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

## TL;DR

This paper presents a unified operator-theoretic framework for kernel analog forecasting (KAF), demonstrating its optimality and consistency in predicting nonlinear time series using Koopman operator theory, with applications to simple dynamical systems.

## Contribution

It synthesizes kernel methods and Koopman theory to provide a comprehensive understanding of KAF, including extensions and error analysis.

## Key findings

- KAF approximates conditional expectations under ergodic dynamics.
- KAF yields optimal predictions with large data.
- Extensions include variance and probability function construction.

## Abstract

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.