/usr/include/openturns/swig/RankMCovarianceModel_doc.i is in libopenturns-dev 1.9-5.
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 | %feature("docstring") OT::RankMCovarianceModel
"Covariance function of finite rank.
Available constructors:
RankMCovarianceModel(*spatialDimension*)
RankMCovarianceModel(*variance, basis*)
RankMCovarianceModel(*covariance, basis*)
Parameters
----------
variance : sequence of float
The marginal variances of the coefficients of the basis.
covariance : :class:`~openturns.CovarianceMatrix`
The covariance of the coefficients of the basis.
Notes
-----
Let :math:`X` be a stochastic process defined by:
.. math::
X(\\\\omega,\\\\vect{t})=\\\\sum_{i=1}^M\\\\xi_i(\\\\omega)\\\\phi_i(\\\\vect{t}), \\\\quad \\\\forall \\\\omega \\\\in \\\\Omega, \\\\vect{t} \\\\in \\\\cD
where :math:`(\\\\xi_1,\\\\dots,\\\\xi_M)` is a random vector of dimension :math:`M` and :math:`(\\\\phi_i)_{i=1,\\\\dots,M}` are the :math:`M` first elements of a given basis.
Its covariance function, a *rank-M* covariance function, is given by:
.. math::
C(\\\\vect{s}, \\\\vect{t}) = \\\\sum_{i=1}^M \\\\sum_{j=1}^M \\\\Sigma_{ij}\\\\phi_i(\\\\vect{s})\\\\Tr{\\\\phi_j(\\\\vect{t})}, \\\\quad \\\\forall (\\\\vect{s}, \\\\vect{t}) \\\\in \\\\cD
where :math:`\\\\mat{\\\\Sigma}` is the covariance matrix of :math:`(\\\\xi_1,\\\\dots,\\\\xi_M)`.
When :math:`\\\\mat{\\\\Sigma}` is diagonal, it reduces to:
.. math::
C(\\\\vect{s}, \\\\vect{t}) = \\\\sum_{i=1}^M \\\\sigma_i^2\\\\phi_i(\\\\vect{s})\\\\Tr{\\\\phi_i(\\\\vect{t})}, \\\\quad \\\\forall (\\\\vect{s}, \\\\vect{t}) \\\\in \\\\cD
where :math:`\\\\sigma_i^2` is the variance of :math:`\\\\xi_i`.
The name *rank-M* is here to recall that the discretization of such a model will always lead to a covariance matrix of rank *at most* :math:`M`.
Examples
--------
>>> import openturns as ot
>>> variance = [1.0, 2.0]
>>> basis = ot.LinearBasisFactory().build()
>>> myCovarianceModel = ot.RankMCovarianceModel(variance, basis)
>>> covariance = ot.CovarianceMatrix(2, [1.0, 0.5, 0.5, 2.0])
>>> myCovarianceModel = ot.RankMCovarianceModel(covariance, basis)
"
// ---------------------------------------------------------------------
%feature("docstring") OT::RankMCovarianceModel::getCovariance
"Covariance accessor.
Returns
-------
covariance : :class:`~openturns.CovarianceMatrix`
Covariance matrix of :math:`(\\\\xi_1,\\\\dots,\\\\xi_M)`. Its dimension is zero if the coefficients are uncorrelated, in which case the marginal variances are given by *getVariance()*.
"
// ---------------------------------------------------------------------
%feature("docstring") OT::RankMCovarianceModel::getVariance
"Variance accessor.
Returns
-------
variance : :class:`~openturns.Point`
Vector of marginal variances of :math:`(\\\\xi_1,\\\\dots,\\\\xi_M)`. Its dimension is zero if the coefficients are correlated, in which case the covariance matrix is given by *getCovariance()*.
"
// ---------------------------------------------------------------------
%feature("docstring") OT::RankMCovarianceModel::getBasis
"Basis accessor.
Returns
-------
basis : :class:`~openturns.Basis`
Basis to which the functions :math:`(\\\\phi_i)_{i=1,\\\\dots,M}` belong.
"
// ---------------------------------------------------------------------
%feature("docstring") OT::RankMCovarianceModel::getFunctions
"Function collection accessor.
Returns
-------
functions : :class:`~openturns.FunctionCollection`
The collection of functions :math:`(\\\\phi_i)_{i=1,\\\\dots,M}` defining the covariance model.
"
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