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Lognormal Distribution

Hi, Good day !
A random variable with lognormal distribution, I get its PCEs but , I need to do some further analysis to build an augmented matrix as shown below in the code form. Problem is: I need to know the weighting function and orthogonal polynomials associated with lognormal distribution, I tried to use Hermite but results were not accurate. Please help me resolve this issue. Thanks
Kind Regards,
Akif
---- Code for further Analysis ----

image

Hi @akifnadeem,

Maybe I don’t get your question, but is issue somehow related to UQLab or more about how to implement things in MATLAB? If it’s the former, the Community Q&A of UQLab section would be a better place to put this (with an additional background info on what you want to achieve beyond what’s written in the code).

If it’s the latter, the question is quite specific and, I guess, it would be difficult at this point to get a response from other members :sweat_smile:

Thanks!

Thanks for the reply, can you please clarify that the PCE coefficients obtained using lognormal distribution marginal , would follow which distribution i.e. normal or lognormal? These coefficients ,should they needed to be re-transformed or not ?

Hi @akifnadeem,

Thanks for the post here. For lognormal distributions, the default PCE type is Hermite polynomials (please refer to the PCE manual). This indicates that the surrogate model is not a polynomial of the input X but of \ln(X). The reason for this transformation is related to the fundamental theory of PCE (see this paper for details). As a result, if you build a PCE of X\sim \mathcal{LN}(a,b) with coefficients \boldsymbol{c}, the model is expressed as

\sum_i c_i h_i\left(\frac{\ln(X)-a}{b}\right)

with h_i's being Hermite polynomials.

Nevertheless, you can force UQLab to use polynomials that are orthogonal with respect to lognormal distributions by specifying the polynomial type Arbitrary (please refer to the PCE manual).

I hope this helps.

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