The easiest nonlinear method of local prediction was developed by Lorenz [4] and called method of analogies. Let be xt a point within the d-dimensional phase space, the predicted value a time T later, xt+T, will belong to some kind of interpolation between known points xt1+T, …, xtr+T, where x;7, …, xtr are the nearest r points to xt. Farmer and Sidorowich [12] propose an enhanced method by making a least-squared fit of a local linear map of xt into xt+T. Doing T = 1, we can obtain the predicted value as x’t+1 = atxt + bt by minimising:

W = SI lx+i- atxt — bt |2 (1)

XtGNi

in regard to at and bt. Nt is the s-neighborhood of xt.

Casdagli [10] suggests to use these models as a test for nonlinearity prediction, obtaining the average forecast error given by:

where <•> is the arithmetic average. If E = 0 then the prediction is perfect, and if E = 1, the prediction is no better than a constant predictor in the time series average. If E is calculated for different s-values, occurring the optimum for large neighbourhood sizes, this will indicate that the data are better described by a linear stochastic process in this embedding space. On the other hand, an optimum at rather small sizes will point out the existence of a nonlinear equation of motion.

The average forecast error of local linear models is plotted as a function of the neighbourhood size s in Fig. 5. Note the high value of E and how the minimum value of E occurs at higher neighborhood sizes. This result is associated with stochastic processes.