Permutation complexity via duality between values and orderings

We study the permutation complexity of finite-state stationary stochastic processes based on a duality between values and orderings between values. First, we establish a duality between the set of all words of a fixed length and the set of all permutations of the same length. Second, on this basis, we give an elementary alternative proof of the equality between the permutation entropy rate and the entropy rate for a finite-state stationary stochastic processes first proved in [J.M. Amigó, M.B. Kennel, L. Kocarev, The permutation entropy rate equals the metric entropy rate for ergodic information sources and ergodic dynamical systems, Physica D 210 (2005) 77-95]. Third, we show that further information on the relationship between the structure of values and the structure of orderings for finite-state stationary stochastic processes beyond the entropy rate can be obtained from the established duality. In particular, we prove that the permutation excess entropy is equal to the excess entropy, which is a measure of global correlation present in a stationary stochastic process, for finite-state stationary ergodic Markov processes.     

Permutation entropy and detrend fluctuation analysis for the natural complexity of cardiac heart interbeat signals

We compute fractal dimension and permutation entropy for healthy and people who have experienced heart failure. Our result shows that permutation entropy is a suitable approach as well as detrend fluctuation analysis (DFA). The result of DFA shows that the fractal dimensions for healthy and heart failure are different as well as the permutation entropy result. The fluctuation value for permutation entropy for an individual who has experienced heart failure is bigger than for a healthy person. There is some specific change in the interbeat signal of a person who has experienced heart failure, but there is not previous trend for a healthy person.     

Entropy as a Topological Operad Derivation

We share a small connection between information theory, algebra, and topology—namely, a correspondence between Shannon entropy and derivations of the operad of topological simplices. We begin with a brief review of operads and their representations with topological simplices and the real line as the main example. We then give a general definition for a derivation of an operad in any category with values in an abelian bimodule over the operad. The main result is that Shannon entropy defines a derivation of the operad of topological simplices, and that for every derivation of this operad there exists a point at which it is given by a constant multiple of Shannon entropy. We show this is compatible with, and relies heavily on, a well-known characterization of entropy given by Faddeev in 1956 and a recent variation given by Leinster.     

Scaling properties of the measure of constant topological entropy

The topological entropy for some families of one-dimensional unimodal maps is studied. By arranging the windows of constant topological entropy in a binary tree, we have obtained the total measure of these windows. The scaling properties of this measure are studied.     

Evolution patterns and iterative maps

A simple mathematical scheme for the time evolution of systems in the space of possible structures, with selection of structures according to pre-determined code rules, is proposed. Patterns of evolution are obtained. As a result of selection of structures the entropy (topological entropy) decreases with time. An application to iterative maps is described.     

On the dynamics of the q-deformed logistic map

We analyze the q-deformed logistic map, where the q-deformation follows the scheme inspired in the Tsallis q-exponential function. We compute the topological entropy of the dynamical system, obtaining the parametric region in which the topological entropy is positive and hence the region in which chaos in the sense of Li and Yorke exists. In addition, it is shown the existence of the so-called Parrondo's paradox where two simple maps are combined to give a complicated dynamical behavior.     

Rigorous bounds on the fast dynamo growth rate involving topological entropy

The fast dynamo growth rate for aC ^k+1 map or flow is bounded above by topological entropy plus a 1/k correction. The proof uses techniques of random maps combined with a result of Yomdin relating curve growth to topological entropy. This upper bound implies the following anti-dynamo theorem: inC systems fast dynamo action is not possible without the presence of chaos. In addition topological entropy is used to construct a lower bound for the fast dynamo growth rate in the caseR _m =.This author is supported by an NSF postdoctoral fellowshipThis author is partially supported by an NSF grant     

Symbolic transfer entropy rate is equal to transfer entropy rate for bivariate finite-alphabet stationary ergodic Markov processes

Transfer entropy is a measure of the magnitude and the direction of information flow between jointly distributed stochastic processes. In recent years, its permutation analogues are considered in the literature to estimate the transfer entropy by counting the number of occurrences of orderings of values, not the values themselves. It has been suggested that the method of permutation is easy to implement, computationally low cost and robust to noise when applying to real world time series data. In this paper, we initiate a theoretical treatment of the corresponding rates. In particular, we consider the transfer entropy rate and its permutation analogue, the symbolic transfer entropy rate, and show that they are equal for any bivariate finite-alphabet stationary ergodic Markov process. This result is an illustration of the duality method introduced in [T. Haruna, K. Nakajima, Physica D 240, 1370 (2011)]. We also discuss the relationship among the transfer entropy rate, the time-delayed mutual information rate and their permutation analogues.     

Forbidden ordinal patterns in higher dimensional dynamics

Forbidden ordinal patterns are ordinal patterns (or rank blocks) that cannot appear in the orbits generated by a map taking values on a linearly ordered space, in which case we say that the map has forbidden patterns. Once a map has a forbidden pattern of a given length L₀, it has forbidden patterns of any length L≥L₀ and their number grows superexponentially with L. Using recent results on topological permutation entropy, in this paper we study the existence and some basic properties of forbidden ordinal patterns for self-maps on n-dimensional intervals. Our most applicable conclusion is that expansive interval maps with finite topological entropy have necessarily forbidden patterns, although we conjecture that this is also the case under more general conditions. The theoretical results are nicely illustrated for n=2 both using the naive counting estimator for forbidden patterns and Chao’s estimator for the number of classes in a population. The robustness of forbidden ordinal patterns against observational white noise is also illustrated.     

Fine-grained permutation entropy as a measure of natural complexity for time series

In a recent paper [2002 Phys. Rev. Lett. 88 174102], Bandt and Pompe propose permutation entropy (PE) as a natural complexity measure for arbitrary time series which may be stationary or nonstationary, deterministic or stochastic. Their method is based on a comparison of neighbouring values. This paper further develops PE, and proposes the concept of fine-grained PE (FGPE) defined by the order pattern and magnitude of the difference between neighbouring values. This measure excludes the case where vectors with a distinct appearance are mistakenly mapped onto the same permutation type, and consequently FGPE becomes more sensitive to the dynamical change of time series than does PE, according to our simulation and experimental results.     

一个二次多项式混沌系统的均匀化及其熵分析

利用已有理论给出了一个二次多项式混沌系统,证明了该系统与Tent映射拓扑共轭,给出了该混沌系统的概率密度函数;并根据此概率密度函数,得到了轨道均匀分布的反三角函数映射;对均匀化前后的混沌系统在不同参数下产生序列的信息熵、Kolmogorov熵、离散熵的特性进行了分析,结果显示均匀化后产生的混沌序列混沌程度不改变且具有更好的均匀性.     

Universal encoding for unimodal maps

The standard encoding procedure to describe the chaotic orbits of unimodal maps is accurately investigated. We show that the grammatical rules of the underlying language can be easily classified in a compact form by means of a universal parameter . Two procedures to construct finite graphs which approximate non-Markovian cases are discussed, showing also the intimate relation with the corresponding construction of approximate Markov partitions. The convergence of the partial estimates of the topological entropy is discussed, proving that the error decreases exponentially with the length of the sequences considered. The rate is shown to coincide with the topological entropyh itself. Finally, a superconvergent method to estimateh is introduced.     

Quantifying chaos for ecological stoichiometry

The theory of ecological stoichiometry considers ecological interactions among species with different chemical compositions. Both experimental and theoretical investigations have shown the importance of species composition in the outcome of the population dynamics. A recent study of a theoretical three-species food chain model considering stoichiometry [B. Deng and I. Loladze, Chaos 17, 033108 (2007)] shows that coexistence between two consumers predating on the same prey is possible via chaos. In this work we study the topological and dynamical measures of the chaotic attractors found in such a model under ecological relevant parameters. By using the theory of symbolic dynamics, we first compute the topological entropy associated with unimodal Poincare? return maps obtained by Deng and Loladze from a dimension reduction. With this measure we numerically prove chaotic competitive coexistence, which is characterized by positive topological entropy and positive Lyapunov exponents, achieved when the first predator reduces its maximum growth rate, as happens at increasing δ1. However, for higher values of δ1 the dynamics become again stable due to an asymmetric bubble-like bifurcation scenario. We also show that a decrease in the efficiency of the predator sensitive to prey's quality (increasing parameter ζ) stabilizes the dynamics. Finally, we estimate the fractal dimension of the chaotic attractors for the stoichiometric ecological model.     

Permutation approach to finite-alphabet stationary stochastic processes based on the duality between values and orderings

The duality between values and orderings is a powerful tool to discuss relationships between various information-theoretic measures and their permutation analogues for discrete-time finite-alphabet stationary stochastic processes (SSPs). Applying it to output processes of hidden Markov models with ergodic internal processes, we have shown in our previous work that the excess entropy and the transfer entropy rate coincide with their permutation analogues. In this paper, we discuss two permutation characterizations of the two measures for general ergodic SSPs not necessarily having the Markov property assumed in our previous work. In the first approach, we show that the excess entropy and the transfer entropy rate of an ergodic SSP can be obtained as the limits of permutation analogues of them for the N-th order approximation by hidden Markov models, respectively. In the second approach, we employ the modified permutation partition of the set of words which considers equalities of symbols in addition to permutations of words. We show that the excess entropy and the transfer entropy rate of an ergodic SSP are equal to their modified permutation analogues, respectively.     

On the Continuity of the SRB Entropy for Endomorphisms

We consider classes of dynamical systems admitting Markov induced maps. Under general assumptions, which in particular guarantee the existence of SRB measures, we prove that the entropy of the SRB measure varies continuously with the dynamics. We apply our result to a vast class of non-uniformly expanding maps of a compact manifold and prove the continuity of the entropy of the SRB measure. In particular, we show that the SRB entropy of Viana maps varies continuously with the map.     

An information-based tool for inferring the nature of deterministic sources in real data

The scope of the paper is to find signatures of the forces controlling complex systems modeled by Langevin equations, by recourse to information-theory quantifiers. We evaluate in detail the permutation entropy (PE) and the permutation statistical complexity (PSC) measures for two similarity classes of stochastic models, characterized by either drifting or reversion properties, and employ them as a reference basis for the inspection of real series. New relevant model parameters arise as compared to standard entropy measures. We determine the normalized PE and PSC curves according to them over a range of permutation orders n$n$ and infer the limiting measures for arbitrary large order. We found that the PSC measure is strongly scale-dependent, with systems of the drifting class showing crossovers as n$n$ increases. This result gives warning signs about the proper interpretation of finite-scale analysis of complexity in general processes. Conversely, a key n$n$-invariant outcome arises, that is, the normalized PE values for both classes of models keep complementary for any n$n$. We argue that both PE and PSC measures enable one to unravel the nature (drifting or restoring) of the deterministic sources underlying complexity. We conclude by investigating the presence of local trends in stock price series.     

乙醇含量拉曼光谱检测中噪声与背景同时消除方法研究

乙醇含量拉曼光谱检测中,拉曼光谱信号中的各种噪声及光谱荧光造成的基线漂移和样品池背景等,影响了校正模型的预测精度。利用总体平均经验模态分解,将光谱信号分解成若干无模态混叠的内在模式分量,根据排列熵的信号随机性检测判据判断出代表背景信息和噪声信息的内在模式分量,将其置零即可同时消除拉曼光谱中的噪声与背景。将总体平均经验模态分解与排列熵相结合的预处理方法应用于乙醇含量的拉曼光谱检测中,并与小波变换和平均平滑滤波做了对比。实验结果表明：应用总体平均经验模态分解与排列熵相结合的方法能够有效的同时消除乙醇含量拉曼光谱检测中的噪声和背景信息,提高校正模型的预测精度,且使用简便,无需参数设置,对乙醇含量拉曼光谱检测具有实用价值。     

TOLOMEO,a Novel Machine Learning Algorithm to Measure Information and Order in Correlated Networks and Predict Their State

We present ToloMEo (TOpoLogical netwOrk Maximum Entropy Optimization), a program implemented in C and Python that exploits a maximum entropy algorithm to evaluate network topological information. ToloMEo can study any system defined on a connected network where nodes can assume N discrete values by approximating the system probability distribution with a Pottz Hamiltonian on a graph. The software computes entropy through a thermodynamic integration from the mean-field solution to the final distribution. The nature of the algorithm guarantees that the evaluated entropy is variational (i.e., it always provides an upper bound to the exact entropy). The program also performs machine learning, inferring the system’s behavior providing the probability of unknown states of the network. These features make our method very general and applicable to a broad class of problems. Here, we focus on three different cases of study: (i) an agent-based model of a minimal ecosystem defined on a square lattice, where we show how topological entropy captures a crossover between hunting behaviors; (ii) an example of image processing, where starting from discretized pictures of cell populations we extract information about the ordering and interactions between cell types and reconstruct the most likely positions of cells when data are missing; and (iii) an application to recurrent neural networks, in which we measure the information stored in different realizations of the Hopfield model, extending our method to describe dynamical out-of-equilibrium processes.     

Calculating Topological Entropy

The attempt to find effective algorithms for calculating the topological entropy of piecewise monotone maps of the interval having more than three monotone pieces has proved to be a difficult problem. The algorithm introduced here is motivated by the fact that if f: [0, 1] → [0, 1] is a piecewise monotone map of the unit interval into itself, thenh(f)=lim_n→∞ (1/n) log Var(f ⁿ), where h(f) is the topological entropy off, and Var(f ⁿ) is the total variation off ⁿ. We show that it is not feasible to use this formula directly to calculate numerically the topological entropy of a piecewise monotone function, because of the slow convergence. However, a close examination of the reasons for this failure leads ultimately to the modified algorithm which is presented in this paper. We prove that this algorithm is equivalent to the standard power method for finding eigenvalues of matrices (with shift of origin) in those cases for which the function is Markov, and present encouraging experimental evidence for the usefulness of the algorithm in general by applying it to several one-parameter families of test functions.