/usr/include/openturns/swig/ProjectionStrategyImplementation_doc.i is in libopenturns-dev 1.7-3.
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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 | %define OT_ProjectionStrategy_doc
"Base class for the evaluation strategies of the approximation coefficients.
Available constructors:
ProjectionStrategy(*projectionStrategyImplementation*)
Parameters
----------
projectionStrategyImplementation : :class:`~openturns.ProjectionStrategy`
A projection strategy which is a :class:`~openturns.LeastSquaresStrategy` or
an :class:`~openturns.IntegrationStrategy`.
See also
--------
FunctionalChaosAlgorithm, LeastSquaresStrategy, IntegrationStrategy
Notes
-----
The polynomial chaos (PC) expansion allows one to obtain an explicit
representation of the random response :math:`\\\\vect{Y} = g(\\\\vect{X})` of the
model under consideration. More precisely, the response *Y* is cast as a
converging series featuring orthonormal polynomials. For computational purpose,
it is necessary though to retain a finite number of terms by truncating the
expansion using an :class:`~openturns.AdaptiveStrategy`. Then it is necessary to
estimate the PC coefficients in order to characterize *Y* . This may be achieved
using a projection strategy.
The model response is assumed to be scalar for the sake of simplicity, i.e.
:math:`\\\\vect{Y} = Y`. However the following derivations hold componentwise in
case of a vector-valued random response. Let us consider the following truncated
PC representation of the model response:
.. math::
Y \\\\equiv g(\\\\vect{X}) \\\\approx \\\\sum_{k \\\\in K} a_k \\\\Psi_k \\\\circ T(\\\\vect{X})
where *K* is a non empty finite set of indices, whose cardinality is denoted
by *P* and where :math:`\\\\vect{U} = T(\\\\vect{X})`. Using the matrix notation
:math:`\\\\vect{a} = \\\\{a_0, \\\\ldots, a_{P-1} \\\\}` and
:math:`\\\\vect{\\\\Psi}(\\\\vect{U}) = \\\\{\\\\Psi_0 (\\\\vect{U}), \\\\ldots, \\\\Psi_{P-1}(\\\\vect{U})\\\\}^{\\\\intercal}`,
the PC representation rewrites:
.. math::
Y \\\\equiv g \\\\circ T^{-1}(\\\\vect{U}) \\\\approx \\\\vect{a}^{\\\\intercal} \\\\vect{\\\\Psi}(\\\\vect{U})
The coefficients may be computed by a :math:`L^2`-projection onto the PC basis
as follows:
.. math::
\\\\vect{a} = \\\\argmin_{\\\\vect{b} \\\\in \\\\Rset^P} E_{\\\\mu} \\\\left[ \\\\left( g \\\\circ T^{-1}
(\\\\vect{U}) - \\\\vect{b}^{\\\\intercal} \\\\vect{\\\\Psi}(\\\\vect{U}) \\\\right)^2 \\\\right]
This relation means that the coefficients :math:`(a_k)_{k \\\\in K}` minimize the
quadratic error between the model and the polynomial approximation. OpenTURNS
implements it through the :class:`~openturns.LeastSquaresStrategy`.
Due to the orthonormality of the PC basis, it may be shown that the previous
equation rewrites:
.. math::
\\\\vect{a} = E_{\\\\mu} \\\\left[ g \\\\circ T^{-1} (\\\\vect{U}) \\\\vect{\\\\Psi}(\\\\vect{U}) \\\\right]
This relation means that :math:`a_k` is the scalar product of the model with the
:math:`k^{th}` element of the orthonormal basis :math:`(\\\\Psi_k)_{k \\\\in K}`.
OpenTURNS implements it through the :class:`~openturns.IntegrationStrategy`.
In both cases, the mean expectation :math:`E_{\\\\mu}` is approximated by a linear
quadrature formula:
.. math::
E_{\\\\mu} \\\\left[ f(\\\\vect{U}) \\\\right] \\\\approx \\\\sum_{i \\\\in I} \\\\omega_i f(\\\\Xi_i)
where is a function :math:`L_1(\\\\mu)`. In the approximation of the mean expectation, 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 defined in
:class:`~openturns.WeightedExperiment`."
%enddef
%feature("docstring") OT::ProjectionStrategyImplementation
OT_ProjectionStrategy_doc
// ---------------------------------------------------------------------
%define OT_ProjectionStrategy_getCoefficients_doc
"Accessor to the coefficients.
Returns
-------
coef : :class:`~openturns.NumericalPoint`
Coefficients of the basis."
%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."
%enddef
%feature("docstring") OT::ProjectionStrategyImplementation::getExperiment
OT_ProjectionStrategy_getExperiment_doc
// ---------------------------------------------------------------------
%define OT_ProjectionStrategy_getInputSample_doc
"Accessor to the input sample.
Returns
-------
X : :class:`~openturns.NumericalSample`
Input Sample."
%enddef
%feature("docstring") OT::ProjectionStrategyImplementation::getInputSample
OT_ProjectionStrategy_getInputSample_doc
// ---------------------------------------------------------------------
%define OT_ProjectionStrategy_getMeasure_doc
"Accessor to the measure.
Returns
-------
m : Distribution
Measure."
%enddef
%feature("docstring") OT::ProjectionStrategyImplementation::getMeasure
OT_ProjectionStrategy_getMeasure_doc
// ---------------------------------------------------------------------
%define OT_ProjectionStrategy_getOutputSample_doc
"Accessor to the output sample.
Returns
-------
Y : :class:`~openturns.NumericalSample`
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."
%enddef
%feature("docstring") OT::ProjectionStrategyImplementation::getResidual
OT_ProjectionStrategy_getResidual_doc
// ---------------------------------------------------------------------
%define OT_ProjectionStrategy_getWeights_doc
"Accessor to the weights.
Returns
-------
w : :class:`~openturns.NumericalPoint`
Weights."
%enddef
%feature("docstring") OT::ProjectionStrategyImplementation::getWeights
OT_ProjectionStrategy_getWeights_doc
// ---------------------------------------------------------------------
%define OT_ProjectionStrategy_setExperiment_doc
"Accessor to the experiments.
Parameters
----------
exp : :class:`~openturns.WeightedExperiment`
Weighted experiment."
%enddef
%feature("docstring") OT::ProjectionStrategyImplementation::setExperiment
OT_ProjectionStrategy_setExperiment_doc
// ---------------------------------------------------------------------
%define OT_ProjectionStrategy_setMeasure_doc
"Accessor to the measure.
Parameters
----------
m : Distribution
Measure."
%enddef
%feature("docstring") OT::ProjectionStrategyImplementation::setMeasure
OT_ProjectionStrategy_setMeasure_doc
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