mahalanobis

Mahalanobis refers to Prasanta Chandra Mahalanobis (1893–1972), an Indian statistician who helped establish modern statistics in India. He founded the Indian Statistical Institute in Calcutta in 1931 and contributed to survey sampling, biostatistics, and data analysis. The name is also attached to the Mahalanobis distance, a multivariate measure of distance named after him.

The Mahalanobis distance between a vector x and a distribution with mean μ and covariance Σ is defined as D_M(x, μ, Σ) = sqrt((x − μ)^T Σ^{-1} (x − μ)). In practice, for a sample with mean x̄ and covariance S, the distance is D_M(x) = sqrt((x − x̄)^T S^{-1} (x − x̄)). It generalizes Euclidean distance by incorporating the correlations among variables and the different scales of measurement, so that deviations along directions with high variance or strong correlation are downscaled appropriately.

Properties and uses: The statistic is widely used in multivariate analysis for outlier detection, pattern recognition, clustering, and classification. It is also employed in discrimination tasks, quality control, and to study population differences in anthropology and genetics. In statistics education and practice, the Mahalanobis distance remains a standard tool for measuring multivariate similarity and for transforming data with covariance-based normalization.