Flexible integrated functional depths

Germain Van Bever; Pauliina Ilmonen; Stanislav Nagy; Sami Helander; Lauri Viitasaari

doi:10.3150/20-bej1254

Flexible integrated functional depths

Germain Van Bever
, Pauliina Ilmonen
, Stanislav Nagy
, Sami Helander
, Lauri Viitasaari

Namur Institute for Complex Systems

Research output: Contribution to journal › Article › peer-review

Abstract

This paper develops a new class of functional depths. A generic member of this class is coined Jth 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.

Original language	English
Pages (from-to)	673-701
Number of pages	29
Journal	Bernoulli : a journal of mathematical statistics and probability
Volume	27
Issue number	1
DOIs	https://doi.org/10.3150/20-bej1254
Publication status	Published - Feb 2021

Funding

The authors greatly appreciate the insightful comments of an Associate Editor and a referee, which led to distinct improvements in the paper. The authors would also like to acknowledge the computational resources provided by the Aalto University School of Science “Science-IT” project. S. Nagy wishes to thank the Czech Science Foundation (grant 19-16097Y) and Charles University (grant PRIMUS/17/SCI/3). G. Van Bever would like to thank the Belgian FNRS (grant Crédit de recherche C 60/5–CDR/OL). P. Ilmonen and L. Viitasaari wish to thank the Väisälä foundation for its support.

Funders	Funder number
Aalto University School of Science
Fonds de la Recherche Scientifique F.R.S.-FNRS
Univerzita Karlova v Praze	PRIMUS/17/SCI/3
Grantová Agentura České Republiky	19-16097Y
Väisälän Rahasto

Keywords

Asymptotics
Data depth
Functional data analysis
Integrated depths
Supervised classification

Access to Document

10.3150/20-bej1254

Cite this

@article{2196729fd4b44f34a869c4c7a3216647,

title = "Flexible integrated functional depths",

abstract = "This paper develops a new class of functional depths. A generic member of this class is coined Jth 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.",

keywords = "Asymptotics, Data depth, Functional data analysis, Integrated depths, Supervised classification",

author = "\{Van Bever\}, Germain and Pauliina Ilmonen and Stanislav Nagy and Sami Helander and Lauri Viitasaari",

note = "Publisher Copyright: {\textcopyright} 2021 ISI/BS.",

year = "2021",

month = feb,

doi = "10.3150/20-bej1254",

language = "English",

volume = "27",

pages = "673--701",

journal = "Bernoulli : a journal of mathematical statistics and probability",

issn = "1350-7265",

publisher = "Bernoulli Society for Mathematical Statistics and Probability",

number = "1",

}

TY - JOUR

T1 - Flexible integrated functional depths

AU - Van Bever, Germain

AU - Ilmonen, Pauliina

AU - Nagy, Stanislav

AU - Helander, Sami

AU - Viitasaari, Lauri

PY - 2021/2

Y1 - 2021/2

N2 - This paper develops a new class of functional depths. A generic member of this class is coined Jth 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.

AB - This paper develops a new class of functional depths. A generic member of this class is coined Jth 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.

KW - Asymptotics

KW - Data depth

KW - Functional data analysis

KW - Integrated depths

KW - Supervised classification

UR - https://www.scopus.com/pages/publications/85097471887

U2 - 10.3150/20-bej1254

DO - 10.3150/20-bej1254

M3 - Article

SN - 1350-7265

VL - 27

SP - 673

EP - 701

JO - Bernoulli : a journal of mathematical statistics and probability

JF - Bernoulli : a journal of mathematical statistics and probability

IS - 1

ER -

Flexible integrated functional depths

Abstract

Funding

Keywords

Access to Document

Other files and links

Fingerprint

Cite this