Hi Lisa,
Nice to see you back on the list. I will first give you some insight on the
role of
FunctionalChaosAlgorithm/FunctionalChaosResult/FunctionalChaosRandomVector, but
first a little summary on functional chaos expansion.
Given a function f:R^n->R^p, a random vector X with distribution D and a basis
function (\Phi_n) orthonormal wrt an auxilliary distribution L, we are looking
for an approximate stochastic representation of Y=f(X) in the form of:
\tilde{Y}=\tilde{g}(Z) where g=\sum_n a_n\Phi_n and Z~L
If (\Phi_n) are multivariate polynomials, it is called a polynomial chaos
expansion. Note that if L is different from D (think about D being Weibull, L
Uniform and \Phi_n Legendre polynomials) then, introducing an iso probabilistic
transformation T such that T(X)~L, then:
\tilde{Y}=\tilde{g}(T(X))=\sum_n a_n\Phi_n(T(X))
which is another approximate representation of Y, this time in terms of X and
not Z, from which we get an approximation of f:
\tilde{f}(x)=\sum_n a_n\Phi_n(T(x))
This time, it is no more a polynomial approximation as T is highly nonlinear
and non-polynomial in general.
With this in mind, it is easier to understand the logic behind the OpenTURNS
objects:
First, the pair FunctionalChaosAlgorithm/FunctionalChaosResult. It is a usual
pattern in OpenTURNS (see OptimizationAlgorithm/OptimizationResult). When a
computation is complex, it is done by an Algorithm class in 3 steps:
+ the creation of the algorithm with the relevant data
+ the computation by itself, using the run() method
+ the extraction of the result, using the getResult() method
The resulting object is a FunctionalChaosResult object, which contains a lot of
information including first the meta-model \tilde{f} (the object you have to
use as an approximation of your initial function) as it is the main result of
the algorithm, and a lot of additional by-products (the elements to build
\tilde{g}, T,...). As you said, \tilde{f} is in general more complex than a
simple polynomial.
The FunctionalChaosRandomVector is the class in charge of the more advanced
post-processing of the result:
+ it allows to sample the output of your initial function efficiently, using
Y=\tilde{g}(Z) as L is fast to sample and \tilde{g} is a polynomial while
\tilde{f} includes a potentially costly transformation T
+ it allows to compute Sobol indices and Sobol total indices of any order.
-> it is this object you have to use in order to compute these indices, and not
the analytical expression of \tilde{f}.
The relevant documentation is here:
http://openturns.github.io/user_manual/response_surface/_generated/openturns.FunctionalChaosRandomVector.html?highlight=functionalchaosrandomvector
So for the meta-model, it is the getMetaModel() method of the
FunctionalChaosResult class, and for the Sobol indices it is the
getSobolIndex() method of the FunctionalChaosRandomVector class.
Best regards,
Régis
>________________________________
> De : Lisa RIVALIN <[email protected]>
>À : users <[email protected]>
>Envoyé le : Lundi 27 juin 2016 18h08
>Objet : [ot-users] Polynomial chaos Results?
>
>
>
>Hello OpenTURNS users,
>
>
>I would like to get a simple form of a degre 1 Polynomial Chaos Expansion as,
>for instance: f(x1, x2, x3) = a x1 +b x2 + c x3 +d
>
>
>It is known that Sobol indices can be calculated from the polynomial chaos.
>
>
>
>When I use ".getMetaModel()", I get a complex composite function and I can't
>connect Sobol indices to my inputs.
>
>
>My goal is to re-use this simple function afterwards.
>
>
>
>
>Could you help me with that?
>
>
>
>
>All the best,
>
>
>Lisa
>
>
>
>
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