I have two stochastic variables, for which I know the distributions, and I wish to obtain the convolution between these variables. The problem is that they are dependent in a complicated way. What would be the best way to model this dependency?
The question is how you have the two variables described? If you have them as time series then it is just to add the time series.
There is another possibility which I used some (many) years ago. You could calculate the moments and cross moments up to a higher degree and from these you could calculate the moments of the sum of the two s.v. The challenge is then to estimate the pdf of the sum. And here you certainly have to do approximations. Described in my PhD thesis from 1988 available from KTH DIVA database.
A fundamental challenge is that it is only for Gaussian Distributions with an "equal" correlation between all parts of the pdf's that adding of correlated variables is clearly defined. If dependence is more complicated then one can use the theory of "coupolas" which, e.g., can imply that there is a high correlation in some situations (e.g. when one variable is on its peak, then there is nearly always a peak on the other one, but for non-peak situations they are independent)).
Lennart Söder, KTH, Stockholm
Via mutual information?
http://ijcai.org/papers13/Papers/IJCAI13-251.pdf
You can ask Prof. Benchetteh Prof Lamine Becheick Le hacine, He is very expert in this field, how allama
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