Entropy of finite random binary sequences with weak long-range correlations

Abstract

We study the N -step binary stationary ergodic Markov chain and analyze its differential entropy. Supposing that the correlations are weak we express the conditional probability function of the chain through the pair correlation function and represent the entropy as a functional of the pair correlator. Since the model uses the two-point correlators instead of the block probability, it makes it possible to calculate the entropy of strings at much longer distances than using standard methods. A fluctuation contribution to the entropy due to finiteness of random chains is examined. This contribution can be of the same order as its regular part even at the relatively short lengths of subsequences. A self-similar structure of entropy with respect to the decimation transformations is revealed for some specific forms of the pair correlation function. Application of the theory to the DNA sequence of the R3 chromosome of Drosophila melanogaster is presented.

A Parametric Study of Binary Sequence

“…A binary sequence [1,2,3,4] is vector x defined by its number of elements or equivalently its length L such that each element n x for 1,..., nL  can be equal to one of two possible measurable values, the sequence can refer to measurements or simulated data with respect to the space or time dimension. In computer science [5], the data are treated in binary form which consists of strings of zeros and ones.…”

Symbolic Markov chains with multilinear memory function

“…Это неудовлетворительное заключение предыдущего подраздела побудило нас предло-жить метод [29,30], основанный на аддитивных цепях Маркова высших порядков, где функция условной вероятности принимает вид…”