  Analog forecasting is a nonparametric technique introduced by Lorenz in 1969
which predicts the evolution of states of a dynamical system (or observables
defined on the states) by following the evolution of the sample in a historical
record of observations which most closely resembles the current initial data.
Here, we introduce a suite of forecasting methods which improve traditional
analog forecasting by combining ideas from kernel methods developed in harmonic
analysis and machine learning and state-space reconstruction for dynamical
systems. A key ingredient of our approach is to replace single-analog
forecasting with weighted ensembles of analogs constructed using local
similarity kernels. The kernels used here employ a number of dynamics-dependent
features designed to improve forecast skill, including Takens' delay-coordinate
maps (to recover information in the initial data lost through partial
observations) and a directional dependence on the dynamical vector field
generating the data. Mathematically, our approach is closely related to kernel
methods for out-of-sample extension of functions, and we discuss alternative
strategies based on the Nystr\"om method and the multiscale Laplacian pyramids
technique. We illustrate these techniques in applications to forecasting in a
low-order deterministic model for atmospheric dynamics with chaotic
metastability, and interannual-scale forecasting in the North Pacific sector of
a comprehensive climate model. We find that forecasts based on kernel-weighted
ensembles have significantly higher skill than the conventional approach
following a single analog.
