Flexible integrated functional depths

Abstract

This paper develops a new class of functional depths. A generic member of this class is coined J th order kth moment integrated depth. It is based on the distribution of the cross-sectional halfspace depth of a function in the marginal evaluations (in time) of the random process. Asymptotic properties of the proposed depths are provided: we show that they are uniformly consistent and satisfy an inequality related to the law of the iterated logarithm. Moreover, limiting distributions are derived under mild regularity assumptions. The versatility displayed by the new class of depths makes them particularly amenable for capturing important features of functional distributions. This is illustrated in supervised learning, where we show that the corresponding maximum depth classifiers outperform classical competitors.

From Geometric Quantiles to Halfspace Depths: A Geometric Approach for External Behaviour.

“…This is a very important problem in view of applications, questioning the relevance of those tools when working on samples. It is worth recalling that geometric quantiles uniquely identify the underlying probability measure, but this property is not always true for halfspace depth, as recently proven in Nagy et al (2021). Nevertheless, the characterisation is unique when having measures with finite support, as is the case of empirical measures.…”

“…Numerous depth functions have been introduced and studied, starting with Mahalanobis distance depths (Liu and Singh (1993), Mahalanobis (1936), Zuo and Serfling (2000)), the well-known and used Tukey or halfspace depth (Tukey (1975)), going on, for instance, with simplicial (volume) depths (Liu (1990), Oja (1983)), onion depths (Barnett (1976), Eddy (1982)), all notions of spatial depths (Chaudhury (1996), Dudley and Koltchinskii (1992), Koltchinskii (1997), Möttönen et al (2005), Vardi and Zhang (2000)), the projection depth (Donoho and Gasko (1992), Dutta and Ghosh (2012), Nagy et al (2020), Zuo (2003)), the zonoid depth (Dyckerhoff et al (1996), Koshevoy (2003), Koshevoy and Mosler (1997)), local depths (Agostinelli and Romanazzi (2011), Paindaveine and Van Bever (2013)). We refer to Chernozhukov et al (2017), Hallin et al (2010), Kuelbs and Zinn (2016), Mosler (2002Mosler ( , 2013, Mosler and Mozharovskyi (2021), Nagy et al (2020Nagy et al ( , 2021, Nagy (2022) and references therein, for theoretical and practical aspects (as well as computational) of depth functions, in particular halfspace depths.…”

Statistical Depth Meets Machine Learning: Kernel Mean Embeddings and Depth in Functional Data Analysis

“…It is the first result of its kind for a functional depth available in the literature. The only comparable results are [15,Theorem 4] and [73,Theorem 1] where much weaker, non-uniform versions of CLTs are derived for specific integrated functional depths, under restrictive conditions on the distribution P ∈ P(X ). Theorem 9.…”

Integrated shape-sensitive functional metrics

“…This can be seen, for instance, in the functional depth literature, where recent advances have begun emphasizing shape features as the focal point of analysis. See, for example Sguera et al [31], Claeskens et al [13], Harris et al [20] and Nagy et al [27,28] for recent approaches to shape-sensitive functional depths. * Correspondence: Stanislav Nagy, Email:…”