Gene

Chaotic

Volatility (finance)

Biology

Discrete mathematics

Statistics

Support vector machine

Applied mathematics

Conditional expectation

Biochemistry

Computer science

Initialization

Paleontology

Kernel (algebra)

Database

Mathematics

Kernel method

Nonparametric statistics

Econometrics

Transcription factor

Mathematical analysis

Artificial intelligence

Conditional variance

Series (stratigraphy)

Measure (data warehouse)

Kernel regression

Repressor

Chemistry

Programming language

Invariant measure

Operator (biology)

Autoregressive conditional heteroskedasticity

Ergodic theory

Physica D Nonlinear Phenomena

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.

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