Assessing the performance of dynamical trajectory estimates

Estimating trajectories and parameters of dynamical systems from observations is a problem frequently encountered in various branches of science; geophysicists for example refer to this problem as data assimilation. Unlike as in estimation problems with exchangeable observations, in data assimilation the observations cannot easily be divided into separate sets for estimation and validation; this creates serious problems, since simply using the same observations for estimation and validation might result in overly optimistic performance assessments. To circumvent this problem, a result is presented which allows us to estimate this optimism, thus allowing for a more realistic performance assessment in data assimilation. The presented approach becomes particularly simple for data assimilation methods employing a linear error feedback (such as synchronization schemes, nudging, incremental 3DVAR and 4DVar, and various Kalman filter approaches). Numerical examples considering a high gain observer confirm the theory.

Renormalized time scale for anticipating and lagging synchronization

Anticipating synchronization has been recently proposed as a mechanism of interaction in dynamical systems which are able to bring about predictions of future states of a driver system. We suggest that an interesting insight into the anticipating synchronization can be obtained by the renormalization of the time scale in the driven system. Our approach directly links the feedback delay of the driven system with the renormalized time scale of the driven system, identifying the main component in the anticipating synchronization paradigm and suggesting an alternative method to generate the anticipating and the lagging synchronization.