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Transferring information from observations to models of complex systems may meet impediments when the number of observations at any observation time is not sufficient. This is especially so when chaotic behavior is expressed. We show how to use time-delay embedding, familiar from nonlinear dynamics, to provide the information required to obtain accurate state and parameter estimates. Good estimates of parameters and unobserved states are necessary for good predictions of the future state of a model system. This method may be critical in allowing the understanding of prediction in complex systems as varied as nervous systems and weather prediction where insufficient measurements are typical.

We discuss the problem of determining unknown fixed parameters and unobserved state variables in nonlinear models of a dynamical system using observed time series data from that system. In dynamica...

Most data based state and parameter estimation methods require suitable initial values or guesses to achieve convergence to the desired solution, which typically is a global minimum of some cost function. Unfortunately, however, other stable solutions (e.g., local minima) may exist and provide suboptimal or even wrong estimates. Here we demonstrate for a 9-dimensional Lorenz-96 model how to characterize the basin size of the global minimum when applying some particular optimization based estimation algorithm. We compare three different strategies for generating suitable initial guesses and we investigate the dependence of the solution on the given trajectory segment (underlying the measured time series). To address the question of how many state variables have to be measured for optimal performance, different types of multivariate time series are considered consisting of 1, 2, or 3 variables. Based on these time series the local observability of state variables and parameters of the Lorenz-96 model is investigated and confirmed using delay coordinates. This result is in good agreement with the observation that correct state and parameter estimation results are obtained if the optimization algorithm is initialized with initial guesses close to the true solution. In contrast, initialization with other exact solutions of the model equations (different from the true solution used to generate the time series) typically fails, i.e. the optimization procedure ends up in local minima different from the true solution. Initialization using random values in a box around the attractor exhibits success rates depending on the number of observables and the available time series (trajectory segment)., 15 pages, 2 figures

In variational formulations of data assimilation, the estimation of parameters or initial state values by a search for a minimum of a cost function can be hindered by the numerous local minima in the dependence of the cost function on those quantities. We argue that this is a result of instability on the synchronization manifold where the observations are required to match the model outputs in the situation where the data and the model are chaotic. The solution to this impediment to estimation is given as controls moving the positive conditional Lyapunov exponents on the synchronization manifold to negative values and adding to the cost function a penalty that drives those controls to zero as a result of the optimization process implementing the assimilation. This is seen as the solution to the proper size of ‘nudging’ terms: they are zero once the estimation has been completed, leaving only the physics of the problem to govern forecasts after the assimilation window. We show how this procedure, called Dynamical State and Parameter Estimation (DSPE), works in the case of the Lorenz96 model with nine dynamical variables. Using DSPE, we are able to accurately estimate the fixed parameter of this model and all of the state variables, observed and unobserved, over an assimilation time interval [0, T]. Using the state variables at T and the estimated fixed parameter, we are able to accurately forecast the state of the model for t > T to those times where the chaotic behaviour of the system interferes with forecast accuracy. Copyright © 2010 Royal Meteorological Society