Fig. 9

Prediction in the Lorentz system. The top panel is the signal from the Lorentz system calculated from the differential equations that describe the system. We use the first 3000 steps as a training set for a nonparametric prediction over the next 500 steps in the range 3001 to 3500 steps shown in the lower panel. One can see that the prediction (in red) hugs the real signal (in black) fairly well over the first 250 steps of the prediction (normalised error < 0.0008). The error is normalised relative to the prediction obtained from a linear prediction or random walk model. From then on the error begins to rise exponentially (as is to be expected from a chaotic series) and is 0.0064 (or very close to 100% of the variance in the data) at 500 steps. If one wanted to improve the accuracy of the prediction past this range then the collection of data within those last 250 steps is necessary. One sees that the predictions for this simple chaotic model are very good