libopenturns-dev 1.7-3 / usr / include / openturns / swig / SpectralModelImplementation_doc.i

%define OT_SpectralModel_doc
"Spectral density model.

Notes
-----
We consider :math:`X: \\\\Omega \\\\times\\\\cD \\\\mapsto \\\\Rset^d` a multivariate
stochastic process of dimension :math:`d`, where :math:`\\\\omega \\\\in \\\\Omega`
is an event, :math:`\\\\cD` is a domain of :math:`\\\\Rset^n`,
:math:`\\\\vect{t}\\\\in \\\\cD` is a multivariate index and
:math:`X(\\\\omega, \\\\vect{t}) \\\\in \\\\Rset^d`.

We note :math:`X_{\\\\vect{t}}: \\\\Omega \\\\rightarrow \\\\Rset^d` the random variable at
index :math:`\\\\vect{t} \\\\in \\\\cD` defined by
:math:`X_{\\\\vect{t}}(\\\\omega)=X(\\\\omega, \\\\vect{t})` and
:math:`X(\\\\omega): \\\\cD  \\\\mapsto \\\\Rset^d` a realization of the process
:math:`X`, for a given :math:`\\\\omega \\\\in \\\\Omega` defined by
:math:`X(\\\\omega)(\\\\vect{t})=X(\\\\omega, \\\\vect{t})`.

If the process is a second order process, zero-mean and weakly **stationary**, we define its *bilateral spectral density function*
:math:`S : \\\\Rset^n \\\\mapsto  \\\\cH^+_{d}` with:

- :math:`\\\\cH^+(d) \\\\in \\\\cM^+(d)(\\\\Cset)` is the set of *d-dimensional* positive hermitian matrices

Using the *stationary covariance function* :math:`C^{stat}` with :math:`C^{stat} : \\\\cD \\\\mapsto  \\\\cM_{d \\\\times d}(\\\\Rset)` and the Fourier transform, the spectral density writes:


.. math::

   \\\\forall \\\\vect{f} \\\\in \\\\Rset^n, \\\\cS(\\\\vect{f}) = \\\\int_{\\\\Rset^n} \\\\exp\\\\left(-2\\\\imath \\\\pi <\\\\vect{f}, \\\\vect{\\\\tau}>\\\\right) C^{stat}(\\\\vect{\\\\tau}) d\\\\vect{\\\\tau}

A SpectralModel object can be created only through its derived classes:
:class:`~openturns.CauchyModel`"

%enddef
%feature("docstring") OT::SpectralModelImplementation
OT_SpectralModel_doc

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%define OT_SpectralModel_computeStandardRepresentative_doc
"Compute the standard representant of the spectral density function.

Parameters
----------
tau : float
    Frequency value.

Returns
-------
rho : Complex
     Standard representant factor of the spectral density function.

Notes
-----
According to definitions in :class:`~openturns.CovarianceModel`, as the spectral density function is the Fourier transform of the stationary covariance function and using the expression of the last one, the spectral density function writes as a matrix-complex product where the matrix is the constant spatial covariance structure and the complex represents the standard representative:

Thus,

.. math::

  \\\\forall \\\\vect{f} \\\\in \\\\Rset^n, \\\\cS(\\\\vect{f}) = \\\\prod_{k=1}^{n} \\\\vect{\\\\theta}_k \\\\mat{\\\\Sigma} \\\\rho(\\\\vect{f} \\\\odot \\\\vect{\\\\theta})

where :math:`\\\\mat{\\\\Sigma}` is a covariance matrix that explains the covariance structure and
:math:`(\\\\vect{f} \\\\odot \\\\vect{\\\\theta})_k = \\\\vect{f}_k \\\\vect{\\\\theta}_k`"
%enddef
%feature("docstring") OT::SpectralModelImplementation::computeStandardRepresentative
OT_SpectralModel_computeStandardRepresentative_doc

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%define OT_SpectralModel_draw_doc
"Draw a specific component of the spectral density function.

Parameters
----------
rowIndex : int
    i-th row component to draw.
    Default value is 0

columnIndex : int
    j-th column component to draw .
    Default value is 0

module : bool
    Tells if module has to be drawn(True) or if it is
    the argument to be drawn (False).
    Default value is True.

Returns
-------
graph : Graph
     Graphic of the specified component

Notes
-----
The method plots a specific component of the spectral density function over a frequency grid. Thus, this last one should be specified thanks to the `setFrequencyGrid` method."
%enddef
%feature("docstring") OT::SpectralModelImplementation::draw
OT_SpectralModel_draw_doc

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

%define OT_SpectralModel_get_frequency_grid_doc
"Return the RegularGrid associated to the SpectralModel.

Returns
-------
grid : :class:`~openturns.RegularGrid`
    Associated frequency grid of the SpectralModel."

%enddef
%feature("docstring") OT::SpectralModelImplementation::getFrequencyGrid
OT_SpectralModel_get_frequency_grid_doc

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

%define OT_SpectralModel_getAmplitude_doc
"Get the amplitude parameter of the spectral density function.

Returns
-------
amplitude : :class:`~openturns.NumericalPoint`
    The used amplitude parameter."
%enddef
%feature("docstring") OT::SpectralModelImplementation::getAmplitude
OT_SpectralModel_getAmplitude_doc

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

%define OT_SpectralModel_getDimension_doc
"Get the dimension of the *SpectralModel*.

Returns
-------
dimension : int
    Dimension of the *SpectralModel*."
%enddef
%feature("docstring") OT::SpectralModelImplementation::getDimension
OT_SpectralModel_getDimension_doc

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

%define OT_SpectralModel_getScale_doc
"Get the scale parameter of the spectral density function.

Returns
-------
scale : :class:`~openturns.NumericalPoint`
    The used scale parameter."
%enddef
%feature("docstring") OT::SpectralModelImplementation::getScale
OT_SpectralModel_getScale_doc

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

%define OT_SpectralModel_getSpatialCorrelation_doc
"Get the spatial correlation matrix of the spectral density function.

Returns
-------
spatialCorrelation : :class:`~openturns.CorrelationMatrix`
    Correlation matrix :math:`\\\\mat{R} \\\\in \\\\mathcal{M}_{dimension \\\\times dimension}([-1, 1])`."
%enddef
%feature("docstring") OT::SpectralModelImplementation::getSpatialCorrelation
OT_SpectralModel_getSpatialCorrelation_doc

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

%define OT_SpectralModel_get_spatial_dimension_doc
"Get the spatial dimension of the spectral density function.

Returns
-------
spatialDimension : int
    SpatialDimension of the *SpectralModel*."

%enddef
%feature("docstring") OT::SpectralModelImplementation::getSpatialDimension
OT_SpectralModel_get_spatial_dimension_doc

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

%define OT_SpectralModel_setAmplitude_doc
"Set the amplitude parameter of the spectral density function.

Parameters
----------
amplitude : :class:`~openturns.NumericalPoint`
    The amplitude parameter to be used in the spectral density function."
%enddef
%feature("docstring") OT::SpectralModelImplementation::setAmplitude
OT_SpectralModel_setAmplitude_doc

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

%define OT_SpectralModel_set_frequency_grid_doc
"Set the RegularGrid associated to the SpectralModel.

Parameters
----------
grid : :class:`~openturns.RegularGrid`
    Associated frequency grid of the SpectralModel."

%enddef
%feature("docstring") OT::SpectralModelImplementation::setFrequencyGrid
OT_SpectralModel_set_frequency_grid_doc

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


%define OT_SpectralModel_setScale_doc
"Set the scale parameter of the spectral density function.

Parameters
----------
scale : :class:`~openturns.NumericalPoint`
    The scale parameter to be used in the spectral density function.
    It should be of size dimension."
%enddef
%feature("docstring") OT::SpectralModelImplementation::setScale
OT_SpectralModel_setScale_doc

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

%define OT_SpectralModel_operator_doc
"Evaluate the spectral density function for a specific frequency.

Parameters
----------
f : float
    Frequency value.

Returns
-------
spd : HermitianMatrixs
    The evaluation of spectral density function at frequency f."

%enddef
%feature("docstring") OT::SpectralModelImplementation::operator()
OT_SpectralModel_operator_doc