Entropic Dynamics on Gibbs Statistical Manifolds

Entropic dynamics is a framework in which the laws of dynamics are derived as an application of entropic methods of inference. Its successes include the derivation of quantum mechanics and quantum field theory from probabilistic principles. Here, we develop the entropic dynamics of a system, the state of which is described by a probability distribution. Thus, the dynamics unfolds on a statistical manifold that is automatically endowed by a metric structure provided by information geometry. The curvature of the manifold has a significant influence. We focus our dynamics on the statistical manifold of Gibbs distributions (also known as canonical distributions or the exponential family). The model includes an “entropic” notion of time that is tailored to the system under study; the system is its own clock. As one might expect that entropic time is intrinsically directional; there is a natural arrow of time that is led by entropic considerations. As illustrative examples, we discuss dynamics on a space of Gaussians and the discrete three-state system.     

A New Family of Continuous Probability Distributions

In this paper, a new parametric compound G family of continuous probability distributions called the Poisson generalized exponential G (PGEG) family is derived and studied. Relevant mathematical properties are derived. Some new bivariate G families using the theorems of “Farlie-Gumbel-Morgenstern copula”, “the modified Farlie-Gumbel-Morgenstern copula”, “the Clayton copula”, and “the Renyi’s entropy copula” are presented. Many special members are derived, and a special attention is devoted to the exponential and the one parameter Pareto type II model. The maximum likelihood method is used to estimate the model parameters. A graphical simulation is performed to assess the finite sample behavior of the estimators of the maximum likelihood method. Two real-life data applications are proposed to illustrate the importance of the new family.     

Financial Return Distributions: Past,Present, and COVID-19

We analyze the price return distributions of currency exchange rates, cryptocurrencies, and contracts for differences (CFDs) representing stock indices, stock shares, and commodities. Based on recent data from the years 2017–2020, we model tails of the return distributions at different time scales by using power-law, stretched exponential, and q-Gaussian functions. We focus on the fitted function parameters and how they change over the years by comparing our results with those from earlier studies and find that, on the time horizons of up to a few minutes, the so-called “inverse-cubic power-law” still constitutes an appropriate global reference. However, we no longer observe the hypothesized universal constant acceleration of the market time flow that was manifested before in an ever faster convergence of empirical return distributions towards the normal distribution. Our results do not exclude such a scenario but, rather, suggest that some other short-term processes related to a current market situation alter market dynamics and may mask this scenario. Real market dynamics is associated with a continuous alternation of different regimes with different statistical properties. An example is the COVID-19 pandemic outburst, which had an enormous yet short-time impact on financial markets. We also point out that two factors—speed of the market time flow and the asset cross-correlation magnitude—while related (the larger the speed, the larger the cross-correlations on a given time scale), act in opposite directions with regard to the return distribution tails, which can affect the expected distribution convergence to the normal distribution.     

A class of asymmetric pathway distributions and an entropy interpretation

Asymmetric distributions are widely used in probability modeling and statistical analysis. Recently, various asymmetric distributions are being developed by many researchers for modeling various data sets in real life contexts. In the present paper, we introduce a new class of q-type asymmetric distributions which include q-analogues of asymmetric Laplace, exponential power, Weibull etc. and corresponding standard distributions as special cases. Also we show that this pathway model can be obtained by optimizing Mathai’s generalized entropy with more general setup, which is a generalization of various entropy measures due to Shannon and others.     

A Locally Both Leptokurtic and Fat-Tailed Distribution with Application in a Bayesian Stochastic Volatility Model

In the paper, we begin with introducing a novel scale mixture of normal distribution such that its leptokurticity and fat-tailedness are only local, with this “locality” being separately controlled by two censoring parameters. This new, locally leptokurtic and fat-tailed (LLFT) distribution makes a viable alternative for other, globally leptokurtic, fat-tailed and symmetric distributions, typically entertained in financial volatility modelling. Then, we incorporate the LLFT distribution into a basic stochastic volatility (SV) model to yield a flexible alternative for common heavy-tailed SV models. For the resulting LLFT-SV model, we develop a Bayesian statistical framework and effective MCMC methods to enable posterior sampling of the parameters and latent variables. Empirical results indicate the validity of the LLFT-SV specification for modelling both “non-standard” financial time series with repeating zero returns, as well as more “typical” data on the S&P 500 and DAX indices. For the former, the LLFT-SV model is also shown to markedly outperform a common, globally heavy-tailed, t-SV alternative in terms of density forecasting. Applications of the proposed distribution in more advanced SV models seem to be easily attainable.     

Information Geometry of the Exponential Family of Distributions with Progressive Type-II Censoring

In geometry and topology, a family of probability distributions can be analyzed as the points on a manifold, known as statistical manifold, with intrinsic coordinates corresponding to the parameters of the distribution. Consider the exponential family of distributions with progressive Type-II censoring as the manifold of a statistical model, we use the information geometry methods to investigate the geometric quantities such as the tangent space, the Fisher metric tensors, the affine connection and the α-connection of the manifold. As an application of the geometric quantities, the asymptotic expansions of the posterior density function and the posterior Bayesian predictive density function of the manifold are discussed. The results show that the asymptotic expansions are related to the coefficients of the α-connections and metric tensors, and the predictive density function is the estimated density function in an asymptotic sense. The main results are illustrated by considering the Rayleigh distribution.     

Heterogeneous Graphical Granger Causality by Minimum Message Length

The heterogeneous graphical Granger model (HGGM) for causal inference among processes with distributions from an exponential family is efficient in scenarios when the number of time observations is much greater than the number of time series, normally by several orders of magnitude. However, in the case of “short” time series, the inference in HGGM often suffers from overestimation. To remedy this, we use the minimum message length principle (MML) to determinate the causal connections in the HGGM. The minimum message length as a Bayesian information-theoretic method for statistical model selection applies Occam’s razor in the following way: even when models are equal in their measure of fit-accuracy to the observed data, the one generating the most concise explanation of data is more likely to be correct. Based on the dispersion coefficient of the target time series and on the initial maximum likelihood estimates of the regression coefficients, we propose a minimum message length criterion to select the subset of causally connected time series with each target time series and derive its form for various exponential distributions. We propose two algorithms—the genetic-type algorithm (HMMLGA) and exHMML to find the subset. We demonstrated the superiority of both algorithms in synthetic experiments with respect to the comparison methods Lingam, HGGM and statistical framework Granger causality (SFGC). In the real data experiments, we used the methods to discriminate between pregnancy and labor phase using electrohysterogram data of Islandic mothers from Physionet databasis. We further analysed the Austrian climatological time measurements and their temporal interactions in rain and sunny days scenarios. In both experiments, the results of HMMLGA had the most realistic interpretation with respect to the comparison methods. We provide our code in Matlab. To our best knowledge, this is the first work using the MML principle for causal inference in HGGM.     

Bayesian and Classical Inference under Type-II Censored Samples of the Extended Inverse Gompertz Distribution with Engineering Applications

In this article, we introduce a new three-parameter distribution called the extended inverse-Gompertz (EIGo) distribution. The implementation of three parameters provides a good reconstruction for some applications. The EIGo distribution can be seen as an extension of the inverted exponential, inverse Gompertz, and generalized inverted exponential distributions. Its failure rate function has an upside-down bathtub shape. Various statistical and reliability properties of the EIGo distribution are discussed. The model parameters are estimated by the maximum-likelihood and Bayesian methods under Type-II censored samples, where the parameters are explained using gamma priors. The performance of the proposed approaches is examined using simulation results. Finally, two real-life engineering data sets are analyzed to illustrate the applicability of the EIGo distribution, showing that it provides better fits than competing inverted models such as inverse-Gompertz, inverse-Weibull, inverse-gamma, generalized inverse-Weibull, exponentiated inverted-Weibull, generalized inverted half-logistic, inverted-Kumaraswamy, inverted Nadarajah–Haghighi, and alpha-power inverse-Weibull distributions.     

q exponential is a Burr distribution

It is pointed out that the q-exponential distribution introduced in the ground-breaking paper by Tsallis [C. Tsallis, J. Stat. Phys. 52 (1988) 479] is contained by a family of distributions known since the 1940s. Some elementary statistical properties of this family are discussed. Six data sets on fracture roughness are used to demonstrate that this family provides much better models for fracture roughness than the q exponential distribution.     

Study on probability distributions for evolution in modified extremal optimization

It is widely believed that the power-law is a proper probability distribution being effectively applied for evolution in τ-EO (extremal optimization), a general-purpose stochastic local-search approach inspired by self-organized criticality, and its applications in some NP-hard problems, e.g., graph partitioning, graph coloring, spin glass, etc. In this study, we discover that the exponential distributions or hybrid ones (e.g., power-laws with exponential cutoff) being popularly used in the research of network sciences may replace the original power-laws in a modified τ-EO method called self-organized algorithm (SOA), and provide better performances than other statistical physics oriented methods, such as simulated annealing, τ-EO and SOA etc., from the experimental results on random Euclidean traveling salesman problems (TSP) and non-uniform instances. From the perspective of optimization, our results appear to demonstrate that the power-law is not the only proper probability distribution for evolution in EO-similar methods at least for TSP, the exponential and hybrid distributions may be other choices.     

Power-law and exponential rank distributions: A panoramic Gibbsian perspective

Rank distributions are collections of positive sizes ordered either increasingly or decreasingly. Many decreasing rank distributions, formed by the collective collaboration of human actions, follow an inverse power-law relation between ranks and sizes. This remarkable empirical fact is termed Zipf’s law, and one of its quintessential manifestations is the demography of human settlements — which exhibits a harmonic relation between ranks and sizes. In this paper we present a comprehensive statistical-physics analysis of rank distributions, establish that power-law and exponential rank distributions stand out as optimal in various entropy-based senses, and unveil the special role of the harmonic relation between ranks and sizes. Our results extend the contemporary entropy-maximization view of Zipf’s law to a broader, panoramic, Gibbsian perspective of increasing and decreasing power-law and exponential rank distributions — of which Zipf’s law is one out of four pillars.     

Alternative Entropy Measures and Generalized Khinchin–Shannon Inequalities

The Khinchin–Shannon generalized inequalities for entropy measures in Information Theory, are a paradigm which can be used to test the Synergy of the distributions of probabilities of occurrence in physical systems. The rich algebraic structure associated with the introduction of escort probabilities seems to be essential for deriving these inequalities for the two-parameter Sharma–Mittal set of entropy measures. We also emphasize the derivation of these inequalities for the special cases of one-parameter Havrda–Charvat’s, Rényi’s and Landsberg–Vedral’s entropy measures.     

Weblog patterns and human dynamics with decreasing interest

In order to describe the phenomenon that people’s interest in doing something always keep high in the beginning while gradually decreases until reaching the balance, a model which describes the attenuation of interest is proposed to reflect the fact that people’s interest becomes more stable after a long time. We give a rigorous analysis on this model by non-homogeneous Poisson processes. Our analysis indicates that the interval distribution of arrival-time is a mixed distribution with exponential and power-law feature, which is a power law with an exponential cutoff. After that, we collect blogs in ScienceNet.cn and carry on empirical study on the interarrival time distribution. The empirical results agree well with the theoretical analysis, obeying a special power law with the exponential cutoff, that is, a special kind of Gamma distribution. These empirical results verify the model by providing an evidence for a new class of phenomena in human dynamics. It can be concluded that besides power-law distributions, there are other distributions in human dynamics. These findings demonstrate the variety of human behavior dynamics.     

Introduction and Analysis of a Method for the Investigation of QCD-like Tree Data

The properties of decays that take place during jet formation cannot be easily deduced from the final distribution of particles in a detector. In this work, we first simulate a system of particles with well-defined masses, decay channels, and decay probabilities. This presents the “true system” for which we want to reproduce the decay probability distributions. Assuming we only have the data that this system produces in the detector, we decided to employ an iterative method which uses a neural network as a classifier between events produced in the detector by the “true system” and some arbitrary “test system”. In the end, we compare the distributions obtained with the iterative method to the “true” distributions.     

On Representations of Divergence Measures and Related Quantities in Exponential Families

Within exponential families, which may consist of multi-parameter and multivariate distributions, a variety of divergence measures, such as the Kullback–Leibler divergence, the Cressie–Read divergence, the Rényi divergence, and the Hellinger metric, can be explicitly expressed in terms of the respective cumulant function and mean value function. Moreover, the same applies to related entropy and affinity measures. We compile representations scattered in the literature and present a unified approach to the derivation in exponential families. As a statistical application, we highlight their use in the construction of confidence regions in a multi-sample setup.     

Relationship between degree-rank distributions and degree distributions of complex networks

Both the degree distribution and the degree-rank distribution, which is a relationship function between the degree and the rank of a vertex in the degree sequence obtained from sorting all vertices in decreasing order of degree, are important statistical properties to characterize complex networks. We derive an exact mathematical relationship between degree-rank distributions and degree distributions of complex networks. That is, for arbitrary complex networks, the degree-rank distribution can be derived from the degree distribution, and the reverse is true. Using the mathematical relationship, we study the degree-rank distributions of scale-free networks and exponential networks. We demonstrate that the degree-rank distributions of scale-free networks follow a power law only if scaling exponent λ>2. We also demonstrate that the degree-rank distributions of exponential networks follow a logarithmic law. The simulation results in the BA model and the exponential BA model verify our results.     

基于拖尾分布的高分辨率合成孔径雷达图像建模

基于中心极限定理的合成孔径雷达(SAR)图像统计分布不能反映高分辨率SAR图像尖峰和厚尾的统计特征.文中使用广义中心极限定理,由雷达回波的实部和虚部的对称稳定分布,得到SAR图像的拖尾分布(幅值图像的拖尾Rayleigh分布以及强度图像的拖尾指数分布),并以拖尾Rayleigh分布为例,讨论了拖尾分布的代数拖尾特征以及尖峰厚尾的统计特性.为了实现拖尾分布对高分辨率SAR图像的精确建模,基于第二类统计量,提出了对数累积量的参数估计方法,从而高效估计出拖尾分布的参数.真实SAR图像的建模实例表明,基于广义中心极限定理的拖尾分布可以精确描述高分辨率SAR图像的尖峰和厚尾的统计特征.     

Local Analysis of Heterogeneous Intracellular Transport: Slow and Fast Moving Endosomes

Trajectories of endosomes inside living eukaryotic cells are highly heterogeneous in space and time and diffuse anomalously due to a combination of viscoelasticity, caging, aggregation and active transport. Some of the trajectories display switching between persistent and anti-persistent motion, while others jiggle around in one position for the whole measurement time. By splitting the ensemble of endosome trajectories into slow moving subdiffusive and fast moving superdiffusive endosomes, we analyzed them separately. The mean squared displacements and velocity auto-correlation functions confirm the effectiveness of the splitting methods. Applying the local analysis, we show that both ensembles are characterized by a spectrum of local anomalous exponents and local generalized diffusion coefficients. Slow and fast endosomes have exponential distributions of local anomalous exponents and power law distributions of generalized diffusion coefficients. This suggests that heterogeneous fractional Brownian motion is an appropriate model for both fast and slow moving endosomes. This article is part of a Special Issue entitled: “Recent Advances In Single-Particle Tracking: Experiment and Analysis” edited by Janusz Szwabiński and Aleksander Weron.