  We develop a framework for dimension reduction, mode decomposition, and
nonparametric forecasting of data generated by ergodic dynamical systems. This
framework is based on a representation of the Koopman and Perron-Frobenius
groups of unitary operators in a smooth orthonormal basis of the L2 space of
the dynamical system, acquired from time-ordered data through the diffusion
maps algorithm. Using this representation, we compute Koopman eigenfunctions
through a regularized advection-diffusion operator, and employ these
eigenfunctions in dimension reduction maps with projectible dynamics and high
smoothness for the given observation modality. In systems with pure point
spectra, we construct a decomposition of the generator of the Koopman group
into mutually commuting vector fields that transform naturally under changes of
observation modality, which we reconstruct in data space through a
representation of the pushforward map in the Koopman eigenfunction basis. We
also establish a correspondence between Koopman operators and Laplace-Beltrami
operators constructed from data in Takens delay-coordinate space, and use this
correspondence to provide an interpretation of diffusion-mapped delay
coordinates for this class of systems. Moreover, we take advantage of a special
property of the Koopman eigenfunction basis, namely that the basis elements
evolve as simple harmonic oscillators, to build nonparametric forecast models
for probability densities and observables. In systems with more complex
spectral behavior, including mixing systems, we develop a method inspired from
time change in dynamical systems to transform the generator to a new operator
with potentially improved spectral properties, and use that operator for vector
field decomposition and nonparametric forecasting.
