It has been noted that there are numerous similarities between the behaviour
of chaotic and stochastic systems. The theoretical links between chaotic and
stochastic systems are investigated based on the evolution of the density of
dynamics and an equivalency relationship based on the invariant measure of
an ergodic system. It is shown that for simple chaotic systems an equivalent
stochastic model can be analytically derived when the initial position in state
space is only known to a limited precision.
Based on this a new methodology for the modelling of complex nonlinear
time series displaying chaotic behaviour with stochastic models is proposed.
This consists of using a stochastic model to learn the evolution of the density
of the dynamics of the chaotic system by estimating initial and transitional
density functions directly from a time series.
A number of models utilising this methodology are proposed, based on
Markov chains and hidden Markov models. These are implemented and their
performance and characteristics compared using computer simulation with several
standard techniques
