On the structure of a neuro-fuzzy system to forecast chaotic time series

The process of time series forecasting is described in the context of chaotic deterministic complex systems. The Takens-Mane theorem is used to ground the choices of the forecasting function, the number of past values d used and the time interval τ between them. We argue that a neuro-fuzzy system (NFS) has the mathematical properties requested by the cited theorem. Moreover, it offers 2 more advantages: 1) a fast convergence, in CPU-time, from a very approximate to a (quasi) perfect forecasting function; 2) the possibility to actually understand, in a linguistic manner, the actual rules learned. These theoretical considerations are applied to the Mackey-Glass synthetic chaotic system (1977) in order to study the sensitivity of the NFS in function of d and τ. A brief discussion is made on some effects of noise in time series forecasting, and on topological invariants