LogNormalFactory¶
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class
LogNormalFactory
(*args)¶ Lognormal factory distribution.
 Available constructors:
LogNormalFactory()
See also
Notes
Several estimators to build a LogNormal distribution from a scalar sample are proposed.
Moments based estimator:
Lets denote:
the empirical mean of the sample,
its empirical variance,
its empirical skewness.
We note . The estimator of is the positive root of the relation:
(1)¶
Then we estimate using:
(2)¶
where .
Modified moments based estimator:
Using and previously defined, the third equation is:
(3)¶
The quantity is the mean of the first order statistics of a standard normal sample of size . We have:
(4)¶
where and are the PDF and CDF of the standard normal distribution. The estimator of is obtained as the solution of:
(5)¶
where . Then we have using the relations defined for the moments based estimator (2).
Local maximum likelihood estimator:
The following sums are defined:
The Maximum Likelihood estimator of is defined by:
(6)¶
Thus, satisfies the relation:
(7)¶
under the constraint .
Least squares method estimator:
The parameter is numerically optimized by nonlinear least squares.
When is known and the x_i follow a LogNormal distribution then we use linear leastsquares to solve the relation:
(8)¶
And the remaining parameters are estimated with:
Methods
build
(self, \*args)Build the distribution.
buildAsLogNormal
(self, \*args)Build the distribution as a LogNormal type.
buildEstimator
(self, \*args)Build the distribution and the parameter distribution.
buildMethodOfLeastSquares
(self, sample)Build the distribution based on the leastsquares estimator.
Build the distribution based on the local likelihood maximum estimator.
buildMethodOfModifiedMoments
(self, sample)Build the distribution based on the modified moments estimator.
buildMethodOfMoments
(self, sample)Build the distribution based on the method of moments estimator.
getBootstrapSize
(self)Accessor to the bootstrap size.
getClassName
(self)Accessor to the object’s name.
getId
(self)Accessor to the object’s id.
getName
(self)Accessor to the object’s name.
getShadowedId
(self)Accessor to the object’s shadowed id.
getVisibility
(self)Accessor to the object’s visibility state.
hasName
(self)Test if the object is named.
hasVisibleName
(self)Test if the object has a distinguishable name.
setBootstrapSize
(self, bootstrapSize)Accessor to the bootstrap size.
setName
(self, name)Accessor to the object’s name.
setShadowedId
(self, id)Accessor to the object’s shadowed id.
setVisibility
(self, visible)Accessor to the object’s visibility state.

__init__
(self, *args)¶ Initialize self. See help(type(self)) for accurate signature.

build
(self, *args)¶ Build the distribution.
Available usages:
build()
build(sample)
build(sample, method)
build(param)
 Parameters
 sample2d sequence of float, of dimension 1
The sample from which the distribution parameters are estimated.
 methodinteger
An integer corresponding to a specific estimator method:
0 : Local likelihood maximum estimator
1 : Modified moment estimator
2 : Method of moment estimator
3 : Least squares method.
 paramCollection of
PointWithDescription
A vector of parameters of the distribution.
Notes
See the buildAsLogNormal method.

buildAsLogNormal
(self, *args)¶ Build the distribution as a LogNormal type.
Available usages:
build()
build(sample)
build(sample, method)
build(param)
 Parameters
 sample2d sequence of float, of dimension 1
The sample from which the distribution parameters are estimated.
 methodinteger
An integer ranges from 0 to 2 corresponding to a specific estimator method:  0 : Local likelihood maximum estimator (default)  1 : Modified moment estimator  2 : Method of moment estimator  3 : Least squares method.
The default value is from the
ResourceMap
key LogNormalFactoryEstimationMethod. paramCollection 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:
It first uses the local likelihood maximum based estimator.
It uses the modified moments based estimator if the resolution of (7) is not possible.
It 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
(self, *args)¶ Build the distribution and the parameter distribution.
 Parameters
 sample2d sequence of float
Sample from which the distribution parameters are estimated.
 parameters
DistributionParameters
Optional, the parametrization.
 Returns
 resDist
DistributionFactoryResult
The results.
 resDist
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('DistributionFactoryDefaultBootstrapSize', 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 :
>>> fittedRes2 = ot.BetaFactory().buildEstimator(sample, ot.BetaMuSigma())

buildMethodOfLeastSquares
(self, sample)¶ Build the distribution based on the leastsquares estimator.
 Parameters
 sample2d sequence of float, of dimension 1
The sample from which the distribution parameters are estimated.

buildMethodOfLocalLikelihoodMaximization
(self, sample)¶ Build the distribution based on the local likelihood maximum estimator.
 Parameters
 sample2d sequence of float, of dimension 1
The sample from which the distribution parameters are estimated.

buildMethodOfModifiedMoments
(self, sample)¶ Build the distribution based on the modified moments estimator.
 Parameters
 sample2d sequence of float, of dimension 1
The sample from which the distribution parameters are estimated.

buildMethodOfMoments
(self, sample)¶ Build the distribution based on the method of moments estimator.
 Parameters
 sample2d sequence of float, of dimension 1
The sample from which the distribution parameters are estimated.

getBootstrapSize
(self)¶ Accessor to the bootstrap size.
 Returns
 sizeinteger
Size of the bootstrap.

getClassName
(self)¶ Accessor to the object’s name.
 Returns
 class_namestr
The object class name (object.__class__.__name__).

getId
(self)¶ Accessor to the object’s id.
 Returns
 idint
Internal unique identifier.

getName
(self)¶ Accessor to the object’s name.
 Returns
 namestr
The name of the object.

getShadowedId
(self)¶ Accessor to the object’s shadowed id.
 Returns
 idint
Internal unique identifier.

getVisibility
(self)¶ Accessor to the object’s visibility state.
 Returns
 visiblebool
Visibility flag.

hasName
(self)¶ Test if the object is named.
 Returns
 hasNamebool
True if the name is not empty.

hasVisibleName
(self)¶ Test if the object has a distinguishable name.
 Returns
 hasVisibleNamebool
True if the name is not empty and not the default one.

setBootstrapSize
(self, bootstrapSize)¶ Accessor to the bootstrap size.
 Parameters
 sizeinteger
Size of the bootstrap.

setName
(self, name)¶ Accessor to the object’s name.
 Parameters
 namestr
The name of the object.

setShadowedId
(self, id)¶ Accessor to the object’s shadowed id.
 Parameters
 idint
Internal unique identifier.

setVisibility
(self, visible)¶ Accessor to the object’s visibility state.
 Parameters
 visiblebool
Visibility flag.