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So I recently learned about Polynomial Chaos Expansion (aka PCE), and it seemed to me that its purpose was to propagate uncertainty from the inputs to the outputs more efficiently (via closed-form moments). But I am no longer sure that makes sense (only considering a non-intrusive case)...

Given: $Y=\eta(X)$ s.t. $\eta(.)$ is the Physical model. We have PCE: $$Y\approx \tilde Y = \sum^p_{j=0}y_j\psi_j(\Xi) \approx \eta (X \approx \tilde X = \sum^p_{i=0}x_i\psi_i(\Xi))$$

& the moments are:

  • $E[\tilde Y]=y_0$
  • $Var(\tilde Y)=\sum^p_{i=0}y_i^2||\psi_i||_2- y_0^2$

But these moments are constants once a PCE is fit! And the non-intrusive fitting process requires Monte-Carlo sampling-based uncertainty propagation through $\eta(.)$ anyway. So why not just compute the moments directly from the sample attained from Monte-Carlo & fit some other kind of surrogate? What is the advantage of the closed-form moments in the non-intrusive case?

P.S. Non-intrusive is most commonly used in practice...

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    $\begingroup$ Thanks for posting here, it is certainly on-topic. But you might be interested as well in a handful of "polynomial chaos" posts at the statistics SE Community, Cross Validated. In particular I found this related to your Question, How to explain simply that the set of runs for Non Intrusive Polynomial Chaos cannot be used as a Monte Carlo sample. $\endgroup$
    – hardmath ♦
    Commented Jan 14 at 16:16
  • $\begingroup$ @hardmath Thanks for the link! It appears relevant. However I disagree with the poster's assumption that it only makes sense to take sample moments with random samples. I would argue that the entire point in space filling sampling methods is to be more efficient than monte-carlo. So for any decent space-filling method I'd guess it's reasonable to assume Monte-Carlo sampling as a lower-bound on it's efficiency. $\endgroup$
    – profPlum
    Commented Jan 15 at 19:27
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    $\begingroup$ @profPlum using quasi-Monte Carlo samples drawn from a uniform box produces biased estimates. They need to be at least weighted by the probability density of the inputs. If the probability distribution of the inputs is uniform, then using quasi-Monte Carlo sampling is a good strategy to uncover regions where $f$ is badly behaved. $\endgroup$
    – Brian Borchers
    Commented Jan 15 at 21:24
  • $\begingroup$ @BrianBorchers Ok well do you really expect PCE's closed form moments to be unbiased if fit on a sample with biased sample moments? $\endgroup$
    – profPlum
    Commented Jan 16 at 17:02
  • $\begingroup$ Also technically he didn't say space-filling. He said "The design matrix is completely determined once you specify the input distribution, thus it's not a Monte Carlo ensemble, as you can see from its very regular structure." $\endgroup$
    – profPlum
    Commented Jan 16 at 17:04

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