MaximumLikelihoodFactory¶

class MaximumLikelihoodFactory(*args)¶

Maximum likelihood factory.

Parameters:

distributionDistribution: The parametric distribution $p_{\vect{\theta}}$ .

Methods

`BuildEstimator`(factory, sample[, isRegular])	Estimate the parameters and the asymptotic distribution.
`BuildGaussianEstimator`(distribution, sample)	Compute the asymptotic distribution of the parameters.
`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.
`getKnownParameterIndices`()	Accessor to the known parameters indices.
`getKnownParameterValues`()	Accessor to the known parameters values.
`getName`()	Accessor to the object's name.
`getOptimizationAlgorithm`()	Accessor to the optimization solver.
`getOptimizationBounds`()	Accessor to the optimization bounds.
`hasName`()	Test if the object is named.
`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 optimization solver.
`setOptimizationBounds`(optimizationBounds)	Accessor to the optimization bounds.
`setOptimizationInequalityConstraint`(...)	Accessor to the optimization inequality constraint.

See also

DistributionFactory

Notes

This class implements the generic maximum likelihood estimation which is detailed in Maximum Likelihood Principle.

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

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

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

The log-likelihood is defined as:

$\ell(\vect{x}_1, \dots, \vect{x}_n; \vect{\theta}) = \sum_{i=1}^n \log p_{\vect{\theta}}(\vect{x}_i)$

The estimator of $\vect{\theta}$ maximizes the log-likelihood:

$\hat{\vect{\theta}} = \argmax_{\vect{\theta} \in \Theta} \log \ell (\vect{x}_1, \dots, \vect{x}_n; \vect{\theta})$

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)

__init__(*args)¶

static BuildEstimator(factory, sample, isRegular=False)¶

Estimate the parameters and the asymptotic distribution.

Parameters:

factoryDistributionFactory: Distribution factory to infer the data
sample2-d sequence of float: Data to infer
is_regularbool: Indicates whether the parametric distribution is regular.

Returns:

resultDistributionFactoryResult: Result class providing the estimate and the asymptotic distribution.

Notes

If the model is regular, the asymptotic distribution of the estimator is normal and we get it from the Delta method.

If the model is not regular, we use the Bootstrap method and the kernel smoothing method to get the asymptotic distribution of the estimator.

static BuildGaussianEstimator(distribution, sample)¶

Compute the asymptotic distribution of the parameters.

Parameters:

distributionDistribution: Parametric distribution.
sample2-d sequence of float: Data to infer.

Returns:

distributionNormal: Asymptotic normal distribution of $\hat{\vect{\theta}}$ .

Notes

We assume that the parametric model is regular: then, the asymptotic distribution of $\hat{\vect{\theta}}$ is normal.

build(*args)¶

Build the distribution.

Available usages:

build()

build(sample)

build(param)

Parameters:

sample2-d sequence of float: Data.
paramsequence of float: The parameters of the distribution.

Returns:

distDistribution

The estimated distribution.

In the first usage, the default native distribution is built.

buildEstimator(*args)¶

Build the distribution and the parameter distribution.

Parameters:

sample2-d sequence of float: Data.
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:

sizeint: Size of the bootstrap.

getClassName()¶

Accessor to the object’s name.

Returns:

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

getKnownParameterIndices()¶

Accessor to the known parameters indices.

Returns:

indicesIndices: Indices of the known parameters.

getKnownParameterValues()¶

Accessor to the known parameters values.

Returns:

valuesPoint: Values of known parameters.

getName()¶

Accessor to the object’s name.

Returns:

namestr: The name of the object.

getOptimizationAlgorithm()¶

Accessor to the optimization solver.

Returns:

solverOptimizationAlgorithm: The solver used for the optimization of the log-likelihood.

getOptimizationBounds()¶

Accessor to the optimization bounds.

Returns:

boundsInterval: The bounds used for the optimization of the log-likelihood.

hasName()¶

Test if the object is named.

Returns:

hasNamebool: True if the name is not empty.

setBootstrapSize(bootstrapSize)¶

Accessor to the bootstrap size.

Parameters:

sizeint: The size of the bootstrap.

setKnownParameter(values, positions)¶

Accessor to the known parameters.

Parameters:

valuessequence of float: Values of known parameters.
positionssequence of int: Indices of known parameters.

Examples

When a subset of the parameter vector is known, the other parameters only have to 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.BetaFactory()
>>> # set (a,b) out of (r, t, a, b)
>>> factory.setKnownParameter([-1.0, 1.0], [2, 3])
>>> inf_distribution = factory.build(sample)