MethodOfMomentsFactory¶
- class MethodOfMomentsFactory(*args)¶
Estimation by method of moments.
- Parameters:
- distribution
Distribution
The distribution defining the parametric model to be adjusted to data.
- momentsOrdersequence of int
The orders of moments to estimate (1 for mean, 2 for variance, etc)
- bounds
Interval
, optional Parameter bounds
- distribution
See also
Notes
This method fits a distribution to data of dimension 1, using the method of moments.
Let be the sample, the cumulative distribution function we want to fit to the sample, and its parameter vector where is the number of parameters of the parametric model.
We assume that the first moments of the distribution exist. Let be the mean and the first centered moments of the parametric model. They can be can be expressed as a function of the :
Let be the empirical mean and the first empirical centered moments evaluated on the sample :
Then the estimator built by the method of moments is solution of the following nonlinear system:
(1)¶
which is equivalent to the solution of the following optimization problem:
Examples
>>> import openturns as ot >>> ot.RandomGenerator.SetSeed(0) >>> distribution = ot.Normal(0.9, 1.7) >>> sample = distribution.getSample(10) >>> orders = [1, 2] # mean, variance >>> factory = ot.MethodOfMomentsFactory(ot.Normal(), orders) >>> inf_distribution = factory.build(sample)
With parameter bounds:
>>> bounds = ot.Interval([0.8, 1.6], [1.0, 1.8]) >>> factory = ot.MethodOfMomentsFactory(ot.Normal(), orders, bounds) >>> inf_distribution = factory.build(sample)
Methods
build
(*args)Build the distribution.
buildEstimator
(*args)Build the distribution and the parameter distribution.
buildFromMoments
(moments)Build from moments.
Accessor to the bootstrap size.
Accessor to the object's name.
Accessor to the known parameters indices.
Accessor to the known parameters indices.
Accessor to the moment orders.
getName
()Accessor to the object's name.
Accessor to the solver.
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.
setMomentOrders
(momentsOrders)Accessor to the moment orders.
setName
(name)Accessor to the object's name.
setOptimizationAlgorithm
(solver)Accessor to the solver.
setOptimizationBounds
(optimizationBounds)Accessor to the optimization bounds.
- __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:
- dist
Distribution
The estimated distribution.
In the first usage, the default native distribution is built.
- dist
- buildEstimator(*args)¶
Build the distribution and the parameter distribution.
- Parameters:
- sample2-d sequence of float
Data.
- 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.
- buildFromMoments(moments)¶
Build from moments.
- Parameters:
- momentssequence of float
Consists in the mean followed by consecutive central moments from order 2 (variance), of total size at least the distribution parameter dimension.
- Returns:
- dist
Distribution
Estimated distribution.
- dist
Notes
Depending on the parametric model choosed, not all moments define a valid distribution, so it should only used with empirical moments from the same model.
Examples
>>> import openturns as ot >>> distribution = ot.Beta(2.3, 2.2, -1.0, 1.0) >>> factory = ot.MethodOfMomentsFactory(ot.Beta(), [1, 2, 3, 4]) >>> cm = [distribution.getCentralMoment(i + 2)[0] for i in range(3)] >>> moments = [distribution.getMean()[0]] + cm >>> inf_distribution = factory.buildFromMoments(moments)
- 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:
- indices
Indices
Indices of fixed parameters.
- indices
- getKnownParameterValues()¶
Accessor to the known parameters indices.
- Returns:
- values
Point
Values of fixed parameters.
- values
- getMomentOrders()¶
Accessor to the moment orders.
- Returns:
- momentsOrdersequence of int
The orders of moments to estimate (1 for mean, 2 for variance, etc)
- getName()¶
Accessor to the object’s name.
- Returns:
- namestr
The name of the object.
- getOptimizationAlgorithm()¶
Accessor to the solver.
- Returns:
- solver
OptimizationAlgorithm
The solver used for numerical optimization of the moments.
- solver
- getOptimizationBounds()¶
Accessor to the optimization bounds.
- Returns:
- bounds
Interval
The bounds used for numerical optimization of the likelihood.
- bounds
- 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, 2.2, -1.0, 1.0) >>> sample = distribution.getSample(10) >>> orders = [3, 4] # skewness, kurtosis >>> factory = ot.MethodOfMomentsFactory(ot.Beta(), orders) >>> # set (a,b) out of (r, t, a, b) >>> factory.setKnownParameter([-1.0, 1.0], [2, 3]) >>> inf_distribution = factory.build(sample)
- setMomentOrders(momentsOrders)¶
Accessor to the moment orders.
- Parameters:
- momentsOrdersequence of int
The orders of moments to estimate (1 for mean, 2 for variance, etc)
- setName(name)¶
Accessor to the object’s name.
- Parameters:
- namestr
The name of the object.
- setOptimizationAlgorithm(solver)¶
Accessor to the solver.
- Parameters:
- solver
OptimizationAlgorithm
The solver used for numerical optimization of the moments.
- solver