I was thinking of the following problem: Imagine I am given two lists of points on a 2D plane. These lists have the same size, i.e. both lists have the same number of points.
Now, I want to be able to compare these two patterns of points. How could I do that mathematically/statistically?
My try
I calculated the distance (Euclidean) from each point to every other point (pairwise distance). Then, I've ordered these distances. After that, I pick the first pair which will be the distance between two points a and b. At this point, I will ignore any other distance containing a or b (if a is in the first pattern and b in the second pattern). Thus at the end I will have a "matching" that creates a minimum weight match.
Finally, I just sum up these distances and this is my distance coefficient.
Any other ideas?
An example Suppose I have: $[ (0,0), (0,1), (1,0), (1,1)]$ and $[(0,0), (2,1), (0,1), (1,0)]$ and $[(2,3), (2,0), (0,0), (0,2)]$
These are three different patterns of points. I want to assess how similar they are.
How similar are they? Which are the most similar pairs? I want to answer questions like these.
The first idea which came to my head was Hausdorff distance.
If you are interested in the similarity not of the placements of the points of the given sets (say $A$ and $B$), but only of their patterns (which are invariant with respect to isometries of the plane), you can use the counterparts of $\ell_p$-metric with $1\le p\le\infty$ (most popular values of $p$ are $1$, $2$, and $\infty$)
$$d(A,B)=\min_\sigma \left(\sum_{a,a’\in A} |d(a,a’)-d(\sigma(a),\sigma(a’))|^p \right)^{1/p},$$
where the minimum is taken with respect to all bijections $\sigma$ between the sets $A$ and $B$. The power $1/p$ is taken with hope to assure the trinagle inequality
$$ d(A,B)\le d(A,C)+ d(C,B).$$