  Identifying coherent spatiotemporal patterns generated by complex dynamical
systems is a central problem in many science and engineering disciplines. Here,
we combine ideas from the theory of operator-valued kernels with
delay-embedding techniques in dynamical systems to develop of a method for
objective identification of spatiotemporal coherent patterns, without prior
knowledge of the state space and/or the dynamical laws of the system generating
the data. A key aspect of this method is that it operates on a space of
vector-valued observables using a kernel measure of similarity that takes into
account both temporal and spatial degrees of freedom (in contrast, classical
kernel techniques such as PCA utilize aggregate measures of similarity between
'snapshots'). As a result, spectral decomposition of data via our approach
yields a significantly more efficient and physically meaningful representation
of complex spatiotemporal signals than conventional methods based on
scalar-valued kernels. We demonstrate this behavior with applications to an
oscillator model and sea surface temperature data from a climate model.
