77 research outputs found
Binary N-Step Markov Chain as an Exactly Solvable Model of Long-Range Correlated Systems
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 i-th symbol in the chain equals zero (or unity) is a
linear function of the number of unities among the preceding N symbols. The
model allows exact analytical treatment. 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. A self-similarity
of the studied stochastic process is revealed and the similarity transformation
of the chain parameters is presented. The diffusion 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.Comment: LaTeX2e, 16 pages, 9 figure
Entropy of random symbolic high-order bilinear Markov chains
The main goal of this paper is to develop an estimate for the entropy of
random stationary ergodic symbolic sequences with elements belonging to a
finite alphabet. We present here the detailed analytical study of the entropy
for the high-order Markov chain in the bilinear approximation. The appendix
contains a short comprehensive introduction into the subject of study.Comment: 8 pages. arXiv admin note: text overlap with arXiv:1412.369
Decomposition of conditional probability for high-order symbolic Markov chains
The main goal of the 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 a decomposition procedure
for the conditional probability function of sequences considered as the
high-order Markov chains. We represent the conditional probability function as
the sum of multi-linear memory function monomials of different orders (from
zero up to the chain order). This allows us to construct artificial sequences
by method of successive iterations taking into account at each step of
iterations increasingly more 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 up the gap
between two approaches: the likelihood estimation and the additive Markov
chains. The obtained results might be used for sequential approximation of
artificial neural networks training.Comment: 9 pages, 3 figure
Entropy of finite random binary sequences with weak long-range correlations
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.Comment: 9 pages, 4 figures. arXiv admin note: substantial text overlap with
arXiv:1411.2761, arXiv:1412.369
Integrable order parameter dynamics of globally coupled oscillators
We study the nonlinear dynamics of globally coupled nonidentical oscillators
in the framework of two order parameter (mean field and amplitude-frequency
correlator) reduction. The main result of the paper is the exact solution of
the corresponding nonlinear system on an attracting manifold. We present a
complete classification of phase portraits and bifurcations, obtain explicit
expressions for invariant manifolds (a limit cycle among them) and derive
analytical solutions for arbitrary initial data and different regimes
Rank distributions of words in additive many-step Markov chains and the Zipf law
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
stochastic symbolic dynamical systems. We prove that the envelope curve for
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.Comment: 4pages, 3 figure
Adiabatic dynamics of one-dimensional classical Hamiltonian dissipative systems
We give an example of a simple mechanical system described by the generalized
harmonic oscillator equation, which is a basic model in discussion of the
adiabatic dynamics and geometric phase. This system is a linearized plane
pendulum with the slowly varying mass and length of string and the suspension
point moving at a slowly varying speed, the simplest system with broken
-invariance. The paradoxical character of the presented results is that the
same Hamiltonian system, the generalized harmonic oscillator in our case, is
canonically equivalent to two different systems: the usual plane mathematical
pendulum and the damped harmonic oscillator. This once again supports the
important mathematical conclusion, not widely accepted in physical community,
of no difference between the dissipative and Hamiltonian 1D systems, which
stems from the Sonin theorem that any Newtonian second order differential
equation with a friction of general nature may be presented in the form of the
Lagrange equation.Comment: 12 pages, 1 figu
Continuous stochastic processes with non-local memory
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 expression for the higher-order transition
probability function and 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 prescribed pair correlation function. As illustration, some
examples of numerical simulation of the processes with non-local memory are
presented.Comment: 7 pages, 2 figure
Memory Functions of the Additive Markov chains: Applications to Complex Dynamic Systems
A new approach to describing correlation properties of complex dynamic
systems with long-range memory based on a concept of additive Markov chains
(Phys. Rev. E 68, 061107 (2003)) is developed. An equation connecting a memory
function of the chain and its correlation function is presented. This equation
allows reconstructing the memory function using the correlation function of the
system. Thus, we have elaborated a novel method to generate a sequence with
prescribed correlation function. Effectiveness and robustness of the proposed
method is demonstrated by simple model examples. Memory functions of concrete
coarse-grained literary texts are found and their universal power-law behavior
at long distances is revealed.Comment: 5 pages, 5 figures, changes of minor nature, 1 figure adde
Equivalence of Markov's Symbolic Sequences to Two-Sided Chains
A new object of the probability theory, two-sided chain of events (symbols),
is introduced. A theory of multi-steps Markov chains with long-range memory,
proposed earlier in Phys. Rev. E 68, 06117 (2003), is developed and used to
establish the correspondence between these chains and two-sided ones. The
Markov chain is proved to be statistically equivalent to the definite two-sided
one and vice versa. The results obtained for the binary chains are generalized
to the chains taking on the arbitrary number of states.Comment: 5 page
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