ARMALikelihoodFactory¶

class ARMALikelihoodFactory(*args)¶

Maximum likelihood estimator of a multivariate ARMA Gaussian process.

Available constructors:

ARMALikelihoodFactory()

ARMALikelihoodFactory(p, q, d, invertible)

ARMALikelihoodFactory(indP, indQ, d, 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 $d$ .

dint, $d \geq 1$

Dimension of the process.

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 covariance matrix $\mat{\Sigma}_{\varepsilon} = \sigma^2\mat{Q}$ where $|\mat{Q}| = 1$ . It implies that the ARMA process estimated is normal.

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.

If we note $\vect{W} = (\vect{X}_0, \hdots, \vect{X}_{n-1})$ , then $\vect{W}$ is normal with zero mean. Its covariance matrix writes $\mathbb{E}(\vect{W}\Tr{\vect{W}})= \sigma^2 \Sigma_{\vect{W}}$ which depends on the coefficients $(\mat{A}_k, \mat{B}_l)$ for $k = 1,\ldots,p$ and $l = 1,\ldots, q$ and on the matrix $\mat{Q}$ .

The likelihood of $\vect{W}$ writes :

$L(\vect{\beta}, \sigma^2 | \vect{W}) = (2 \pi \sigma^2) ^{-\frac{d n}{2}} |\Sigma_{w}|^{-\frac{1}{2}} \exp\left(- (2\sigma^2)^{-1} \Tr{\vect{W}} \Sigma_{\vect{W}}^{-1} \vect{W} \right)$

where $\vect{\beta} = (\mat{A}_{k}, \mat{B}_{l}, \mat{Q}),\ k = 1,\ldots,p$ , $l = 1,\ldots, q$ and where $|.|$ denotes the determinant.

No evaluation of selection criteria such as AIC or BIC is done.

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, 50)
>>> 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()

Estimate the ARMA process with the maximum likelihood estimator:

>>> myFactory = ot.ARMALikelihoodFactory(4, 2, 1)
>>> myARMA = myFactory.build(ot.TimeSeries(myTimeSeries))

Methods

`build`(self, \*args)	Estimate the ARMA process.
`getClassName`(self)	Accessor to the object’s name.
`getCurrentP`(self)	Accessor to the current P order.
`getCurrentQ`(self)	Accessor to the current Q order.
`getId`(self)	Accessor to the object’s id.
`getInitialARCoefficients`(self)	Accessor to the initial AR coefficients.
`getInitialCovarianceMatrix`(self)	Accessor to the initial covariance matrix of the white noise.
`getInitialMACoefficients`(self)	Accessor to the initial MA coefficients.
`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.
`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.
`setInitialARCoefficients`(self, phi)	Accessor to the initial AR coefficients.
`setInitialConditions`(self, arCoefficients, …)	Accessor to the initial AR coefficients.
`setInitialCovarianceMatrix`(self, …)	Accessor to the initial covariance matrix of the white noise.
`setInitialMACoefficients`(self, theta)	Accessor to the initial MA coefficients.
`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.
`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 maximum likelihood estimator.

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 $d$ .

getId(self)¶

Accessor to the object’s id.

Returns

idint: Internal unique identifier.

getInitialARCoefficients(self)¶

Accessor to the initial AR coefficients.

Returns

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

getInitialCovarianceMatrix(self)¶

Accessor to the initial covariance matrix of the white noise.

Returns

initCovMatCovarianceMatrix: The initial covariance matrix of the white noise used for the optimization algorithm.

getInitialMACoefficients(self)¶

Accessor to the initial MA coefficients.

Returns

initMACoeffARMACoefficients: The initial MA coefficients used for the optimization algorithm.

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.

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.

setInitialARCoefficients(self, phi)¶

Accessor to the initial AR coefficients.

Parameters

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

setInitialConditions(self, arCoefficients, maCoefficients, covarianceMatrix)¶

Accessor to the initial AR coefficients.

Parameters

initARCoeffARMACoefficients: The initial AR coefficients used for the optimization algorithm.
initMACoeffARMACoefficients: The initial AR coefficients used for the optimization algorithm.
initCovMatrCovarianceMatrix: The initial covariance matrix of the white noise used for the optimization algorithm.

setInitialCovarianceMatrix(self, covarianceMatrix)¶

Accessor to the initial covariance matrix of the white noise.

Parameters

initCovMatCovarianceMatrix: The initial covariance matrix of the white noise used for the optimization algorithm.

setInitialMACoefficients(self, theta)¶

Accessor to the initial MA coefficients.

Parameters

initMACoeffARMACoefficients: The initial MA coefficients used for the optimization algorithm.

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.

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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