I'm reading up on 'energy distance' to use possibly as an analysis tool for images of different textures. I'm reading the paper "Energy statistics: A class of statistics based on distances" by Gábor J. Székely and Maria L. Rizzo. They mention the Cramér distance, and the extension, Cramér–von Mises Smirnov distance and how they are not rotationally invariant for spaces of dim(F) > 1. This is the basic argument for why the energy distance is introduced.
The univariate Cramér-distance is defined as $$\int_{-\infty}^\infty (F(x)_n-F(x)) dx$$ The univariate Cramér–von Mises Smirnov as $$\int_{-\infty}^\infty (F(x)_n-F(x)) dF(x)$$
My questions regard the rotational invariance:
- What does rotationally invariant (in the multivariate case) mean here? If we identically rotate both cdfs. Fn (the empirical cdf) and F why would the C-MS distance not be the same? Do they mean only rotating one distribution at the time? And if so, what would this mean in practice? Fn is supposed to be approaching F asymptotically, right?
Rotational invariance of a distance $D$ means that for random variables $X,Y$ and a unitary matrix $U$, that $D(X,Y)=D(UX,UY)$. Note that you are rotating the random variables, not the cdf. The cdfs $F_X$ of $X$ and $F_{UX}$ of $UX$ are not related in any simple way.
For example, suppose that $X$ is a point mass at $(1,2)$. Then $F_X(x)=1$ if $x$ is to the north east of $(1,2)$, and $F_X(x)=0$ otherwise. On the other hand, let $U$ be rotation by $90^\circ$. Then $UX$ is a point mass at $(-2,1)$, and the cdf $F_{UX}$ is one to the north and east of $(-2,1)$. Notice that this is not a rotation of $F_X$.