Explicit expression of the PCE surrogate

UQ with UQLab Community Q&A and How To
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Hi @WindEng,

Thank you for asking here.

  1. If you want to evaluate a PCE model, please use uq_evalModel (see the model manual for more details).

  2. The standard space for uniform distributed variables is [-1,1]. More precisely, a PCE model for X\sim\mathcal{U}(a,b) is evaluated by

    \mathcal{M}^{\rm PCE}(x) = \sum_{i} c_i \Psi_i\left( \frac{2x - b - a}{b-a}\right).

    where \frac{2x - b - a}{b-a} is an affine transformation that maps x\in[a,b] to [-1,1], \Psi_i's are standard Legendre polynomials defined on [-1,1], and c_i's are the associated coefficients.

  3. The built-in function polyval of Matlab uses monomials (i.e. 1,x,x^2,x^3,\ldots) as a basis, which are not orthogonal polynomials (please refer to the the input manual for a more detailed presentation of orthogonal polynomials). Consequently, if you want to keep using polyval, it is important to represent orthogonal polynomials by monomials.

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