Estimation of parameters in one-dimensional maps from noisy chaotic time series

The problem of parameter estimation in model maps from noisy time series is addressed. We suggest a new technique for a special case of one-dimensional maps and chaotic signals. It is based on the maximum likelihood (ML) principle and evaluation of the cost function via backward iterations of a model map. We demonstrate in numerical experiments and, in part, justify theoretically that this “backward ML technique” gives more accurate estimates than previously known techniques for low and moderate noise levels. In particular, global optimisation of the cost function becomes much easier; biases in the estimates vanish as the time series length N increases; variances of the estimates decrease as fast as N−α where α depends on the original system, typical values being about α=2.0 under mild conditions on the original systems.