**Pre-meeting (DeMO) at 3 pm****Chair & Spe
aker: **Yoav Zemel**Authors: **Dubey and Müller

**About the event:**

**Speaker: Paromita Dubey and Hans-Georg Müller(University of California at Davis,
USA)**

**Functional data analysis provides a popular toolbox of functiona
l models for the analysis of samples of random functions that are real valu
ed. In recent years, samples of time-varying object data such as time-varyi
ng 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. We propose metr
ic covariance, a novel association measure for paired object data lying in
a metric space (?, d) that we use to define a metric autocovariance functio
n for a sample of random ?-valued curves, where ? generally will not have a
vector space or manifold structure. The proposed metric autocovariance fun
ction is non-negative definite when the squared semimetric d2 is of negativ
e type. Then the eigenfunctions of the linear operator with the autocovaria
nce function as kernel can be used as building blocks for an object functio
nal principal component analysis for ?-valued functional data, including ti
me-varying probability distributions, covariance matrices and time dynamic
networks. Analogues of functional principal components for time-varying obj
ects are obtained by applying Fréchet means and projections of distance fun
ctions of the random object trajectories in the directions of the eigenfunc
tions, leading to real-valued Fréchet scores. Using the notion of generaliz
ed Fréchet integrals, we construct object functional principal components t
hat lie in the metric space ?. We establish asymptotic consistency of the s
ample-based estimators for the corresponding population targets under mild
metric entropy conditions on ? and continuity of the ?-valued random curves
. These concepts are illustrated with samples of time-varying probability d
istributions for human mortality, time-varying covariance matrices derived
from trading patterns and time-varying networks that arise from New York ta
xi trips.**

**Members, non-Members
, all welcome.**