MethodOfMomentsFactory

class MethodOfMomentsFactory(*args)

Estimation by method of moments.

Parameters:
distributionDistribution

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)

boundsInterval, optional

Parameter bounds

Notes

This method fits a distribution to data of dimension 1, using the method of moments.

Let (x_1, \dots, x_n) be the sample, F_{\vect{\theta}} the cumulative distribution function we want to fit to the sample, and \vect{\theta} = (\theta_1, \dots, \theta_{d_{\theta}}) \in \Rset^{d_\theta} its parameter vector where {d_\theta} \in \Nset is the number of parameters of the parametric model.

We assume that the d_\theta first moments of the distribution exist. Let (\mu_1, \dots, \mu_{d_\theta}) be the mean and the (d_\theta -1) first centered moments of the parametric model. They can be can be expressed as a function of the \vect{\theta}:

\mu_1 & =  \Expect{X} = g_1(\theta_1, \dots, \theta_{d_\theta})  \\
\mu_k & =  \Expect{(X - \mu_1)^k} = g_k(\theta_1, \dots, \theta_{d_\theta}), \quad 2 \leq k \leq d_\theta.

Let (\widehat{\mu}_1, \dots, \widehat{\mu}_{d_\theta}) be the empirical mean and the (d_\theta -1) first empirical centered moments evaluated on the sample (x_1, \dots, x_n):

\widehat{\mu}_1 & = \dfrac{1}{n} \sum_{i=1}^n x_i \\
\widehat{\mu}_k & = \dfrac{1}{n} \sum_{i=1}^n (x_i - \widehat{\mu}_1)^k, \quad 2 \leq k \leq d_\theta.

Then the estimator \widehat{\theta} = \left(\widehat{\theta}_1,
\dots, \widehat{\theta}_{d_\theta}\right) built by the method of moments is solution of the following nonlinear system:

(1)\widehat{\mu}_1 & =  g_1(\theta_1, \dots, \theta_{d_\theta}) \\
\widehat{\mu}_2 & = g_2(\theta_1, \dots, \theta_{d_\theta}) \\
\vdots &  \\
\widehat{\mu}_{d_\theta} & = g_{d_\theta}(\theta_1, \dots, \theta_{d_\theta})

which is equivalent to the solution of the following optimization problem:

\widehat{\theta} = \argmin_{\vect{\theta} \in \Rset^{d_\theta}}  \sum_{k=1}^{d_\theta} \left[ \left( g_k(\vect{\theta}) \right)^{1/k}-
\widehat{\mu}_k^{1/k} \right]^2

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.

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 indices.

getMomentOrders()

Accessor to the moment orders.

getName()

Accessor to the object's name.

getOptimizationAlgorithm()

Accessor to the solver.

getOptimizationBounds()

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:
distDistribution

The estimated distribution.

In the first usage, the default native 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.

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:
distDistribution

Estimated distribution.

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:
indicesIndices

Indices of fixed parameters.

getKnownParameterValues()

Accessor to the known parameters indices.

Returns:
valuesPoint

Values of fixed parameters.

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:
solverOptimizationAlgorithm

The solver used for numerical optimization of the moments.

getOptimizationBounds()

Accessor to the optimization bounds.

Returns:
boundsInterval

The bounds used for numerical optimization of the likelihood.

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:
solverOptimizationAlgorithm

The solver used for numerical optimization of the moments.

setOptimizationBounds(optimizationBounds)

Accessor to the optimization bounds.

Parameters:
boundsInterval

The bounds used for numerical optimization of the likelihood.