# MaximumLikelihoodFactory¶

class MaximumLikelihoodFactory(*args)

Maximum likelihood factory.

Refer to Maximum Likelihood Principle.

Parameters: distribution : Distribution The distribution defining the parametric model to be adjusted to data.

Notes

Implements generic maximum likelihood estimation.

Let us denote the sample, the particular distribution of probability density function we want to fit to the sample, and its the parameter vector.

The likelihood of the sample according to is:

The parameters are numerically optimized using an optimization algorithm:

Examples

>>> 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)

Initialize self. See help(type(self)) for accurate signature.

build(*args)

Build the distribution.

Available usages:

build(sample)

build(param)

Parameters: sample : 2-d sequence of float Sample from which the distribution parameters are estimated. param : Collection of PointWithDescription A vector of parameters of the distribution. dist : Distribution The built distribution.
buildEstimator(*args)

Build the distribution and the parameter distribution.

Parameters: sample : 2-d sequence of float Sample from which the distribution parameters are estimated. parameters : DistributionParameters Optional, the parametrization. resDist : DistributionFactoryResult 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.

Examples

Create a sample from a Beta distribution:

>>> import openturns as ot
>>> sample = ot.Beta().getSample(10)
>>> ot.ResourceMap.SetAsUnsignedInteger('DistributionFactory-DefaultBootstrapSize', 100)


Fit a Beta distribution in the native parameters and create a DistributionFactory:

>>> fittedRes = ot.BetaFactory().buildEstimator(sample)


Fit a Beta distribution in the alternative parametrization :

>>> fittedRes2 = ot.BetaFactory().buildEstimator(sample, ot.BetaMuSigma())

getBootstrapSize()

Accessor to the bootstrap size.

Returns: size : integer Size of the bootstrap.
getClassName()

Accessor to the object’s name.

Returns: class_name : str The object class name (object.__class__.__name__).
getId()

Accessor to the object’s id.

Returns: id : int Internal unique identifier.
getKnownParameterIndices()

Accessor to the known parameters indices.

Returns: indices : Indices Indices of fixed parameters.
getKnownParameterValues()

Accessor to the known parameters indices.

Returns: values : Point Values of fixed parameters.
getName()

Accessor to the object’s name.

Returns: name : str The name of the object.
getOptimizationAlgorithm()

Accessor to the solver.

Returns: solver : OptimizationAlgorithm The solver used for numerical optimization of the likelihood.
getOptimizationBounds()

Accessor to the optimization bounds.

Returns: problem : Interval The bounds used for numerical optimization of the likelihood.
getShadowedId()

Accessor to the object’s shadowed id.

Returns: id : int Internal unique identifier.
getVisibility()

Accessor to the object’s visibility state.

Returns: visible : bool Visibility flag.
hasName()

Test if the object is named.

Returns: hasName : bool True if the name is not empty.
hasVisibleName()

Test if the object has a distinguishable name.

Returns: hasVisibleName : bool True if the name is not empty and not the default one.
setBootstrapSize(bootstrapSize)

Accessor to the bootstrap size.

Parameters: size : integer Size of the bootstrap.
setKnownParameter(values, positions)

Accessor to the known parameters.

Parameters: values : sequence of float Values of fixed parameters. indices : sequence of int Indices of fixed parameters.

Examples

>>> import openturns as ot
>>> ot.RandomGenerator.SetSeed(0)
>>> distribution = ot.Beta(2.3, 4.5, -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: name : str The name of the object.
setOptimizationAlgorithm(solver)

Accessor to the solver.

Parameters: solver : OptimizationAlgorithm The solver used for numerical optimization of the likelihood.
setOptimizationBounds(optimizationBounds)

Accessor to the optimization bounds.

Parameters: problem : Interval The bounds used for numerical optimization of the likelihood.
setOptimizationInequalityConstraint(optimizationInequalityConstraint)

Accessor to the optimization inequality constraint.

Parameters: inequalityConstraint : Function The inequality constraint used for numerical optimization of the likelihood.
setShadowedId(id)

Accessor to the object’s shadowed id.

Parameters: id : int Internal unique identifier.
setVisibility(visible)

Accessor to the object’s visibility state.

Parameters: visible : bool Visibility flag.