The Mahalanobis Distance is a statistical measure that quantifies the distance between a point and a distribution. Unlike the more common Distancia Euclidiana, which calculates the straight-line distance between two points in a Cartesian space, the Mahalanobis Distance takes into account the correlations of the conjunto de datos y la varianza a lo largo de cada dimensión.
Matemáticamente, la distancia de Mahalanobis se define como:
D_M = sqrt((x – μ)ᵀ S⁻¹ (x – μ))
donde:
- D_M es la distancia de Mahalanobis.
- x es el vector del punto que se mide.
- μ es el vector medio de la distribución.
- S is the matriz de covarianza de la distribución.
- S⁻¹ es la inversa de la matriz de covarianza.
Esta medida es particularmente útil en estadísticas multivariadas, 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 aprendizaje automático, pattern recognition, and detección de anomalías, due to its ability to handle correlated variables effectively.
Por ejemplo, en una 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.