Mahalanobis distance

Mahalanobis distance formula: D² = (x - μ)'Σ^(-1)(x - μ)

The Mahalanobis distance formula is a multivariate measure that quantifies the distance between a point and a probability distribution. It generalizes the concept of standard score distance, taking into account the covariance among variables. This distance is zero for points within the distribution and increases with the distance from the mean.

Example

Given a point P with coordinates (x1, x2) and a distribution D with mean μ = (μ1, μ2) and covariance matrix Σ, the Mahalanobis distance D^2 is calculated as D^2 = [(x1 - μ1), (x2 - μ2)]'Σ^(-1)[(x1 - μ1), (x2 - μ2)].

Understanding the Mahalanobis distance formula is crucial for applications in multivariate statistics, such as clustering and classification, where it helps measure similarity and differentiate between groups.

Related concepts

Euclidean distance

Euclidean distance formula: √((x2 - x1)² + (y2 - y1)²)

Distance transform

Manhattan distance formula: |x1 - x2| + |y1 - y2|

Minkowski spacetime

Minkowski distance formula: D = (Σ |x_i - y_i|^p)^(1/p)

Mean squared error

Mean squared error (MSE) formula: MSE = (1/n) * Σ(y_i - ŷ_i)²

Standard deviation

Standard deviation (σ) is the square root of variance

Batch normalization

Batch normalization formula: Y = (X - μ) / σ * γ + β

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