The Mahalanobis Distance is a statistical measure that quantifies the distance between a point and a distribution. Unlike the more common ユークリッド距離, which calculates the straight-line distance between two points in a Cartesian space, the Mahalanobis Distance takes into account the correlations of the データセット そして、各次元に沿った分散。
数学的には、マハラノビス距離は次のように定義されます：
D_M = sqrt((x – μ)ᵀ S⁻¹ (x – μ))
ただし：
- D_M はマハラノビス距離です。
- x 測定対象の点のベクトルです。
- μ 分布の平均ベクトルです。
- S is the 共分散行列 の分布です。
- S⁻¹ は共分散行列の逆行列です。
この測定は特に有用です 多変量統計学, 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 機械学習, pattern recognition, and 異常検知, due to its ability to handle correlated variables effectively.
例えば、において 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.