Bivariate location depth. (English) Zbl 0905.62002

Rousseeuw, Peter J.; Ruts, Ida

J. R. Stat. Soc., Ser. C 45, No. 4, 516-526 (1996).

Summary: The half-space depth of a point \(\theta\) relative to a bivariate data set \(\{x_1, \dots, x_n\}\) is given by the smallest number of data points contained in a closed half-plane of which the boundary line passes through \(\theta\). A straightforward algorithm for the half-space depth needs \(O(n^2)\) steps. The simplicial depth of \(\theta\) relative to \(\{x_1, \dots, x_n\}\) is given by the number of data triangles \(\Delta (x_i,x_j,x_k)\) that contain \(\theta\); this appears to require \(O(n^3)\) steps. The algorithm proposed here computes both depths in \(O(n \log n)\) time, by combining geometric properties with certain sorting and updating mechanisms. Both types of depth can be used for data description, bivariate confidence regions, \(p\)-values, quality indices and control charts. Moreover, the algorithm can be extended to the computation of depth contours and bivariate sign test statistics.

MSC:

62-07	Data analysis (statistics) (MSC2010)
62-04	Software, source code, etc. for problems pertaining to statistics
65C99	Probabilistic methods, stochastic differential equations

Keywords:

bivariate sign tests; data triangles; Fortran 77; algorithm AS 307; half-space depth; simplicial depth

Software:

AS 307