Editing Mahalanobis distance (section)

==Applications==
Mahalanobis distance is widely used in [[Data clustering|cluster analysis]] and [[Statistical classification|classification]] techniques. It is closely related to [[Hotelling's T-square distribution]] used for multivariate statistical testing and Fisher's [[linear discriminant analysis]] that is used for [[supervised classification]].<ref>{{cite book |url={{google books |plainurl=y |id=O_qHDLaWpDUC&pg=PR13}} |title=Discriminant Analysis and Statistical Pattern Recognition |last=McLachlan |first=Geoffrey |date=4 August 2004 |publisher=John Wiley & Sons |isbn=978-0-471-69115-0 |pages=13–}}</ref>

In order to use the Mahalanobis distance to classify a test point as belonging to one of ''N'' classes, one first [[Estimation of covariance matrices|estimates the covariance matrix]] of each class, usually based on samples known to belong to each class. Then, given a test sample, one computes the Mahalanobis distance to each class, and classifies the test point as belonging to that class for which the Mahalanobis distance is minimal.

Mahalanobis distance and leverage are often used to detect [[outlier]]s, especially in the development of [[linear regression]] models. A point that has a greater Mahalanobis distance from the rest of the sample population of points is said to have higher leverage since it has a greater influence on the slope or coefficients of the regression equation. Mahalanobis distance is also used to determine multivariate outliers. Regression techniques can be used to determine if a specific case within a sample population is an outlier via the combination of two or more variable scores. Even for normal distributions, a point can be a multivariate outlier even if it is not a univariate outlier for any variable (consider a probability density concentrated along the line <math>x_1 = x_2</math>, for example), making Mahalanobis distance a more sensitive measure than checking dimensions individually.

Mahalanobis distance has also been used in [[ecological niche modelling]],<ref>{{Cite journal|last=Etherington|first=Thomas R.|date=2019-04-02|title=Mahalanobis distances and ecological niche modelling: correcting a chi-squared probability error|journal=PeerJ|language=en|volume=7|pages=e6678|doi=10.7717/peerj.6678|issn=2167-8359|pmc=6450376|pmid=30972255 |doi-access=free }}</ref><ref>{{Cite journal|last1=Farber|first1=Oren|last2=Kadmon|first2=Ronen|date=2003|title=Assessment of alternative approaches for bioclimatic modeling with special emphasis on the Mahalanobis distance|journal=Ecological Modelling|language=en|volume=160|issue=1–2|pages=115–130|doi=10.1016/S0304-3800(02)00327-7|doi-access=}}</ref> as the convex elliptical shape of the distances relates well to the concept of the [[fundamental niche]].

Another example of usage is in finance, where Mahalanobis distance has been used to compute an indicator called the "turbulence index",<ref>{{Cite journal|last1=Kritzman|first1=M.|last2=Li|first2=Y.|date=2019-04-02|title=Skulls, Financial Turbulence, and Risk Management|url=https://www.tandfonline.com/doi/abs/10.2469/faj.v66.n5.3|journal=Financial Analysts Journal|language=en|volume=66|issue=5|pages=30–41|doi=10.2469/faj.v66.n5.3 |s2cid=53478656 |url-access=subscription}}</ref> which is a statistical measure of financial markets abnormal behaviour. An implementation as a Web API of this indicator is available online.<ref>{{Cite web|url=https://portfoliooptimizer.io/|title=Portfolio Optimizer |website=portfoliooptimizer.io/|access-date=2022-04-23}}</ref>