I want to calculate the Mahalanobis distance between cluster $a$ and cluster $b$, each consisting from a set of multidimensional points. Assuming no correlation, calculating the distance between a random point $p$ and cluster $b$ can be achieved using the normalized Euclidean distance formula:

$$ d(p,b)=\sqrt{ \sum_{i=1}^d \frac{(p_i-b_i)^2}{\sigma_i^2}}$$

where $d$ is the number of dimensions and $\sigma_i^2$ is the squared standard deviation for each dimension in cluster B.

Now, I want to estimate the Mahalanobis distance between clusters $a$ and $b$. Should I assume that cluster $a$ is a single point (i.e. $a$'s centroid)? Or should I normalize using the standard deviation of both clusters? In the latter case that would translate in the following formula:

$$ d(a,b)=\sqrt{ \sum_{i=1}^d \frac{(a_i-b_i)^2}{(\sigma_i^a)^2 \cdot (\sigma_i^b)^2}}$$

where $\sigma_i^b$ is the standard deviation for dimension $i$ in cluster $b$ and $\sigma_i^a$ is the standard deviation for dimension $i$ in cluster $a$.

Thank you in advance.

I want to calculate the Mahalanobis distance between cluster $a$ and cluster $b$, each consisting from a set of multidimensional points. Assuming no correlation, calculating the distance between a random point $p$ and cluster $b$ can be achieved using the normalized Euclidean distance formula:

$$ d(p,b)=\sqrt{ \sum_{i=1}^d \frac{(p_i-b_i)^2}{\sigma_i^2}}$$

where $d$ is the number of dimensions and $\sigma_i^2$ is the squared standard deviation for each dimension in cluster $b$.

Now, I want to estimate the Mahalanobis distance between clusters $a$ and $b$. Should I assume that cluster $a$ is a single point (i.e. $a$'s centroid)? Or should I normalize using the standard deviation of both clusters? In the latter case that would translate in the following formula:

$$ d(a,b)=\sqrt{ \sum_{i=1}^d \frac{(a_i-b_i)^2}{(\sigma_i^a)^2 \cdot (\sigma_i^b)^2}}$$

where $\sigma_i^b$ is the standard deviation for dimension $i$ in cluster $b$ and $\sigma_i^a$ is the standard deviation for dimension $i$ in cluster $a$.

Thank you in advance.