Physiological time-series analysis using approximate entropy and sample entropy

Entropy, as it relates to dynamical systems, is the rate of information production. Methods for estimation of the entropy of a system represented by a time series are not, however, well suited to analysis of the short and noisy data sets encountered in cardiovascular and other biological studies. Pincus introduced approximate entropy (ApEn), a set of measures of system complexity closely related to entropy, which is easily applied to clinical cardiovascular and other time series. ApEn statistics, however, lead to inconsistent results. We have developed a new and related complexity measure, sample entropy (SampEn), and have compared ApEn and SampEn by using them to analyze sets of random numbers with known probabilistic character. We have also evaluated cross-ApEn and cross-SampEn, which use cardiovascular data sets to measure the similarity of two distinct time series. SampEn agreed with theory much more closely than ApEn over a broad range of conditions. The improved accuracy of SampEn statistics should make them useful in the study of experimental clinical cardiovascular and other biological time series.     

基于熵理论的城市生态系统服务流定量评估——以北京市为例

量化评价生态系统的服务功能是合理开发生态资源、实现可持续发展的重要前提.生态系统服务流能够为生态系统服务功能的量化评估提供解决方案.本文以四大生态系统服务类型为框架,构建了城市生态系统服务流评估体系;将生态系统服务流重新分类,引入熵理论,对指标及系统无序度、发展演变方向进行定量评价;并以北京市为研究对象,选择24个因子开展案例研究.结果表明:2004—2015年间,北京城市生态系统服务总熵值为0.794,总熵流为-0.024,系统无序度较大,濒临非健康状态;系统熵值3次取得极值,3个变化周期内系统熵值的年平均变化幅度逐渐增大,说明人为活动对城市生态系统干预强度增大;2007年熵流达到最小值,生态环境质量最高;北京市常住人口总数与城市生态系统熵流拟合函数的决定系数为0.921,说明人口数量与城市生态环境状态有很强的相关性.     

Why use population entropy? It determines the rate of convergence

Summary The population entropy introduced by Demetrius is shown to have a precise dynamical meaning as a measure of convergence rate to the stable age distribution. First the Leslie population model is transformed exactly into a Markov chain on a state space of age-classes. Next the dynamics of convergence from a nonequilibrium state to the stable state are analyzed. The results provide the first clear biological reason why entropy is a broadly useful population statistic.     

干旱半干旱区灌丛斑块与降水量响应关系的熵模型模拟

灌丛斑块分布格局是灌木在干旱缺水条件下对生存环境的自我调节和适应的具体表现。应用熵理论和Klausmier模型,解释了灌丛斑块水分聚集原理并模拟了不同年降水条件下灌丛斑块的最佳面积比值(即最佳灌丛盖度)。研究结果表明:灌丛斑块生物量与其土壤含水量呈反比例函数关系,当生态系统处于稳定状态时(即熵最大状况下),年降水量与灌丛斑块面积比值符合一定的线性关系。研究采用内蒙古草原地区的野外调查数据,获得模型所需参数,进而模拟了不同年降水量条件下灌丛斑块最佳面积比值,研究结果可为半干旱地区植被保护与恢复提供参考。     

The concept of structural entropy in tissue-based diagnosis

The concept of entropy is described and its characteristics discussed as applied in tissue-based diagnosis. The concept of entropy includes at least 2 points of view--thermodynamic and informatics perspectives. Entropy can be defined by various methods: a measure of nonreversible energy or of system heterogeneity or as information content of a message. It is a statistical measure and system feature composed of macrosystems and microsystems. The structural entropy of macrosystems relies on definition of individual events and built-in microsystems. It depends on interaction of events and probability distribution (e.g., Gibbs-Boltzmann). The more generalized q-entropy involves account interaction of neighboring events. The thermodynamic concept of structural entropy can be expanded according to the theorem of Prigogine, introducing entropy flow. In biology, cells usually serve for events in the thermodynamic entropy approach. Entropy has been successfully used to describe tissue sections, nuclei and nuclear substructures such as DNA content, chromosomes and AgNORs. The concept of entropy reveals a close relationship of structural entropy and prognosis-associated diagnosis of malignancies. It is useful in prognosis-associated, tissue-based diagnosis in breast, prostate, bladder and lung cancer and is a promising expansion of image analysis in diagnostic agnosis in breast, prostate, bladder and lung cancer and is a promising expansion of image analysis in diagnostic pathology.     

时空信息熵在延河流域生态可持续性分析中的应用

视流域为一个有生命力的不可分割的有机整体,为表征其生态可持续性,从熵的视角出发,提出结合空间信息熵和时间信息熵的时空信息熵方法。其中空间信息熵用于表征生态系统格局在空间分布的有序程度,时间信息熵用于度量生态系统的动态演变是否有序,时空信息熵方法将格局和动态有机结合,定量分析流域生态系统的可持续性。以延河流域为研究区,基于土地利用数据和归一化植被指数数据,利用时空信息熵方法分析2000-2018年延河流域生态可持续性。结果表明:(1)延河流域生态系统格局朝着有序的方向变化,此间整体处于生长期或恢复期;(2)时间信息熵结果呈现空间异质性,耕地、中低覆盖度草地和其他林地的时间信息熵值较高,生态弹性能力更强;(3)研究区生态可持续性以"强"和"较强"为主(61%),广泛分布在其中部和北部地区,表明流域的生态弹性能力总体增强,生态可持续状况明显改善。对基于熵视角研究生态可持续问题的有益探索,为延河流域及黄土高原其他类似流域的生态保护和修复提供借鉴和参考。     

Sample entropy analysis of cervical neoplasia gene-expression signatures

Background  

We introduce Approximate Entropy as a mathematical method of analysis for microarray data. Approximate entropy is applied here as a method to classify the complex gene expression patterns resultant of a clinical sample set. Since Entropy is a measure of disorder in a system, we believe that by choosing genes which display minimum entropy in normal controls and maximum entropy in the cancerous sample set we will be able to distinguish those genes which display the greatest variability in the cancerous set. Here we describe a method of utilizing Approximate Sample Entropy (ApSE) analysis to identify genes of interest with the highest probability of producing an accurate, predictive, classification model from our data set.     

Maximum-entropy principle: ecological organization and evolution

In the present paper, we have first studied the role of the maximum-entropy principle to explain the concept of organization of a physical system in the decreasing law of entropy with the increase of external constraints imposed on the system. We have then considered an open ecosystem (living) and determined a quantitative measure of ecological organization from the consideration of the thermodynamics of irreversible processes. Finally, we have tried to explain the evolution of the ecosystem in the light of Prigogine’s principle of “order through fluctuation.”     

Parametric scaling from species relative abundances to absolute abundances in the computation of biological diversity: a first proposal using Shannon's entropy

Traditional diversity measures such as the Shannon entropy are generally computed from the species' relative abundance vector of a given community to the exclusion of species' absolute abundances. In this paper, I first mention some examples where the total information content associated with a given community may be more adequate than Shannon's average information content for a better understanding of ecosystem functioning. Next, I propose a parametric measure of statistical information that contains both Shannon's entropy and total information content as special cases of this more general function.     

Vegetational response to change in environment and change in species tolerance with time

Conclusion Applications of system analysis and model simulation can provide for criteria of preservation regulating the processes of entropy and negentropy in ecosystems. Changes in floristic composition and species tolerance may provide information for the necessary dynamics and the cause-effect relationships of vegetation with its environment. Information about the autecology and synecology of species can be utilized in computer programs to measure the impact of man upon the ecosystem. This information will contribute increasingly to wilderness management. Change seems to be of great importance and habitat manipulation through planned large-scale or small-scale disturbance will contribute to new concepts in phytosociology. Research will have to be directed to the total ecosystem and its own environment, and to the most desirable disequilibrium of such an ecosystem. In such research, system analysis and model simulation will play key roles in the future.     

Estimating causal interaction between prefrontal cortex and striatum by transfer entropy

Transfer entropy (TE) is an information-theoretic measure for the investigation of causal interaction between two systems without a requirement of pre-specific interaction model (such as: linear or nonlinear). We introduced an efficient algorithm to calculate TE values between two systems based on observed time signals. By this method, we demonstrated that the TE correctly estimated the coupling strength and the direction of information transmission of two nonlinearly coupled systems. We also calculated TE values of real local field potentials (LFPs) recorded simultaneously in the lateral prefrontal cortex (LPFC) and the striatum of the behavioral monkey, and observed that the TE value from the LPFC to the striatum was stronger than that from the striatum to the LPFC, consistent with anatomical structure between the two areas. Moreover, the TE value dynamically varied dependent on behavior stages of the monkey. These results from simulated and real LFPs data suggested that the TE was able to effectively estimate functional connectivity between different brain regions and characterized their dynamical properties.     

熵权模糊综合评价法在城市生态安全评价中的应用

为了拓新城市复合生态系统生态安全评价方法的研究,根据压力-状态-响应模型,构建了一个有3个要素和33项具体指标的城市生态安全水平度量的指标体系,并运用客观的熵权法赋权、模糊综合评价法以及划分的等级标准对我国5个经济发达城市的生态安全水平进行量化分析.结果表明,苏州市和北京市对“较安全”级别的隶属度最大,分别为0.376和0.286;深圳市、上海市和广州市则处于“临界安全”状态.另外,深圳市具有较大的系统压力,苏州市的系统状态和系统响应表现最优.与其它评价方法相比,该方法评价过程简易,结果定量和相对客观可信.     

Measures of entropy and complexity in altered states of consciousness

Quantification of complexity in neurophysiological signals has been studied using different methods, especially those from information or dynamical system theory. These studies have revealed a dependence on different states of consciousness, and in particular that wakefulness is characterized by a greater complexity of brain signals, perhaps due to the necessity for the brain to handle varied sensorimotor information. Thus, these frameworks are very useful in attempts to quantify cognitive states. We set out to analyze different types of signals obtained from scalp electroencephalography (EEG), intracranial EEG and magnetoencephalography recording in subjects during different states of consciousness: resting wakefulness, different sleep stages and epileptic seizures. The signals were analyzed using a statistical (permutation entropy) and a deterministic (permutation Lempel–Ziv complexity) analytical method. The results are presented in complexity versus entropy graphs, showing that the values of entropy and complexity of the signals tend to be greatest when the subjects are in fully alert states, falling in states with loss of awareness or consciousness. These findings were robust for all three types of recordings. We propose that the investigation of the structure of cognition using the frameworks of complexity will reveal mechanistic aspects of brain dynamics associated not only with altered states of consciousness but also with normal and pathological conditions.     

基于主成分分析和熵权的水库生态系统健康评价——以海南省万宁水库为例

采用主成分分析(PCA)与熵权相结合的新方法,对万宁水库水生态系统健康进行评价,旨在检验该方法是否能解决传统的基于熵权法的水生态系统健康评价方法所存在的赋权重复问题.结果表明: 2010-2012年,万宁水库的水生态系统健康状况整体呈变好趋势;年均水生态系统健康综合指数(EHCI)分别为0.534、0.617、0.634,健康状态评级分别为Ⅲ类(中等)、Ⅱ类(较好)、Ⅱ类(较好).该水库水生态系统健康状况存在季节性差异,但并没有明显的季节性变化规律.从EHCI的整体波动程度来看,其波幅逐渐变小,表明近年来万宁水库的水生态系统趋于相对稳定.新方法与传统方法的指标赋权对比表明,传统方法中相关性较强的溶解氧、化学需氧量、五日生化需要量、铵态氮4项指标的累计权重为0.382,而新方法中仅为0.179;说明PCA的引入有效解决了赋权重复的问题.营养状态指数与EHCI呈显著的负相关关系,说明PCA与熵权结合的新方法在解决了赋权重复的基础上,很好地保证了评价结果的准确性,适用于该水库水生态系统健康评价.
     

Maximum entropy,logistic regression,and species abundance

There is considerable debate about the utility of statistical mechanics in predicting diversity patterns in terms of life history traits. Here, I reflect on this debate and show that a community is controlled by the balance of two opposite forces: the entropic part (the natural tendency of the system to be in the configuration with the highest possible entropy) and environmental, ecological and evolutionary constraints maintaining order (reducing entropy). The Boltzmann distribution law that can be derived from the maximum entropy formalism provides a fundamental model for linking species abundance to life history traits and environmental constraining factors. This model predicts a global pattern of diversity evenness along a latitudinal gradient. Although the Boltzmann distribution and the logistic regression models represent two fundamentally different approaches, the two models have an identical mathematical form. Their identical formalisms facilitate the interpretation of logistic regression models with statistical mechanics, and reveal several limitations of the maximum entropy formalism. I argued that although maximum entropy formalism is a promising tool for modeling species abundances and for linking microscopic quantities of individual life history traits to macroscopic patterns of diversity, it is necessary to revise the Boltzmann distribution law for successful prediction of species abundance.     

Cross entropy minimization in uninvadable states of complex populations

Selection is often. viewed as a process that maximizes the average fitness of a population. However, there are often constraints even on the phenotypic level which may prevent fitness optimization. Consequently, in evolutionary game theory, models of frequency dependent selection are investigated, which focus on equilibrium states that are characterized by stability (or uninvadability) rather than by optimality. The aim of this article is to show that nevertheless there is a biologically meaningful quantity, namely cross (fitness) entropy, which is optimized during the course of evolution: a dynamical model adapted to evolutionary games is presented which has the property that relative entropy decreases monotonically, if the state of a (complex) population is close to an uninvadable state. This result may be interpreted as if evolution has an order stabilizing effect.     

An automatic image registration scheme using Tsallis entropy

We have investigated the registration of mammograms based on the Tsallis entropy using mutual information measure. Tsallis entropy has one more parameter ‘q’ and the values of ‘q’ decide the quality of the registration. Existing Tsallis entropy based algorithms are not automatic as they claimed to be. In this article, an automatic affine image registration based on Tsallis entropy is proposed and its performance is analyzed for clinically acquired mammograms for globally registering them. The accuracy is compared with traditionally used mutual information and normalized mutual information based on Shannon entropy. Our algorithm shows promising results with increased accuracy with reduction in number of evaluations. Further, the need for pre-registration in mammogram is discussed in detail. Through this experiment, it is found that the proposed algorithm is effective enough to replace Shannon and existing Tsallis entropy based affine registration schemes.     

A thermodynamic perspective on food webs: quantifying entropy production within detrital-based ecosystems

Because ecosystems fit so nicely the framework of a "dissipative system", a better integration of thermodynamic and ecological perspectives could benefit the quantitative analysis of ecosystems. One obstacle is that traditional food web models are solely based upon the principles of mass and energy conservation, while the theory of non-equilibrium thermodynamics principally focuses on the concept of entropy. To properly cast classical food web models within a thermodynamic framework, one requires a proper quantification of the entropy production that accompanies resource processing of the food web. Here we present such a procedure, which emphasizes a rigorous definition of thermodynamic concepts (e.g. thermodynamic gradient, disequilibrium distance, entropy production, physical environment) and their correct translation into ecological terms. Our analysis provides a generic way to assess the thermodynamic operation of a food web: all information on resource processing is condensed into a single resource processing constant. By varying this constant, one can investigate the range of possible food web behavior within a given fixed physical environment. To illustrate the concepts and methods, we apply our analysis to a very simple example ecosystem: the detrital-based food web of marine sediments. We examine whether entropy production maximization has any ecological relevance in terms of food web functioning.     

基于熵值法与变异系数的大熊猫分布区生态系统评价

近年来大熊猫栖息地、竹林的面积与野生种群数量均有较大增长,同时栖息地破碎化与局域种群隔离也有加剧趋势.正确认知当前大熊猫分布区生态系统的状态,对于大熊猫保护至关重要.以大熊猫分布区为研究区域,3次全国大熊猫调查为时间节点,依据等级系统理论对分布区及六大山系生态系统进行分解,联合应用熵值法、变异系数、相关性分析,基于全国大熊猫调查数据与相关文献数据,对分布区及六大山系生态系统演化规律进行了研究.发现在生态系统持续改善的背景下,不同山系生态系统的演化与现状存在差异性,六大山系可以分为3个组别,组内山系的生态系统具有较大共性,组间区别较大;同时,栖息地破碎化与局域种群隔离,造成生态系统质量的普遍下降.应因地制宜地制定差异化的生态保护措施,才能更好地实现各山系大熊猫生态系统的持续改善.     

Evolutionary entropy determines invasion success in emergent epidemics

Background

Standard epidemiological theory claims that in structured populations competition between multiple pathogen strains is a deterministic process which is mediated by the basic reproduction number () of the individual strains. A new theory based on analysis, simulation and empirical study challenges this predictor of success.

Principal Findings

We show that the quantity is a valid predictor in structured populations only when size is infinite. In this article we show that when population size is finite the dynamics of infection by multi-strain pathogens is a stochastic process whose outcome can be predicted by evolutionary entropy, S, an information theoretic measure which describes the uncertainty in the infectious age of an infected parent of a randomly chosen new infective. Evolutionary entropy characterises the demographic stability or robustness of the population of infectives. This statistical parameter determines the duration of infection and thus provides a quantitative index of the pathogenicity of a strain. Standard epidemiological theory based on as a measure of selective advantage is the limit as the population size tends to infinity of the entropic selection theory. The standard model is an approximation to the entropic selection theory whose validity increases with population size.

Conclusion

An epidemiological analysis based on entropy is shown to explain empirical observations regarding the emergence of less pathogenic strains of human influenza during the antigenic drift phase. Furthermore, we exploit the entropy perspective to discuss certain epidemiological patterns of the current H1N1 swine ''flu outbreak.