Hi @YuanXi_Wu,

Very interesting questions! In PCE, we usually want to use polynomials that are orthonormal with respect to the joint distribution of the input variables. Then we get convergence results (Ernst et al 2012) and, if we do regression-based PCE, the regression matrix has the favorable property of having in expectation orthogonal columns. This helps in computing the coefficients (see e.g. Alemazkoor and Meidani (2018) for a nice overview of relevant literature). Also, mean and variance of the output can be computed from the PCE coefficients using the well-known formulas.

If we just neglect the dependence, why can we still obtain good approximations?

We can also view PCE as “just” performing polynomial regression, where we usually use some fancy polynomial families. In principle, it would even work if we simply used monomials (1, x, x^2, …), just that then the regression matrix can become very ill-conditioned and we might run into numerical issues when trying to compute the coefficients.

**Heuristic explanation why neglecting the dependence might work well:** The idea with “ignoring the dependence and building arbitrary PCE w.r.t. each variable” is that this specific independent distribution always contains the support of the dependent distribution and, if the dependence is not too strong, might actually be quite close to the dependent distribution. In that sense, we don’t lose too much orthogonality by using the “wrong” polynomials. This approach is called *aPCEonX* by Torre et al. (2018) and *dominating* by Jakeman et al. (2019).

**Heuristic explanation of why considering the dependence might not work that well:** On the other hand, when the basis is constructed to be orthonormal w.r.t. the dependent distribution, some difficulties might arise. There could be numerical issues e.g. from Gram-Schmidt orthogonalization. Furthermore, the orthogonalized polynomials don’t have tensor-product structure anymore, but they mix different input variables, which can be unphysical and can lead to less sparse solutions, which then require more data points to be computed accurately.

Whether the dominating approach works well or not also depends on the case study. P. Schnabel recently wrote a report summarizing and comparing several methods for building dependent PCE (available here), it could be of interest to you.

And if so, why should many papers today focus on how to take the dependence into consideration when building PCE?

I would say, because it’s an interesting open problem, and it has the potential of improving the resulting PCE (if the numerical issues during construction of the basis do not overshadow the improvements due to orthogonality). But for sure, more comparisons should be performed to find out whether (and in which cases) there is a benefit over the dominating approach.

If you know of other papers about PCE for dependent inputs than the ones mentioned in P. Schnabel’s report, let us know!