Simplicial Depth

The *Simplicial Depth*, as proposed by Liu [2], 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*.

In [3] we proposed an alternative definition for simplicial depth which continues to remain valid over a continuous probability field, but also fixes some of the problems in the finite sample case, including those discussed by Zuo and Serfling [2]. Additional problems with the revised definition still remain. One of them depicted in the figure.

We are currently working on computing tight bounds on the values of the simplicial depth based on the half-space depth

Figure

A problem with the revised definition. The data points A,B, and C all have depth 587/1120 and the data point D, which is at the unique center of the data set has depth 355/1120.
For clarity reasons not all cells are drawn.

## Related Papers

[1] **"****On a notion of data depth based on random
simplices"***, L*iu, R. *The Annals of Statistics (18)
405-414,1990*

[2] **"****General notions of statistical depth function**R. Serfling and Y. Zuo, *{The Annals of Statistics (28) 461-482, 2000*

[3] **"Simplicial Depth:
An Improved Definition, Analysis, and Efficiency for the Discrete Case"
**, M. Burr, E. Rafalin, D. Souvaine,
*DIMACS Technical Report, 2003-28*.

**"Simplicial Depth:
An Improved Definition, Analysis, and Efficiency for the Discrete Case"
**, M. Burr, E. Rafalin, D. Souvaine,
CCCG 04, Concordia Montreal, Canada
ps file.

**"Simplicial Depth:
An Improved Definition, Analysis, and Efficiency for the Discrete Case"**, Power Point Presentation.