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Assume a bivariate data that has some sort of dependency between the two variables is generated by unknown distribution, but not bivariate normal. Is it possible to remove the dependency of the two variables, i.e, is that possible to transform the dependent bivariate data into independent bivariate data ? If so, would you please point me to a paper or book that explains how the transformation work?

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You could try independent component analysis, see Making sense of independent component analysis. More information.

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