TruncatedNormalFactory

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../../_images/openturns-TruncatedNormalFactory-1.png
class TruncatedNormalFactory(*args)

Truncated Normal factory.

Available constructor:

TruncatedNormalFactory()

See also

DistributionFactory, TruncatedNormal

Notes

Several estimators to build a TruncatedNormal distribution from a scalar sample are available.

Maximum likelihood estimator:

The parameters are estimated by numerical maximum likelihood estimation with scaling. The starting point of the optimization algorithm is based on the moment based estimator.

Let n be the sample sample size. Let x_{min} be the sample minimum and x_{max} be the sample maximum.

We compute the scaling parameters \alpha and \beta from the equations:

\begin{eqnarray*} \displaystyle \alpha = \frac{2}{x_{min} - x_{max}}, \\ \displaystyle \beta = \frac{1}{2} (x_{min} + x_{max}). \end{eqnarray*}

Then the sample \{x_i\}_{i=1,...,n} is scaled into \{u_i\}_{i=1,...,n} from the equation:

\begin{eqnarray*} \displaystyle u_i = \alpha (x_i - \beta) \end{eqnarray*}

for i=1,...,n. Hence, the scaled sample is so that u_i\in[-1,1] for i=1,...,n.

The starting point of the likelihood maximization algorithm is based on the scaled sample. Let

\begin{eqnarray*} \displaystyle \mu_0^u = \bar{u}, \\ \displaystyle \sigma_0^u = \sigma_{u, n} \end{eqnarray*}

where \bar{u} is the sample mean of the scaled sample and \sigma_{u, n} is the sample standard deviation of the scaled sample.

Then the likelihood maximization optimization algorithm is used to fit the scaled truncated normal distribution. The TruncatedNormalFactory-SigmaLowerBound key in the ResourceMap is used as a lower bound for the scaled standard deviation.

Let \epsilon be computed from the sample size:

\begin{eqnarray*} \displaystyle \epsilon = 1 + \frac{1}{n}. \end{eqnarray*}

The lower and upper bounds of the scaled truncated normal distribution are set to -\epsilon and \epsilon and are not optimized. This leads to a maximum likelihood optimization problem in 2 dimensions only, where the solution is the optimum scaled mean \mu_u^\star and the optimum scaled standard deviation \sigma_u^\star.

Finally, the parameters of the truncated normal distribution are computed from the parameters of the scaled truncated normal distribution. The inverse scaling equation is x = \beta + \frac{u}{\beta}, which leads to:

\begin{eqnarray*} \displaystyle \mu = \beta + \frac{\mu_u^\star}{\alpha}, \\ \displaystyle \sigma = \frac{\sigma_u^\star}{\alpha}, \\ \displaystyle a = \beta - \frac{\epsilon}{\alpha}, \\ \displaystyle b = \beta + \frac{\epsilon}{\alpha}. \end{eqnarray*}

Moment based estimator:

Let x_{min} be the sample minimum and x_{max} be the sample maximum. Let \delta = x_{max} - x_{min} be the sample range.

The distribution bounds are computed from the equations:

\begin{eqnarray*} \displaystyle\Hat{a}_n = x_{min} - \frac{\delta}{n + 2}\\ \displaystyle\Hat{b}_n = x_{max} + \frac{\delta}{n + 2} \end{eqnarray*}

Then the \mu and \sigma parameters are estimated from the methods of moments.

Examples

In the following example, the parameters of a TruncatedNormal are estimated from a sample.

>>> import openturns as ot
>>> ot.RandomGenerator.SetSeed(0)
>>> size = 10000
>>> distribution = ot.TruncatedNormal(2.0, 3.0, -1.0, 4.0)
>>> sample = distribution.getSample(size)
>>> factory = ot.TruncatedNormalFactory()
>>> estimated = factory.build(sample)
>>> estimated = factory.buildMethodOfMoments(sample)
>>> estimated = factory.buildMethodOfLikelihoodMaximization(sample)

Methods

build(*args)

Estimate the distribution using the default strategy.

buildAsTruncatedNormal(*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.

getId()

Accessor to the object’s id.

getName()

Accessor to the object’s name.

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.

setName(name)

Accessor to the object’s name.

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)

Estimate the distribution using the default strategy.

Parameters
sampleSample

Data

Returns
distributionDistribution

The estimated distribution

Notes

The default strategy is using the maximum likelihood estimators with scaling.

buildAsTruncatedNormal(*args)

Estimate the distribution as native distribution.

Parameters
sampleSample

Data

Returns
distributionTruncatedNormal

The estimated 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.

buildMethodOfLikelihoodMaximization(sample)

Method of likelihood maximization.

Parameters
sampleSample

Data

Returns
distributionTruncatedNormal

The estimated distribution

buildMethodOfMoments(sample)

Method of moments estimator.

Parameters
sampleSample

Data

Returns
distributionTruncatedNormal

The estimated distribution

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.

getName()

Accessor to the object’s name.

Returns
namestr

The name of the object.

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.

setName(name)

Accessor to the object’s name.

Parameters
namestr

The name of the object.

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.