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1.
Phys Rev E ; 110(2-1): 024303, 2024 Aug.
Artículo en Inglés | MEDLINE | ID: mdl-39295034

RESUMEN

We study two-state (dichotomous, telegraph) random ergodic continuous-time processes with dynamics depending on their past. We take into account the history of the process in an explicit form by introducing integral nonlocal memory term into conditional probability function. We start from an expression for the conditional transition probability function describing additive multistep binary random chain and show that the telegraph processes can be considered as continuous-time interpolations of discrete-time dichotomous random sequences. An equation involving the memory function and the two-point correlation function of the telegraph process is analytically obtained. This integral equation defines the correlation properties of the processes with given memory functions. It also serves as a tool for solving the inverse problem, namely for generation of a telegraph process with a prescribed pair correlation function. We obtain analytically the correlation functions of the telegraph processes with two exactly solvable examples of memory functions and support these results by numerical simulations of the corresponding telegraph processes.

2.
Phys Rev E ; 106(3-1): 034127, 2022 Sep.
Artículo en Inglés | MEDLINE | ID: mdl-36266815

RESUMEN

We propose two different approaches for introducing the information temperature of binary Nth-order Markov chains. The first approach is based on a comparison of Markov sequences with equilibrium Ising chains at given temperatures. The second approach uses probabilities of finite-length subsequences of symbols occurring, which determine their entropies. The derivative of the entropy with respect to the energy gives the information temperature measured on the scale of introduced energy. For the case of a nearest-neighbor spin-symbol interaction, both approaches give similar results. However, the method based on the correspondence of the N-step Markov and Ising chains appears to be very cumbersome for N>3. We also introduce the information temperature for the weakly correlated one-parametric Markov chains and present results for the stepwise and power memory functions. An application of the developed method to obtain the information temperature of some literary texts is given.

3.
Phys Rev E ; 103(3-1): 032139, 2021 Mar.
Artículo en Inglés | MEDLINE | ID: mdl-33862761

RESUMEN

We study random processes with nonlocal memory and obtain solutions of the Mori-Zwanzig equation describing non-Markovian systems. We analyze the system dynamics depending on the amplitudes ν and µ_{0} of the local and nonlocal memory and pay attention to the line in the (ν, µ_{0}) plane separating the regions with asymptotically stationary and nonstationary behavior. We obtain general equations for such boundaries and consider them for three examples of nonlocal memory functions. We show that there exist two types of boundaries with fundamentally different system dynamics. On the boundaries of the first type, diffusion with memory takes place, whereas on borderlines of the second type the phenomenon of noise-induced resonance can be observed. A distinctive feature of noise-induced resonance in the systems under consideration is that it occurs in the absence of an external regular periodic force. It takes place due to the presence of frequencies in the noise spectrum, which are close to the self-frequency of the system. We analyze also the variance of the process and compare its behavior for regions of asymptotic stationarity and nonstationarity, as well as for diffusive and noise-induced-resonance borderlines between them.

4.
Phys Rev E ; 102(2-1): 022119, 2020 Aug.
Artículo en Inglés | MEDLINE | ID: mdl-32942436

RESUMEN

Considering symbolic and numerical random sequences in the framework of the additive Markov chain approach, we establish a relation between their correlation functions and conditional entropies. We express the entropy by means of the two-point probability distribution functions and then evaluate the entropy for the numerical random chain in terms of the correlation function. We show that such approximation gives a satisfactory result only for special types of random sequences. In general case the conditional entropy of numerical sequences obtained in the two-point distribution function approach is lower. We derive the conditional entropy of the additive Markov chain as a sum of the Kullback-Leibler mutual information and give an example of random sequence with the exactly zero correlation function and the nonzero correlations.

5.
Phys Rev E ; 100(5-1): 052141, 2019 Nov.
Artículo en Inglés | MEDLINE | ID: mdl-31869899

RESUMEN

We study the non-Markovian random continuous processes described by the Mori-Zwanzig equation. As a starting point, we use the Markovian Gaussian Ornstein-Uhlenbeck process and introduce an integral memory term depending on the past of the process into an expression for the higher-order transition probability function and the stochastic differential equation. We show that the proposed processes can be considered as continuous-time interpolations of discrete-time higher-order autoregressive sequences. An equation connecting the memory function (the kernel of integral term) and the two-point correlation function is obtained. A condition for stationarity of the process is established. We suggest a method to generate stationary continuous stochastic processes with a prescribed pair correlation function. As an illustration, some examples of numerical simulation of the processes with nonlocal memory are presented.

6.
Phys Rev E Stat Nonlin Soft Matter Phys ; 76(2 Pt 2): 027701, 2007 Aug.
Artículo en Inglés | MEDLINE | ID: mdl-17930183

RESUMEN

We suggest a method for generation of random binary sequences of elements 0 and 1, with prescribed correlation properties. It is based on a modification of the widely used convolution method of constructing continuous random processes. Using this method, a binary sequence with a power-law decaying pair correlator can be easily generated.

7.
Phys Rev E ; 96(1-1): 012158, 2017 Jul.
Artículo en Inglés | MEDLINE | ID: mdl-29347267

RESUMEN

The main goal of this paper is to develop an estimate for the conditional probability function of random stationary ergodic symbolic sequences with elements belonging to a finite alphabet. We elaborate on a decomposition procedure for the conditional probability function of sequences considered to be high-order Markov chains. We represent the conditional probability function as the sum of multilinear memory function monomials of different orders (from zero up to the chain order). This allows us to introduce a family of Markov chain models and to construct artificial sequences via a method of successive iterations, taking into account at each step increasingly high correlations among random elements. At weak correlations, the memory functions are uniquely expressed in terms of the high-order symbolic correlation functions. The proposed method fills the gap between two approaches, namely the likelihood estimation and the additive Markov chains. The obtained results may have applications for sequential approximation of artificial neural network training.

8.
Phys Rev E ; 93(6): 062144, 2016 06.
Artículo en Inglés | MEDLINE | ID: mdl-27415245

RESUMEN

The goal of this paper is to develop an estimate for the entropy of random symbolic sequences with elements belonging to a finite alphabet. As a plausible model, we use the high-order additive stationary ergodic Markov chain with long-range memory. Supposing that the correlations between random elements of the chain are weak, we express the conditional entropy of the sequence by means of the symbolic pair correlation function. We also examine an algorithm for estimating the conditional entropy of finite symbolic sequences. We show that the entropy contains two contributions, i.e., the correlation and the fluctuation. The obtained analytical results are used for numerical evaluation of the entropy of written English texts and DNA nucleotide sequences. The developed theory opens the way for constructing a more consistent and sophisticated approach to describe the systems with strong short-range and weak long-range memory.

9.
Phys Rev E Stat Nonlin Soft Matter Phys ; 72(4 Pt 2): 046138, 2005 Oct.
Artículo en Inglés | MEDLINE | ID: mdl-16383499

RESUMEN

The binary many-step Markov chain with the step-like memory function is considered as a model for the analysis of rank distributions of words in correlated stochastic symbolic systems. We prove that this distribution obeys the power law with the exponent of the order of unity in the case of rather strong persistent correlations. The Zipf law is shown to be valid for the rank distribution of words with lengths about and shorter than the correlation length in the Markov sequence. A self-similarity in the rank distribution with respect to the decimation procedure is observed.

10.
Phys Rev E Stat Nonlin Soft Matter Phys ; 72(2 Pt 2): 026140, 2005 Aug.
Artículo en Inglés | MEDLINE | ID: mdl-16196677

RESUMEN

A theory of additive Markov chains with long-range memory is used to describe the correlation properties of coarse-grained literary texts. The complex structure of the correlations in the texts is revealed. Anti-persistent correlations at small distances, L approximately < 300, and persistent ones at L approximately > 300 define this non-trivial structure. For some concrete examples of literary texts, the memory functions are obtained and their power-law behavior at long distances is disclosed. This property is shown to be a cause of self-similarity of texts with respect to the decimation procedure.

11.
Phys Rev E Stat Nonlin Soft Matter Phys ; 68(6 Pt 1): 061107, 2003 Dec.
Artículo en Inglés | MEDLINE | ID: mdl-14754180

RESUMEN

A theory of systems with long-range correlations based on the consideration of binary N-step Markov chains is developed. In the model, the conditional probability that the ith symbol in the chain equals zero (or unity) is a linear function of the number of unities among the preceding N symbols. The correlation and distribution functions as well as the variance of the number of symbols in the words of arbitrary length L are obtained analytically and numerically. A self-similarity of the studied stochastic process is revealed and the similarity group transformation of the chain parameters is presented. The diffusion Fokker-Planck equation governing the distribution function of the L words is explored. If the persistent correlations are not extremely strong, the distribution function is shown to be the Gaussian with the variance being nonlinearly dependent on L. The applicability of the developed theory to the coarse-grained written and DNA texts is discussed.

12.
Comput Biol Chem ; 53 Pt A: 26-31, 2014 Dec.
Artículo en Inglés | MEDLINE | ID: mdl-25213853

RESUMEN

We analyze the structure of DNA molecules of different organisms by using the additive Markov chain approach. Transforming nucleotide sequences into binary strings, we perform statistical analysis of the corresponding "texts". We develop the theory of N-step additive binary stationary ergodic Markov chains and analyze their differential entropy. Supposing that the correlations are weak we express the conditional probability function of the chain by means of the pair correlation function and represent the entropy as a functional of the pair correlator. Since the model uses two point correlators instead of probability of block occurring, it makes possible to calculate the entropy of subsequences at much longer distances than with the use of the standard methods. We utilize the obtained analytical result for numerical evaluation of the entropy of coarse-grained DNA texts. We believe that the entropy study can be used for biological classification of living species.


Asunto(s)
Bacillus subtilis/genética , Mapeo Cromosómico/estadística & datos numéricos , Drosophila melanogaster/genética , Genoma , Cadenas de Markov , Animales , Secuencia de Bases , Entropía , Datos de Secuencia Molecular , Análisis de Secuencia de ADN
13.
Phys Rev E Stat Nonlin Soft Matter Phys ; 90(5-1): 052106, 2014 Nov.
Artículo en Inglés | MEDLINE | ID: mdl-25493739

RESUMEN

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.

14.
Phys Rev E Stat Nonlin Soft Matter Phys ; 90(5-1): 053305, 2014 Nov.
Artículo en Inglés | MEDLINE | ID: mdl-25493902

RESUMEN

We propose an efficient iterative method for generating random correlated binary sequences with a prescribed correlation function. The method is based on consecutive linear modulations of an initially uncorrelated sequence into a correlated one. Each step of modulation increases the correlations until the desired level has been reached. The robustness and efficiency of the proposed algorithm are tested by generating sequences with inverse power-law correlations. The substantial increase in the strength of correlation in the iterative method with respect to single-step filtering generation is shown for all studied correlation functions. Our results can be used for design of disordered superlattices, waveguides, and surfaces with selective transport properties.

15.
Phys Rev Lett ; 90(11): 110601, 2003 Mar 21.
Artículo en Inglés | MEDLINE | ID: mdl-12688921

RESUMEN

A theory of systems with long-range correlations based on the consideration of binary N-step Markov chains is developed. In our model, the conditional probability that the ith symbol in the chain equals zero (or unity) is a linear function of the number of unities among the preceding N symbols. The correlation and distribution functions as well as the variance of number of symbols in the words of arbitrary length L are obtained analytically and numerically. If the persistent correlations are not extremely strong, the variance is shown to be nonlinearly dependent on L. A self-similarity of the studied stochastic process is revealed. The applicability of the developed theory to the coarse-grained written and DNA texts is discussed.


Asunto(s)
Cadenas de Markov , Modelos Teóricos , Bacillus subtilis/genética , ADN/genética , Genoma Bacteriano
16.
Artículo en Inglés | MEDLINE | ID: mdl-11970027

RESUMEN

We investigate the problem of nonlinear wave propagation in periodic media. Four different classes of periodic nonlinear media are taken into consideration: a nonlinear diatomic elastic chain, modulated nonlinear optical media, a diatomic easy-axis ferromagnetic chain, and an easy-plane antiferromagnet in an external magnetic field. The main result of our work is a qualitative analysis of all kinds of small amplitude soliton excitations with frequencies lying in the gap and near the gap of the linear wave spectrum. We also study the evolution of the system phase portrait and the bifurcation picture of the soliton solutions under changes of the medium parameters.

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