The Mahalanobis distance is defined for a point (x) with respect to a distribution characterized by its mean vector (\(\mu\)) and covariance matrix (\(\Sigma\)) as follows:
Here:
- (\(D_M(x)\)) is the Mahalanobis distance for the point (x).
- (x) is the vector representing the data point.
- (\(\mu\)) is the mean vector of the distribution.
- (\(\Sigma\)) is the covariance matrix of the distribution.
- (\(\Sigma^{-1}\)) is the inverse of the covariance matrix.
The Mahalanobis distance accounts for the correlations between different features in the data, which makes it particularly useful when dealing with multivariate data. It is a normalized distance metric, providing a measure of how many standard deviations a data point is from the mean along each dimension, considering the correlations.
Applications of Mahalanobis distance include outlier detection, clustering, and classification. In outlier detection, data points with unusually large Mahalanobis distances from the mean of a distribution are considered potential outliers. The Mahalanobis distance is also used in the Mahalanobis-Taguchi System, a technique for quality engineering and process optimization.
