How to find Mahalanobis distance

Mahalanobis distance

d=(x-\mu)^T\Sigma^{-1}(x-\mu)

$ x $ is the vector for which you want to find the distance to the data group. $ \ Mu $ is the average value of the data group. $ \ Sigma ^ {-1} $ is the inverse of the covariance matrix of the data group. Using the Cholesky factorization, the equation can be transformed as follows.

\begin{eqnarray}
d &=& (x-\mu)^T\Sigma^{-1}(x-\mu) \\
  &=& (x-\mu)^T(LL^T)^{-1}(x-\mu) \\
  &=& (L^{-1}(x-\mu))^T(L^{-1}(x-\mu)) \\
  &=& z^Tz
\end{eqnarray}

$ L $ is the lower triangular matrix obtained by the Cholesky decomposition. If you set $ z = (L ^ {-1} (x- \ mu)) $ and find this, all you have to do is calculate the inner product.

Implementation

Implement the above in python.

import numpy as np
from scipy.linalg import solve_triangular

def mahalanobis(x, mu, sigma):
    L = np.linalg.cholesky(sigma)
    d = x - mu
    z = solve_triangular(
        L, d.T, lower=True, check_finite=False,
        overwrite_b=True)
    squared_maha = np.sum(z * z, axis=0)
    return squared_maha

$ L $ can be found in numpy's linalg.cholesky. $ z $ can be found in scipy's linalg.solve_triangular.

reference

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