  The estimation of covariance operators of spatio-temporal data is in many
applications only computationally feasible under simplifying assumptions, such
as separability of the covariance into strictly temporal and spatial
factors.Powerful tests for this assumption have been proposed in the
literature. However, as real world systems, such as climate data are
notoriously inseparable, validating this assumption by statistical tests, seems
inherently questionable. In this paper we present an alternative approach: By
virtue of separability measures, we quantify how strongly the data's covariance
operator diverges from a separable approximation. Confidence intervals localize
these measures with statistical guarantees. This method provides users with a
flexible tool, to weigh the computational gains of a separable model against
the associated increase in bias. As separable approximations we consider the
established methods of partial traces and partial products, and develop weak
convergence principles for the corresponding estimators. Moreover, we also
prove such results for estimators of optimal, separable approximations, which
are arguably of most interest in applications. In particular we present for the
first time statistical inference for this object, which has been confined to
estimation previously. Besides confidence intervals, our results encompass
tests for approximate separability. All methods proposed in this paper are free
of nuisance parameters and do neither require computationally expensive
resampling procedures nor the estimation of nuisance parameters. A simulation
study underlines the advantages of our approach and its applicability is
demonstrated by the investigation of German annual temperature data.
