Methods for forecasting time series are a critical aspect of the understanding and control of complex networks. When the model of the network is unknown, nonparametric methods for prediction have been developed, based on concepts of attractor reconstruction pioneered by Takens and others. In this Rapid Communication we consider how to make use of a subset of the system equations, if they are known, to improve the predictive capability of forecasting methods. A counterintuitive implication of the results is that knowledge of the evolution equation of even one variable, if known, can improve forecasting of all variables. The method is illustrated on data from the Lorenz attractor and from a small network with chaotic dynamics. One of the hallmarks of chaotic systems is the breakdown of accurate prediction. Due to exponential divergence of trajectories, long-term forecasting is seldom possible. On the other hand, short-term prediction is often feasible and significant progress has been achieved. In particular, when the system is known through time series observations alone, nonparametric methods have been developed to forecast chaotic trajectories. For relatively low-dimensional chaotic dynamics, Takens' method of attractor reconstruction [1-4] has long been the foundation of nonparametric time series prediction methods. Under suitable genericity hypotheses, the attractor may be represented by delay coordinate vectors built from the time series observations, and methods of prediction, control, and other applications from chaotic time series have been developed [5] [6] [7] . In particular, time series prediction algorithms locate the current position in the delay coordinate representation and use analogs from previously recorded data to establish a local, low-order predictive statistical model, which can be accomplished in several ways [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] . At the other end of the spectrum are parametric forecasting methods, applicable when a physically motivated, complete model for the network is available. Nonlinear approaches to filtering [22] [23] [24] [25] allow forecasting models to use the model equations to develop close to optimal predictions. Even if some variables are not observable, they may be reconstructed, provided that their model equations are known. However, when the model is not completely known, there has been little progress, and few methods are known that are able to span the gap between nonparametric and data assimilation methods. In this Rapid Communication we consider the problem where not only is time series data available from a dynamical network, but in addition we possess a subset of the dynamical equations for the network. This situation reflects the case of a "partial model," in which evolution equations for some, but not all, of the measured variables are available. We address the question of how best to extend nonparametric methods with this added, partial information for system prediction purposes.
They require a pre-knowledge of the dynamical system REF .
Predicting chaotic time series with a partial model.
Physical review. E, Statistical, nonlinear, and soft matter physics
Physical review. E, Statistical, nonlinear, and soft matter physics
Medicine Physics