I'm working on a project where I want to find similarities between groups of events. So far I have expressed groups of events as vectors of event counts and computing similarities between them. I'm looking for a similarity metric that, at the same time, captures the proportion between events and the number of events itself. For example, say I have the following vectors:
a = [0, 1, 2, 3, 0]$a = [0, 1, 2, 3, 0]$
b = [0, 3, 6, 9, 0]$b = [0, 3, 6, 9, 0]$
c = [1, 1, 1, 1, 1]$c = [1, 1, 1, 1, 1]$
d = [0, 100, 200, 300, 0]$d = [0, 100, 200, 300, 0]$
e = [0, 110, 210, 310, 0]$e = [0, 110, 210, 310, 0]$
I would like to have something like
sim(d,e) > sim(a,b) > sim(a,c) > sim(a,d) > sim(a,e)$sim(d,e) > sim(a,b) > sim(a,c) > sim(a,d) > sim(a,e)$
From one side I have metrics like cosine similaritycosine similarity, which is good to find that sim(a,b)$sim(a,b)$ is big, but will also make sim(a,d)$sim(a,d)$ big. From another side I have metrics such as manhattanmanhattan distance, which will give a big distance to (d,e)$(d,e)$, which shouldn't be the case. I could normalize the vectors, but then I would be saying that sim(a,b) ~= sim(a,c)$sim(a,b) \sim sim(a,c)$, which is also wrong for what I want.