Multivariate Orthonormal basis

In that file, we detail the construction of an orthonormal basis with respect to a multivariate measure denoted by \mu, addressing both the independent and dependent marginal settings.

In the functional expansion setting (refer to Functional Chaos Expansion), the distribution \mu may be the input distribution \mu_{\inputRV}, the distribution \mu_{\vect{U}} which is the push-forward distribution of \mu_{\inputRV} through the isoprobabilistic transformation T, or the instrumental distribution \tilde{p} of the domination method.

The set of functions (\psi_k)_{k \geq 0} with \psi_k : \Rset^{\inputDim} \rightarrow \Rset is orthonormal with respect to \mu if:

(1)\scalarproduct{\psi_k}{\psi_{\ell}}_{L^2\left(\mu\right)} = \delta_{k,\ell}

for any k, \ell \geq 0 where \delta_{k, \ell} is the Kronecker symbol:

\delta_{k, \ell} = \begin{cases} 1 & \textrm{ if } k = \ell, \\ 0 & \textrm{otherwise.} \end{cases}

Case of independent marginals: Tensorized univariate basis

We assume here that \mu is a multivariate distribution with independent marginals denoted by \mu_i:

\mu = \otimes_{i=0}^{\inputDim-1} \mu_i

In that case, a multivariate basis orthonormal with respect to \mu can be built as the tensorization of univariate basis orthonormal with respect to its marginals.

For any 0 \leq i \leq \inputDim-1, let \left(\varphi_k^{(i)}\right)_{k \geq 0} be the univariate basis orthonormal with respect to the marginal distribution \mu_i. The tensorized multivariate basis is defined by:

\Psi_\vect{\alpha}(\vect{x}) = \prod_{i=0}^{\inputDim-1} \varphi_{\alpha_i}^{(i)}(x_i)

where \vect{\alpha} = (\alpha_0, \dots, \alpha_{\inputDim-1}) \in \Nset^{\inputDim} is a multi-index that enables to define each element of the multivariate basis from the elements of the univariate marginal basis (see Tensorized multivariate basis enumeration functions and the class EnumerateFunction to get details on that bijection).

The univariate bases may be:

  • polynomials: the associated distribution \mu_i can be continuous or discrete. The orthonormal polynomial basis is represented by its three-term recurrence and the reverse Clenshaw algorithm enables fast, stable evaluation of the polynomials at any point (see OrthogonalUniVariatePolynomial). For some distributions, the orthonormal polynomial basis is known (see Univariate orthonormal polynomials). For all the other arbitrary distributions, the three-term recurrence is computed (see AdaptiveStieltjesAlgorithm) and used to build its orthonormal polynomial family (see StandardDistributionPolynomialFactory).

  • Haar wavelets: the Haar wavelets basis is orthonormal with respect to the \cU(0,1) measure. This basis is used to approximate functions with discontinuities. For details on this basis, see HaarWaveletFactory.

  • Fourier series: the Fourier series is orthonormal with respect to the \cU(-\pi, \pi) measure. For more details on this basis, see FourierSeriesFactory.

Case of dependent marginals

When \mu is a multivariate distribution with dependent marginals, several methods enable to build a basis orthonormal with respect to \mu.

One method is to use an isoprobabilistic transformation denoted by T: \Rset^{\inputDim} \rightarrow \Rset^{\inputDim} that maps the measure \mu into a measure \tilde{\mu} with independent marginals. Therefore, if (\varphi_k)_k is a basis orthonormal with respect to \tilde{\mu}, then (\varphi_k \circ T)_k is a basis orthonormal with respect to \mu. It is important to note that if (\varphi_k)_k is a polynomial basis, then (\varphi_k \circ T)_k is not a polynomial basis as soon as T is not affine.

Another method consists in the following functional basis, which was first introduced by Soize-Ghanem in [soizeghanem2004] defined by:

\psi_{\vect{\alpha}}(\vect{x}) = \dfrac{1}{\sqrt{c(F_0(x_0), \dots, F_{\inputDim-1}(x_{\inputDim-1}))}} \prod_{i=0}^{\inputDim-1} \varphi^{(i)}_{\alpha_{i}}(x_{i})

where F_i is the cumulative distribution function of the i-th marginal of \mu, c the probability density function of its copula and (\varphi^{(i)}_{\alpha_i})_{\alpha_i} the polynomial basis orthonormal with respect to \mu_i.

It is important to note that although (\varphi_{\alpha_i})_{\alpha_i} is a polynomial basis, (\psi_{\vect{\alpha}})_\vect{\alpha} is not a polynomial basis. In addition, (\psi_\vect{0}) \neq 1 if the copula is not the independent copula.

Furthermore, this basis generates approximation subspaces of poor quality. Indeed, for most copulas, the density c tends to infinity at \vect{0} or \vect{1} (and possibly at other points). As a consequence, all the basis functions (\psi_{\vect{\alpha}})_\vect{\alpha} vanish at these points and take small values in their neighborhoods. Therefore, the coefficients in the expansion of a function that takes non zero values at these points must necessarily be large. This leads to numerical instabilities, both for the computation of the expansion coefficients and for the evaluation of the metamodel.

That is the reason why the use of such a multivariate orthonormal basis is not recommended.