# Functional Models for Time-Varying Random Objects

## TL;DR

This paper introduces a new method called metric covariance for analyzing time-varying object data in complex metric spaces, enabling functional principal component analysis without requiring linear structure.

## Contribution

It proposes a novel metric auto-covariance function and object functional principal component analysis applicable to non-linear, non-vector space data such as networks and distributions.

## Key findings

- Defines metric covariance for object data in metric spaces
- Develops object functional PCA using eigenfunctions of auto-covariance
- Enables analysis of complex time-varying objects like networks and distributions

## Abstract

In recent years, samples of time-varying object data such as time-varying networks that are not in a vector space have been increasingly collected. These data can be viewed as elements of a general metric space that lacks local or global linear structure and therefore common approaches that have been used with great success for the analysis of functional data, such as functional principal component analysis, cannot be applied. In this paper we propose metric covariance, a novel association measure for paired object data lying in a metric space $(\Omega,d)$ that we use to define a metric auto-covariance function for a sample of random $\Omega$-valued curves, where $\Omega$ generally will not have a vector space or manifold structure. The proposed metric auto-covariance function is non-negative definite when the squared semimetric $d^2$ is of negative type. Then the eigenfunctions of the linear operator with the auto-covariance function as kernel can be used as building blocks for an object functional principal component analysis for $\Omega$-valued functional data, including time-varying probability distributions, covariance matrices and time-dynamic networks. Analogues of functional principal components for time-varying objects are obtained by applying Fr\'echet means and projections of distance functions of the random object trajectories in the directions of the eigenfunctions, leading to real-valued Fr\'echet scores. Using the notion of generalized Fr\'echet integrals, we construct object functional principal components that lie in the metric space $\Omega$.