  Kernel analog forecasting (KAF), alternatively known as kernel principal
component regression, is a kernel method used for nonparametric statistical
forecasting of dynamically generated time series data. This paper synthesizes
descriptions of kernel methods and Koopman operator theory in order to provide
a single consistent account of KAF. The framework presented here illuminates
the property of the KAF method that, under measure-preserving and ergodic
dynamics, it consistently approximates the conditional expectation of
observables that are acted upon by the Koopman operator of the dynamical system
and are conditioned on the observed data at forecast initialization. More
precisely, KAF yields optimal predictions, in the sense of minimal root mean
square error with respect to the invariant measure, in the asymptotic limit of
large data. The presented framework facilitates, moreover, the analysis of
generalization error and quantification of uncertainty. Extensions of KAF to
the construction of conditional variance and conditional probability functions,
as well as to non-symmetric kernels, are also shown. Illustrations of various
aspects of KAF are provided with applications to simple examples, namely a
periodic flow on the circle and the chaotic Lorenz 63 system.
