Measurement scale in maximum entropy models of species abundance

The consistency of the species abundance distribution across diverse communities has attracted widespread attention. In this paper, I argue that the consistency of pattern arises because diverse ecological mechanisms share a common symmetry with regard to measurement scale. By symmetry, I mean that different ecological processes preserve the same measure of information and lose all other information in the aggregation of various perturbations. I frame these explanations of symmetry, measurement, and aggregation in terms of a recently developed extension to the theory of maximum entropy. I show that the natural measurement scale for the species abundance distribution is log-linear: the information in observations at small population sizes scales logarithmically and, as population size increases, the scaling of information grades from logarithmic to linear. Such log-linear scaling leads naturally to a gamma distribution for species abundance, which matches well with the observed patterns. Much of the variation between samples can be explained by the magnitude at which the measurement scale grades from logarithmic to linear. This measurement approach can be applied to the similar problem of allelic diversity in population genetics and to a wide variety of other patterns in biology.     

The common patterns of nature

We typically observe large‐scale outcomes that arise from the interactions of many hidden, small‐scale processes. Examples include age of disease onset, rates of amino acid substitutions and composition of ecological communities. The macroscopic patterns in each problem often vary around a characteristic shape that can be generated by neutral processes. A neutral generative model assumes that each microscopic process follows unbiased or random stochastic fluctuations: random connections of network nodes; amino acid substitutions with no effect on fitness; species that arise or disappear from communities randomly. These neutral generative models often match common patterns of nature. In this paper, I present the theoretical background by which we can understand why these neutral generative models are so successful. I show where the classic patterns come from, such as the Poisson pattern, the normal or Gaussian pattern and many others. Each classic pattern was often discovered by a simple neutral generative model. The neutral patterns share a special characteristic: they describe the patterns of nature that follow from simple constraints on information. For example, any aggregation of processes that preserves information only about the mean and variance attracts to the Gaussian pattern; any aggregation that preserves information only about the mean attracts to the exponential pattern; any aggregation that preserves information only about the geometric mean attracts to the power law pattern. I present a simple and consistent informational framework of the common patterns of nature based on the method of maximum entropy. This framework shows that each neutral generative model is a special case that helps to discover a particular set of informational constraints; those informational constraints define a much wider domain of non‐neutral generative processes that attract to the same neutral pattern.     

混交林测树因子概率分布模型的构建及应用

基于最大熵原理,针对目前对混交林测树因子概率分布模型研究的不足,提出了联合最大熵概率密度函数,该函数具有如下特点：1）函数的每一组成部分都是相互联系的最大熵函数,故可以综合混交林各主要组成树种测树因子的概率分布信息;2）函数是具有双权重的概率表达式,能体现混交林结构复杂的特点,在最大限度地利用混交林每一主要树种测树因子概率分布信息的同时,还能精确地全面反映混交林测树因子概率分布规律;3）函数的结构简洁、性能优良.用天目山自然保护区的混交林样地对混交林测树因子概率分布模型进行了应用与检验,结果表明：模型的拟合精度(R²=0.9655)与检验精度(R²=
0.9772)都较高.说明联合最大熵概率密度函数可以作为混交林测树因子概率分布模型,为全面了解混交林林分结构提供了一种可行的方法.     

Stimulus-dependent Maximum Entropy Models of Neural Population Codes

Neural populations encode information about their stimulus in a collective fashion, by joint activity patterns of spiking and silence. A full account of this mapping from stimulus to neural activity is given by the conditional probability distribution over neural codewords given the sensory input. For large populations, direct sampling of these distributions is impossible, and so we must rely on constructing appropriate models. We show here that in a population of 100 retinal ganglion cells in the salamander retina responding to temporal white-noise stimuli, dependencies between cells play an important encoding role. We introduce the stimulus-dependent maximum entropy (SDME) model—a minimal extension of the canonical linear-nonlinear model of a single neuron, to a pairwise-coupled neural population. We find that the SDME model gives a more accurate account of single cell responses and in particular significantly outperforms uncoupled models in reproducing the distributions of population codewords emitted in response to a stimulus. We show how the SDME model, in conjunction with static maximum entropy models of population vocabulary, can be used to estimate information-theoretic quantities like average surprise and information transmission in a neural population.     

Invariant properties in coevolutionary networks of plant–animal interactions

Plant–animal mutualistic networks are interaction webs consisting of two sets of entities, plant and animal species, whose evolutionary dynamics are deeply influenced by the outcomes of the interactions, yielding a diverse array of coevolutionary processes. These networks are two‐mode networks sharing many common properties with others such as food webs, social, and abiotic networks. Here we describe generalized patterns in the topology of 29 plant–pollinator and 24 plant–frugivore networks in natural communities. Scale‐free properties have been described for a number of biological, social, and abiotic networks; in contrast, most of the plant–animal mutualistic networks (65.6%) show species connectivity distributions (number of links per species) with a power‐law regime but decaying as a marked cut‐off, i.e. truncated power‐law or broad‐scale networks and few (22.2%) show scale‐invariance. We hypothesize that plant–animal mutualistic networks follow a build‐up process similar to complex abiotic nets, based on the preferential attachment of species. However, constraints in the addition of links such as morphological mismatching or phenological uncoupling between mutualistic partners, restrict the number of interactions established, causing deviations from scale‐invariance. This reveals generalized topological patterns characteristic of self‐organized complex systems. Relative to scale‐invariant networks, such constraints may confer higher robustness to the loss of keystone species that are the backbone of these webs.     

Integrating macroecological metrics and community taxonomic structure

We extend macroecological theory based on the maximum entropy principle from species level to higher taxonomic categories, thereby predicting distributions of species richness across genera or families and the dependence of abundance and metabolic rate distributions on taxonomic tree structure. Predictions agree with qualitative trends reported in studies on hyper‐dominance in tropical tree species, mammalian body size distributions and patterns of rarity in worldwide plant communities. Predicted distributions of species richness over genera or families for birds, arthropods, plants and microorganisms are in excellent agreement with data. Data from an intertidal invertebrate community, but not from a dispersal‐limited forest, are in excellent agreement with a predicted new relationship between body size and abundance. Successful predictions of the original species level theory are unmodified in the extended theory. By integrating macroecology and taxonomic tree structure, maximum entropy may point the way towards a unified framework for understanding phylogenetic community structure.     

Combining mechanism and drift in community ecology: a novel statistical mechanics approach

A key challenge for models of community ecology is to combine deterministic mechanism and stochastic drift in a systematic, transparent and tractable manner. Another challenge is to explain and unify different ecological patterns, hitherto modelled in isolation, within a single modelling framework. Here, we show that statistical mechanics provides an effective way to meet both challenges. We apply the statistical principle of maximum entropy (MaxEnt) to a simple resource-based, non-neutral model of a plant community. In contrast to previous ecological applications of MaxEnt, our use of MaxEnt emphasises its theoretical basis in the combinatorics of sampling frequencies, an approach that clarifies its ecological interpretation. In this approach, mechanism and drift are identified, respectively, with ecological resource constraints and entropy maximization. We obtain realistic predictions for species abundance distributions as well as contrasting stability-diversity relationships at community and population levels. The model also predicts critical behaviour that may provide a basis for understanding desertification and other ecological tipping points. Our results complement and extend previous ecological applications of MaxEnt to new areas of community ecology, and further illustrate MaxEnt as a powerful yet simple modelling tool for combining mechanism and drift in a way that unifies disparate ecological patterns.     

Employing relative entropy techniques for assessing modifications in animal behavior

In order to make quantitative statements regarding behavior patterns in animals, it is important to establish whether new observations are statistically consistent with the animal's equilibrium behavior. For example, traumatic stress from the presence of a telemetry transmitter may modify the baseline behavior of an animal, which in turn can lead to a bias in results. From the perspective of information theory such a bias can be interpreted as the amount of information gained from a new measurement, relative to an existing equilibrium distribution. One important concept in information theory is the relative entropy, from which we develop a framework for quantifying time-dependent differences between new observations and equilibrium. We demonstrate the utility of the relative entropy by analyzing observed speed distributions of Pacific bluefin tuna, recorded within a 48-hour time span after capture and release. When the observed and equilibrium distributions are gaussian, we show that the tuna's behavior is modified by traumatic stress, and that the resulting modification is dominated by the difference in central tendencies of the two distributions. Within a 95% confidence level, we find that the tuna's behavior is significantly altered for approximately 5 hours after release. Our analysis reveals a periodic fluctuation in speed corresponding to the moment just before sunrise on each day, a phenomenon related to the tuna's daily diving pattern that occurs in response to changes in ambient light.     

河南省旬日照时数的标度不变性分析

利用河南省107个气象观测站1961-2012年逐旬日照时数资料,分析河南省旬日照时数的时空变化特征,并利用去趋势波动分析方法探讨了河南省旬日照时数的标度不变性.结果表明： 研究期间,河南省逐旬日照时数及其站间均方差的平均值分别为57.90和9.18h,其概率分布均不服从正态分布.河南省逐旬日照时数累积离差具有显著上升趋势,而逐旬日照时数站间均方差的累积离差具有阶段性变化特征,在阶段内均具有显著的线性变化趋势.河南省逐站旬日照时数的标度指数均在0.5以上,表明时间序列具有持久性.河南省旬日照时数标度指数空间变异性较小,具有正态分布的特征.     

Consequences of the fractal architecture of trees on their structural measures

While the mechanics of trees are well known, a systematic and comprehensive study of the mechanical consequences of a tree's fractal structure has been lacking. Here, we analyze the structure of botanical trees using computer modeling and show that many relevant measures of support throughout all the branches of a tree follow specific patterns which can be described by characteristic probability distributions and well-defined spatial relationships. Most notably, moments, forces, and axial and shear stresses throughout the different branches all exhibit power-law distributions. These results suggest a new approach to the study of the mechanics of trees, one accounting for the implications of the above results.     

Locally Weighted Score Estimation for Quantile Classification in Binary Regression Models

One common use of binary response regression methods is classification based on an arbitrary probability threshold dictated by the particular application. Since this is given to us a priori, it is sensible to incorporate the threshold into our estimation procedure. Specifically, for the linear logistic model, we solve a set of locally weighted score equations, using a kernel-like weight function centered at the threshold. The bandwidth for the weight function is selected by cross validation of a novel hybrid loss function that combines classification error and a continuous measure of divergence between observed and fitted values; other possible cross-validation functions based on more common binary classification metrics are also examined. This work has much in common with robust estimation, but differs from previous approaches in this area in its focus on prediction, specifically classification into high- and low-risk groups. Simulation results are given showing the reduction in error rates that can be obtained with this method when compared with maximum likelihood estimation, especially under certain forms of model misspecification. Analysis of a melanoma dataset is presented to illustrate the use of the method in practice.     

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.     

Size-distribution analysis of macromolecules by sedimentation velocity ultracentrifugation and lamm equation modeling

A new method for the size-distribution analysis of polymers by sedimentation velocity analytical ultracentrifugation is described. It exploits the ability of Lamm equation modeling to discriminate between the spreading of the sedimentation boundary arising from sample heterogeneity and from diffusion. Finite element solutions of the Lamm equation for a large number of discrete noninteracting species are combined with maximum entropy regularization to represent a continuous size-distribution. As in the program CONTIN, the parameter governing the regularization constraint is adjusted by variance analysis to a predefined confidence level. Estimates of the partial specific volume and the frictional ratio of the macromolecules are used to calculate the diffusion coefficients, resulting in relatively high-resolution sedimentation coefficient distributions c(s) or molar mass distributions c(M). It can be applied to interference optical data that exhibit systematic noise components, and it does not require solution or solvent plateaus to be established. More details on the size-distribution can be obtained than from van Holde-Weischet analysis. The sensitivity to the values of the regularization parameter and to the shape parameters is explored with the help of simulated sedimentation data of discrete and continuous model size distributions, and by applications to experimental data of continuous and discrete protein mixtures.     

The historical assembly of continental biotas: Late Quaternary range-shifting, areas of endemism, and bio-geographic structure in the North American mammal fauna

A controversial question in biogeography and ecology involves the extent to which vicariance and dispersal interact to determine the structure of continental biotic assemblages, Accumulating evidence of" distributional changes during the past 40 000 yr (Late Quaternary) has suggested to ecologists that changes in geographic ranges during the Pleistocene were of sufficient magnitude to erode prior associations between earth and biotic evolution in continental biotas. This paper first argues that this question can only he addressed by examining the magnitude of Late Quaternary range-shifting at the spatial scale established within the framework of historical historical geography (e.g., areas of endemism) rather than that of ecology (e.g., local community assemblages); and second reassesses patterns of range-shifting in the FAUNMAP data base recording Late Quaternary distributions of North American mammals. At the scale of geo-morphological provinces. North American rodents have exhibited highly stable distributions during this time frame, suggesting that previous inferences drawn from analyses of stability at a local community scale are not relevant to questions of congruence between earth and biotic history at regional or continental scales, A comprehensive understanding of processes underlying the assembly of continental biotas still requires incorporation of biogeographic patterns developed well before episodes of Late Quaternary climatic turbulence.     

Masting,mixtures and modes: are two models better than one?

A key hypothesis in population ecology is that synchronous and intermittent seed production, known as mast seeding, is driven by the alternating allocation of carbohydrates and mineral nutrients between growth and reproduction in different years, i.e. ‘resource switching’. Such behaviour may ultimately generate bimodal distributions of long‐term flower and seed production, and evidence of these patterns has been taken to support the resource switching hypothesis. Here, we show how a widely‐used statistical test of bimodality applied by many studies in different ecological contexts may fail to reject the null hypothesis that focal probability distributions are unimodal. Using data from five tussock grass species in South Island, New Zealand, we find clear evidence of bimodality only when flowering patterns are analyzed with probabilistic mixture models. Mixture models provide a theory oriented framework for testing hypotheses of mast seeding patterns, enabling the different responses underlying medium‐ and high‐ versus non‐ and low‐flowering years to be modelled more realistically by associating these with distinct probability distributions. Coupling theoretical expectations with more rigorous statistical approaches will empower ecologists to reject null hypotheses more often.     

Semiparametric transformation models for multiple continuous biomarkers in ROC analysis

Recent technological advances continue to provide noninvasive and more accurate biomarkers for evaluating disease status. One standard tool for assessing the accuracy of diagnostic tests is the receiver operating characteristic (ROC) curve. Few statistical methods exist to accommodate multiple continuous‐scale biomarkers in the framework of ROC analysis. In this paper, we propose a method to integrate continuous‐scale biomarkers to optimize classification accuracy. Specifically, we develop semiparametric transformation models for multiple biomarkers. We assume that unknown and marker‐specific transformations of biomarkers follow a multivariate normal distribution. Our models accommodate biomarkers subject to limits of detection and account for the dependence among biomarkers by including a subject‐specific random effect. We also propose a diagnostic measure using an optimal linear combination of the transformed biomarkers. Our diagnostic rule does not depend on any monotone transformation of biomarkers and is not sensitive to extreme biomarker values. Nonparametric maximum likelihood estimation (NPMLE) is used for inference. We show that the parameter estimators are asymptotically normal and efficient. We illustrate our semiparametric approach using data from the Endometriosis, Natural History, Diagnosis, and Outcomes (ENDO) study.     

The maximum entropy formalism and the idiosyncratic theory of biodiversity

Why does the neutral theory, which is based on unrealistic assumptions, predict diversity patterns so accurately? Answering questions like this requires a radical change in the way we tackle them. The large number of degrees of freedom of ecosystems pose a fundamental obstacle to mechanistic modelling. However, there are tools of statistical physics, such as the maximum entropy formalism (MaxEnt), that allow transcending particular models to simultaneously work with immense families of models with different rules and parameters, sharing only well‐established features. We applied MaxEnt allowing species to be ecologically idiosyncratic, instead of constraining them to be equivalent as the neutral theory does. The answer we found is that neutral models are just a subset of the majority of plausible models that lead to the same patterns. Small variations in these patterns naturally lead to the main classical species abundance distributions, which are thus unified in a single framework.     

生态功能区划理论研究进展

作为生态系统管理的重要手段,开展科学合理的生态功能区划,已成为世界各国走向可持续发展所面临的关键挑战之一。生态功能区划针对一定区域内自然地理环境分异性、生态系统多样性、以及经济与社会发展不均衡性的现状,结合自然资源保护和可持续开发利用的思想,整合与分异生态系统服务功能对区域人类活动影响的生态敏感性,将区域空间划分为不同生态功能区的研究过程。生态功能区划反映了基于景观特征的主要生态模式,强调了不同时空尺度的景观异质性。通过梳理生态功能区划的概念与内涵、形成与发展及其理论基础,提出了生态功能区划是以恢复区域持续性、完整性的生态系统健康为目标,基于区域的自然地理背景,界定生态功能分区及其子系统的边界,结合区域水陆生态系统、社会经济与土地利用的现状评价与问题诊断,识别生态系统空间格局的分布特征、生态过程的关键因子以及动态演替的驱动因子,明确影响生态系统服务功能的景观格局与结构、景观过程与功能以及景观动态变化,构建生态功能区划的指标体系与技术体系,实现生态功能多级区划,并为决策者更为全面和综合地开展生态系统管理提供科学依据。