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.

See also

DistributionFactory

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.

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