This paper studies one application of mutual information to symbolic sequences: the mutual information functionM(d). This function is compared with the more frequently used correlation functionΓ(d). An exact relation betweenM(d) andΓ(d) is derived for binary sequences. For sequences with more than two symbols, no such general relation exists; in particular,Γ(d)=0 may or may not lead toM(d)=0. This linear, but not general, independence between symbols separated by a distance is studied for ternary sequences. Also included is the estimation of the finite-size effect on calculating mutual information. Finally, the concept of “symbolic noise” is discussed.

Mutual information functions versus correlation functions

Mutual information function, which is an alternative to correlation function for symbolic sequences, and a "symbolic spectrum" are calculated for a human DNA sequence containing mostly intron segments, those that do not code for proteins. It is observed that the mutual information function of this sequence decays very slowly, and the correlation length is extremely long (at least 800 bases). The symbolic spectrum of the sequence at very low frequencies can be approximated by 1/fα, where f is the frequency and α ranges from 0.5 to 0.85. It is suggested that the existence of the repetitive patterns in the sequence is mainly responsible for the observed long-range correlation. A possible connection between this long-range correlation and those in music notes is also briefly discussed.

https://iopscience.iop.org/article/10.1209/0295-5075/17/7/014/pdf

Long-range correlation and partial 1/fα spectrum in a noncoding DNA sequence

In this paper, we review the literature on statistical long-range correlation in DNA sequences. We examine the current evidence for these correlations, and conclude that a mixture of many length scales (including some relatively long ones) in DNA sequences is responsible for the observed 1/f-like spectral component. We note the complexity of the correlation structure in DNA sequences. The observed complexity often makes it hard, or impossible, to decompose the sequence into a few statistically stationary regions. We suggest that, based on the complexity of DNA sequences, a fruitful approach to understand long-range correlation is to model duplication, and other rearrangement processes, in DNA sequences. One model, called ``expansion-modification system", contains only point duplication and point mutation. Though simplistic, this model is able to generate sequences with 1/f spectra. We emphasize the importance of DNA duplication in its contribution to the observed long-range correlation in DNA sequences.

Understanding Long-range Correlations in DNA Sequences

Pink (1/f-) noise is one of the most common behaviors of biosystems. This paper is devoted to clarifying the stochastic answer given to white-noise excitation of bio-systems.It is assumed that a li...

Bio-response to white noise excitation

Compositional heterogeneity within and among isochores in mammalian genomes. I. CsCl and sequence analyses.

A new mechanism for spatial 1/f power spectra is proposed for a type of open dynamical system. This mechanism has two competing ingredients. One tends to preserve the long-range correlations while the other tends to destroy it. A specific model with this mechanism based on a class of extremely simple context-free grammars is analyzed. In this model replication and transposition are repeated applied to a string of discrete elements. Computer experiments show that for a wide range of parameter values, the model yields spatial power spectra of the form 1/fα with α close to 1. The frequency range spanned by the 1/fα form as well as the exponent α depend on the probability with which replication and transposition operations are applied.

Spatial 1/f spectra in open dynamical systems

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Spatial 1/f spectra in open dynamical systems