PCE is a sum of truncated terms to estimate response of a dynamic system when uncertainties are involved, specifically in its design or parameters. Imagine you have a dynamic system such as a mass and spring in which your mass follows a normal distribution. If you want to find the mean and standard deviation of your system's response, let's say your mass acceleration, when such uncertainty exists, you can build a polynomial chaos expansion for your system and get your mean and standard deviation from it.

To build your PCE you need a set of basis functions which their types depend on your random variables' type, here your mass is normally distributed so based on the literature you should choose Hermite polynomials as your basis functions.

There are plenty of papers out there on this topic that explain this AWESOME tool in detail.

Good luck.

Hi Esmaeil Rezaei , I think that might be possible for a single parameter being random, however, for a system with multiple random variables, because of the interactions of the parameters and their randomness, a proper stochastic analysis like gPC is required.

Esmaeil Rezaei you are very welcome! I think we are not on the same page so let me explain a little. If you have a dynamic system and some parameters of your system are random, then your system's response will be stochastic. Meaning that it will be different from a deterministic system in which the parameters have no randomness. Now, if you want to analyze your system, say you want to calculate the mean or variance of your system's response, you cannot do it since the effects of those random parameters on your system's response are unknown. So what we do is that we create a polynomial of the system and NOT the parameters. This polynomial when created by gPC can be used to get some useful information about the system's response. I hope it helps.