Using Local Dynamics to Explain Analog Forecasting of Chaotic Systems

Analogs are nearest neighbors of the state of a system. By using analogs and their successors in time, one is able to produce empirical forecasts. Several analog forecasting methods have been used in atmospheric applications and tested on well-known dynamical systems. Such methods are often used without reference to theoretical connections with dynamical systems. Yet, analog forecasting can be related to the dynamical equations of the system of interest. This study investigates the properties of different analog forecasting strategies by taking local approximations of the system’s dynamics. We find that analog forecasting performances are highly linked to the local Jacobian matrix of the flow map, and that analog forecasting combined with linear regression allows us to capture projections of this Jacobian matrix. Additionally, the proposed methodology allows us to efficiently estimate analog forecasting errors, an important component in many applications. Carrying out this analysis also makes it possible to compare different analog forecasting operators, helping us to choose which operator is best suited depending on the situation. These results are derived analytically and tested numerically on two simple chaotic dynamical systems. The impact of observational noise and of the number of analogs is evaluated theoretically and numerically.