Lets assume one has the following response surface: $y(x,\xi) = \sum^N_{i=0} c_i H_i(\xi)$. Where $\xi$ is Gaussian and $H_i$ is the $i^{th}$ Hermite polynomial. I've seen a lot of papers show the PDF of y - but how are they calculating it? One way would be to randomly sample $y$ I guess - but many papers give the analytical pdf. i.e. no sampling and I can't see how?
Cheers Guys!
One could sample $y$ and approximate the coefficients $c_i$. This is the only tenable solution when the distribution of $y$ is unknown.
However, consider the crux of the polynomial chaos approach: we compute the deterministic coefficients using a Galkerkin method, viz.
$$c_i = \frac{1}{\langle H_i^2 \rangle} \langle y H_i \rangle.$$ Using the integral representation of the inner product, the latter term becomes
$$ \langle y H_i \rangle = \int_D y H_i(\xi) w(\xi)\ d\xi. \label{eq}\tag{1} $$
However, this is meaningless as it is -- the support of $y$ and $\xi$ are almost certainly different.
To reconcile this, we recognize that $y$ and $\xi$ are fully-correlated and use an inverse transform. That is, write the cumulative distribution function of $y$ and $\xi$ as the same uniform random variable $u \sim \mathcal{U}(0,1)$:
$$F(y) = u = G(\xi).$$
Therefore, we may define
$$h(u) \stackrel{\textrm{def}}{=} F^{-1}(u) = y \\ l(u) \stackrel{\textrm{def}}{=} G^{-1}(u) = \xi$$
and substitute this into $\ref{eq}$ above.
Note that the CDF is much easier to get from sampled data than the PDF.
In the event that the distribution of $y$ is known... well, then we have the PDF anyhow!