Distances and kernels based on cumulative distribution functions - Emerging Trends in Image Processing, Computer Vision, and Pattern Recognition - page 468

This metric is induced from the norm of the Hilbert space L 2 ([ a , b ]). Consequently, the kernel
matrix M is positive semidefinite, since it is the kernel matrix of the Gaussian kernel for
L 2 ([ a , b ]). Therefore, k is a kernel.
The formula for d p ( F , G ) resembles the metric induced by the norm in L p ( R ). However, a CDF
F cannot be an element of L p ( R ) because . The condition of bounded support
will guarantee the convergence of the integral. In practical applications, this will not likely be a
limitation. Theoretically, the integral could be divergent without this constraint. For example,
let F be the step function at 0 and G ( x ) = x /( x + 1), x ≥ 0. Then
Given a data sample, ( X 1 , X 2 , …, X n ), an empirical CDF can be constructed as:
which can be used to approximate the distance d p ( F , G ).
When p = ∞, we have
The distance d ∞ is similar to the Kolmogorov-Smirnov statistic ([ 8 ] ).
The distance measures defined above satisfy certain desirable properties of invariance.