Rank Dynamics for Functional Data

TL;DR

This paper introduces a new statistical framework for analyzing the dynamic behavior of ranks in functional data over time, decomposing rank changes into population and individual components, with applications demonstrated on diverse longitudinal datasets.

Contribution

It proposes novel methods for estimating and modeling rank dynamics in functional data, including a rank derivative model and asymptotic properties, advancing the analysis of temporal rank evolution.

Findings

Effective rank estimation methods demonstrated on real datasets

Decomposition of rank changes into population and individual effects

Model validation through simulations and empirical data

Abstract

The study of the dynamic behavior of cross-sectional ranks over time for functional data and the ranks of the observed curves at each time point and their temporal evolution can yield valuable insights into the time dynamics of functional data. This approach is of interest in various application areas. For the analysis of the dynamics of ranks, estimation of the cross-sectional ranks of functional data is a first step. Several statistics of interest for ranked functional data are proposed. To quantify the evolution of ranks over time, a model for rank derivatives is introduced, where rank dynamics are decomposed into two components. One component corresponds to population changes and the other to individual changes that both affect the rank trajectories of individuals. The joint asymptotic normality for suitable estimates of these two components is established. The proposed approaches are illustrated with simulations and three longitudinal data sets: Growth curves obtained from the Z\"urich Longitudinal Growth Study, monthly house price data in the US from 1996 to 2015 and Major League Baseball offensive data for the 2017 season.