/usr/include/openturns/swig/LeastSquaresStrategy_doc.i is in libopenturns-dev 1.7-3.
This file is owned by root:root, with mode 0o644.
The actual contents of the file can be viewed below.
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 | %feature("docstring") OT::LeastSquaresStrategy
"Least squares strategy for the approximation coefficients.
Available constructors:
LeastSquaresStrategy(*weightedExp*)
LeastSquaresStrategy(*weightedExp, approxAlgoImpFact*)
LeastSquaresStrategy(*measure, approxAlgoImpFact*)
LeastSquaresStrategy(*measure, weightedExp, approxAlgoImpFact*)
LeastSquaresStrategy(*inputSample, outputSample, approxAlgoImpFact*)
LeastSquaresStrategy(*inputSample, weights, outputSample, approxAlgoImpFact*)
Parameters
----------
weightedExp : :class:`~openturns.WeightedExperiment`
Experimental design used for the transformed input data. When not precised,
OpenTURNS uses a :class:`~openturns.MonteCarloExperiment`.
approxAlgoImpFact : ApproximationAlgorithmImplementationFactory
The factory that builds the desired :class:`~openturns.ApproximationAlgorithm`.
When not precised, OpenTURNS uses the
:class:`~openturns.PenalizedLeastSquaresAlgorithmFactory`.
measure : :class:`~openturns.Distribution`
Distribution :math:`\\\\mu` with respect to which the basis is orthonormal.
When not precised, OpenTURNS uses the limit measure defined within the
:class:`~openturns.WeightedExperiment`.
inputSample, outputSample : 2-d sequence of float
The input random variables :math:`\\\\vect{X}=(X_1, \\\\dots, X_{n_X})^T`
and the output samples :math:`\\\\vect{Y}` that describe the model.
weights : sequence of float
Numerical point that are the weights associated to the input sample points
such that the corresponding weighted experiment is a good approximation of
:math:`\\\\mu`. If not precised, all weights are equals to
:math:`\\\\omega_i = \\\\frac{1}{size}`, where :math:`size` is the size of the
sample.
See also
--------
FunctionalChaosAlgorithm, ProjectionStrategy, IntegrationStrategy
Notes
-----
This class is not usable because it has sense only within the
:class:`~openturns.FunctionalChaosAlgorithm` : the least squares strategy
evaluates the coefficients :math:`(a_k)_{k \\\\in K}` of the polynomials
decomposition 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]
where :math:`\\\\vect{U} = T(\\\\vect{X})`.
The mean expectation :math:`E_{\\\\mu}` is approximated by a relation of type:
.. 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)` defined as:
.. math::
f(\\\\vect{U} = \\\\left( g \\\\circ T^{-1} (\\\\vect{U}) - \\\\vect{b}^{\\\\intercal}
\\\\vect{\\\\Psi}(\\\\vect{U}) \\\\right)^2
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 methods implemented in the :class:`~openturns.WeightedExperiment`."
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