The Mahalanobis Distance is a statistical measure that quantifies the distance between a point and a distribution. Unlike the more common Distance Euclidienne, which calculates the straight-line distance between two points in a Cartesian space, the Mahalanobis Distance takes into account the correlations of the ensemble de données et la variance le long de chaque dimension.
Mathématiquement, la distance de Mahalanobis est définie comme :
D_M = sqrt((x – μ)ᵀ S⁻¹ (x – μ))
où :
- D_M est la distance de Mahalanobis.
- x est le vecteur du point mesuré.
- μ est le vecteur moyen de la distribution.
- S is the matrice de covariance de la distribution.
- S⁻¹ est l'inverse de la matrice de covariance.
Cette mesure est particulièrement utile en statistiques multivariées, as it allows for identifying outliers in multivariate data and understanding the relative position of a point within a distribution. It is widely applied in various fields, including apprentissage automatique, pattern recognition, and la détection d'anomalies, due to its ability to handle correlated variables effectively.
Par exemple, dans un classification problem, using Mahalanobis Distance can improve the accuracy of the model by considering the underlying structure of the data rather than treating each feature as independent. This makes it a valuable tool in the arsenal of data scientists and statisticians.