Abstract
In this paper, we study the behaviour of the so-called k-simplicial distances and k-minimal-variance distances between a point and a sample. The family of k-simplicial distances includes the Euclidean distance, the Mahalanobis distance, Oja’s simplex distance and many others. We give recommendations about the choice of parameters used to calculate the distances, including the size of the sub-sample of simplices used to improve computation time, if needed. We introduce a new family of distances which we call k-minimal-variance distances. Each of these distances is constructed using polynomials in the sample covariance matrix, with the aim of providing an alternative to the inverse covariance matrix, that is applicable when data is degenerate. We explore some applications of the considered distances, including outlier detection and clustering, and compare how the behaviour of the distances is affected for different parameter choices.
Highlights
The Mahalanobis distance is one of the most useful tools in multivariate data science, underpinning a huge variety of practical data analysis methods
We explore the choice of the parameter k, and show that k can be relatively low to produce good results, making the k-minimal-variance distance a quick and viable alternative to the Mahalanobis distance
We prove the following theorem comparing the variance of the squared Euclidean distance, Mahalanobis distance and k-simplicial distance with k = 2 and δ = 2
Summary
The Mahalanobis distance is one of the most useful tools in multivariate data science, underpinning a huge variety of practical data analysis methods. This distance measures the proximity of a point x ∈ Rd to a d-dimensional set of points X = {x1, . It was introduced in Mahalanobis [27]. The Mahalanobis distance corresponds to the Euclidean distance in the standardized space where variables are uncorrelated
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