document: Generalized polynomial chaos expansions. One approach to model densities with stochastically dependent components numerically, is to reformulate the uncertainty problem as a set of independent components through generalised polynomial chaos expansion [34]. As described in detail in Section 3.1, a Rosenblatt transformation allows for the mapping between any domain and the unit hypercube [0, 1]D. With a double transformation we can reformulate the response function f asf(x,t,Q)=f(x,t,TQ−1(TR(R)))≈fˆ(x,t,R)=∑n∈INcn(x,t)Φn(R),where R is any random variable drawn from pR, which for simplicity is chosen to consists of independent components. Also, {Φn}n∈IN is constructed to be orthogonal with respect to LR, not LQ. In any case, R is either selected from the Askey-Wilson scheme, or calculated using the discretized Stieltjes procedure. We remark that the accuracy of the approximation deteriorate if the transformation composition TQ−1∘TR is not smooth [34]. Dakota, Turns, and Chaospy all support generalized polynomial chaos expansions for independent stochastic variables and the Normal/Nataf copula listed in Table 2. Since Chaospy has the Rosenblatt transformation underlying the computational framework, generalized polynomial chaos expansions are in fact available for all densities.
extractive_keyphrases: computational framework discretized stieltjes procedure generalised polynomial chaos expansion generalized polynomial chaos expansions model densities rosenblatt transformation uncertainty problem
abstractive_keyphrases: askey-wilson scheme