Predicting chaotic time series with a partial model

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.

Figure 1

Using one equation from the Lorenz system, the data set is interpolated to reconstruct all of the system variables at subsample intervals. (a) Measured data $y$ values (black circles) are interpolated by the multistep equation solver (gray circles), approximately matching the exact solution of the Lorenz equations (gray line). At each interpolation step, we use delay coordinates of the $y$ variable to reconstruct the corresponding (unmodeled) (b) $x$ and (c) $z$ variables. In (b) and (c), the reconstructed $x$ and $z$ variables (gray circles) are compared to the exact solution (gray line).

Figure 2

Forecasting an individual trajectory of the Lorenz-63 $y$ variable. Using the equation to supplement the forecast (gray circles) is more accurate than the direct forecast method alone (open circles). The black curve is the true $y$ trajectory and the black circles on the black curve denote the sampling rate ($h=0.2$).

Figure 3

Forecasting error statistics versus prediction horizon $T$ for the Lorenz-63 (a) $y$, (b) $x$, and (c) $z$ trajectories over 5000 realizations (standard error is less than $0.15$ at all forecast horizons). The data set consists of 2000 points sampled at $h=0.2$. Using our knowledge of the $y$ equation, we can interpolate the training data at subsample step size $k=0.02$ and supplement the original training set. Direct forecasting using this interpolated data set (lower trace) outperforms direct forecasting with access to only the original data set (upper trace). Note the finer forecasting resolution of the lower trace. (d) Even under a moderate amount of observational noise (8% of the standard deviation), our hybrid forecasting offers better short-term prediction of the $y$ variable than the nonparametric forecast.

Figure 4

Improved short-term forecasting of a small network. (a) Example interpolation of one node from the four node chaotic network driven by Lorenz-63. The data set (black circles) is used to interpolate Lorenz-96 (gray circles). (b) At each step of the interpolation, delay coordinates reconstruct the unmodeled Lorenz-63 forcing term (gray circles). (c) Mean forecasting error of the chaotic network over 300 realizations. Standard error is less than $0.01$ at all forecast horizons. (Fewer realizations were necessary compared with Fig. 3 due to smaller variation in the dynamics.) Direct forecasting using this interpolated data set (lower trace) outperforms direct forecasting with access to only the original data set (upper trace).