neliöMahalanobisetäisyys

neliöMahalanobisetäisyys, often translated as squared Mahalanobis distance, is a statistical measure used to quantify the distance between a point and a distribution, taking into account the correlations between variables. Unlike Euclidean distance, which treats all variables equally and assumes they are independent, squared Mahalanobis distance accounts for the shape and orientation of the data's covariance. It is particularly useful in multivariate analysis for identifying outliers and classifying data points.

The calculation of squared Mahalanobis distance involves the inverse of the covariance matrix of the data. For a data point x and a distribution with mean vector μ and covariance matrix Σ, the squared Mahalanobis distance is given by the formula (x - μ)ᵀ Σ⁻¹ (x - μ). The transpose of the difference vector is multiplied by the inverse covariance matrix and then by the difference vector itself. This process effectively "whitens" the data, transforming it so that the variables are uncorrelated and have unit variance.

A key advantage of using the squared Mahalanobis distance is its invariance to linear transformations of the data. This means that if you rescale or rotate your data, the calculated squared Mahalanobis distances will remain the same. This property makes it robust in various applications. It is commonly employed in anomaly detection, where points with a large squared Mahalanobis distance are considered potential outliers. In classification tasks, it can be used to determine which class a new data point is most likely to belong to, based on its distance to the class's mean and covariance.