I was wondering what are some good distance measures for binary data that have the following properties. I know that there are measures like the Jaccard index and the Dice Index, but they don't exactly suit my needs.
v1 = [1 1 1 1 0 0 0 0 0 0 0 0 0]
v2 = [1 1 1 0 1 0 0 0 0 0 0 0 0]
v1' = [1 1 1 1 0 0 0 0 0 0 0 0]
v2' = [1 1 1 0 0 0 0 0 0 0 0 1]
v1 and v2 I would call 'similar', whereas v1' and v2' are very different. Basically, I want the distance measure to be able to describe the change distribution of the 1s throughout the array. if that changes substantially, then the distance should be high, otherwise low.
Is there a distance measure like this that exists? Thanks!
Look into Earth mover's distance (or Wasserstein metric).