pythia package

Submodules

pythia.basis module

Assemble sparse univariate and multivariate basis polynomials.

Build univariate or multivariate normalized basis polynomials depending on the domain and distribution (and other degrees of freedom) of the parameter(s).

Currently supported are the following distribution types:

• uniform

• normal

• Gamma

• Beta

pythia.basis.multivariate_basis(paramBasis, mdx, partial=None)

Assemble multivariate polynomial basis.

Set the (partial derivative of the) multivariate (product) polynomial basis functions.

Parameters:

• paramBasis (list of list of callable) – Univariate basis functions for parameters. Is called by paramBasis[paramIdx][deg]().

• mdx (array_like) – Array of multiindices for multivariate basis functions.

• partial (list of int) – Number of partial derivatives for each dimension. Length is same as paramBasis.

Returns:

pcBasis – List of normalized multivariate polynomials w.r.t. parameter domain and distribution and univariate degrees as specified in mdx.

Return type:

list of functions

pythia.basis.normalize_polynomial(weight, p_list, param)

Normalize orthogonal polynomials.

Normalize a polynomial of an orthogonal system with respect to the scalar product

\[a(u,v)_\mathrm{pdf} = \int u(p) v(p) \mathrm{pdf}(p) \mathrm{d}p.\]

The normalized polynomial \(\phi_j\) for any given polynomial \(P_j\) is given by \(\phi_j = P_j / \sqrt{c_j}\) for the constant \(c_j = \int \mathrm{pdf}(p) * P_j(p)^2 \mathrm{d}p\).

Parameters:

• weight (callable) – Probability density function.

• p_list (list of numpy.polynomial.Polynomial) – Polynomials to normalize w.r.t. weight.

• param (pythia.parameter.Parameter) – Parameter used for distribution and domain information.

Returns:

List of normalized univariate polynomials.

Return type:

list of function

pythia.basis.set_hermite_basis(param, deg)

Generate list of probabilists Hermite polynomials.

Generate the Hermite Polynomials up to certain degree according to the mean and variance of the specified parameter.

Parameters:

• param (pythia.parameters.Parameter) – Parameter for basis function. Needs to be normal distributed.

• deg (int) – Maximum degree for polynomials.

Returns:

Probabilists Hermite polynomials up to (including) degree specified in deg.

Return type:

list of function

pythia.basis.set_jacobi_basis(param, deg)

Generate list of Jacobi polynomials.

Generate the Jacobi Polynomials up to certain degree on the interval and DoFs specified by the parameter.

Parameters:

• param (pythia.parameters.Parameter) – Parameter for basis function. Needs to be Beta-distributed.

• deg (int) – Maximum degree for polynomials.

Returns:

Jacobi polynomials up to (including) degree specified in deg.

Return type:

list of function

pythia.basis.set_laguerre_basis(param, deg)

Generate list of Leguerre polynomials.

Generate the generalized Laguerre polynomials up to certain degree on the interval and DoFs specified by the parameter.

Parameters:

• param (pythia.parameters.Parameter) – Parameter for basis function. Needs to be Gamma-distributed.

• deg (int) – Maximum degree for polynomials.

Returns:

Laguerre polynomials up to (including) degree specified in deg.

Return type:

list of function

pythia.basis.set_legendre_basis(param, deg)

Generate list of the Legendre Polynomials.

Generate the Legendre Polynomials up to certain degree on the interval specified by the parameter.

Parameters:

• param (pythia.parameters.Parameter) – Parameter for basis function. Needs to be uniformly distributed.

• deg (int) – Maximum degree for polynomials.

Returns:

Legendre polynomials up to (including) degree specified in deg.

Return type:

list of function

pythia.basis.univariate_basis(params, degList)

Assemble a univariate polynomial basis.

Set polynomial basis up to deg for each parameter in params according to the parameter distribution and area of definition.

Parameters:

• params (list of pythia.parameter.Parameter) – Parameters to compute univariate basis function for.

• degList (array_like) – Max. degrees of univariate polynomials for each parameter.

Returns:

basis – List of normalized univariate polynomials w.r.t. parameter domain and distribution up to specified degree for each parameter in params.

Return type:

list of list of functions

pythia.chaos module

Sample-based computation of polynomial chaos expansion.

This module provides classes to compute the polynomial chaos approximation of an unknown function given via input/output training data pairs via linear least-squares regression.

class pythia.chaos.PolynomialChaos(params, mdx, regPoints, weights, fEval, xVal=None, infoMatrix=None, basisMat=None, regLength=0, paramPDF=None, pdf=None, paramBasis=None, basis=None, coeff=None, verbose=False)

Bases: object

Computation of sparse polynomial chaos expansion.

Parameters:

• params (dict/list of pythia.parameter.RandomParameter) – Parameter to specify random variables for chaos expansion.

• mdx (pythia.index.PCMultiIndex) – Multiindices of coefficients of the polynomial chaos expansion.

• regPoints (array_like) – Regression points for linar regression (Parameter realizations).

• weights (array_like) – Regression weights for regPoints.

• fEval (array_like) – Function evaluation of regPoints and xVal.

• xVal (array_like, optional) – Spacial/physical point of function evaluation.

• infoMatrix (array_like, optional) – Information matrix (Gramian) of the basis functions and samples.

• basisMat (array_like, optional) – Basis evaluation matrix of samples.

• regLength (int, default=0) – Number of regression points used in linear regression. Needs to be less or equal to length of mdx. Zero results in automatic choice of amount of sample points.

• paramPDF (list of function, optional) – List of univariate PDFs for each parameter.

• pdf (function, optional) – Multivariate PDF of parameters.

• paramBasis (list of list of function, optional) – Univariate basis functions for each parameter.

• basis (list of function, optional) – Multivariate basis functions as specified in mdx.

• coeff (array_like, optional) – PC coefficients.

• verbose (bool, default=False) – Makes PCE talk.

approximation(x, partial=None)

Evaluate the (partial derivative of the) PC approximation.

Parameters:

• x (array_like) – Parameter realizations in which the approximation is evaluated.

• partial (list of int) – List that specifies the number of derivatives in each component. Length is the number of parameters.

Returns:

Evaluation of PC expansion in parameter realizations.

Return type:

array_like

property basis

Multivariate basis functions used for the PC expansion.

property basisMat

Basis evaluation matrix \(P_j(y_i)\).

property coeff

Coefficients of the PC expansion.

property fDim

Image dimension of the function.

property fEval

Function evaluations used for training.

property infoMatrix

Information matrix (unscaled Gramian) for the samples.

property mdx

Multiindices object.

property mean

Mean of the PC expansion.

property pDim

Number of random parameters.

property paramBasis

List of univariate basis functions for each parameter.

property paramPDF

List of univariate PDFs for each parameter.

property parameter

List of parameters.

property pdf

Multivariate PDF of parameters.

property regLength

Number of used training data to compute PC expansion.

property regPoints

Parameter realization used for training.

property sobolCoeff

Approximated Sobol indices for the PC expansion.

property std

Standard deviation of the PC expansion.

property variance

Variance of the PC expansion.

property weights

Regression weights.

property xDim

Spatial dimension.

property xVal

Spatial component.

class pythia.chaos.PolynomialChaosFrame(params, mdx, regPoints, weights, fEval, xVal, infoMatrix=None, basisMat=None, coeff=None, regLength=0, verbose=False)

Bases: object

Computation of sparse PC expansion for each spatial component.

Deprecated since version 2.0.0: PolynomialChaosFrame will be removed in PyThia 3.0.0, it is replaced by PolynomialChaos because the latter performs more efficiently.

Compute the PC expansion of a function for given parameters for specified multiindices with linear regression. The coefficients are computed for each spatial component individually.

Parameters:

• params (dict/list of pythia.parameter.RandomParameter) – Parameter to specify random variables for chaos expansion.

• mdx (pythia.index.PCMultiIndex) – Multiindices of coefficients of the polynomial chaos expansion.

• regPoints (array_like) – Regression points for linar regression (Parameter realizations).

• weights (array_like) – Regression weights for regPoints.

• fEval (array_like) – Function evaluation of regPoints and xVal.

• xVal (array_like, optional) – Spacial/physical point of function evaluation.

• infoMatrix (array_like, optional) – Information matrix (Gramian) of the basis functions and samples.

• basisMat (array_like, optional) – Basis evaluation matrix of samples.

• coeff (array_like, optional) – PC coefficients.

• regLength (int, default=0) – Number of regression points used in linear regression. Needs to be less or equal to length of mdx. Zero results in automatic choice of amount of sample points.

• verbose (bool, default=False) – Makes PCE talk.

approximation(x, partial=None)

Evaluate the (partial derivative of the) PC approximation.

Parameters:

• x (array_like) – Parameter realizations in which the approximation is evaluated.

• partial (list of int) – List that specifies the number of derivatives in each component. Length is the number of parameters.

Returns:

Evaluation of PC expansion in parameter realizations.

Return type:

array_like

property basis

Multivariate basis functions used for the PC expansion.

property basisMat

Basis evaluation matrix \(P_j(y_i)\).

property coeff

Coefficients of the PC expansion.

property fDim

Image dimension of the function.

property fEval

Function evaluations used for training.

property infoMatrix

Information matrix (unscaled Gramian) for the samples.

property mdx

Multiindices object.

property mean

Mean of the PC expansion.

property pDim

Number of random parameters.

property paramBasis

List of univariate basis functions for each parameter.

property paramPDF

List of univariate PDFs for each parameter.

property parameter

List of parameters.

property pdf

Multivariate PDF of parameters.

property regLength

Number of used training data to compute PC expansion.

property regPoints

Parameter realization used for training.

run(coeff=None)

Compute PC expansion for each spatial component individually.

Compute the PC expansion coefficients for each spatial component without repeatedly assembling the information matrix.

Parameters:

coeff (array_like, optional) – Precomputed PC coefficients.

Notes

This method needs to be called separately after initiating a PolynomialChaosFrame object.

property sobolCoeff

Approximated Sobol indices for the PC expansion.

property std

Standard deviation of the PC expansion.

property variance

Variance of the PC expansion.

property weights

Regression weights.

property xDim

Spatial dimension.

property xVal

Spatial component.

pythia.chaos.assemble_mdx(IDXs, VALs, SDXs, nParams, max_terms)

Compute automatic choice of multiindices.

Parameters:

• IDXs (list) – Enumeration of all possible Sobol indices.

• VALs (array_like) – Values of Sobol indices.

• SDXs (list) – Sobol indices (subscripts, not the values).

• nParams (int) – Number of stochastic parameters.

• max_terms (int) – Maximum number of expansion terms.

Returns:

mdx – Sparse multiindices.

Return type:

pythia.index.PCMultiIndex

pythia.chaos.find_optimal_mdx(params, regPoints, weights, fEval, max_terms=0, threshold=0.001, get_sobol_coeff=False)

Compute optimal multiindices of PCE.

Compute the optimal multiindices for a polynomial chaos expansion based on an estimate of the Sobol coefficient values.

Parameters:

• params (list of pythia.Parameters.RandomParameter) – Random parameters of the problem.

• regPoints (array_like) – Sample points for training

• weights (array_like) – Weights for training.

• fEval (array_like) – Function evaluations for training.

• max_terms (int, default=0) – Maximum number of expansion terms. Number of expansion terms is chosen automatically for max_terms=0.

• get_sobol_coeff (bool) – If true, function also return crude intermediate approximation of Sobol coefficient values.

Returns:

• mdx (pythia.Indices.PCMultiIndex) – Adaptive multiindices.

• sobolCoeff (array_like, optional) – Crude intermediate approximation of Sobol indices.

pythia.chaos.get_infoMat_batchsize(dim, save_memory=538445312.5)

Compute memory allocation batch sizes for information matrix.

Compute the maximal number of samples in each batch when assembling the information matrix to be maximally memory efficient and avoid OutOfMemory errors.

Parameters:

• dim (int) – Number of rows/columns of information matrix.

• save_memory (int, default=3*1025/2) – Memory (in bytes), that should be kept free. The default is equivalent to 512 MB.

Returns:

n – Batchsize for assembling of information matrix.

Return type:

int

pythia.chaos.get_sobol_coefficients(params, regPoints, weights, fEval, max_terms)

Compute rough estimate of Sobol coefficient.

Compute an approximation of the Sobol coefficients with as many PC expansion terms as possible.

Parameters:

• params (list of pythia.parameter.RandomParameter) – List of stochastic parameters.

• regPoints (array_like) – Parameter realizations.

• weights (array_like) – Regressio weights.

• fEval (array_like) – Function evaluations in regression points.

• max_terms (int) – Number of considered expansion terms.

Returns:

• pc.sobolCoeff (array_like) – Sobol indices.

• mdx.sdx (list of tuple) – Sobol index identifiers (subscripts).

pythia.density module

Assemble univariate and multivariate PDFs for stochastic parameters.

Build the univariate and multivariate probability density functions (PDFs) for pythia.parameter.RandomParameter parameters.

pythia.density.multivariatePDF(uPDF)

Compute the multivariate (product) PDF of the given parameters.

Parameters:

uPDF (list of function) – List of univariate PDFs.

Returns:

pdf – Multivariate parameter PDF.

Return type:

function

pythia.density.paramPDF(params)

Set the PDF for each parameter according to the parameter distribution.

The PDF of parameter j can be evaluated at some point \(x\in\mathbb{R}\) by pdf[j](x).

Parameters:

params (list of pythia.parameters.RandomParameter) – List of parameters with information of the probability distribution.

Returns:

pdf – List of univariate PDFs for each parameter.

Return type:

list of function

pythia.index module

Store information about sparse multiindices.

Provide information about sparse PC expansion terms in form of multiindices of polynomial degrees for each of the parameter dimensions. Also compute Sobol indices subscripts and provide conversions of PC multiindices (mdx) Sobol multiindices (sdx) and linear enumeration (idx).

class pythia.index.PCMultiIndex(dimension)

Bases: object

Generate multiindices for sparse PC expansion.

Parameters:

dimension (int) – Number of stochastic parameters.

property count

Number of PC expansion terms.

property dimension

Number of stochastic parameters.

generateMDX2IDX()

Generate dict to map tuple(mdx) to corresponding index.

generateMDX2SDX()

Generate dictionary to map mdx to sdx.

Generate dictionary with structre dct[tuple(mdx)] = sdx.

Note

The mdx are numpy ndarrays which cannot be cast to keys in python dictionaries. Hence the individual multiindices need to be cast to tuples before using the dictionary.

generateSDX()

Generate Sobol index subscripts for parameter dimension.

generateSDX2IDX()

Generate dict to map tuple(sdx) to corresponding index.

generateSDX2MDX()

Generate dictionary to map sdx to all corresponding mdx.

property maximum

Maximum univariate polynomial degree.

property mdx

Array of PC expansion multiindices.

property mdx2idx

Dictionary to map mdx to enumeration index.

property mdx2sdx

Dictionary to map mdx to corresponding sdx.

mdxAdd(pos, deg)

Assemble array of multiindices as cartesian product of deg in pos.

Parameters:

• pos (list) – Positions where multiindices are added.

• deg (list of lists) – Multiindex values for position of same length as pos.

Returns:

Sorted array of PC multiindices.

Return type:

array_like

mdxFromShape(shape)

Compute (full tensor) multiindices up to specified degree.

Parameters:

shape (iterable) – Maximal indices for multiindex.

mdxLimitAbs(limit)

Limit maximal dimension of multiindices.

Restrict self.mdx to a sub-array of multiindices with total degree less or equal to the specified limit.

Parameters:

limit (int) – Limit for the total degree of the multiindices.

mdxSort(mdx)

Sort mdx by sum.

Parameters:

mdx (array_like) – Array of PC multiindices.

refresh()

Generate Sobol indices and index dictionaries from mdx array.

property sdx

List of Sobol indices subscripts.

property sdx2mdx

Dictionary to map sdx to all corresponding mdx.

pythia.likelihood module

Provide Gaussian likelihoods for PC surrogates.

class pythia.likelihood.Gaussian(f, sigma, xdim)

Bases: object

Gaussian likelihood function for differentiable forward problem.

Assemble the Gaussian likelihood

\[\mathcal{L}(x) = \frac{1}{(2\pi)^{M/2}\sqrt{\det \Sigma}} \exp\Bigl( -\frac{1}{2} \Vert \Sigma^{-1/2}(f(x)-\delta) \Vert \Bigr)\]

for covariance matrix \(\Sigma = \operatorname{diag}(\sigma_1(y_1),\dots,\sigma_M(y_M))\) and measurement/observation \(\delta\).

Parameters:

• f (function) – Forward model.

• sigma (function) – Error model for standard deviation describing \(\Sigma\).

• xdim (int) – Number of stochastic parameters.

likelihood(x, y_meas)

Evaluate the Gaussian likelihood with specified measurement.

Parameters:

• x (array_like) – Realizations of stochastic parameters.

• y_meas (array_like) – Measurement \(\delta\).

log_likelihood(x, y_meas)

Evaluate the Gaussian log-likelihood with specified measurement.

Parameters:

• x (array_like) – Realizations of stochastic parameters.

• y_meas (array_like) – Measurement \(\delta\).

pythia.likelihood.gauss_likelihood(p, y, error_model)

Compute a gaussian likelihood function.

Deprecated since version 2.0.0: gauss_likelihood will be removed in PyThia 3.0.0, it is replaced by the Gaussian class as the latter is clearer and more comfortable to use.

Parameters:

• p (array_like) – Parameter realizations.

• y (array_like) – Measurement data \(\delta\).

• error_model (function) – Function that maps p to two arrays containing the evaluation of the forward model f(p) and the standard deviation sigma(p).

pythia.loggers module

Logger class to write output to terminal and in log file.

class pythia.loggers.Logger(fileName='logfile.log', filePath='./Log/')

Bases: object

Logger to write function output to terminal and in log file.

Parameters:

• fileName (string, default='logfile.log') – Name of the log file.

• filePath (string, default='./Log/') – Path to the log file.

Examples

>>> sys.stdout = Logger(fileName, filePath)

flush()

Flush screen.

write(message)

Write string to terminal and log file.

pythia.loggers.log_progress(sequence, every=None, size=None, name='Items')

Progress bar for jupyter notebooks.

Deprecated since version 2.0.0: log_progress will be removed in PyThia 3.0.0 as this is not part of the PyThia core functionality.

Note

Code copied from https://github.com/alexanderkuk/log-progress

pythia.mcmc_sampling module

pythia.mcmc_sampling.random_walk_metropolis(density, domain, seeds, CHAIN_LENGTH, DELTA)

Random walk Metropolis-Hastings algorithm for sampling densities.

Since the random walks for multiple seeds are computed in parallel, the seed length of each seed is chosen uniformly.

Deprecated since version 2.0.0: This is just for benchmarking MCMC sampling. Use other python libraries for reasonable computations.

Parameters:

• density (function) – Density (not necessarily normalized) from which the samples are generated.

• seeds (array_like, ndim = 2) – Seed for stochastic sampling. First dimension is the number of seeds used for parallel sampling, second dimension is the dimension of the samples/parameters.

• CHAIN_LENGTH (int) – Length of the Markov chains.

• DELTA (array_like) – Standart deviation of noise direction added in each step for each parameter.

pythia.misc module

pythia.misc.batch(iterable, n=1)

Split iterable into different batches of batchsize n.

Parameters:

• iterable (array_like) – Iterable to split.

• n (int, default=1) – Batch size.

Yields:

iterable – Iterable for different batches.

pythia.misc.cartProd(arrayList)

Compute the outer product of two or more arrays.

Assemble an array containing all possible combinations of the elements of the input vectors \(v_1,\dots,v_n\).

Parameters:

arrayList (list of array_like) – List of vectors \(v_1,\dots,v_n\).

Returns:

x – Cartesian product array.

Return type:

array_like

pythia.misc.distributionDict()

Set aliases for distribution descriptions.

Deprecated since version 2.0.0: distributionDict will be removed in PyThia 3.0.0.

Returns:

distDict – Dictionary with aliases for distribution descriptions.

Return type:

dict

pythia.misc.doerfler_marking(values, idx=None, threshold=0.9)

Dörfler marking for arbitrary values.

Parameters:

• values (array_like) – Values for the Dörfler marking.

• idx (list of int, optional) – List of indices associated with the entries of values. If None, this is set to range(len(values)).

• threshold (float, default=0.9) – Threshold paramter for Dörfler marking.

Returns:

• idx_reordered (list) – Reordered indices given by idx. Ordered from largest to smallest value.

• ordered_values (array_like) – Reordered values. Ordered from largest to smallest.

• marker (int) – Threshold marker such that sum(values[:marker]) > threshold * sum(values).

pythia.misc.formatTime(dt)

Converts time (seconds) to time format string.

Parameters:

dt (float) – Time in seconds.

Returns:

Formatted time string.

Return type:

str

pythia.misc.gelman_rubin_condition(chains)

Compute Gelman-Rubin criterion.

Implementation of the Gelman-Rubin convergence criterion for multiple parameters. A Markov chain is said to be in its convergence, if the final ration is close to one.

Parameters:

chains (array_like, ndim=3) – Array containing the Markov chains of each parameter. All chains are equal in length, the assumed shape is (#chains, chain length, #params).

Returns:

Value computed by Gelman-Rubin criterion.

Return type:

float

pythia.misc.get_confidence_interval(samples, rate=0.95, resolution=500)

Compute confidence intervals of samples.

Compute the confidence intervals of the 1D marginals of the samples (slices). The confidence interval of a given rate is the interval around the median (not mean) of the samples containing roughly rate percent of the total mass. This is computed for the left and right side of the median independently.

Parameters:

• samples (array_like, ndim < 3) – Array containing the (multidimensional) samples.

• rate (float, default=0.95) – Fraction of the total mass the interval should contain.

• resolution (int, default=500) – Number of bins used in histogramming the samples.

Returns:

conf_intervals – Confidence intervals for each component.

Return type:

array_like

pythia.misc.is_contained(val, domain)

Check if a given value (vector) is contained in a domain.

Checks if each component of the vector lies in the one dimensional interval of the corresponding component of the domain.

Parameters:

• val (array_like) – Vector to check containment in domain

• domain (array_like) – Product domain of one dimensional intervals.

Returns:

Bool stating if value is contained in domain.

Return type:

bool

pythia.misc.line(indicator, message=None)

Print a line of 80 characters by repeating indicator.

An additional message can be given.

Parameters:

• indicator (string) – Indicator the line consists of, e.g. ‘-’, ‘+’ or ‘+-‘.

• message (string, optional) – Message integrated in the line.

Returns:

text – String of 80 characters length.

Return type:

str

pythia.misc.load(filename)

Alias for numpy.load().

pythia.misc.now(sep=False)

Get string of current machine date and time.

Parameters:

sep (bool, default=False) – If true, return date and time separately.

Returns:

Tuple containing date and time or single string of both concatenated.

Return type:

tuple of str or str

pythia.misc.paramDictToList(paramDict)

Cast dictionary of pythia parameters to a list sorted by index.

Deprecated since version 2.0.0: paramDictToList will be removed in PyThia 3.0.0, parameters should always be given as a list not a dictionary.

Parameters:

paramDict (dict) – Dictionary of parameter names and corresponding pythia.parameter.RandomParameter.

Returns:

paramList – Parameter list sorted by index of parameter.

Return type:

list of pythia.parameter.RandomParameter

pythia.misc.save(filename, data, path='./')

Wrapper for numpy.save that assures path directory is created if necessary and that backs old data up if existent.

Parameters:

• name (str) – Filename to save data to.

• data (array_like) – Data to save as .npy file.

• path (str, default='./') – Path under which the file should be created.

pythia.misc.shiftCoord(x, T, I)

Map x in interval T to interval I.

Use an affine transformation to shift points \(x\) from the interval \(T = [t_0, t_1]\) to the interval \(I = [a,b]\).

Parameters:

• x (array_like) – Points in interval \(T\).

• T (array_like) – Original interval.

• I (array_like) – Target interval.

Returns:

Shifted values for x.

Return type:

array_like

pythia.misc.str2iter(string, iterType=<class 'list'>, dataType=<class 'int'>)

Cast str(iterable) to iterType of dataType.

Cast a string of lists, tuples, etc to the specified iterable and data type, i.e., for iterType=tuple and dataType=float cast str([1,2,3]) -> (1.0, 2.0, 3.0).

Parameters:

• string (str) – String representation of iterable.

• iterType (iterable, default=list) – Iterable type the string is converted to.

• dataType (type, default=int) – Data type of entries of iterable, e.g. int or float.

pythia.misc.wlsSamplingBound(m, c=4)

Compute the weighted Least-Squares sampling bound.

The number of samples \(n\) is chosen such that

\[\frac{n}{\log(n)} \geq cm,\]

where \(m\) is the dimension of the Gramian matrix (number of PC expansion terms) and \(c\) is an arbitrary constant. In Cohen & Migliorati 2017 the authors observed that the coice \(c=4\) yields a well conditioned Gramian with high probability.

Parameters:

• m (int) – Dimension of Gramian matrix.

• c (float, default=4) – Scaling constant.

Returns:

n – Number of required wLS samples.

Return type:

int

pythia.parameter module

PyThia classes containing Parameter information.

class pythia.parameter.Parameter(name, value=None)

Bases: object

Basic parameter with fixed value.

Parameters:

• name (str) – Name identifier of the parameter.

• value (float, optional) – Parameter value.

property mode

Type (mode) of parameter.

property name

Parameter name.

property value

Parameter value.

class pythia.parameter.RandomParameter(name, domain, distribution, value=None, index=None)

Bases: Parameter

Subclass of Parameter used for stochasic parameters.

Parameters:

• name (str) – Parameter name.

• domain (array_like) – Supported domain of the parameter distribution.

• distribution (str) – Distribution identifier of the parameter.

• value (float or array_like, optional) – Value (expected) of the parameter. Only used to store information never required for computations.

• index (int, optional) – Enumeration index of the parameter. Set automatically if None.

property alpha

Parameter required for Beta or Gamma distribution.

property beta

Parameter required for Beta distribution.

property distribution

Distribution identifier of the parameter.

property domain

Supported domain of the parameter.

property index

Index of the parameter.

property mean

Mean of parameter distribution.

property variance

Variance of parameter distribution.

class pythia.parameter.SteadyParameter(name, value, index=None)

Bases: Parameter

Subclass of Parameter used for deterministic parameter variations.

Parameter representing part of the spatial/physical domain of the problem.

Parameters:

• name (str) – Parameter name.

• value (array_like) – Values the parameter is iterated over.

• index (int, optional) – Enumeration index of the parameter. Set automatically if None.

property index

Index of the parameter.

property name

Parameter name.

property value

Parameter value.

pythia.sampler module

Sampler classes for generating in random samples and PDF evaluations.

class pythia.sampler.BetaSampler(domain, alpha, beta)

Bases: Sampler

Sampler for univariate Beta distributed samples on given domain.

Parameters:

• domain (array_like) – Supported domain of distribution.

• alpha (float) – Parameter for Beta distribution.

• beta (float) – Parameter for Beta distribution.

property alpha

Parameter \(\alpha\) of Gamma distribution.

property beta

Parameter \(\beta\) of Gamma distribution.

density(x)

Evaluate PDF.

Deprecated since version 2.0.0: density will be removed in PyThia 3.0.0, use pdf instead.

Parameters:

x (array_like) – Evaluation points.

Returns:

Values of PDF evaluated in x.

Return type:

array_like

property dimension

Dimension of the parameters.

property domain

Parameter domain of support.

grad_x_log_pdf(x)

Evaluate gradient of log-PDF.

Note

Not yet implemented.

Parameters:

x (array_like) – Evaluation points.

Returns:

Values of gradient (vector valued) of log-PDF evaluated in x.

Return type:

array_like

hess_x_log_pdf(x)

Evaluate Hessian of log-PDF.

Note

Not yet implemented.

Parameters:

x (array_like) – Evaluation points.

Returns:

Values of Hessian (matrix valued) of log-PDF evaluated in x.

Return type:

array_like

log_pdf(x)

Evaluate log-PDF.

Parameters:

x (array_like) – Evaluation points.

Returns:

Values of log-PDF evaluated in x.

Return type:

array_like

property mass

Mass of the PDF.

property maximum

Maximum value of the PDF.

pdf(x)

Evaluate PDF.

Parameters:

x (array_like) – Evaluation points.

Returns:

Values of PDF evaluated in x.

Return type:

array_like

sample(*shape)

Draw samples from distribution.

Parameters:

shape (array_like) – Shape of the samples.

Returns:

Random samples of specified shape.

Return type:

array_like

class pythia.sampler.ConstraintSampler(sampler, constraints)

Bases: Sampler

Sampler that applies algebraic constraints when sampling.

Parameters:

• sampler (Sampler) – Sampler to sample from.

• constraints (list) – List of functions that return True if sample point satisfies the constraint.

Notes

The constaints may lead to a non-normalized density function.

property constraints

Specified constrains.

density(x)

Evaluate PDF.

Deprecated since version 2.0.0: density will be removed in PyThia 3.0.0, use pdf instead.

Parameters:

x (array_like) – Evaluation points.

Returns:

Values of PDF evaluated in x.

Return type:

array_like

property dimension

Dimension of the parameters.

property domain

Parameter domain of support.

property mass

Mass of the PDF.

property maximum

Maximum value of the PDF.

pdf(x)

Evaluate PDF.

Parameters:

x (array_like) – Evaluation points.

Returns:

Values of PDF evaluated in x.

Return type:

array_like

sample(*shape)

Draw samples from distribution.

Parameters:

shape (array_like) – Shape of the samples.

Returns:

Random samples of specified shape.

Return type:

array_like

class pythia.sampler.GammaSampler(domain, alpha, beta)

Bases: Sampler

Sampler for univariate Gamma distributed samples on given domain.

Parameters:

• domain (array_like) – Supported domain of distribution.

• alpha (float) – Parameter for Gamma distribution.

• beta (float) – Parameter for Gamma distribution.

property alpha

Parameter \(\alpha\) of Gamma distribution.

property beta

Parameter \(\beta\) of Gamma distribution.

density(x)

Evaluate PDF.

Deprecated since version 2.0.0: density will be removed in PyThia 3.0.0, use pdf instead.

Parameters:

x (array_like) – Evaluation points.

Returns:

Values of PDF evaluated in x.

Return type:

array_like

property dimension

Dimension of the parameters.

property domain

Parameter domain of support.

grad_x_log_pdf(x)

Evaluate gradient of log-PDF.

Note

Not yet implemented.

Parameters:

x (array_like) – Evaluation points.

Returns:

Values of gradient (vector valued) of log-PDF evaluated in x.

Return type:

array_like

hess_x_log_pdf(x)

Evaluate Hessian of log-PDF.

Note

Not yet implemented.

Parameters:

x (array_like) – Evaluation points.

Returns:

Values of Hessian (matrix valued) of log-PDF evaluated in x.

Return type:

array_like

log_pdf(x)

Evaluate log-PDF.

Parameters:

x (array_like) – Evaluation points.

Returns:

Values of log-PDF evaluated in x.

Return type:

array_like

property mass

Mass of the PDF.

property maximum

Maximum value of the PDF.

pdf(x)

Evaluate PDF.

Parameters:

x (array_like) – Evaluation points.

Returns:

Values of PDF evaluated in x.

Return type:

array_like

sample(*shape)

Draw samples from distribution.

Parameters:

shape (array_like) – Shape of the samples.

Returns:

Random samples of specified shape.

Return type:

array_like

class pythia.sampler.NormalSampler(mean, var)

Bases: Sampler

Sampler for univariate normally distributed samples.

Parameters:

• mean (float) – Mean of the Gaussian distribution.

• var (float) – Variance of the Gaussian distribution.

density(x)

Evaluate PDF.

Deprecated since version 2.0.0: density will be removed in PyThia 3.0.0, use pdf instead.

Parameters:

x (array_like) – Evaluation points.

Returns:

Values of PDF evaluated in x.

Return type:

array_like

property dimension

Dimension of the parameters.

property domain

Parameter domain of support.

grad_x_log_pdf(x)

Evaluate gradient of log-PDF.

Parameters:

x (array_like) – Evaluation points.

Returns:

Values of gradient (vector valued) of log-PDF evaluated in x.

Return type:

array_like

hess_x_log_pdf(x)

Evaluate Hessian of log-PDF.

Parameters:

x (array_like) – Evaluation points.

Returns:

Values of Hessian (matrix valued) of log-PDF evaluated in x.

Return type:

array_like

log_pdf(x)

Evaluate log-PDF.

Parameters:

x (array_like) – Evaluation points.

Returns:

Values of log-PDF evaluated in x.

Return type:

array_like

property mass

Mass of the PDF.

property maximum

Maximum value of the PDF.

property mean

Mean of the distribution.

pdf(x)

Evaluate PDF.

Parameters:

x (array_like) – Evaluation points.

Returns:

Values of PDF evaluated in x.

Return type:

array_like

sample(*shape)

Draw samples from distribution.

Parameters:

shape (array_like) – Shape of the samples.

Returns:

Random samples of specified shape.

Return type:

array_like

property var

Variance of the distribution.

class pythia.sampler.ParameterSampler(params)

Bases: Sampler

Product sampler of given parameters.

Parameters:

params (list of pythia.parameter.RandomParameter) – List or dictionary containing information of parameters.

density(x)

Evaluate PDF.

Deprecated since version 2.0.0: density will be removed in PyThia 3.0.0, use pdf instead.

Parameters:

x (array_like) – Evaluation points.

Returns:

Values of PDF evaluated in x.

Return type:

array_like

property dimension

Dimension of the parameters.

property domain

Parameter domain of support.

property mass

Mass of the PDF.

Note

The PDF is assumed to be normalized. This is not computed here.

property maximum

Maximum value of the PDF.

property parameter

List of parameters.

pdf(x)

Evaluate PDF.

Parameters:

x (array_like) – Evaluation points.

Returns:

Values of PDF evaluated in x.

Return type:

array_like

sample(*shape)

Draw samples from distribution.

Parameters:

shape (array_like) – Shape of the samples.

Returns:

Random samples of specified shape.

Return type:

array_like

weight(x)

Weights for the parameter product PDF sampler.

class pythia.sampler.ProductSampler(samplerList)

Bases: Sampler

Tensor sampler for independent parameters.

Sampler for cartesian product samples of a list of (independent) univariate samplers.

Parameters:

samplerList (list of pythia.sampler.Sampler) – List of (univariate) Sampler objects.

density(x)

Evaluate PDF.

Deprecated since version 2.0.0: density will be removed in PyThia 3.0.0, use pdf instead.

Parameters:

x (array_like) – Evaluation points.

Returns:

Values of PDF evaluated in x.

Return type:

array_like

property dimension

Dimension of the parameters.

property domain

Parameter domain of support.

grad_x_log_pdf(x)

Evaluate gradient of log-PDF.

Parameters:

x (array_like) – Evaluation points.

Returns:

Values of gradient (vector valued) of log-PDF evaluated in x.

Return type:

array_like

hess_x_log_pdf(x)

Evaluate Hessian of log-PDF.

Parameters:

x (array_like) – Evaluation points.

Returns:

Values of Hessian (matrix valued) of log-PDF evaluated in x.

Return type:

array_like

log_pdf(x)

Evaluate log-PDF.

Parameters:

x (array_like) – Evaluation points.

Returns:

Values of log-PDF evaluated in x.

Return type:

array_like

property mass

Mass of the PDF.

property maximum

Maximum value of the PDF.

pdf(x)

Evaluate PDF.

Parameters:

x (array_like) – Evaluation points.

Returns:

Values of PDF evaluated in x.

Return type:

array_like

sample(*shape)

Draw samples from distribution.

Parameters:

shape (array_like) – Shape of the samples.

Returns:

Random samples of specified shape.

Return type:

array_like

class pythia.sampler.RejectionSampler(domain, pdf, trialSampler, maximum=None)

Bases: Sampler

Rejection sampler for sampling arbitrary n-dimensional densities.

Parameters:

• domain (array_like) – Domain of definition of density

• pdf (function) – Target density to generate samples from.

• trialSampler (pythia.sampler.Sampler) – Sampler for trial distribution, should generate samples fast.

• maximum (float, optional) – Threshold parameter with density <= maximum * sampler.pdf

density(x)

Evaluate PDF.

Deprecated since version 2.0.0: density will be removed in PyThia 3.0.0, use pdf instead.

Parameters:

x (array_like) – Evaluation points.

Returns:

Values of PDF evaluated in x.

Return type:

array_like

property dimension

Dimension of the parameters.

property domain

Parameter domain of support.

property mass

Mass of the PDF.

Note

The PDF is assumed to be normalized. This is not computed here.

property maximum

Maximum value of the PDF.

pdf(x)

Evaluate PDF.

Parameters:

x (array_like) – Evaluation points.

Returns:

Values of PDF evaluated in x.

Return type:

array_like

sample(*shape)

Draw samples from distribution.

Parameters:

shape (array_like) – Shape of the samples.

Returns:

Random samples of specified shape.

Return type:

array_like

class pythia.sampler.Sampler

Bases: object

Base class for all continuous samplers.

property dimension

Dimension of the ambient space.

abstract property domain

Supported domain in ambient space.

The domain of a sampler is a list of pairs representing the n-orthotope (hyperrectangle) where it is defined.

abstract property mass

Mass of the sampler distribution.

The integral of the sampler distribution over the domain of definition. If the density is normalised this value should be one.

abstract pdf(point)

Density of the samplers distribution.

Computes the density of the samplers underlying distribution at the given point.

Parameters:

point (array_like of shape (..., D)) – List of points or single point. D is the objects dimension.

Returns:

out – Density at the point.

Return type:

ndarray of shape (…, 1)

abstract sample(*shape)

Random values in a given shape.

Create an array of the given shape and populate it with random samples from the samplers distribution.

Parameters:

shape (array_like, optional) – The dimensions of the returned array, should all be positive. If no argument is given a single Python float is returned.

Returns:

out – Random values of specified shape.

Return type:

array_like

class pythia.sampler.UniformSampler(domain)

Bases: Sampler

Sampler for univariate uniformly distributed samples on given domain.

Parameters:

domain (array_like) – Interval of support of distribution.

density(x)

Evaluate PDF.

Deprecated since version 2.0.0: density will be removed in PyThia 3.0.0, use pdf instead.

Parameters:

x (array_like) – Evaluation points.

Returns:

Values of PDF evaluated in x.

Return type:

array_like

property dimension

Dimension of the parameters.

property domain

Parameter domain of support.

grad_x_log_pdf(x)

Evaluate gradient of log-PDF.

Parameters:

x (array_like) – Evaluation points.

Returns:

Values of gradient (vector valued) of log-PDF evaluated in x.

Return type:

array_like

hess_x_log_pdf(x)

Evaluate Hessian of log-PDF.

Parameters:

x (array_like) – Evaluation points.

Returns:

Values of Hessian (matrix valued) of log-PDF evaluated in x.

Return type:

array_like

log_pdf(x)

Evaluate log-PDF.

Parameters:

x (array_like) – Evaluation points.

Returns:

Values of log-PDF evaluated in x.

Return type:

array_like

property mass

Mass of the PDF.

property maximum

Maximum value of the PDF.

pdf(x)

Evaluate PDF.

Parameters:

x (array_like) – Evaluation points.

Returns:

Values of PDF evaluated in x.

Return type:

array_like

sample(*shape)

Draw samples from distribution.

Parameters:

shape (array_like) – Shape of the samples.

Returns:

Random samples of specified shape.

Return type:

array_like

class pythia.sampler.WLSConditionalSampler(params, basis, alphas, condition=<Mock name='mock.array()' id='139719299411872'>)

Bases: Sampler

WLS univariate conditional Sampler as described by Cohen & Migliorati.

Parameters:

• params (dict) – Dictionary of pythia.Parameters.Parameter.

• basis (dict) – Dictionary of basis functions for all parameters in params.

• alphas (list) – List of tuples representing multiindices.

• condition (array_like) – Array of condition for sampling.

• note: (..) – This is implemented very inefficiently.:

density(x)

Evaluate PDF.

Deprecated since version 2.0.0: density will be removed in PyThia 3.0.0, use pdf instead.

Parameters:

x (array_like) – Evaluation points.

Returns:

Values of PDF evaluated in x.

Return type:

array_like

property dimension

Dimension of the parameters.

property domain

Parameter domain of support.

property mass

Mass of the PDF.

property maximum

Maximum value of the PDF.

property parameter

Parameter list.

pdf(x)

Evaluate PDF.

Parameters:

x (array_like) – Evaluation points.

Returns:

Values of PDF evaluated in x.

Return type:

array_like

property position

Parameter index.

sample(*shape)

Draw samples from distribution.

Parameters:

shape (array_like) – Shape of the samples.

Returns:

Random samples of specified shape.

Return type:

array_like

property trialSampler

Trial sampler used for the rejection sampling.

class pythia.sampler.WLSRejectionSampler(params, basis, tsa)

Bases: Sampler

WLS rejection sampler.

Sampler for weighted Least-Squares sampling as described by Cohen and Migliorati. Uses rejection sampling for the weighted Least-Squares density.

Parameters:

• params (list of pythia.parameter.RandomParameter) – List of parameters.

• basis (dict) – Dictionary of basis functions.

• tsa (bool) – Trial sampler adaptation. If True, a trial sampler is chosen on the distributions of parameters, if false a uniform trial sampler is used.

property basis

Employed basis functions.

density(x)

Evaluate PDF.

Deprecated since version 2.0.0: density will be removed in PyThia 3.0.0, use pdf instead.

Parameters:

x (array_like) – Evaluation points.

Returns:

Values of PDF evaluated in x.

Return type:

array_like

property dimension

Dimension of the parameters.

property domain

Parameter domain of support.

property mass

Mass of the PDF.

property maximum

Maximum value of the PDF.

pdf(x)

Evaluate PDF.

Parameters:

x (array_like) – Evaluation points.

Returns:

Values of PDF evaluated in x.

Return type:

array_like

sample(*shape)

Draw samples from distribution.

Parameters:

shape (array_like) – Shape of the samples.

Returns:

Random samples of specified shape.

Return type:

array_like

property trialSampler

Trial sampler used for the rejection sampling.

weight(x)

Weights for the PDF.

Parameters:

x (array_like) – Points the weight function is evaluated in.

Returns:

w – weights of evaluation points x.

Return type:

array_like

class pythia.sampler.WLSSampler(params, basis, strategy='rejection', paramBasis=None, alphas=None, tsa=True)

Bases: Sampler

Weighted Least-Squares sampler as in Cohen & Migliorati 2017.

Parameters:

• trialSampler (pythia.sampler.Sampler) – Trial sampler to draw samples from.

• basis (dict) – Dictionary of basis functions.

• strategy (string, ('rejection', 'sequential')) – Strategy for Sampling.

• paramBasis (dict of dict of function, optional) – Univariate basis functions for all parameter.

• alphas (list of tuple, optional) – List of Multiindices represented by tuples.

• tsa (bool, default=False) – Trial sampler adaptation. If True, a trial sampler is chosen on the distributions of parameters, if false a uniform trial sampler is used.

property basis

Employed basis functions.

density(x)

Evaluate PDF.

Deprecated since version 2.0.0: density will be removed in PyThia 3.0.0, use pdf instead.

Parameters:

x (array_like) – Evaluation points.

Returns:

Values of PDF evaluated in x.

Return type:

array_like

property dimension

Dimension of the parameters.

property domain

Parameter domain of support.

property mass

Mass of the PDF.

property maximum

Maximum value of the PDF.

pdf(x)

Evaluate PDF.

Parameters:

x (array_like) – Evaluation points.

Returns:

Values of PDF evaluated in x.

Return type:

array_like

sample(*shape)

Draw samples from distribution.

Parameters:

shape (array_like) – Shape of the samples.

Returns:

Random samples of specified shape.

Return type:

array_like

property trialSampler

Trial sampler used for the rejection sampling.

weight(x)

Weights for the PDF sampler.

class pythia.sampler.WLSSequentialSampler(params, basis, paramBasis, alphas)

Bases: Sampler

WLS sequential conditional sampler as in Cohen & Migliorati 2017.

Parameters:

• params (dict of pythia.parameter.RandomParameter) –

• basis (dict of function) – Multivariate basis functions.

• paramBasis (dict of dict of function) – Univariate basis functions for all parameter.

• alphas (list of tuple) – List of Multiindices represented by tuples.

property alphas

List of Multiindices represented by tuples.

density(x)

Evaluate PDF.

Deprecated since version 2.0.0: density will be removed in PyThia 3.0.0, use pdf instead.

Parameters:

x (array_like) – Evaluation points.

Returns:

Values of PDF evaluated in x.

Return type:

array_like

property dimension

Dimension of the parameters.

property domain

Parameter domain of support.

property mass

Mass of the PDF.

property maximum

Maximum value of the PDF.

property parameter

Parameter list.

pdf(x)

Evaluate PDF.

Parameters:

x (array_like) – Evaluation points.

Returns:

Values of PDF evaluated in x.

Return type:

array_like

sample(*shape)

Draw samples from distribution.

Parameters:

shape (array_like) – Shape of the samples.

Returns:

Random samples of specified shape.

Return type:

array_like

weight(x)

Weights for the PDF.

Parameters:

x (array_like) – Points the weight function is evaluated in.

Returns:

w – weights of evaluation points x.

Return type:

array_like

class pythia.sampler.WLSTensorSampler(params, deg, tsa=True)

Bases: Sampler

WLS sampler for tensor multivariate basis.

Sampler for weighted Least-Squares sampling as described by Cohen and Migliorati. Only full tensor space can be sampled. This allows for univariate weighted Least-Squares sampling in each component.

Parameters:

• params (list of pythia.parameter.RandomParameter) – Parameter list.

• deg (int) – Polynomial degree of each component (same for all).

• tsa (bool, default=True) – Trial sampler adaptation. If True, a trial sampler is chosen on the distributions of parameters, if false a uniform trial sampler is used.

property degree

Polynomial degree for each component.

density(x)

Evaluate PDF.

Deprecated since version 2.0.0: density will be removed in PyThia 3.0.0, use pdf instead.

Parameters:

x (array_like) – Evaluation points.

Returns:

Values of PDF evaluated in x.

Return type:

array_like

property dimension

Dimension of the parameters.

property domain

Parameter domain of support.

property mass

Mass of the PDF.

Note

The PDF is assumed to be normalized. This is not computed here.

property maximum

Maximum value of the PDF.

pdf(x)

Evaluate PDF.

Parameters:

x (array_like) – Evaluation points.

Returns:

Values of PDF evaluated in x.

Return type:

array_like

sample(*shape)

Draw samples from distribution.

Parameters:

shape (array_like) – Shape of the samples.

Returns:

Random samples of specified shape.

Return type:

array_like

univariateDensity(x)

Evaluate univariate PDFs.

Parameters:

x (array_like) – Evaluation points.

Returns:

Values of univariate PDFs evaluated in x.

Return type:

array_like

univariateWeight(x)

Weights for the univariate components.

Parameters:

x (array_like) – Points the weight function is evaluated in.

Returns:

w – weights of evaluation points x.

Return type:

array_like

weight(x)

Weights for the PDF.

Parameters:

x (array_like) – Points the weight function is evaluated in.

Returns:

w – weights of evaluation points x.

Return type:

array_like

pythia.sampler.assign_sampler(param)

Assign a univariate sampler to the given parameter.

Parameters:

param (pythia.parameter.RandomParameter) –

Returns:

Univariate sampler.

Return type:

pythia.sampler.Sampler

pythia.sampler.get_maximum(f, domain, nPoints=1000)

Compute essential maximum of function by point evaluations.

Parameters:

• f (function) – Function to evaluate. Needs to map from n-dim space to 1-dim space.

• domain (array_like) – Domain to evaluate function on.

• nPoints (int, default=1000) – Number of function evaluations. Evaluations are done on a uniform grid in domain. Actual number of points may thus be a little greater.

Returns:

Approximation of maximum of function f.

Return type:

float

Module contents