LogNormalFactory

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

Lognormal factory distribution.

Available constructors:
LogNormalFactory()

Notes

OpenTURNS proposes several estimators to build a LogNormal distribution from a scalar sample.

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,
  • \displaystyle a_{3,n} = \sqrt{n} \frac{\sum_{i=1}^n (x_i - \overline{x}_n)^3}{ \left( \sum_{i=1}^n (x_i - \overline{x}_n)^2 \right)^{3/2}} its empirical skewness.

We note \omega = e^{\sigma_l^2}. The estimator \hat{\omega}_n of \omega is the positive root of the relation:

(1)\omega^3 + 3 \omega^2 - (4 + a_{3,n}^2) = 0

Then we estimate (\hat{\mu}_{ln}, \hat{\sigma}_{ln}, \hat{\gamma}_{n}) using:

(2)\hat{\mu}_{ln} &= \log \hat{\beta}_{n} \\
\hat{\sigma}_{ln} &= \sqrt{ \log \hat{\omega}_{n} } \\
\hat{\gamma}_{ln} &= \overline{x}_n - \hat{\beta}_{n} \sqrt{ \hat{\omega}_{n} }

where \displaystyle \hat{\beta}_{n} = \frac{s_n}{\hat{\omega}_{n} (\hat{\omega}_{n} - 1)}.

Modified moments based estimator:

Using \overline{x}_n and s_n^2 previously defined, the third equation is:

(3)\Eset[ \log (X_{(1)} - \gamma)] = \log (x_{(1)} - \gamma)

The quantity \displaystyle EZ_1 (n) = \frac{\Eset[ \log (X_{(1)} - \gamma)] - \mu_l}{\sigma_l} is the mean of the first order statistics of a standard normal sample of size n. We have:

(4)EZ_1(n) = \int_\Rset nz\phi(z) (1 - \Phi(z))^{n-1}\di{z}

where \varphi and \Phi are the PDF and CDF of the standard normal distribution. The estimator \hat{\omega}_{n} of \omega is obtained as the solution of:

(5)\omega (\omega - 1) - \kappa_n \left[ \sqrt{\omega} - e^{EZ_1(n)\sqrt{\log \omega}} \right]^2 = 0

where \displaystyle \kappa_n = \frac{s_n^2}{(\overline{x}_n - x_{(1)})^2}. Then we have (\hat{\mu}_{ln}, \hat{\sigma}_{ln}, \hat{\gamma}_{n}) using the relations defined for the moments based estimator (2).

Local maximum likelihood estimator:

The following sums are defined:

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 \log^2 (x_i - \gamma) \\
S_3 &= \sum_{i=1}^n \frac{\log (x_i - \gamma)}{x_i - \gamma}

The Maximum Likelihood estimator of (\mu_{l}, \sigma_{l}, \gamma) is defined by:

(6)\hat{\mu}_{l,n} &= \frac{S_1(\hat{\gamma})}{n} \\
\hat{\sigma}_{l,n}^2 &= \frac{S_2(\hat{\gamma})}{n} - \hat{\mu}_{l,n}^2

Thus, \hat{\gamma}_n satisfies the relation:

(7)S_0 (\gamma) \left(S_2(\gamma) - S_1(\gamma) \left( 1 + \frac{S_1(\gamma)}{n} \right) \right) + n S_4(\gamma) = 0

under the constraint \gamma \leq \min x_i.

Methods

build(*args) Build the distribution.
buildAsLogNormal(*args) Build the distribution as a LogNormal type.
buildEstimator(*args) Build the distribution and the parameter distribution.
buildMethodOfLocalLikelihoodMaximization(sample) Build the distribution based on the local likelihood maximum estimator.
buildMethodOfModifiedMoments(sample) Build the distribution based on the modified moments estimator.
buildMethodOfMoments(sample) Build the distribution based on the 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)

x.__init__(…) initializes x; see help(type(x)) for signature

build(*args)

Build the distribution.

Available usages:

build()

build(sample)

build(sample, method)

build(param)

Parameters:

sample : 2-d sequence of float, of dimension 1

The sample from which the distribution parameters are estimated.

method : integer

An integer ranges from 0 to 2 corresponding to a specific estimator method: - 0 : Local likelihood maximum estimator - 1 : Modified moment estimator - 2 : method of moment estimator.

param : Collection of PointWithDescription

A vector of parameters of the distribution.

Notes

See the buildAsLogNormal method.

buildAsLogNormal(*args)

Build the distribution as a LogNormal type.

Available usages:

build()

build(sample)

build(sample, method)

build(param)

Parameters:

sample : 2-d sequence of float, of dimension 1

The sample from which the distribution parameters are estimated.

method : integer

An integer ranges from 0 to 2 corresponding to a specific estimator method: - 0 : Local likelihood maximum estimator - 1 : Modified moment estimator - 2 : method of moment estimator.

param : Collection of PointWithDescription

A vector of parameters of the distribution.

Notes

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

In the second usage, the parameters are evaluated according the following strategy:

  • OpenTURNS first uses the local likelihood maximum based estimator.
  • OpenTURNS uses the modified moments based estimator if the resolution of (7) is not possible.
  • OpenTURNS uses the moments based estimator, which are always defined, if the resolution of (5) is not possible.

In the third usage, the parameters of the LogNormal are estimated using the given method.

In the fourth usage, a LogNormal distribution corresponding to the given parameters is built.

buildEstimator(*args)

Build the distribution and the parameter distribution.

Parameters:

sample : 2-d sequence of float

Sample from which the distribution parameters are estimated.

parameters : DistributionParameters

Optional, the parametrization.

Returns:

resDist : DistributionFactoryResult

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.

Examples

Create a sample from a Beta distribution:

>>> import openturns as ot
>>> sample = ot.Beta().getSample(10)
>>> ot.ResourceMap.SetAsUnsignedInteger('DistributionFactory-DefaultBootstrapSize', 100)

Fit a Beta distribution in the native parameters and create a DistributionFactory:

>>> fittedRes = ot.BetaFactory().buildEstimator(sample)

Fit a Beta distribution in the alternative parametrization (\mu, \sigma, a, b):

>>> fittedRes2 = ot.BetaFactory().buildEstimator(sample, ot.BetaMuSigma())
buildMethodOfLocalLikelihoodMaximization(sample)

Build the distribution based on the local likelihood maximum estimator.

Parameters:

sample : 2-d sequence of float, of dimension 1

The sample from which the distribution parameters are estimated.

buildMethodOfModifiedMoments(sample)

Build the distribution based on the modified moments estimator.

Parameters:

sample : 2-d sequence of float, of dimension 1

The sample from which the distribution parameters are estimated.

buildMethodOfMoments(sample)

Build the distribution based on the method of moments estimator.

Parameters:

sample : 2-d sequence of float, of dimension 1

The sample from which the distribution parameters are estimated.

getBootstrapSize()

Accessor to the bootstrap size.

Returns:

size : integer

Size of the bootstrap.

getClassName()

Accessor to the object’s name.

Returns:

class_name : str

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

getId()

Accessor to the object’s id.

Returns:

id : int

Internal unique identifier.

getName()

Accessor to the object’s name.

Returns:

name : str

The name of the object.

getShadowedId()

Accessor to the object’s shadowed id.

Returns:

id : int

Internal unique identifier.

getVisibility()

Accessor to the object’s visibility state.

Returns:

visible : bool

Visibility flag.

hasName()

Test if the object is named.

Returns:

hasName : bool

True if the name is not empty.

hasVisibleName()

Test if the object has a distinguishable name.

Returns:

hasVisibleName : bool

True if the name is not empty and not the default one.

setBootstrapSize(bootstrapSize)

Accessor to the bootstrap size.

Parameters:

size : integer

Size of the bootstrap.

setName(name)

Accessor to the object’s name.

Parameters:

name : str

The name of the object.

setShadowedId(id)

Accessor to the object’s shadowed id.

Parameters:

id : int

Internal unique identifier.

setVisibility(visible)

Accessor to the object’s visibility state.

Parameters:

visible : bool

Visibility flag.