DIMACS TR: 2003-28

Simplicial Depth: An Improved Definition, Analysis, and Efficiency for the Finite Sample Case

Authors: Michael A. Burr, Eynat Raflin and Diane L.


As proposed by Liu 1990 the simplicial depth of a point $x$ with respect to a probability distribution $F$ on $R^d$ is the probability that $x$ belongs to a random simplex in $R^d$. The simplicial depth of $x$ with respect to a data set $S$ in $R^d$ is the fraction of the closed simplices given by $d+1$ of the data points containing the point $x$. We propose an alternative definition for simplicial depth which continues to remain valid over a continuous probability field, but also fixes some of the problems for the finite sample case, including those discussed by Zuo and Serfling 2000. Additionally, we discuss the effect of the revised definition on the efficiency of previously developed algorithms and prove tight bounds on the value of the simplicial depth based on the half-space depth.

Paper available at ftp://dimacs.rutgers.edu/pub/dimacs/TechnicalReports/TechReports/2003/2003-28.ps.gz.
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