%feature("docstring") OT::ARMALikelihoodFactory
"Maximum likelihood estimator of a multivariate ARMA Gaussian process.

Available constructors:
    ARMALikelihoodFactory()

    ARMALikelihoodFactory(*p, q, d, invertible*)

    ARMALikelihoodFactory(*indP, indQ, d, invertible*)


Parameters
----------
p : int
    Order of the AR part of the :math:`ARMA(p,q)` process of dimension :math:`d`.
q : int
    Order of the MA part of the :math:`ARMA(p,q)` process of dimension :math:`d`.
d : int, :math:`d \geq 1`
    Dimension of the process.
invertible : bool, optional
    Restrict the estimation to invertible ARMA processes.

    By default: True.
indP : :class:`~openturns.Indices`
    All the :math:`p` orders that will be investigated.
    Care: not yet implemented.
indQ : :class:`~openturns.Indices`
    All the :math:`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 :math:`\mat{\Sigma}_{\varepsilon} = \sigma^2\mat{Q}` where :math:`|\mat{Q}| = 1`. It implies that the ARMA process estimated is normal.

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

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

The likelihood of :math:`\vect{W}` writes :

.. math::

    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  :math:`\vect{\beta} = (\mat{A}_{k}, \mat{B}_{l}, \mat{Q}),\ k = 1,\ldots,p`, :math:`l = 1,\ldots, q` and where  :math:`|.|` 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))"

// ---------------------------------------------------------------------

%feature("docstring") OT::ARMALikelihoodFactory::build
"Estimate the ARMA process.

Available usages:
    build(*myTimeSeries*)

    build(*myProcessSample*)

Parameters
----------
myTimeSeries : :class:`~openturns.TimeSeries`
    One realization of the  process.
myProcessSample : :class:`~openturns.ProcessSample`
    Several realizations of the  process.

Returns
-------
myARMA : :class:`~openturns.ARMA`
    The  process estimated with the maximum likelihood estimator.
"

// ---------------------------------------------------------------------

%feature("docstring") OT::ARMALikelihoodFactory::setVerbose
"Accessor to the verbose mode.

Parameters
----------
verboseMode : bool
    Set the verbose mode while both the exploration of the possible models and the optimization steps.
"

// ---------------------------------------------------------------------

%feature("docstring") OT::ARMALikelihoodFactory::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.
"

// ---------------------------------------------------------------------

%feature("docstring") OT::ARMALikelihoodFactory::getInitialARCoefficients
"Accessor to the initial AR coefficients.

Returns
-------
initARCoeff : :class:`~openturns.ARMACoefficients`
    The initial AR coefficients used for the optimization algorithm.
"

// ---------------------------------------------------------------------

%feature("docstring") OT::ARMALikelihoodFactory::setInitialARCoefficients
"Accessor to the initial AR coefficients.

Parameters
----------
initARCoeff : :class:`~openturns.ARMACoefficients`
    The initial AR coefficients used for the optimization algorithm.
"

// ---------------------------------------------------------------------
%feature("docstring") OT::ARMALikelihoodFactory::getInitialMACoefficients
"Accessor to the initial MA coefficients.

Returns
-------
initMACoeff : :class:`~openturns.ARMACoefficients`
    The initial MA coefficients used for the optimization algorithm.
"

// ---------------------------------------------------------------------

%feature("docstring") OT::ARMALikelihoodFactory::setInitialMACoefficients
"Accessor to the initial MA coefficients.

Parameters
----------
initMACoeff : :class:`~openturns.ARMACoefficients`
    The initial MA coefficients used for the optimization algorithm.
"

// ---------------------------------------------------------------------
%feature("docstring") OT::ARMALikelihoodFactory::getInitialCovarianceMatrix
"Accessor to the initial covariance matrix of the white noise.

Returns
-------
initCovMat : :class:`~openturns.CovarianceMatrix`
    The initial covariance matrix of the white noise used for the optimization algorithm.
"

// ---------------------------------------------------------------------
%feature("docstring") OT::ARMALikelihoodFactory::setInitialCovarianceMatrix
"Accessor to the initial covariance matrix of the white noise.

Parameters
----------
initCovMat : :class:`~openturns.CovarianceMatrix`
    The initial covariance matrix of the white noise used for the optimization algorithm.
"

// ---------------------------------------------------------------------

%feature("docstring") OT::ARMALikelihoodFactory::setInitialConditions
"Accessor to the initial AR coefficients.

Parameters
----------
initARCoeff : :class:`~openturns.ARMACoefficients`
    The initial AR coefficients used for the optimization algorithm.
initMACoeff : :class:`~openturns.ARMACoefficients`
    The initial AR coefficients used for the optimization algorithm.
initCovMatr : :class:`~openturns.CovarianceMatrix`
    The initial covariance matrix of the white noise used for the optimization algorithm.
"