the gPC is used for uncertainty quantification, I find the gPC coefficients for Z=f(theta),where theta is a random variable(it may have any kind of pdf), I know the first coefficient a0 is the mean of Z and the sum of other coefficient to the power of 2 is variance.
but How can I find the shape of the pdf?
because Z might have Uniform or Gamma or even Multi Modal Gaussian Distribution, in that case mean and variance is not that much useful.
Dear Ali,
It is nice to think how to construct a pdf polynomial function whose coefficients have a statistical interpretation. (regardless if pdf is either continuous or discrete), I suggest the following method:
Consider f(x)=(x- x1)(x-x2).....(x-xn) , obviously x1,x2,....xn are the roots of the polynomial. ( you can choose the order upon your request.
You can state the desired conditions on the roots to obtain the desired coefficients of that polynomial.
You prefer the first term (the constant term a0 = the mean
this is equivalent to take the product x1x2....xn = the mean.
(negative or positive according to n)
the sum of other coefficients ( P(1) - a0 )2 = variance.
which is equivalent to )=(1- x1)(1-x2).....(1-xn) = root(variance)+mean.
In addition the integral (or summation) of f(x) over the given domain is equal to one.
Wish you good luck
Issam
I think, that Ali's problem is to find parameters/shape of pdf of Z=f(theta), where the rv theta possesses a given pdf, and f is a given polynomial. This would be a model differing from the one suggested by Issam.
Are the coefficients random, too?
Dear Dr Issam.
I do not intent to build a pdf polynomial.
Dear Dr Joachim my problem is to find parameters/shape of pdf of Z=f(theta), where the rv theta possesses a given pdf, but f is not a given polynomial. f could be any function. gPC (generalized polynomial chaos) is a method to approximate a random variable (somehow like Fourier series which can approximate a function in deterministic case). it is an spectral method. for example if theta has a Gaussian distribution the Hermite polynomial is chosen for expansion of f(theta). after calculation of this coefficients (like Fourier series which a0 is DC value) a0 is the mean (the first moment) and the second moment is sum of all other coefficients to the power of 2 . now my question is based on this coefficients how to characterize the pdf of Z (because Z might have multi-modal Gaussian distribution and in that case mean and variance is not helpful).
Dear Ali Namadchian, Using Matlab function
Lind = [xx_in:x:xx_fi]; % choose these values according to your mean & varriance values
L0(1) = L(1)/1e-9; % mean --> in mine case the mean was 215e-9 so to normalize i devided by 1e-9
L1(2) = L(2)/1e-9;
L2(3) = L(3)/1e-9;
Lind_mean = L0(1);
Lind_std = sqrt(L2(2)^2 + L3(3)^2);
PDF = normpdf(Lind,Lind_mean,Lind_std);
plot(Lind,PDF)
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