Editing Mahalanobis distance (section)

{{Short description|Statistical distance measure}}
The '''Mahalanobis distance''' is a [[distance measure|measure of the distance]] between a point <math>P</math> and a [[probability distribution]] <math>D</math>, introduced by [[Prasanta Chandra Mahalanobis|P.&nbsp;C. Mahalanobis]] in 1936.<ref>{{Cite journal |date=2018-12-01 |title=Reprint of: Mahalanobis, P.C. (1936) "On the Generalised Distance in Statistics." |url=https://doi.org/10.1007/s13171-019-00164-5 |journal=Sankhya A |language=en |volume=80 |issue=1 |pages=1–7 |doi=10.1007/s13171-019-00164-5 |issn=0976-8378|url-access=subscription }}</ref> The mathematical details of Mahalanobis distance first appeared in the ''Journal of The Asiatic Society of Bengal'' in 1936.<ref>{{Cite book |last= |url=https://archive.org/details/dli.ernet.28728/page/n813/mode/1up |title=Journal and Procedings Of The Asiatic Society Of Bengal Vol-xxvi |date=1933 |publisher=Asiatic Society Of Bengal Calcutta}}</ref> Mahalanobis's definition was prompted by the problem of [[similarity measure|identifying the similarities]] of skulls based on measurements (the earliest work related to similarities of skulls are from 1922 and another later work is from 1927).<ref>{{Cite book |last=Mahalanobis |first=Prasanta Chandra |url=http://archive.org/details/records-indian-museum-23-001-096 |title=Anthropological Observations on the Anglo-Indians of Culcutta---Analysis of Male Stature |date=1922 |language=English}}</ref><ref>{{Cite journal |last=Mahalanobis |first=Prasanta Chandra |date=1927 |title=Analysis of race mixture in Bengal |url=https://archive.org/details/in.ernet.dli.2015.280409/page/n522/mode/1up |journal=Journal and Proceedings of the Asiatic Society of Bengal |volume=23 |pages=301–333}}</ref> [[Raj Chandra Bose|R.C. Bose]] later obtained the sampling distribution of Mahalanobis distance, under the assumption of equal dispersion.<ref>{{Cite book |last= |url=https://archive.org/details/in.ernet.dli.2015.23164/page/n169/mode/1up |title=Science And Culture (1935-36) Vol. 1 |date=1935 |publisher=Indian Science News Association |pages=205–206}}</ref>

It is a multivariate generalization of the square of the [[standard score]] <math>z=(x- \mu)/\sigma</math>: how many [[standard deviations]] away <math>P</math> is from the [[mean]] of <math>D</math>. This distance is zero for <math>P</math> at the mean of <math>D</math> and grows as <math>P</math> moves away from the mean along each [[principal component]] axis. If each of these axes is re-scaled to have unit variance, then the Mahalanobis distance corresponds to standard [[Euclidean distance]] in the transformed space. The Mahalanobis distance is thus [[unitless]], [[Scale invariance|scale-invariant]], and takes into account the [[correlations]] of the [[data set]].