I have seen that copula transformation changes my sample space to the range of $[0 \; 1]^d$ where d is the number of dimensions. Can anyone explain me about copula transformation?
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2 Answers
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I believe you're just referring to transforming each marginal distribution to $U[0,1]$ via the probability integral transform, which when applied to each of the variables individually, transforms a d-dimensional distribution to its copula.
For example, if you had a bivariate normal $(X,Y)$, and transform $U=F_X(X)$ and $V=F_Y(Y)$, then $(U,V)$ is a Gaussian copula.
e.g. see here
There are some recommended introductory readings here
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Sklar's theorem says that (bivariate case), for any given joint distribution function $H$, of two random variables $X$ and $Y$, with uniform univariate margins, $F_x(x)$ and $G_y(y)$, of $X$ and $Y$ respectively, then there exist a copula function $C$, such that:
$$H(x,y)= C(F_x(x),G_y(y)) $$
If all margins is continuous the copula is unique.
The idea of copula transformation is that: Copula models allow to model the margins separately from the dependence structure. Hence, we transform all the variable to be uniform in order to capture the pure dependence structures between variables without any affect of the margins. $u=F(x)$ and hence, $F^{-1}(u)=x$. So we can go back easily to the original data.