How to interpret the PCE display?


I have some very basic questions about the uq_display function and its graphical output:

  • What do the quantities plotted on the x-axis (alpha) and y-axis (log(y)) represent?

  • What kind of information is obtained from such a graphical representation and how can it be interpreted?


If there are any scientific papers explaining this aspect, please let me know as well.

I would be grateful for any insight on this topic, thank you.

N.B. The attached image is just taken form one of the uqlab examples (PCE → Calculation strategies)

Dear Achille

the attached figure plots the magnitude of the PCE coefficients as a function of their enumeration, i.e. the a_j's in the equation Y = \displaystyle{\sum_{j=0}^{P-1} a_j \, \Psi_j(\textbf{X})} as a function of j. As the basis of \Psi_j(\textbf{X})'s is orthonormal, it is licit to compare the magnitude of the coefficients.

Moreover, the coefficients of the low-degree polynomials are to the left in this figure: the mean value is a_0 as you can see. What we see in this figure, where the y-axis is in log scale, is that most coefficients have a magnitude 2 to 3 orders smaller than the largest coefficients. Remember that the variance of the response \textrm{Var}\left[ \textbf{X}\right] = \displaystyle{\sum_{j=1}^{P-1} a_j^2}: a coefficient that is 3 orders of magnitude smaller than a_0 contributes to 10^{-6} to this variance, i.e., is completely negligible. This is because of this strong decay in the coefficients magnitude that PC expansions converge rather fast.

The figure also shows that we cannot truncate the series by the largest polynomial degree: we see with the purple points that several non negligible coefficients (> 0.1 a_0) correspond to \Psi_j's of degree larger than 3: we need sparse polynomial chaos expansions to address this efficiently, that is, a sparse solver which directly finds and computes only the coefficients with largest magnitude, see e.g. Blatman & Sudret (2011); Lüthen et al (2021).

Best regards

Blatman, G., Sudret, B., 2011. Adaptive sparse polynomial chaos expansion based on Least Angle Regression. Journal of Computational Physics 230, 2345–2367.

Lüthen, N., Marelli, S., Sudret, B., 2021. Sparse polynomial chaos expansions: Literature survey and benchmark. SIAM/ASA Journal on Uncertainty Quantification 9, 593–649.


Hello Dr Sudret,

Thank you so much for your help!

I now understand the significance of this plot.

Best regards