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

Result code in IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F25%3A10507225" target="_blank" >RIV/00216208:11320/25:10507225 - isvavai.cz</a>

Result on the web

<a href="https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=pb51J7I1CD" target="_blank" >https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=pb51J7I1CD</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1111/insr.12611" target="_blank" >10.1111/insr.12611</a>

Result language

angličtina

Original language name

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

Original language description

Statistical depth is the act of gauging how representative a point is compared with a reference probability measure. The depth allows introducing rankings and orderings to data living in multivariate, or function spaces. Though widely applied and with much experimental success, little theoretical progress has been made in analysing functional depths. This article highlights how the common h$$ h $$-depth and related depths from functional data analysis can be viewed as a kernel mean embedding, widely used in statistical machine learning. This facilitates answers to several open questions regarding the statistical properties of functional depths. We show that (i) h$$ h $$-depth has the interpretation of a kernel-based method; (ii) several h$$ h $$-depths possess explicit expressions, without the need to estimate them using Monte Carlo procedures; (iii) under minimal assumptions, h$$ h $$-depths and their maximisers are uniformly strongly consistent and asymptotically Gaussian (also in infinite-dimensional spaces and for imperfectly observed functional data); and (iv) several h$$ h $$-depths uniquely characterise probability distributions in separable Hilbert spaces. In addition, we also provide a link between the depth and empirical characteristic function based procedures for functional data. Finally, the unveiled connections enable to design an extension of the h$$ h $$-depth towards regression problems.

Czech name

—

Czech description

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Type

J_imp - Article in a specialist periodical, which is included in the Web of Science database

CEP classification

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OECD FORD branch

10103 - Statistics and probability

Project

Result was created during the realization of more than one project. More information in the Projects tab.

Continuities

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Publication year

2025

Confidentiality

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Name of the periodical

International Statistical Review

ISSN

0306-7734

e-ISSN

1751-5823

Volume of the periodical

93

Issue of the periodical within the volume

2

Country of publishing house

GB - UNITED KINGDOM

Number of pages

32

Pages from-to

317-348

UT code for WoS article

001446505600001

EID of the result in the Scopus database

2-s2.0-105000407721