Estimation

The properties of chaotic systems, specially the fact that their orbits are confined to an attractor, can be used to better our ability to predict their future within the predictability horizon. The fundamental idea is that from an observed time series one can reconstruct the attractor and thus the internal dynamics of the system [26].

Another idea involving dynamical systems and prediction is that of having a series of ``candidate`` dynamical systems, each of which estimate the future of the system being observed. The predictions are compared against the real outcomes and from this comparison a confidence function is built. This function determines the probability of the n-th candidate system yielding the right prediction given actual and past conditions of the system under observation. We take as our estimation the output of the dynamical system for whom this confidence function is a maximum [27] [32].

The set of candidate systems constitutes an iterated function system (IFS) and under relatively relaxed conditions an IFS presents a fractional dimension attractor, so these systems are referred as defining a fractal.

Figure 5: Estimation using an IFS