Building the polynomial basis and using copulas


I am a beginner at building metamodels and I have two theoretical questions about the process.

  1. I have a question in regards to building the polynomial basis and calculation of the coefficients of PCE models. I can’t seem to understand how are the moment (parameters) of the marginal pdfs of the variables used in it. So if I have an experimental data set and I specify the marginals for the input variables:

Input.Marginals(1).Type = ‘Gaussian’;
Input.Marginals(1).Parameters = [0 1];
Input.Marginals(2).Type = ‘Gaussian’;
Input.Marginals(2).Moments = [0 1];

and than also choose the polynomials in regards to the marginals pdfs:
MetaOpts.PolyTypes = {‘hermite’,‘hermite’};

and the maximal degree of the basis:
MetaOpts.Degree = 2;

while I am using the ‘OLS’ method for the calculation of the coefficients, how are the parameters of the marginal distributions used? I have chosen the polynomials and the maximal degree of the basis (so I already know the basis) and I get the coefficients with the OLS method directly from evaluating on the experimental data set, so how do the parameters influence the PCE model/bases?

  1. The second question is in regards to copulas. So if I have the same input for example, how will using a Gaussian copula mathematically influence the basis?

InputOpts.Copula.Type = ‘Gaussian’;
InputOpts.Copula.RankCorr = [1 0.8; 0.8 1];

So where in the calculation does the copula change the model? The polynomials in the basis? The coefficients?

I would really appreciate the help.

Hi @sstore23

For your first question: as soon as you have defined the marginal distributions, UQLab automatically selects the right orthogonal polynomials. The “technical” construction of the polynomials uses the full distribution of each parameter X_1, X_2, etc. and you don’t need to take care of it by yourself (even the line MetaOpts.PolyTypes = {‘hermite’,‘hermite’}; is not necessary in your input file).

In other words, what you should define in the Input.Marginals are the real distributions of the physical parameters of your problem with their real mean and standard deviation, and not standard normal variables. The construction of the PCE in UQLab takes care of those distributions automatically (i.e., also if they are not Gaussian).

For your second question: when some dependence between parameters is introduced through a copula, UQLab will use an isoprobabilistic transform into independent variables (so-called Nataf transform in the case of a Gaussian copula, and Knothe-Rosenblatt transform for more complex general dependence structures). Then the polynomial chaos expansion is computed using the transformed variables. In the end, for predictions (e.g., if you sample the PCE to plot an histogram of the output), the PCE and the inverse transform are automatically combined.

You can find introductory details about the PCE theory and isoprobabilistic transform for instance in this book chapter.

Best regards