MaximumLikelihoodFactory

class MaximumLikelihoodFactory(*args)

Maximum likelihood factory.

Refer to Maximum Likelihood Principle.

Parameters
distributionDistribution

The distribution defining the parametric model p_{\vect{\theta}} to be adjusted to data.

Notes

Implements generic maximum likelihood estimation.

Let us denote (\vect{x}_1, \dots, \vect{x}_n) the sample, p_{\vect{\theta}} the particular distribution of probability density function we want to fit to the sample, and \vect{\theta} \in \Theta \in \Rset^p its the parameter vector.

The likelihood of the sample according to p_{\vect{\theta}} is:

likelihood(\vect{x}_1, \dots, \vect{x}_n,\vect{\theta}) = \prod_{i=1}^n p_{\vect{\theta}}(\vect{x}_i)

The parameters \vect{\theta} are numerically optimized using an optimization algorithm:

\max_{\vect{\theta} \in \Theta} \log likelihood\, (\vect{x}_1, \dots, \vect{x}_n,\vect{\theta}) = \max_{\vect{\theta} \in \Theta} \sum_{i=1}^n log(p_{\vect{\theta}}(\vect{x}_i))

Examples

In the following example, we estimate the parameters of a Normal distribution with maximum likelihood estimation.

>>> import openturns as ot
>>> ot.RandomGenerator.SetSeed(0)
>>> distribution = ot.Normal(0.9, 1.7)
>>> sample = distribution.getSample(10)
>>> factory = ot.MaximumLikelihoodFactory(ot.Normal())
>>> inf_distribution = factory.build(sample)

Methods

build(*args)

Build the distribution.

buildEstimator(*args)

Build the distribution and the parameter distribution.

getBootstrapSize()

Accessor to the bootstrap size.

getClassName()

Accessor to the object's name.

getId()

Accessor to the object's id.

getKnownParameterIndices()

Accessor to the known parameters indices.

getKnownParameterValues()

Accessor to the known parameters indices.

getName()

Accessor to the object's name.

getOptimizationAlgorithm()

Accessor to the solver.

getOptimizationBounds()

Accessor to the optimization bounds.

getShadowedId()

Accessor to the object's shadowed id.

getVisibility()

Accessor to the object's visibility state.

hasName()

Test if the object is named.

hasVisibleName()

Test if the object has a distinguishable name.

setBootstrapSize(bootstrapSize)

Accessor to the bootstrap size.

setKnownParameter(values, positions)

Accessor to the known parameters.

setName(name)

Accessor to the object's name.

setOptimizationAlgorithm(solver)

Accessor to the solver.

setOptimizationBounds(optimizationBounds)

Accessor to the optimization bounds.

setOptimizationInequalityConstraint(...)

Accessor to the optimization inequality constraint.

setShadowedId(id)

Accessor to the object's shadowed id.

setVisibility(visible)

Accessor to the object's visibility state.

__init__(*args)
build(*args)

Build the distribution.

Available usages:

build(sample)

build(param)

Parameters
sample2-d sequence of float

Sample from which the distribution parameters are estimated.

paramCollection of PointWithDescription

A vector of parameters of the distribution.

Returns
distDistribution

The built distribution.

buildEstimator(*args)

Build the distribution and the parameter distribution.

Parameters
sample2-d sequence of float

Sample from which the distribution parameters are estimated.

parametersDistributionParameters

Optional, the parametrization.

Returns
resDistDistributionFactoryResult

The results.

Notes

According to the way the native parameters of the distribution are estimated, the parameters distribution differs:

  • Moments method: the asymptotic parameters distribution is normal and estimated by Bootstrap on the initial data;

  • Maximum likelihood method with a regular model: the asymptotic parameters distribution is normal and its covariance matrix is the inverse Fisher information matrix;

  • Other methods: the asymptotic parameters distribution is estimated by Bootstrap on the initial data and kernel fitting (see KernelSmoothing).

If another set of parameters is specified, the native parameters distribution is first estimated and the new distribution is determined from it:

  • if the native parameters distribution is normal and the transformation regular at the estimated parameters values: the asymptotic parameters distribution is normal and its covariance matrix determined from the inverse Fisher information matrix of the native parameters and the transformation;

  • in the other cases, the asymptotic parameters distribution is estimated by Bootstrap on the initial data and kernel fitting.

getBootstrapSize()

Accessor to the bootstrap size.

Returns
sizeinteger

Size of the bootstrap.

getClassName()

Accessor to the object’s name.

Returns
class_namestr

The object class name (object.__class__.__name__).

getId()

Accessor to the object’s id.

Returns
idint

Internal unique identifier.

getKnownParameterIndices()

Accessor to the known parameters indices.

Returns
indicesIndices

Indices of fixed parameters.

getKnownParameterValues()

Accessor to the known parameters indices.

Returns
valuesPoint

Values of fixed parameters.

getName()

Accessor to the object’s name.

Returns
namestr

The name of the object.

getOptimizationAlgorithm()

Accessor to the solver.

Returns
solverOptimizationAlgorithm

The solver used for numerical optimization of the likelihood.

getOptimizationBounds()

Accessor to the optimization bounds.

Returns
boundsInterval

The bounds used for numerical optimization of the likelihood.

getShadowedId()

Accessor to the object’s shadowed id.

Returns
idint

Internal unique identifier.

getVisibility()

Accessor to the object’s visibility state.

Returns
visiblebool

Visibility flag.

hasName()

Test if the object is named.

Returns
hasNamebool

True if the name is not empty.

hasVisibleName()

Test if the object has a distinguishable name.

Returns
hasVisibleNamebool

True if the name is not empty and not the default one.

setBootstrapSize(bootstrapSize)

Accessor to the bootstrap size.

Parameters
sizeinteger

Size of the bootstrap.

setKnownParameter(values, positions)

Accessor to the known parameters.

Parameters
valuessequence of float

Values of fixed parameters.

positionssequence of int

Indices of fixed parameters.

Examples

There are situations where a subset of the parameters is known. In this case, only the other parameters must be estimated from data.

In the following example, we consider a sample and want to fit a Beta distribution. We assume that the a and b parameters are known beforehand. In this case, we set the third parameter (at index 2) to -1 and the fourth parameter (at index 3) to 1.

>>> import openturns as ot
>>> ot.RandomGenerator.SetSeed(0)
>>> distribution = ot.Beta(2.3, 2.2, -1.0, 1.0)
>>> sample = distribution.getSample(10)
>>> factory = ot.MaximumLikelihoodFactory(ot.Beta())
>>> # set (a,b) out of (r, t, a, b)
>>> factory.setKnownParameter([-1.0, 1.0], [2, 3])
>>> inf_distribution = factory.build(sample)
setName(name)

Accessor to the object’s name.

Parameters
namestr

The name of the object.

setOptimizationAlgorithm(solver)

Accessor to the solver.

Parameters
solverOptimizationAlgorithm

The solver used for numerical optimization of the likelihood.

setOptimizationBounds(optimizationBounds)

Accessor to the optimization bounds.

Parameters
boundsInterval

The bounds used for numerical optimization of the likelihood.

setOptimizationInequalityConstraint(optimizationInequalityConstraint)

Accessor to the optimization inequality constraint.

Parameters
inequalityConstraintFunction

The inequality constraint used for numerical optimization of the likelihood.

setShadowedId(id)

Accessor to the object’s shadowed id.

Parameters
idint

Internal unique identifier.

setVisibility(visible)

Accessor to the object’s visibility state.

Parameters
visiblebool

Visibility flag.