libopenturns-dev 1.5-7build2 / usr / include / openturns / swig / CovarianceModelImplementation_doc.i

%define OT_CovarianceModel_doc
"Covariance 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, we note:

- :math:`m : \\\\cD \\\\mapsto  \\\\Rset^d` its *mean function*, defined by
  :math:`m(\\\\vect{t})=\\\\Expect{X_{\\\\vect{t}}}`,
- :math:`C : \\\\cD \\\\times \\\\cD \\\\mapsto  \\\\cM_{d \\\\times d}(\\\\Rset)` its
  *covariance function*, defined by
  :math:`C(\\\\vect{s}, \\\\vect{t})=\\\\Expect{(X_{\\\\vect{s}}-m(\\\\vect{s}))(X_{\\\\vect{t}}-m(\\\\vect{t}))^t}`,
- :math:`R : \\\\cD \\\\times \\\\cD \\\\mapsto  \\\\mathcal{M}_{d \\\\times d}(\\\\Rset)` its
  *correlation function*, defined for all :math:`(\\\\vect{s}, \\\\vect{t})`,
  by :math:`R(\\\\vect{s}, \\\\vect{t})` such that for all :math:`(i,j)`,
  :math:`R_{ij}(\\\\vect{s}, \\\\vect{t})=C_{ij}(\\\\vect{s}, \\\\vect{t})/\\\\sqrt{C_{ii}(\\\\vect{s}, \\\\vect{t})C_{jj}(\\\\vect{s}, \\\\vect{t})}`.

A CovarianceModel object can be created only through its derived classes:
:class:`~openturns.StationaryCovarianceModel`,
:class:`~openturns.UserDefinedCovarianceModel`,
:class:`~openturns.GeneralizedExponential`,
:class:`~openturns.AbsoluteExponential`,
:class:`~openturns.SquaredExponential`."
%enddef
%feature("docstring") OT::CovarianceModelImplementation
OT_CovarianceModel_doc

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%define OT_CovarianceModel_computeAsScalar_doc
"Compute the covariance function for 1D model.

Available usages:
    computeAsScalar(s, t)

    computeAsScalar(tau)

Parameters
----------
s, t : float sequences
    Inputs.
tau : float sequence
    Input.

Returns
-------
covariance : float
    Covariance.

Notes
-----
*computeAsScalar* evaluates the covariance model
:math:`C : \\\\cD \\\\times \\\\cD \\\\mapsto  \\\\cM_{d \\\\times d}(\\\\Rset)` at
:math:`(s,t)\\\\in \\\\Rset^n`:

.. math::

    C(\\\\vect{s}, \\\\vect{t})=\\\\Expect{(X_{\\\\vect{s}}-m(\\\\vect{s}))(X_{\\\\vect{t}}-m(\\\\vect{t}))^t}

In the second usage, the covariance model must be stationary. Then we note
:math:`C^{stat}(\\\\vect{\\\\tau})` for :math:`C(\\\\vect{s}, \\\\vect{s}+\\\\vect{\\\\tau})` as
this quantity does not depend on :math:`\\\\vect{s}`."
%enddef
%feature("docstring") OT::CovarianceModelImplementation::computeAsScalar
OT_CovarianceModel_computeAsScalar_doc

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%define OT_CovarianceModel_discretize_doc
"Discretize the covariance function on a given RegularGrid/Mesh.

Parameters
----------
meshOrGrid : :class:`~openturns.Mesh` or :class:`~openturns.RegularGrid`
    Mesh or time grid of size :math:`N` associated with the process.

Returns
-------
covarianceMatrix : :class:`~openturns.CovarianceMatrix`
    Covariance matrix :math:`\\\\in\\\\cM_{nd\\\\times nd}(\\\\Rset)` (if the process is of
    dimension :math:`d`).

Notes
-----
This method makes a discretization of the model on *meshOrGrid* composed of
the vertices :math:`(\\\\vect{t}_1, \\\\dots, \\\\vect{t}_{N-1})` and returns the
covariance matrix:

.. math ::

    \\\\mat{C}_{1,\\\\dots,k} = \\\\left(
        \\\\begin{array}{cccc}
        C(\\\\vect{t}_1, \\\\vect{t}_1) &C(\\\\vect{t}_1, \\\\vect{t}_2) & \\\\dots & C(\\\\vect{t}_1, \\\\vect{t}_{k}) \\\\\\\\
        \\\\dots & C(\\\\vect{t}_2, \\\\vect{t}_2)  & \\\\dots & C(\\\\vect{t}_2, \\\\vect{t}_{k}) \\\\\\\\
        \\\\dots & \\\\dots & \\\\dots & \\\\dots \\\\\\\\
        \\\\dots & \\\\dots & \\\\dots & C(\\\\vect{t}_{k}, \\\\vect{t}_{k})
        \\\\end{array} \\\\right)"
%enddef
%feature("docstring") OT::CovarianceModelImplementation::discretize
OT_CovarianceModel_discretize_doc

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

%define OT_CovarianceModel_discretizeRow_doc
"**(TODO)**"
%enddef
%feature("docstring") OT::CovarianceModelImplementation::discretizeRow
OT_CovarianceModel_discretizeRow_doc

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%define OT_CovarianceModel_getDimension_doc
"Get the dimension of the *CovarianceModel*.

Returns
-------
dimension : int
    Dimension of the *CovarianceModel*."
%enddef
%feature("docstring") OT::CovarianceModelImplementation::getDimension
OT_CovarianceModel_getDimension_doc

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

%define OT_CovarianceModel_getParameters_doc
"Get the parameters of the covariance function.

Returns
-------
parameters : float sequence with description
    Parameters of the covariance function."
%enddef
%feature("docstring") OT::CovarianceModelImplementation::getParameters
OT_CovarianceModel_getParameters_doc

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

%define OT_CovarianceModel_isStationary_doc
"Test whether the model is stationary or not.

Returns
-------
isStationary : bool
    *True* if the model is stationary.

Notes
-----
The covariance function :math:`C` is stationary when it is invariant by
translation:

.. math::

    \\\\forall(\\\\vect{s},\\\\vect{t},\\\\vect{h}) \\\\in \\\\cD, & \\\\, \\\\quad
    C(\\\\vect{s}, \\\\vect{s}+\\\\vect{h}) = C(\\\\vect{t}, \\\\vect{t}+\\\\vect{h})

We note :math:`C^{stat}(\\\\vect{\\\\tau})` for :math:`C(\\\\vect{s}, \\\\vect{s}+\\\\vect{\\\\tau})`."
%enddef
%feature("docstring") OT::CovarianceModelImplementation::isStationary
OT_CovarianceModel_isStationary_doc

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%define OT_CovarianceModel_partialGradient_doc
"Compute the gradient of the covariance function.

Parameters
----------
s, t : float sequences
    Inputs.

Returns
-------
gradient : :class:`~openturns.SymmetricTensor`
    Gradient of the covariance function."
%enddef
%feature("docstring") OT::CovarianceModelImplementation::partialGradient
OT_CovarianceModel_partialGradient_doc

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

%define OT_CovarianceModel_setParameters_doc
"Set the parameters of the covariance function.

Returns
-------
parameters : float sequence with description
    Parameters of the covariance function."
%enddef
%feature("docstring") OT::CovarianceModelImplementation::setParameters
OT_CovarianceModel_setParameters_doc