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

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 values.
`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.

See also

DistributionFactory

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)

__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 the known parameters.

getKnownParameterValues()¶

Accessor to the known parameters values.

Returns:

valuesPoint: Values of known 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 known parameters.
positionssequence of int: Indices of known parameters.

Examples

When a subset of the parameter vector is known, the other parameters only have to be estimated from data.

In the following example, we consider a sample and want to fit a Beta distribution. We assume that the $a$ and $b$ parameters are known beforehand. In this case, we set the third parameter (at index 2) to -1 and the fourth parameter (at index 3) to 1.

>>> import openturns as ot
>>> ot.RandomGenerator.SetSeed(0)
>>> distribution = ot.Beta(2.3, 2.2, -1.0, 1.0)
>>> sample = distribution.getSample(10)
>>> factory = ot.BetaFactory()
>>> # 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.

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