Consistent estimation of symmetric tent chaotic sequences with coded itineraries

Maximum-likelihood (ML) estimation of long chaotic sequences is generally statistically inefficient. Therefore, as the length of the sequences increase we do not obtain the usual ML behavior of consistency and normality. We discuss here a specific class of chaotic sequences for which consistency is preserved. The itineraries of the chaotic sequences in this class are derived from a set of binary code words. As a result, we are able to guarantee a minimal Hamming distance between them which increases linearly with the sequences length. In other words, pairs of sequences from this class have a non vanishing normalized Hamming distance between their itineraries as their length goes to infinity. We derive expressions related to the associated Euclidean distance between these chaotic sequences. Using these expressions we show that the condition for consistent estimation of a sequence from the class is satisfied with probability 1. Throughout the paper, we use the discrete-time symmetric tent chaotic sequences as an example and the expressions are derived for this specific case. We argue, however, that our results apply for a wider class of chaotic sequences. We mention the applicability of the work to the field of error-correcting codes for analog signals. However, it may be of interest for readers working on other aspects of chaotic maps as well.