field.

Return type:

array

Module contents#

Contents