The Annals of Statistics
- Ann. Statist.
- Volume 46, Number 4 (2018), 1513-1540.
Pareto quantiles of unlabeled tree objects
In this paper, we consider a set of unlabeled tree objects with topological and geometric properties. For each data object, two curve representations are developed to characterize its topological and geometric aspects. We further define the notions of topological and geometric medians as well as quantiles based on both representations. In addition, we take a novel approach to define the Pareto medians and quantiles through a multi-objective optimization problem. In particular, we study two different objective functions which measure the topological variation and geometric variation, respectively. Analytical solutions are provided for topological and geometric medians and quantiles, and in general, for Pareto medians and quantiles, the genetic algorithm is implemented. The proposed methods are applied to analyze a data set of pyramidal neurons.
Ann. Statist., Volume 46, Number 4 (2018), 1513-1540.
Received: November 2016
Revised: March 2017
First available in Project Euclid: 27 June 2018
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 62G99: None of the above, but in this section
Secondary: 62P10: Applications to biology and medical sciences
Sienkiewicz, Ela; Wang, Haonan. Pareto quantiles of unlabeled tree objects. Ann. Statist. 46 (2018), no. 4, 1513--1540. doi:10.1214/17-AOS1593. https://projecteuclid.org/euclid.aos/1530086424
- Supplement to “Pareto quantiles of unlabeled tree objects”. This document includes the description of the data object construction, proofs, and additional details regarding simulation and data analysis.