LeastSquaresDistributionFactory

class LeastSquaresDistributionFactory(*args)

Least squares factory.

Parameters:
distributionDistribution

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

See also

DistributionFactory

Notes

The method fits a scalar distribution to data of dimension 1, using a least-squares minimization method.

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

let \hat{F} denote the empirical cumulative distribution function built from the sample.

The estimator \hat{\theta} minimizes the mean square error between F_{\vect{\theta}} and \hat{F} on the empirical quantiles.It is defined as:

\hat{\theta} = \argmin_{\vect{\theta} \in \Theta} \sum_{i=1}^{n} \left( F_{\vect{\theta}}(\vect{x}_i) - F_{\hat{\vect{\theta}}}(\vect{x}_i) \right) ^2

Examples

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

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.

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 solver.

setOptimizationBounds(optimizationBounds)

Accessor to the optimization bounds.

setOptimizationInequalityConstraint(...)

Accessor to the optimization inequality constraint.
__init__(*args)
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 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:
problemInterval

The bounds used for numerical optimization of the 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 fixed parameters.

indicessequence 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.LeastSquaresDistributionFactory(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:
problemInterval

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.