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Kernel analog forecasting (KAF), alternatively known as kernel principalcomponent regression, is a kernel method used for nonparametric statisticalforecasting of dynamically generated time series data. This paper synthesizesdescriptions of kernel methods and Koopman operator theory in order to providea single consistent account of KAF. The framework presented here illuminatesthe property of the KAF method that, under measure-preserving and ergodicdynamics, it consistently approximates the conditional expectation ofobservables that are acted upon by the Koopman operator of the dynamical systemand are conditioned on the observed data at forecast initialization. Moreprecisely, KAF yields optimal predictions, in the sense of minimal root meansquare error with respect to the invariant measure, in the asymptotic limit oflarge data. The presented framework facilitates, moreover, the analysis ofgeneralization error and quantification of uncertainty. Extensions of KAF tothe construction of conditional variance and conditional probability functions,as well as to non-symmetric kernels, are also shown. Illustrations of variousaspects of KAF are provided with applications to simple examples, namely aperiodic flow on the circle and the chaotic Lorenz 63 system.

Operator-theoretic framework for forecasting nonlinear time series with kernel analog techniques