WhittleFactory¶

class WhittleFactory(*args)¶

Whittle estimator of a scalar ARMA Gaussian process.

Available constructors:

WhittleFactory()

WhittleFactory(p, q, invert)

WhittleFactory(indP, indQ, invertible)

Parameters

pint

Order of the AR part of the $ARMA(p,q)$ process of dimension $d$ .

qint

Order of the MA part of the $ARMA(p,q)$ process of dimension $d$ .

invertiblebool, optional

Restrict the estimation to invertible ARMA processes.

By default: True.

indPIndices

All the $p$ orders that will be investigated. Care: not yet implemented.

indQIndices

All the $p$ orders that will be investigated. Care: not yet implemented.

Notes

We suppose here that the white noise is normal with zero mean and variance $\sigma^2$ . It implies that the ARMA process estimated is normal.

For each order $(p,q)$ , the estimation of the coefficients $(a_k)_{1\leq k\leq p}$ , $(b_k)_{1\leq k\leq q}$ and the variance $\sigma^2$ is done using the Whittle estimator which is based on the maximization of the likelihood function in the frequency domain.

The principle is detailed hereafter for the case of a time series : in the case of a process sample, the estimator is similar except for the periodogram which is computed differently.

Let $(t_i, \vect{X}_i)_{0\leq i \leq n-1}$ be a multivariate time series of dimension $d$ from an $ARMA(p,q)$ process.

The spectral density function of the $ARMA(p,q)$ process writes :

$f(\lambda, \vect{\theta}, \sigma^2) = \frac{\sigma^2}{2 \pi} \frac{|1 + b_1 \exp(-i \lambda) + \ldots + b_q \exp(-i q \lambda)|^2}{|1 + a_1 \exp(-i \lambda) + \ldots + a_p \exp(-i p \lambda)|^2} = \frac{\sigma^2}{2 \pi} g(\lambda, \vect{\theta})$

where $\vect{\theta} = (a_1, a_2,...,a_p,b_1,...,b_q)$ and $\lambda$ is the frequency value.

The Whittle log-likelihood writes :

$\log L_w(\vect{\theta}, \sigma^2) = - \sum_{j=1}^{m} \log f(\lambda_j, \vect{\theta}, \sigma^2) - \frac{1}{2 \pi} \sum_{j=1}^{m} \frac{I(\lambda_j)}{f(\lambda_j, \vect{\theta}, \sigma^2)}$

where :

$I$ is the non parametric estimate of the spectral density, expressed in the Fourier space (frequencies in $[0,2\pi]$ instead of $[-f_{max}, f_{max}]$ ). OpenTURNS uses by default the Welch estimator.
$\lambda_j$ is the $j-th$ Fourier frequency, $\lambda_j = \frac{2 \pi j}{n}$ , $j=1 \ldots m$ with $m$ the largest integer $\leq \frac{n-1}{2}$ .

We estimate the $(p+q+1)$ scalar coefficients by maximizing the log-likelihood function. The corresponding equations lead to the following relation :

$\sigma^{2*} = \frac{1}{m} \sum_{j=1}^{m} \frac{I(\lambda_j)}{g(\lambda_j, \vect{\theta}^{*})}$

where $\vect{\theta}^{*}$ maximizes :

$\log L_w(\vect{\theta}) = m (\log(2 \pi) - 1) - m \log\left( \frac{1}{m} \sum_{j=1}^{m} \frac{I(\lambda_j)}{g(\lambda_j, \vect{\theta})}\right) - \sum_{j=1}^{m} g(\lambda_j, \vect{\theta})$

The Whitle estimation requires that :

the determinant of the eigenvalues of the companion matrix associated to the polynomial $1 + a_1X + \dots + a_pX^p$ are outside the unit disc, which guarantees the stationarity of the process;
the determinant of the eigenvalues of the companion matrix associated to the polynomial $1 + ba_1X + \dots + b_qX^q$ are outside the unit disc, which guarantees the invertibility of the process.

The criteria AIC, $AIC_c$ (corrected AIC) and BIC are evaluated to help the model selection:

$\begin{eqnarray*} AIC_c & = & -2\log L_w + 2(p + q + 1)\frac{m}{m - p - q - 2}\\ AIC & = & -2\log L_w + 2(p + q + 1)\\ BIC & = & -2\log L_w + 2(p + q + 1)\log(p + q + 1) \end{eqnarray*}$

where $m$ is half the number of points of the time grid of the process sample (if the data are a process sample) or in a block of the time series (if the data are a time series).

The BIC criterion leads to a model that gives a better prediction. The AIC criterion selects the best model that fits the given data. The $AIC_c$ criterion improves the previous one by penalizing a too high order that would artificially fit to the data.

Examples

Create a time series from a scalar ARMA(4,2) and a normal white noise:

>>> import openturns as ot
>>> myTimeGrid = ot.RegularGrid(0.0, 0.1, 100)
>>> myWhiteNoise = ot.WhiteNoise(ot.Triangular(-1.0, 0.0, 1.0), myTimeGrid)
>>> myARCoef = ot.ARMACoefficients([0.4, 0.3, 0.2, 0.1])
>>> myMACoef = ot.ARMACoefficients([0.4, 0.3])
>>> myARMAProcess = ot.ARMA(myARCoef, myMACoef, myWhiteNoise)
>>> myTimeSeries = myARMAProcess.getRealization()
>>> myProcessSample = myARMAProcess.getSample(10)

Estimate the ARMA process specifying the orders:

>>> myFactory_42 = ot.WhittleFactory(4, 2)

Check the default SpectralModelFactory:

>>> print(myFactory_42.getSpectralModelFactory())  

Set a particular spectral model: WelchFactory as SpectralModelFactory with the Hanning filtering window:

>>> myFilteringWindow = ot. Hanning()
>>> mySpectralFactory = ot.WelchFactory(myFilteringWindow, 4, 0)
>>> myFactory_42.setSpectralModelFactory(mySpectralFactory)
>>> print(myFactory_42.getSpectralModelFactory())  

Estimate the ARMA process specifying a range for the orders:

p = [1, 2, 4] and q = [4,5,6]:

>>> pIndices = ot.Indices([1, 2, 4])
>>> qIndices =  ot.Indices(3)
>>> qIndices.fill(4,1)
>>> myFactory_Range = ot.WhittleFactory(pIndices, qIndices)

To get the quantified AICc, AIC and BIC criteria:

>>> myARMA_42, myCriterion = myFactory_42.buildWithCriteria(ot.TimeSeries(myTimeSeries))
>>> AICc, AIC, BIC = myCriterion[0:3]

Methods

`build`(self, \*args)	Estimate the ARMA process.
`buildWithCriteria`(self, \*args)	Estimate the ARMA process.
`clearHistory`(self)	Clear the history of the factory.
`disableHistory`(self)	Desactivate the history of all the estimated models.
`enableHistory`(self)	Activate the history of all the estimated models.
`getClassName`(self)	Accessor to the object’s name.
`getCurrentP`(self)	Accessor to the current P order.
`getCurrentQ`(self)	Accessor to the current Q order.
`getHistory`(self)	Check whether the history mecanism is activated.
`getId`(self)	Accessor to the object’s id.
`getInvertible`(self)	Accessor to the invertible constraint.
`getName`(self)	Accessor to the object’s name.
`getP`(self)	Accessor to the P orders.
`getQ`(self)	Accessor to the Q orders.
`getShadowedId`(self)	Accessor to the object’s shadowed id.
`getSpectralModelFactory`(self)	Accessor to the spectral factory.
`getStartingPoints`(self)	Accessor to the starting points for the optimization step.
`getVerbose`(self)	Accessor to the verbose mode.
`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.
`isHistoryEnabled`(self)	Check whether the history mecanism is activated.
`setInvertible`(self, invertible)	Accessor to the invertible constraint.
`setName`(self, name)	Accessor to the object’s name.
`setShadowedId`(self, id)	Accessor to the object’s shadowed id.
`setSpectralModelFactory`(self, factory)	Accessor to the spectral factory.
`setStartingPoints`(self, startingPoints)	Accessor to the starting points for the optimization step.
`setVerbose`(self, verbose)	Accessor to the verbose mode.
`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)¶

Estimate the ARMA process.

Available usages:

build(myTimeSeries)

build(myProcessSample)

Parameters

myTimeSeriesTimeSeries: One realization of the process.
myProcessSampleProcessSample: Several realizations of the process.

Returns

myARMAARMA: The process estimated with the Whittle estimator.

Notes

The model selection is made using the spectral density built using the given data and theoretical spectral density of the ARMA process.

The best ARMA process is selected according to the corrected AIC criterion.

buildWithCriteria(self, \*args)¶

Estimate the ARMA process.

Available usages:

buildWithCriteria(myTimeSeries)

buildWithCriteria(myProcessSample)

Parameters

myTimeSeriesTimeSeries: One realization of the process.
myProcessSampleProcessSample: Several realizations of the process.

Returns

myARMAARMA: The process estimated with the Whittle estimator.
criterionPoint: Result of the evaluation of the AICc, AIC and BIC criteria

Notes

The model selection is made using the spectral density built using the given data and theoretical spectral density of the ARMA process.

The best ARMA process is selected according to the corrected AIC criterion.

clearHistory(self)¶

Clear the history of the factory.

Notes

Clear the history of the factory.

disableHistory(self)¶

Desactivate the history of all the estimated models.

Notes

Desactivate the history mechanism which is the trace of all the tested models and their associated information criteria.

enableHistory(self)¶

Activate the history of all the estimated models.

Notes

Activate the history mechanism which is the trace of all the tested models and their associated information criteria.

By default, the history mecanism is activated.

getClassName(self)¶

Accessor to the object’s name.

Returns

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

getCurrentP(self)¶

Accessor to the current P order.

Returns

pint: Order of the AR part of the $ARMA(p,q)$ process of dimension $d$ .

getCurrentQ(self)¶

Accessor to the current Q order.

Returns

qint: Order of the MA part of the $ARMA(p,q)$ process of dimension $d$ .

getHistory(self)¶

Check whether the history mecanism is activated.

Returns

histMeca list of WhittleFactoryState: Returns the collection of all the states that have been built for the estimation.

getId(self)¶

Accessor to the object’s id.

Returns

idint: Internal unique identifier.

getInvertible(self)¶

Accessor to the invertible constraint.

Returns

invertiblebool: The initial AR coefficients used for the optimization algorithm.

getName(self)¶

Accessor to the object’s name.

Returns

namestr: The name of the object.

getP(self)¶

Accessor to the P orders.

Returns

indPIndices: All the $p$ orders that will be investigated.

getQ(self)¶

Accessor to the Q orders.

Returns

indQIndices: All the $p$ orders that will be investigated.

getShadowedId(self)¶

Accessor to the object’s shadowed id.

Returns

idint: Internal unique identifier.

getSpectralModelFactory(self)¶

Accessor to the spectral factory.

Returns

initARCoeffSpectralModelFactory: The spectral factory used to estimate the spectral density based on the data.

getStartingPoints(self)¶

Accessor to the starting points for the optimization step.

Returns

startPointsLista list of Point: Starting points for the optimization step, for each pair of orders that will be tested.

getVerbose(self)¶

Accessor to the verbose mode.

Returns

verboseModebool: Get the verbose mode while both the exploration of the possible models and the optimization steps.

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.

isHistoryEnabled(self)¶

Check whether the history mecanism is activated.

Returns

histMecbool

Check whether the history mecanism is activated.

By default, the history mecanism is activated.

setInvertible(self, invertible)¶

Accessor to the invertible constraint.

Parameters

invertiblebool: The initial AR coefficients used for the optimization algorithm.

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.

setSpectralModelFactory(self, factory)¶

Accessor to the spectral factory.

Parameters

spectralModelFactSpectralModelFactory: The spectral factory used to estimate the spectral density based on the data.

setStartingPoints(self, startingPoints)¶

Accessor to the starting points for the optimization step.

Parameters

startPointsLista list of Point: Starting points for the optimization step, for each pair of orders that will be tested.

setVerbose(self, verbose)¶

Accessor to the verbose mode.

Parameters

verboseModebool: Set the verbose mode while both the exploration of the possible models and the optimization steps.

setVisibility(self, visible)¶

Accessor to the object’s visibility state.

Parameters

visiblebool: Visibility flag.

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An Open source initiative for the Treatment of Uncertainties, Risks'N Statistics

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