Energy distance
Energy distance is a statistical distance between probability distributions. If X and Y are independent random vectors in Rd with cumulative distribution functions (cdf) F and G respectively, then the energy distance between the distributions F and G is defined to be the square root of D 2 ( F , G ) = 2 E ‖ X − Y ‖ − E ‖ X − X ′ ‖ − E ‖ Y − Y ′ ‖ ≥ 0 , {\displaystyle D^{2}(F,G)=2\operatorname {E} \|X-Y\|-\operatorname {E} \|X-X'\|-\operatorname {E} \|Y-Y'\|\geq 0,} where (X, X', Y, Y') are independent, the cdf of X and X' is F, the cdf of Y and Y' is G, E {\displaystyle \operatorname {E} } is the expected value, and || .