Can Bhattacharyya distance be greater than one?

Question

I have two vectors, say $P$ and $Q$. I want to find the statistical overlap between two given that $P$ is my reference which I have modeled after Normal distribution and I have parameters for it. $Q$ is the distribution whose overlap with $P$ I want to find.

For this problem, I took two vectors ($P$ with 8000 elements and $Q$ with 60 elements). $P$ will always remain fixed whereas $Q$ will keep on changing. Now, I find Bhattacharyya distance between two to find its statistical overlap. The formula I have used to find distance is given here. Also the previous link suggests that Bhattacharyya distance is always between 0 and 1. I am getting it to be 4.0978. Am I doing something wrong? I am sure my implementation of formula is correct.

Also, is this the best method to solve the problem I have described?

P.S. To include specific numbers so that anybody will be able to replicate my results and tell me what is wrong. Notations are according to the formula described here:

$\sigma_p = 0.0524$

$\sigma_q = 0.0623$

$\mu_p = 0.9244$

$\mu_q = 0.5952$

Answer 1

Your vector elements must be between 0 and 1 and sum up to 1 (this is what the Wikipedia link you share means, when it says P and Q are probability distributions)

Answer 2

You're indeed calculating the Bhattacharyya distance (BD), but it is not limited to the [0,1] interval. What you need in this case is the Bhattacharyya coefficient (BC), also in the wikipedia link. It says that BD = -ln(BC), thus, BC = 1 / exp(BD), which lies between 0 and 1.