# Distance between two random vectors

I have two random vectors, $A$ and $B$ with each consisting of $n$ geographical co-ordinates $(x_1,y_1),(x_2,y_2)\dots (x_n,y_n)$ and $(\tilde{x}_1,\tilde{y}_1),(\tilde{x}_2,\tilde{y}_2)\dots (\tilde{x}_n,\tilde{y}_n)$, respectively. Suppose we have a given distribution; let $p_1,p_2,\dots,p_n$ be the probabilities associated with points in $A$ and $\tilde{p}_1,\tilde{p}_2,\dots,\tilde{p}_n$ be the respective probabilities for points in $B$.

My question is: What is the best way to find the distance between $A$ and $B$? I am looking for a distance measure which would give me zero value if co-ordinates in $A$ and $B$ are the same (appearing in any order) and it returns a large value when the co-ordinates in $A$ and $B$ are far apart. I considered using a covariance matrix but I don't get the intuition behind using that. In wikipedia, http://en.wikipedia.org/wiki/Statistical_distance, there are a lot of statistical measures listed. Can anybody help me with insights on how to choose a proper distance measure to suit the specifics of my problem?

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I do not see how the probabilities play a role in your question. Could you edit your question to explain where they come from and what you plan to do with it? –  gui11aume Jul 27 '14 at 19:18
Agreed with @gui11aume. Why don't you just consider the geodesic distance between the geodesic means? –  Emre Jul 28 '14 at 7:27
Hey Emre! Geodesic distance is fine but I have no idea how do you go about calculating the geodesic means? Any help is appreciated. –  Samantha Hawking Jul 29 '14 at 19:01
I don't know if there's closed form expression, but it's easy to find the distance between two points on a sphere, so you could use numerical optimization to find the answer in the worst case. For an initial value you could use the projection of the center of mass onto the surface. –  Emre Jul 30 '14 at 8:20

$d(x_1, y_1, \tilde x_1, \tilde x_2) = \sqrt{(x_1 - \tilde x_1)^2 + (y_1 - \tilde y_1)^2}$