/usr/include/openturns/swig/ProjectionStrategyImplementation_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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 | %define OT_ProjectionStrategy_doc
"Base class for the evaluation strategies of the approximation coefficients.
Available constructors:
ProjectionStrategy(*projectionStrategy*)
Parameters
----------
projectionStrategy : :class:`~openturns.ProjectionStrategy`
A projection strategy which is a :class:`~openturns.LeastSquaresStrategy` or
an :class:`~openturns.IntegrationStrategy`.
See also
--------
FunctionalChaosAlgorithm, LeastSquaresStrategy, IntegrationStrategy
Notes
-----
Consider :math:`\\\\vect{Y} = g(\\\\vect{X})` with :math:`g: \\\\Rset^d \\\\rightarrow \\\\Rset^p`,
:math:`\\\\vect{X} \\\\sim \\\\cL_{\\\\vect{X}}` and :math:`\\\\vect{Y}` with finite variance:
:math:`g\\\\in L_{\\\\cL_{\\\\vect{X}}}^2(\\\\Rset^d, \\\\Rset^p)`.
The functional chaos expansion approximates :math:`\\\\vect{Y}` using an isoprobabilistic
transformation *T* and an orthonormal multivariate basis :math:`(\\\\Psi_k)_{k \\\\in \\\\Nset}`
of :math:`L^2_{\\\\mu}(\\\\Rset^d,\\\\Rset)`. See :class:`~openturns.FunctionalChaosAlgorithm`
to get more details.
The meta model of :math:`g`, based on the functional chaos decomposition of
:math:`f = g \\\\circ T^{-1}` writes:
.. math::
\\\\tilde{g} = \\\\sum_{k \\\\in K} \\\\vect{\\\\alpha}_k \\\\Psi_k \\\\circ T
where *K* is a non empty finite set of indices, whose cardinality is denoted by *P*.
We detail the case where :math:`p=1`.
The vector :math:`\\\\vect{\\\\alpha} = (\\\\alpha_k)_{k \\\\in K}` is equivalently defined by:
.. math::
:label: defArgMin
\\\\vect{\\\\alpha} = \\\\argmin_{\\\\vect{\\\\alpha} \\\\in \\\\Rset^K} \\\\Expect{ \\\\left( g \\\\circ T^{-1}(\\\\vect{Z}) - \\\\sum_{k \\\\in K} \\\\alpha_k \\\\Psi_k (\\\\vect{Z})\\\\right)^2 }
and:
.. math::
:label: defEsp
\\\\alpha_k = <g \\\\circ T^{-1}(\\\\vect{Z}), \\\\Psi_k (\\\\vect{Z})>_{\\\\mu} = \\\\Expect{ g \\\\circ T^{-1}(\\\\vect{Z}) \\\\Psi_k (\\\\vect{Z}) }
where :math:`\\\\vect{Z} = T(\\\\vect{X})` and the mean :math:`\\\\Expect{.}` is evaluated with respect to the measure :math:`\\\\mu`.
It corresponds to two points of view:
- relation :eq:`defArgMin` means that the coefficients
:math:`(\\\\alpha_k)_{k \\\\in K}` minimize the quadratic error between the model and
the polynomial approximation. Use :class:`~openturns.LeastSquaresStrategy`.
- relation :eq:`defEsp` means that :math:`\\\\alpha_k` is the scalar product of the
model with the *k-th* element of the orthonormal basis :math:`(\\\\Psi_k)_{k \\\\in \\\\Nset}`.
Use :class:`~openturns.IntegrationStrategy`.
In both cases, the mean :math:`\\\\Expect{.}` is approximated by a linear quadrature formula:
.. math::
:label: approxEsp
\\\\Expect{ f(\\\\vect{Z})} \\\\simeq \\\\sum_{i \\\\in I} \\\\omega_i f(\\\\Xi_i)
where *f* is a function in :math:`L^1(\\\\mu)`.
In the approximation :eq:`approxEsp`, the set *I*, the points :math:`(\\\\Xi_i)_{i \\\\in I}`
and the weights :math:`(\\\\omega_i)_{i \\\\in I}` are evaluated from different methods
implemented in OpenTURNS in the :class:`~openturns.WeightedExperiment`.
The convergence criterion used to evaluate the coefficients is based on the residual value
defined in the :class:`~openturns.FunctionalChaosAlgorithm`."
%enddef
%feature("docstring") OT::ProjectionStrategyImplementation
OT_ProjectionStrategy_doc
// ---------------------------------------------------------------------
%define OT_ProjectionStrategy_getCoefficients_doc
"Accessor to the coefficients.
Returns
-------
coef : :class:`~openturns.Point`
Coefficients :math:`(\\\\alpha_k)_{k \\\\in K}`."
%enddef
%feature("docstring") OT::ProjectionStrategyImplementation::getCoefficients
OT_ProjectionStrategy_getCoefficients_doc
// ---------------------------------------------------------------------
%define OT_ProjectionStrategy_getExperiment_doc
"Accessor to the experiments.
Returns
-------
exp : :class:`~openturns.WeightedExperiment`
Weighted experiment used to evaluate the coefficients."
%enddef
%feature("docstring") OT::ProjectionStrategyImplementation::getExperiment
OT_ProjectionStrategy_getExperiment_doc
// ---------------------------------------------------------------------
%define OT_ProjectionStrategy_getInputSample_doc
"Accessor to the input sample.
Returns
-------
X : :class:`~openturns.Sample`
Input Sample."
%enddef
%feature("docstring") OT::ProjectionStrategyImplementation::getInputSample
OT_ProjectionStrategy_getInputSample_doc
// ---------------------------------------------------------------------
%define OT_ProjectionStrategy_getMeasure_doc
"Accessor to the measure.
Returns
-------
mu : Distribution
Measure :math:`\\\\mu` defining the scalar product."
%enddef
%feature("docstring") OT::ProjectionStrategyImplementation::getMeasure
OT_ProjectionStrategy_getMeasure_doc
// ---------------------------------------------------------------------
%define OT_ProjectionStrategy_getOutputSample_doc
"Accessor to the output sample.
Returns
-------
Y : :class:`~openturns.Sample`
Output Sample."
%enddef
%feature("docstring") OT::ProjectionStrategyImplementation::getOutputSample
OT_ProjectionStrategy_getOutputSample_doc
// ---------------------------------------------------------------------
%define OT_ProjectionStrategy_getRelativeError_doc
"Accessor to the relative error.
Returns
-------
e : float
Relative error."
%enddef
%feature("docstring") OT::ProjectionStrategyImplementation::getRelativeError
OT_ProjectionStrategy_getRelativeError_doc
// ---------------------------------------------------------------------
%define OT_ProjectionStrategy_getResidual_doc
"Accessor to the residual.
Returns
-------
er : float
Residual error."
%enddef
%feature("docstring") OT::ProjectionStrategyImplementation::getResidual
OT_ProjectionStrategy_getResidual_doc
// ---------------------------------------------------------------------
%define OT_ProjectionStrategy_getWeights_doc
"Accessor to the weights.
Returns
-------
w : :class:`~openturns.Point`
Weights of the design of experiments."
%enddef
%feature("docstring") OT::ProjectionStrategyImplementation::getWeights
OT_ProjectionStrategy_getWeights_doc
// ---------------------------------------------------------------------
%define OT_ProjectionStrategy_setExperiment_doc
"Accessor to the design of experiment.
Parameters
----------
exp : :class:`~openturns.WeightedExperiment`
Weighted design of experiment."
%enddef
%feature("docstring") OT::ProjectionStrategyImplementation::setExperiment
OT_ProjectionStrategy_setExperiment_doc
// ---------------------------------------------------------------------
%define OT_ProjectionStrategy_setMeasure_doc
"Accessor to the measure.
Parameters
----------
m : Distribution
Measure :math:`\\\\mu` defining the scalar product."
%enddef
%feature("docstring") OT::ProjectionStrategyImplementation::setMeasure
OT_ProjectionStrategy_setMeasure_doc
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