Mahalanobis Distance using Eigen-Values of the Covariance Matrix

orajput

Given the formula of Mahalanobis Distance:

[itex]D^2_M = (\mathbf{x} - \mathbf{\mu})^T \mathbf{S}^{-1} (\mathbf{x} - \mathbf{\mu})[/itex]

If I simplify the above expression using Eigen-value decomposition (EVD) of the Covariance Matrix:

[itex]S = \mathbf{P} \Lambda \mathbf{P}^T[/itex]

Then,

[itex]D^2_M = (\mathbf{x} - \mathbf{\mu})^T \mathbf{P} \Lambda^{-1} \mathbf{P}^T (\mathbf{x} - \mathbf{\mu})[/itex]

Let, the projections of [itex](\mathbf{x}-\mu)[/itex] on all eigen-vectors present in [itex]\mathbf{P}[/itex] be [itex]\mathbf{b}[/itex], then:

[itex]\mathbf{b} = \mathbf{P}^T(\mathbf{x} - \mathbf{\mu})[/itex]

And,

[itex]D^2_M = \mathbf{b}^T \Lambda^{-1} \mathbf{b}[/itex]

[itex]D^2_M = \sum_i{\frac{b^2_i}{\lambda_i}}[/itex]

The problem that I am facing right now is as follows:

The covariance matrix [itex]\mathbf{S}[/itex] is calculated on a dataset, in which no. of observations are less than the no. of variables. This causes some zero-valued eigen-values after EVD of [itex]\mathbf{S}[/itex].

In these cases the above simplified expression does not result in the same Mahalanobis Distance as the original expression, i.e.:

[itex](\mathbf{x} - \mathbf{\mu})^T \mathbf{S}^{-1} (\mathbf{x} - \mathbf{\mu}) \neq \sum_i{\frac{b^2_i}{\lambda_i}}[/itex] (for non-zero [itex]\lambda_i[/itex])

My question is: Is the simplified expression still functionally represents the Mahalanobis Distance?

P.S.: Motivation to use the simplified expression of Mahalanbis Distance is to calculate its gradient wrt [itex]b[/itex].

GoodSpirit

Hello,

In order to be invertible, S mustn't have zero eigen values, that is , must be positive definite or negative definite. Apart from that , that expression must work...

All the best

GoodSpirit

chiro

Hey orajput and welcome to the forums.

For your problem, if you do have a singular or ill-conditioned covariance matrix, I would try and do something like Principal Components, or to remove the offending variable from your system and re-do the analysis.