Mahalanobis distance

Question

I have two vectors. I want to evaluate Mahalanobis distance between them. None of software packages are able to evaluate the distance producing a warning that the number of rows should be greater than than number of columns, so as to obtain inverse of covariance matrix. Can a covariance matrix be set up manually to evaluate Mahalanobis distance? For a simple example.

v1 = [172.14    193.43  155.99  106.92  127.37  142.18
      110.03    56.92   36.27   0.48    2.53    3.05
      2.18  0.17    0.10    1.61    80.70   89.70]
v2 = [170.61    192.41  156.66  106.50  125.50  142.43
      110.18    57.02   35.94   0.42    2.54    2.94
      2.16  0.12    0.16    1.61    81.50   88.20]

Answer 1

Mahalanobis distance is just

$$ D(v_1, v_2) = \sqrt{ (v_1-v_2)^T \Sigma^{-1} (v_1-v_2) } $$

Where $\Sigma$ is a covariance matrix (it has to be strictly positive definite).

If you don't know this matrix (as often is the case), you can estimate it from data. But then you need to get as many examples as there are features (columns), otherwise the estimate will not be of full rank (and, thus, be non-invertable). Also, this is necessary condition, but not sufficient.

As long as your software allows you to invert matrices and multiply them, you can calculate this distance.