Two systems give uncorrerlated or less correlated outputs while their inputs show some correlated behavior. So how to transform and find some complex relationship between inputs-outputs which can possibly give good correlation and copula dependence for outputs.
Perhaps, the following proposition is far from the problem, but it can be illuminating :)
Let U and Y be independent, with uniform on [0,1] and standard gamma with exponent equal 2, respectively. Then U*Y and (1-U)*Y are iid standard exponential.
Not sure I have understood the question as it seems to self contradict (which may be more my lack of insight) so i will rephrase how I think you mean.
You start with data D1 and D2 as inputs into systems S1 and S2 respectively and you obtain outputs O1 and O2 respectively.
D1 and D2 are highly correlated. O1 and O2 are not well correlated. So S1 and S2 are diverging in their behaviour.
You expect S1 and S2 to behave similarly, therefore you were expecting O1 and O2 to be highly correlated.
You want to know how to mathematically transform S1 and S2 so that the produce better agreement.
Is this correct?
If not apologies for misunderstanding.
If yes, is your D1 and D2 signal based or at least consist of ordered variables where neighbouring variables have a strong covariance?
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