Research in Reversed Chaotic and Stochastic Time Series
This area of research concerns curved chaotic map models and their stochastic reversals. Various simple linear or quadratic map models have been the basis of chaotic models for many years, but only recently have families of curved models been considered which include earlier models as special cases. The more general class reveals a range of behaviour not seen earlier in the tent, binary-shift and logistic maps. The generalised maps have been explored by theoretical and numerical analysis, according to mathematical tractability. In particular, invariant distributions and time series dependency, both linear and quadratic, have exhibited a variety of behaviour characteristics. A novel link of these deterministic models to corresponding stochastic ones has been set up by time-reversal, and stochastically reversed models have been constructed. These can match to adjustable extents the statistical behaviour of the deterministic models, and provide a class of invariant distributions not hitherto exhibited; properties of these distributions have been explored.
References
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Uniformly distributed first order autoregressive time series models and multiplicative congruential random number generators. J. Appl. Prob., 29 (1992), 896-903.
Stochastically reversed chaotic map models. In Complex Stochastic Systems Engineering. M. Titterington (ed.), (1995), 55-58. Oxford University Press; with N.M. Spencer.
Statistical aspects of curved chaotic map models and their stochastic reversals. Scandinavian J. Statistics 25 (1998), 371-382; with N.M. Spencer.
Chaos: But not in both directions ! Statistics and Computing 11 (2001), 213-216.