Illumination Depth

Halfspace depth for general measures: The ray basis theorem and its consequences

“…We introduce again s = t − tan θ 1 − 1/ √ 3 ≥ 0 and investigate G(s), analogously as in (18) and (19), instead of F 2 (t). As in (20), we obtain the derivative of G(s) of the form…”

“…Note that F ∈ F(x, A) implies dim(F ) < d, and that F(x, ∅) = {∅}. The concept of visible faces relates to the theory of illumination of convex bodies by external light sources [1]; for its application to the statistics of the depth see [19]. Note that any point x / ∈ int (A) illuminates at least one non-empty face of a non-empty convex set A; for details see Lemma 17 presented in the Appendix.…”

Clover plot: Versatile visualization in nonparametric classification*

“…Consequently, the illuminations of the osculating points (6) of these two approximating ellipsoids are evaluated with respect to  1 and  2 , respectively, via the formula in ref. [21,Lemma 1] for the illumination on an ellipsoid. The resulting parametric curve of these illuminations is plotted in orange.…”

“…[22]. One recent remedy to the outsider problem and ties in depth‐analysis has been proposed in the form of the illumination [21]. Let B ( P n ) be the bag of a random sample P n defined in ) with some threshold value α specified, and let x ∈ ℝ d .…”

“…As argued in ref. [21], for any α positive the illumination is an outlyingness function that appears to be well suited for the purposes of depth‐based analysis, as it complements the depth both in terms of theory and practice. Inside the bag (in the center of the data cloud) the illumination equals one, and the depth is preferred to be used to assess the centrality of x ; outside the bag and in the tails of the data, illumination is nonconstant and increasing as we move away from the bag, and provides a reliable source of information about the position of x w.r.t.…”