April 2024 Metric statistics: Exploration and inference for random objects with distance profiles
Paromita Dubey, Yaqing Chen, Hans-Georg Müller
Author Affiliations +
Ann. Statist. 52(2): 757-792 (April 2024). DOI: 10.1214/24-AOS2368


This article provides an overview on the statistical modeling of complex data as increasingly encountered in modern data analysis. It is argued that such data can often be described as elements of a metric space that satisfies certain structural conditions and features a probability measure. We refer to the random elements of such spaces as random objects and to the emerging field that deals with their statistical analysis as metric statistics. Metric statistics provides methodology, theory and visualization tools for the statistical description, quantification of variation, centrality and quantiles, regression and inference for populations of random objects, inferring these quantities from available data and samples. In addition to a brief review of current concepts, we focus on distance profiles as a major tool for object data in conjunction with the pairwise Wasserstein transports of the underlying one-dimensional distance distributions. These pairwise transports lead to the definition of intuitive and interpretable notions of transport ranks and transport quantiles as well as two-sample inference. An associated profile metric complements the original metric of the object space and may reveal important features of the object data in data analysis. We demonstrate these tools for the analysis of complex data through various examples and visualizations.

Funding Statement

This research was supported in part by NSF Grants DMS-2311034 (PD), DMS-2311035 (YC), DMS-2014626 and DMS-2310450 (HGM).


We wish to thank two anonymous referees, an Associate Editor, and the Editor for their helpful and constructive comments which led to numerous improvements in the paper.

Data used in preparation of Section 7.5.2 in this article were obtained from the Alzheimer’s Disease Neuroimaging Initiative (ADNI) database (http://adni.loni.usc.edu). As such, the investigators within the ADNI contributed to the design and implementation of ADNI and/or provided data but did not participate in analysis or writing of this report. A complete listing of ADNI investigators and databases can be found at http://adni.loni.usc.edu/wp-content/uploads/how_to_apply/ADNI_Acknowledgement_List.pdf.

This article is an expanded version of the Rietz Lecture delivered on June 27, 2022, by H.G.M. at the IMS Meeting in London and is an updated version of the paper posted at arXiv:2202.06117.

The first two authors contributed equally to the paper.


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Paromita Dubey. Yaqing Chen. Hans-Georg Müller. "Metric statistics: Exploration and inference for random objects with distance profiles." Ann. Statist. 52 (2) 757 - 792, April 2024. https://doi.org/10.1214/24-AOS2368


Received: 1 November 2023; Revised: 1 February 2024; Published: April 2024
First available in Project Euclid: 9 May 2024

Digital Object Identifier: 10.1214/24-AOS2368

Keywords: Distributional data , Fréchet mean , Fréchet regression , Functional data analysis , metric variance , profile metric , transport quantile , transport rank , visualization , Wasserstein metric

Rights: Copyright © 2024 Institute of Mathematical Statistics


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Vol.52 • No. 2 • April 2024
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