Functional Chaos Expansion¶
Introduction¶
Accounting for the joint probability density function (PDF)
of the input random vector
, one seeks the joint PDF of output random vector
. This may be achieved using
Monte Carlo (MC) simulation (see Monte Carlo simulation). However, the MC
method may require a large number of model evaluations, i.e. a great
computational cost, in order to obtain accurate results.
A possible solution to overcome this problem is to project the model
in a suitable functional space, such as
the Hilbert space
of square-integrable functions with
respect to
.
More precisely, we may consider an expansion of the model onto an orthonormal
basis of
.
As an example of this type of expansion, one can mention expansions by
wavelets, polynomials, etc.
The principles of the building of a functional chaos expansion are described in the sequel.
Model¶
We consider the output random vector:
where is the model,
is the input random vector which distribution is
,
is the input dimension,
is the output dimension.
We assume that
has finite variance i.e.
.
When , the functional chaos algorithm is used on each marginal
of
, using the same multivariate orthonormal basis for
all the marginals.
Thus, the method is detailed here for a scalar output
and
.
Iso-probabilistic transformation¶
Let be an isoprobabilistic transformation
(see Isoprobabilistic transformations) such that
where
is the distribution of the standardized random vector
.
The distribution is called the measure below.
As we will see soon, this distribution defines the scalar product that defines
the orthogonality property of the functional basis.
Let
be the function defined by the equation:
Therefore .
Hilbert space¶
We introduce the scalar product:
for any .
For a continuous random variable, the scalar product is:
For a discrete random variable, the scalar product is:
The associated norm is defined by:
for any .
Based on this scalar product, the functional space
is a Hilbert space.
Orthonormal basis¶
In this section, we introduce an orthonormal basis of the
previous Hilbert space.
Let be
a set of functions.
This set is orthonormal with respect to
if:
(1)¶
for any where
is the Kronecker symbol:
See StandardDistributionPolynomialFactory
for more details on the available
orthonormal bases.
In the library, we choose a basis which is orthonormal
with respect to
, so that the equation (1) is
satisfied.
Furthermore, we require that the first element be:
(2)¶
The orthogonality of the functions imply:
for any non-zero .
The equation (2) implies:
for any .
Functional chaos expansion¶
The functional chaos expansion of h is (see [lemaitre2010] page 39):
where is a set of coefficients.
We cannot compute an infinite set of coefficients: we can only compute a finite
subset of these.
The truncated functional chaos expansion is:
where .
Thus
is represented by a finite subset of coefficients
in a truncated basis
.
A specific choice of can be done using one enumeration rule,
as presented in Chaos basis enumeration strategies.
If the number of coefficients,
, is too large,
this can lead to overfitting.
This may happen e.g. if the total polynomial order we choose is too large.
In order to limit this effect, one method is to select the coefficients which
best predict the output, as presented in Sparse least squares metamodel.
Convergence of the expansion¶
In this section, we introduce the conditions which ensures that the expansion converges to the function.
The orthonormal expansion of any function
converges in norm to
, i.e.:
if and only if the basis is a complete
orthonormal system (see [sullivan2015], page 139, [dahlquist2008],
theorem 4.5.16 page 456 and [rudin1987], section 4.24 page 85).
In this case, the closure of the vector space spanned by the orthogonal
functions is equal to the whole set of square integrable functions with
respect to
:
(3)¶
There are known sufficient conditions which ensure this property.
For example, if the support of is bounded, then
the basis is a complete orthonormal system.
There exists some infinite set of orthonormal polynomials
which are not complete, e.g. those derived from the log-normal distribution
(see [ernst2012]).
In this case, the expansion may not converge to the function.
Nevertheless, even without any guarantee, it
is possible that the meta model built using the basis
may be a good approximation of
.
Define and estimate the coefficients¶
In this section, we review two equivalent methods to define the coefficients of the expansion:
using a least squares problem,
using integration.
Both methods can be introduced and then discretized using a sample.
The vector of coefficients is the solution of the linear least-squares problem:
(4)¶
The equation (4) means that the coefficients
minimize the quadratic error between the model
and the functional approximation.
For more details of the approximation based on least squares, see the
LeastSquaresStrategy
class.
Let us discretize the solution of the linear least squares problem.
Let be the sample size.
Let
be an i.i.d.
sample from the random vector
.
Let
be the standardized input sample.
Let
be the corresponding output sample.
Let
be the
vector of output observations of the model.
Let
be the design matrix,
defined by:
for and
.
Assume that the design matrix has full rank.
The discretized linear least squares problem is:
The solution is:
The choice of basis has a major impact on the conditioning of the
least-squares problem (4).
Indeed, if the basis is
orthonormal, then the design matrix of the least squares problem is
well-conditioned.
The problem can be equivalently solved using the scalar product (see [dahlquist2008] theorem 4.5.13 page 454):
(5)¶
for .
These equations express the coefficients of the orthogonal projection of the
function
onto the vector space spanned by the orthogonal functions
in the basis.
Since the definition of the scalar product is based on an expectation,
this amounts to approximate an integral using a quadrature rule.
The equation (5) means that each coefficient is the
scalar product of the model with the k-th element of the orthonormal basis
.
For more details on the PCE based on quadrature, see the
IntegrationStrategy
class.
Let us discretize the solution of the problem based on the scalar product.
This can be done by considering a quadrature rule that makes it possible
to approximate the integral.
Let be the sample size.
Let
be the nodes of the quadrature rule and let
be the weights.
The quadrature rule is:
for .
Several algorithms are available to compute the coefficients
:
see
IntegrationExpansion
for an algorithm based on quadrature,see
LeastSquaresExpansion
for an algorithm based on the least squares problem,see
FunctionalChaosAlgorithm
for an algorithm that can manage both methods.
The two methods to define the coefficients of the expansion are equivalent:
the solution of the equations (4) and (5)
produce the same coefficients .
This is different when we estimate these coefficients based on a sample.
In this discretized framework, the solution of the two methods can be
different.
It can be shown, however, that the limit of the two estimators are equal when
the sample size tends to infinity (see [lemaitre2010] eq. 3.48 page 66).
Moreover, the two discretized methods are equivalent if the sample points
satisfy an empirical orthogonality condition (see [lemaitre2010] eq. 3.49
page 66).
A step-by-step method¶
Three steps are required in order to create a functional chaos algorithm:
define the multivariate orthonormal basis;
truncate the multivariate orthonormal basis;
evaluate the coefficients.
These steps are presented in more detail below.
Step 1 - Define the multivariate orthonormal basis: the
multivariate orthonornal basis is built
as the tensor product of orthonormal univariate families.
The univariate bases may be:
polynomials: the associated distribution
can be continuous or discrete. Note that it is possible to build the polynomial family orthonormal to any arbitrary univariate distribution
under some conditions. For more details on this basis, see
StandardDistributionPolynomialFactory
;Haar wavelets: they approximate functions with discontinuities. For details on this basis, see
HaarWaveletFactory
;Fourier series: for more details on this basis, see
FourierSeriesFactory
.
Furthermore, the numbering of the multivariate orthonormal basis
is given by an enumerate function
which defines a way to generate the collection of polynomial degrees used
for the univariate polynomials: an enumerate function
represents a bijection
.
See
LinearEnumerateFunction
or
HyperbolicAnisotropicEnumerateFunction
for more details
on this topic.
Step 2 - Truncate the multivariate orthonormal basis: a
strategy must be chosen for the selection of the different terms of the
multivariate basis. The selected terms are gathered in the subset .
For information about the possible strategies, see
FixedStrategy
and CleaningStrategy
.
Step 3 - Evaluate the coefficients: a projectionStrategy must be chosen
for the estimation of the coefficients .
The meta model¶
The meta model of g can be defined using the isoprobabilistic transformation :
(6)¶
More details are available on these topics.
See
StandardDistributionPolynomialFactory
for more details on the available constructions of the truncated multivariate orthogonal basisSee
FunctionalChaosAlgorithm
for more details on the computation of the coefficients.
There are many ways to use the functional chaos expansion. In the next two sections, we present two examples:
using the expansion as a random vector generator,
performing the sensitivity analysis of the expansion.
Using the expansion as a random vector generator¶
The approximation can be used to build an efficient
random generator of
based on the random vector
,
using the equation:
This equation can be used to simulate independent random observations
from the PCE.
This can be done by first simulating independent observations from
the distribution of the standardized random vector ,
then by pushing forward these observations through the expansion.
See the
FunctionalChaosRandomVector
class
for more details on this topic.
Sensitivity analysis¶
Assume that the input random vector has independent marginals and
that the basis is computed using
the tensor product of univariate orthonormal functions.
In that case, the Sobol’ indices can easily be deduced from the coefficients
.
Please see
FunctionalChaosSobolIndices
for more details on this topic.
Polynomial chaos expansion for independent variables¶
The library enables one to build the meta model called polynomial chaos expansion based on an orthonormal basis of polynomials. See Polynomial chaos basis for more details on polynomial chaos expansion.
Other chaos expansions for independent variables¶
While the polynomial chaos expansion is a classical method, the functions
in the basis do not necessarily have to be polynomials: provided the functions
are orthogonal with respect to the measure , most of
the theory still holds.
The library enables one to use the Haar wavelet functions or the Fourier series
as orthonormal basis with respect to each margin
.
The Haar wavelets basis is orthonormal with respect to the the
measure (see
HaarWaveletFactory
) and the Fourier series basis is orthonormal with respect to
the measure (see
FourierSeriesFactory
).
Some functional chaos expansions for dependent variables¶
If the components of the input random vector are not
independent, we can use an iso-probabilistic transformation to map
into
with independent components.
Whatever the dependency in the standardized random vector ,
the following multivariate functions are orthonormal with respect to
:
where is the
-th marginal of
and
is the degree
orthonormal
family of polynomial for the
-th marginal.
If the random vector
has a non-trivial dependency, the
previous functions are not necessarily polynomials.
Notice that:
(7)¶
where is the density of the copula of
.
Link with classical deterministic polynomial approximation¶
In a deterministic setting (i.e. when the input parameters are
considered to be deterministic), it is of common practice to substitute
the model function by a polynomial approximation over its
whole domain of definition. Actually this approach is
equivalent to:
regarding the input parameters as random uniform random variables,
expanding any quantity of interest provided by the model onto a PC expansion made of Legendre polynomials.