Polynomial chaos expansion with vine copula input

UQ with UQLab Community Q&A and How To
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Hi Charol, welcome to UQWorld! :slight_smile:

That’s a very interesting question! When PCE are used, most of the time it is assumed that the input variables are independent. But what to do if they are dependent? There are several options:

  • Ignore that there are dependencies and build a PCE with basis polynomials that are orthonormal wrt. the input marginals.
  • Transform the input into independent variables (isoprobabilistic transform), then build a PCE on these independent variables.
  • Build arbitrary PCE with basis polynomials that are orthonormal wrt. the (dependent) probability density function.

The first two approaches have been compared in the recent paper
Torre, Marelli, Embrechts, Sudret (2019): Data-driven PCE for machine learning regression
under the names aPCEonX (first approach) and lPCEonZ (second approach). They find that the first approach aPCEonX yields better pointwise estimates than lPCEonZ and explain this with the non-linear mapping that is needed to transform the input into independent variables, which can make it difficult to approximate the resulting model by PCE.

For the third approach, there exists an analytical formula for the orthonormal basis polynomials functions, but investigations of Emiliano (@torree) have shown that this approach does not perform well in comparison with the others. I faintly remember that there is another recent publication on PCE for dependent inputs, maybe @torree knows more?

By default, when you use UQLab to construct a PCE for dependent inputs, the second approach is used (transformation to standard marginals and independent copula using uq_GeneralIsopTransform).

Now to your second question

You describe that you compared PCE metamodelling for the Ishigami function with independent and with dependent inputs.

What are the desired results that you expected to see, and what are you seeing instead? Could you maybe provide us with the plots that surprised you, and the copula that you used? It might be that you are observing precisely this issue with the highly non-linear mapping to independent variables that Torre et. al. describe in their paper.

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By the way, in case you haven’t seen it yet, here is a nice post by Emiliano on the importance of using the right dependence structure for your inputs when doing UQ.

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