WhittleFactory

class WhittleFactory(*args)

Whittle estimator of a scalar ARMA normal process.

Available constructors:

WhittleFactory()

WhittleFactory(p, q, invert)

WhittleFactory(indP, indQ, invertible)

Parameters:

p : int

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

q : int

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

invertible : bool, optional

Restrict the estimation to invertible ARMA processes.

By default: True.

indP : Indices

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

indQ : Indices

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 eigen values of the companion matrix associated to the polynomial 1 + a_1X + \dots + a_pX^p are outside the unit disc, which garantees the stationarity of the process;
  • the determinant of the eigen values of thez companion matrix associated to the polynomial 1 + ba_1X + \dots + b_qX^q are outside the unit disc, which garantees 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:

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

Methods

build(*args) Estimate the ARMA process.
clearHistory() Clear the history of the factory.
disableHistory() Desactivate the history of all the estimated models.
enableHistory() Activate the history of all the estimated models.
getClassName() Accessor to the object’s name.
getCurrentP()
getCurrentQ()
getHistory() Check whether the history mecanism is activated.
getId() Accessor to the object’s id.
getInvertible()
getName() Accessor to the object’s name.
getP()
getQ()
getShadowedId() Accessor to the object’s shadowed id.
getSpectralModelFactory() Accessor to the spectral factory.
getStartingPoints() Accessor to the starting points for the optimization step.
getVerbose() Accessor to the verbose mode.
getVisibility() Accessor to the object’s visibility state.
hasName() Test if the object is named.
hasVisibleName() Test if the object has a distinguishable name.
isHistoryEnabled() Check whether the history mecanism is activated.
setInvertible(invertible)
setName(name) Accessor to the object’s name.
setShadowedId(id) Accessor to the object’s shadowed id.
setSpectralModelFactory(factory) Accessor to the spectral factory.
setStartingPoints(startingPoints) Accessor to the starting points for the optimization step.
setVerbose(verbose) Accessor to the verbose mode.
setVisibility(visible) Accessor to the object’s visibility state.
__init__(*args)
build(*args)

Estimate the ARMA process.

Available usages:

build(myTimeSeries, criterion)

build(myProcessSample, criterion)

Parameters:

myTimeSeries : TimeSeries

One realization of the process.

criterion : Point with the default implementation, optional

To get the evaluation of the AICc, AIC and BIC criteria

myProcessSample : ProcessSample

Several realizations of the process.

Returns:

myARMA : ARMA

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.

clearHistory()

Clear the history of the factory.

Notes

Clear the history of the factory.

disableHistory()

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()

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()

Accessor to the object’s name.

Returns:

class_name : str

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

getHistory()

Check whether the history mecanism is activated.

Returns:

histMec : a list of WhittleFactoryState

Returns the collection of all the states that have been built for the estimation.

getId()

Accessor to the object’s id.

Returns:

id : int

Internal unique identifier.

getName()

Accessor to the object’s name.

Returns:

name : str

The name of the object.

getShadowedId()

Accessor to the object’s shadowed id.

Returns:

id : int

Internal unique identifier.

getSpectralModelFactory()

Accessor to the spectral factory.

Returns:

initARCoeff : SpectralModelFactory

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

getStartingPoints()

Accessor to the starting points for the optimization step.

Returns:

startPointsList : a list of Point

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

getVerbose()

Accessor to the verbose mode.

Returns:

verboseMode : bool

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

getVisibility()

Accessor to the object’s visibility state.

Returns:

visible : bool

Visibility flag.

hasName()

Test if the object is named.

Returns:

hasName : bool

True if the name is not empty.

hasVisibleName()

Test if the object has a distinguishable name.

Returns:

hasVisibleName : bool

True if the name is not empty and not the default one.

isHistoryEnabled()

Check whether the history mecanism is activated.

Returns:

histMec : bool

Check whether the history mecanism is activated.

By default, the history mecanism is activated.

setName(name)

Accessor to the object’s name.

Parameters:

name : str

The name of the object.

setShadowedId(id)

Accessor to the object’s shadowed id.

Parameters:

id : int

Internal unique identifier.

setSpectralModelFactory(factory)

Accessor to the spectral factory.

Parameters:

spectralModelFact : SpectralModelFactory

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

setStartingPoints(startingPoints)

Accessor to the starting points for the optimization step.

Parameters:

startPointsList : a list of Point

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

setVerbose(verbose)

Accessor to the verbose mode.

Parameters:

verboseMode : bool

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

setVisibility(visible)

Accessor to the object’s visibility state.

Parameters:

visible : bool

Visibility flag.