WeibullMinFactory¶

(Source code, png)

class WeibullMinFactory(*args)¶

WeibullMin factory.

Methods

`build`(*args)	Build the distribution.
`buildAsWeibullMin`(*args)	Estimate the distribution as native distribution.
`buildEstimator`(*args)	Build the distribution and the parameter distribution.
`buildMethodOfLikelihoodMaximization`(sample)	Method of likelihood maximization.
`buildMethodOfMoments`(sample)	Method of moments estimator.
`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.
`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.

See also

DistributionFactory, WeibullMin

Notes

Several estimators to build a WeibullMin distribution from a scalar sample are proposed. The default strategy is using the maximum likelihood estimators.

Moments based estimator:

Lets denote:

$\displaystyle \overline{x}_n = \frac{1}{n} \sum_{i=1}^n x_i$ the empirical mean of the sample,
$\displaystyle s_n^2 = \frac{1}{n-1} \sum_{i=1}^n (x_i - \overline{x}_n)^2$ its empirical variance

The estimator $(\hat{\beta}_n, \hat{\alpha}_n, \hat{\gamma}_n)$ of $(\beta, \alpha, \gamma)$ is defined by :

(1)¶ $\begin{eqnarray*} \displaystyle \Hat{\gamma}_n & = & (1-\mathrm{sign}(x_{(1,n)})/(2+n))x_{(1,n)} \end{eqnarray*} \begin{eqnarray*} \overline{x}_n & = & \hat{\beta}_n \,\Gamma\left(1 + \frac{1}{\hat{\alpha}_n}\right) + \hat{\gamma}_n \\ s_n^2 & = & \hat{\beta}_n^2 \left( \Gamma \left( 1 + \frac{2}{\hat{\alpha}_n} \right) - \Gamma^2 \left( 1 + \frac{1}{\hat{\alpha}_n} \right) \right) \end{eqnarray*}$

Maximum likelihood based estimator:

The following sums are defined by:

$\begin{eqnarray*} S_0 &=& \sum_{i=1}^n \frac{1}{x_i - \gamma} \\ S_1 &=& \sum_{i=1}^n \log (x_i - \gamma) \\ S_2 &=& \sum_{i=1}^n (x_i - \gamma)^{\alpha} \log (x_i - \gamma) \\ S_3 &=& \sum_{i=1}^n (x_i - \gamma)^{\alpha}\\ S_4 &=& \sum_{i=1}^n (x_i - \gamma)^{\alpha-1} \end{eqnarray*}$

The Maximum Likelihood estimator of $(\beta, \alpha, \gamma)$ is defined by $(\hat{\beta}_n, \hat{\alpha}_n, \hat{\gamma}_n)$ verifying:

(2)¶ $\begin{eqnarray*} S_3(\hat{\alpha}_n,\hat{\gamma}_n) - n\hat{\beta}_n^{\hat{\alpha}_n} = 0 \\ \hat{\alpha}_n \left[S_0(\hat{\gamma}_n) - n\dfrac{S_4(\hat{\alpha}_n,\hat{\gamma}_n)}{S_3(\hat{\alpha}_n,\hat{\gamma}_n)} \right] - S_0(\hat{\gamma}_n) = 0 \\ S_0(\hat{\gamma}_n)(S_3(\hat{\alpha}_n,\hat{\gamma}_n)(n+S_1(\hat{\gamma}_n))-nS_2(\hat{\alpha}_n,\hat{\gamma}_n))-n^2S_4(\hat{\alpha}_n,\hat{\gamma}_n) = 0 \end{eqnarray*}$

__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.

buildAsWeibullMin(*args)¶

Estimate the distribution as native distribution.

Available usages:

buildAsWeibullMin()

buildAsWeibullMin(sample)

buildAsWeibullMin(param)

Parameters:

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

Returns:

distWeibullMin

The estimated distribution as a WeibullMin.

In the first usage, the default WeibullMin 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.

buildMethodOfLikelihoodMaximization(sample)¶

Method of likelihood maximization.

Refer to MaximumLikelihoodFactory.

Parameters:

sample2-d sequence of float: Data.

Returns:

distributionWeibullMin: The estimated distribution.

Notes

The maximization of the likelihood is initialized with the value of the estimator calculated with the method of moments.

buildMethodOfMoments(sample)¶

Method of moments estimator.

Parameters:

sample2-d sequence of float: Data.

Returns:

distributionWeibullMin: The estimated distribution.

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.

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)

setName(name)¶

Accessor to the object’s name.

Parameters:

namestr: The name of the object.

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An Open source initiative for the Treatment of Uncertainties, Risks'N Statistics

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