  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 $\Omega$.
