PCE: Bootstrap

Dear Professors,
What is the physical significance of bootstraping in PCE?

Best wishes!
Tony Smith

Dear Tony

When you use a randomly sampled experimental design and the corresponding outputs, say \left\{(\mathbf{x}_i, y_i = \mathcal{M} (\mathbf{x}_i)), \quad i=1,\dots, n \right\}, the computed PCE coefficients depend on the very \mathbf{x}_i's. They would be slightly different (as well as the mean and variance that you compute from these coefficients) if you run again the procedure with another experimental design.

To account for this statistical uncertainty that is only due to the finite size n of the ED, you could repeat the analysis many times, and this way get a full distribution for each PCE coefficients. This would be too costly, however. Bootstrap allows you to mimic this replication using the same input/output points, yet resampled with replacement (meaning that a particular point can appear several times in the bootstrap replicate set). Assuming that the n model runs is the costly part, handling the bootstrap replicates is then usually very rapid.

I hope this answers your question
Best regards

Dear Professor Bruno
Thank you very much. It helps me a lot.
Best wishes!
Tony Smith