I am looking for a way to measure and interpret the similarity between 2 multi-dimensional samples. Thus, I remember the two-sample Student's test method and was wondering if this was extandable to multi-dimensional data (which seems natural to me).
In 1D, this looks like the following: $t = \frac{\bar{X}_1 - \bar{X}_2}{\sqrt{\sigma_1^2 + \sigma_2^2}}$.
I guess I can extand this to ND but what should I do with the covariance matrixes? Should I simply square each elements and sum up both matrixes afterwards?
It would look like something like this $t = \sqrt{(\bar{X}_1 - \bar{X}_2)^T (\Sigma_1^2 + \Sigma_2^2)^{-1}(\bar{X}_1 - \bar{X}_2)}$.
Plus, is this a reasonable way to measure the similarity between samples or should I use another method for multi-dimensions. For example, the Wasserstein distance? But I would not know how to interpret the metric as for the Student's test.
I thank you in advance for your help.