PREDICTION AND ENTROPY OF NONLINEAR DYNAMICAL-SYSTEMS AND SYMBOLIC SEQUENCES WITH LRO

Following Shannon we introduce higher order entropies and derive dynam ic entropies. The nth order dynamic entropy (conditional entropy) is a measure of the uncertainty of the next state which follows after the observation of n foregoing states. The asymptotic behaviour of the dyn amic entropies at large n is studied for several nonlinear model syste ms and for symbolic sequences with long-range order (LRO). For example we investigate 1D-maps, texts, DNA-strings and time series. It is sho wn that the existence of long correlations improves the possibility of predictions. Characteristic scaling laws for the higher order Shannon entropies and the conditional entropies are derived and a new interpo lation formula is tested. Finally instead of the dynamic entropies whi ch yield mean values of the uncertainty/predictability we investigate the local values of the uncertainty/predictability.