Power-law Probability Distribution Research Articles

Self-similar Outflows at the Source of the Fast Solar Wind: A Smoking Gun of Multiscale Impulsive Reconnection?

We present results of a quantitative analysis of structured plasma outflows above a polar coronal hole observed by the Atmospheric Imaging Assembly (AIA) on board the Solar Dynamics Observatory (SDO) spacecraft. In a 6 hr interval of continuous high-cadence SDO/AIA images, we identified more than 2300 episodes of small-scale plasma flows in the polar corona. The mean upward flow speed measured by the surfing transform technique is estimated to be 122 ± 34 km s−1, which is comparable to the local sound speed. The typical recurrence period of the flow episodes is 10–30 minutes, and the mean duration and transverse size of each episode are about 3–5 minutes and 3–4 Mm, respectively. The largest identifiable episodes last for tens of minutes and reach widths up to 40 Mm. For the first time, we demonstrate that the polar coronal-hole outflows obey a family of power-law probability distributions characteristic of impulsive interchange magnetic reconnection. Turbulent photospheric driving may play a crucial role in releasing magnetically confined plasma onto open field. The estimated occurrence rate of the detected self-similar coronal outflows is sufficient for them to make a dominant contribution to the fast-wind mass and energy fluxes and to account for the wind’s small-scale structure.

Revealing the fractal and self-similarity of realistic collective human mobility

Revealing the fractal scaling laws of collective human mobility is benefit to understand the complexity of collective human mobility behaviours and has broad application prospects, such as infectious disease prevention, because fractal networks are potentially more robust against targeted attacks than non-fractal networks. Although many statistical indicators of individual human mobility had been observed in power-law distributions, there is still no evidence so far to verify the fractal or self-similarity of collective human mobility behaviours. This study uncovered the fractal and self-similarity properties on realistic weighted human mobility networks at world-, nation-, and city-wide scales. After observing the necessary conditions of network fractal, i.e., the evident power-law probability distributions of weight-related metrics, the fractal dimensions of weighted human mobility networks were measured to be about 0.67∼0.88. Furthermore, experimental results demonstrated their distinct statistical self-similarity and approximately structural self-similarity. These exciting findings can enhance the fire-new understandings of collective human mobility behaviours, and provide novel solutions for many real-world applications.

Mathematical Characterization of Private and Public Immune Receptor Sequences

Diverse T and B cell repertoires play an important role in mounting effective immune responses against a wide range of pathogens and malignant cells. The number of unique T and B cell clones is characterized by T and B cell receptors (TCRs and BCRs), respectively. Although receptor sequences are generated probabilistically by recombination processes, clinical studies found a high degree of sharing of TCRs and BCRs among different individuals. In this work, we use a general probabilistic model for T/B cell receptor clone abundances to define “publicness” or “privateness” and information-theoretic measures for comparing the frequency of sampled sequences observed across different individuals. We derive mathematical formulae to quantify the mean and the variances of clone richness and overlap. Our results can be used to evaluate the effect of different sampling protocols on abundances of clones within an individual as well as the commonality of clones across individuals. Using synthetic and empirical TCR amino acid sequence data, we perform simulations to study expected clonal commonalities across multiple individuals. Based on our formulae, we compare these simulated results with the analytically predicted mean and variances of the repertoire overlap. Complementing the results on simulated repertoires, we derive explicit expressions for the richness and its uncertainty for specific, single-parameter truncated power-law probability distributions. Finally, the information loss associated with grouping together certain receptor sequences, as is done in spectratyping, is also evaluated. Our approach can be, in principle, applied under more general and mechanistically realistic clone generation models.

Statistical modelling of aquatic size spectra: integrating data from multiple taxa and sampling methods

Abstract Size spectra are used to assess the status and functioning of marine and freshwater ecosystems worldwide. Their use is underpinned by theory linking the dynamics of trophic interactions to a power-law decline of abundance with body size in ecological communities. Recent papers on empirical size spectrum estimation have argued for Maximum Likelihood Estimation of power-law probability distributions as a more accurate alternative to traditional linear regression approaches. One major limitation of currently used size spectrum estimators from Maximum Likelihood Estimation is that they cannot account for the use of multiple sampling protocols, nor the distortions caused by gear size selectivity, and therefore they become restricted to a relatively narrow taxonomic group and size range. Further progress in the field requires new methods that are flexible enough to combine multiple trophic groups and sampling gears into a single size spectrum estimate, while taking advantage of more accurate distributional approaches. The method we propose in this paper fills this gap by deriving the distribution of observed sizes explicitly from the underlying power-law spectrum and gear selectivity functions. It specifies likelihoods as a product of two components: (i) the probability of belonging to a given group and (ii) the probability distribution within the group. Using Bayesian estimation, we applied the method to surveys of phytoplankton, zooplankton, and fishes in lakes of Quetico Provincial Park, northwestern Ontario, using Van Dorn samplers, zooplankton nets, gillnets, and hydroacoustics. The results show that the spectra estimated from subsets of trophic groups or gears are weak predictors of more complete spectra, highlighting the importance of using more inclusive community data. The two-component partitioning of likelihoods also helped demonstrate the existence of between-group spectrum slopes that were overall steeper than within-group slopes, indicating that heterogeneity of trophic transfers across the size spectrum is an important factor structuring these ecosystems.

Seismic Analysis of a Multiscale Discrete Fracture System

The work is devoted to the development and study of a statistical model of a discrete network of cracks at different spatial scales. The distribution of fracture centers is a fractal set, which is determined by a given value of the correlation fractal dimension D 2 . The fracture lengths are described by a power-law probability distribution with the corresponding value of the exponent α . The possibility to estimate these model parameters based on the corresponding seismic images is investigated.

Unveiling the complex canopy spatial structure of a Mediterranean old-growth beech (Fagus sylvatica L.) forest from UAV observations

In front of climate change scenarios and global loss of biodiversity, it is essential to monitor the structure of old-growth forests to study ecosystem status and dynamics to inform future conservation and restoration programmes. We propose an Unmanned Aerial Vehicle (UAV)-based framework to monitor fine-grained forest top canopy structure in a primary old-growth beech (Fagus sylvatica L.) forest in Pollino National Park, Italy, which belongs to the UNESCO World Heritage (UNESCO WH) serial site “Ancient and Primeval beech forests of the Carpathians and other regions of Europe”. Canopy profile, gap properties and their spatial distribution patterns were analysed using the canopy height model (CHM) derived from UAV surveys. Very high-resolution orthomosaic images coupled with direct field measurement data were used to assess gap detection accuracy and CHM validation. Forest canopy properties along with the vertical layering of the canopy were further explored using second-order statistics. The reconstructed canopy profile revealed a bimodal top height frequency distribution. The upper canopy layer (h > 14 m) was the most represented canopy height, with the remaining 50% split between the medium and lowest layer; 551 gaps were identified within 11.5 ha. Gap size varied between 2 m2 and 353 m2, and 19 m2was the mean gap size; the gap size-frequency relationship reflected a power-law probability distribution. About 97 % of the gaps were <100 m2 in size, showing a significant tendency to cluster. Most gaps were located in the upper and medium canopy layers; however, the highest relative gap area was found in the lowest layer. These results confirmed the high natural integrity of the ecosystem processes that distinguish the old-growth beech stands in respect to managed woodlands. Our findings demonstrate that the low-cost UAV-DAP (Digital Aerial Photogrammetry) workflow has the potential to generate realistic old-growth forest canopy attributes at a very fine scale. The proposed protocol can be adopted for monitoring the structural dynamics of high-value natural forest ecosystems as in the case of UNESCO WH sites or other old-growth stands. This approach is also helpful for mapping and deriving spatially explicit canopy structure information over confined forest areas and determining where conservation actions should be directed to preserve or restore natural ecosystem function.

The effect of sampling volume size on the apparent stiffness of jointed rock mass

Overall mechanical properties of a jointed rock mass are strongly affected by discontinuities – fractures – that naturally occur in rocks. Stochastically-generated discrete fracture network (DFN) modeling, which uses a probabilistic approach to describe the spatial distribution of fractures, such as position, size, or orientation, offers an explicit way to describe geometry of the fracture system. Many in-situ measurements and analyses presented in literature indicate that fractures’ sizes can be adequately represented by the power law probability distribution. The parallel plate model of individual fractures combined with an averaging technique makes it possible to estimate the overall compliance or stiffness of jointed rock mass (Oda et al. [1]). In the present study, a series of numerical simulations of jointed rock mass modeled by DFN and Oda’s approach were conducted to analyze the effect of different sizes of the sampling volume on the overall elastic moduli. The results of the numerical study show that the variance as well as the average of the apparent stiffness decrease as the size of sampling element grows.

Fragment size distributions in brittle deformed rocks

The production of breccias and cataclasites is commonly proposed to result in power-law or log-normal probability distributions for fragment (grain) size. We show that in both natural and experimental examples, the common best fit probability distributions for the complete distributions are members of the Generalised Gamma (GG), Extreme Value (GEV) and Pareto (GP) families; power-law and log-normal distributions are commonly, but not always, poor fits to the data. An hierarchical sequence, GG → GEV → GP, emerges as the sample mean of the fragment size decreases. The physical foundations (self-similar fragmentation, collisional fragmentation, shattering) for these distributions are discussed. Particularly important is the shattering continuous phase transition that results in the simultaneous development of both coarse fragments and ultra-fine particles (dust). This phase transition leads to Generalised Pareto fragment size distributions for the coarse fragments. Also included is a discussion of the relations between fragment size distribution, processes and deformation history in the context of monomineralic rocks. The overall reported size distributions are compatible with theoretical developments but the topic would benefit from observations and experiments conducted with the theories in mind.

STUDY OF A MEASURE OF EFFICIENCY AS A TOOL FOR APPLYING THE PRINCIPLE OF LEAST EFFORT TO THE DERIVATION OF THE ZIPF AND THE PARETO LAWS

The principle of least effort (PLE) is believed to be a universal rule for living systems. Its application to the derivation of the power law probability distributions of living systems has long been challenging. Recently, a measure of efficiency was proposed as a tool of deriving Zipf’s and Pareto’s laws directly from the PLE. This work is a further investigation of this efficiency measure from a mathematical point of view. The aim is to get further insight into its properties and usefulness as a metric of performance. We address some key mathematical properties of this efficiency such as its sign, uniqueness and robustness. We also look at the relationship between this measure and other properties of the system of interest such as inequality and uncertainty, by introducing a new method for calculating nonnegative continuous entropy.

On a heavy-tailed distribution and the stability of an equilibrium in a distributed delay symmetric network

We consider a static artificial neural network model endowed with multiple unbounded S-type distributed time delays. The delay kernels are described by the Pareto distribution, which is a heavy-tailed power-law probability distribution frequently employed in the characterisation of many observable phenomena. We give a characterisation of the effects of the shape and the scale of the Pareto delay distribution on the stability of an equilibrium of the network.

Unified view of avalanche criticality in sheared glasses.

Plastic events in sheared glasses are considered an example of so-called avalanches, whose sizes obey a power-law probability distribution with the avalanche critical exponent τ. Although the so-called mean-field depinning (MFD) theory predicts a universal value of this exponent, τ_{MFD}=1.5, such a simplification is now known to connote qualitative disagreement with realistic systems. Numerically and experimentally, different values of τ have been reported depending on the literature. Moreover, in the elastic regime, it has been noted that the critical exponent can be different from that in the steady state, and even criticality itself is a matter of debate. Because these confusingly varying results have been reported under different setups, our knowledge of avalanche criticality in sheared glasses is greatly limited. To gain a unified understanding, in this work, we conduct a comprehensive numerical investigation of avalanches in Lennard-Jones glasses under athermal quasistatic shear. In particular, by excluding the ambiguity and arbitrariness that has crept into the conventional measurement schemes, we achieve high-precision measurement and demonstrate that the exponent τ in the steady state follows the prediction of MFD theory, τ_{MFD}=1.5. Our results also suggest that there are two qualitatively different avalanche events. This binariness leads to the nonuniversal behavior of the avalanche size distribution and is likely to be the cause of the varying values of τ reported thus far. To investigate the dependence of criticality and universality on applied shear, we further study the statistics of avalanches in the elastic regime and the ensemble of the first avalanche event in different samples, which provide information about the unperturbed system. We show that while the unperturbed system is indeed off-critical, criticality gradually develops as shear is applied. The degree of criticality is encoded in the fractal dimension of the avalanches, which starts from zero in the off-critical unperturbed state and saturates in the steady state. Moreover, the critical exponent τ is consistent with the prediction of the MFD τ_{MFD} universally, regardless of the amount of applied shear, once the system becomes critical.

Speckle statistics of biological tissues in optical coherence tomography.

The speckle statistics of optical coherence tomography images of biological tissue have been studied using several historical probability density functions. Here, we propose a new theoretical framework based on power-law functions, where we hypothesize that an underlying power-law distribution governs scattering from tissues. Thus, multi-scale scattering sites including the fractal branching vasculature will contribute to power-law probability distributions of speckle statistics. Specifically, these are the Burr type XII distribution for speckle amplitude, the Lomax distribution for intensity, and the generalized logistic distribution for log amplitude. Experimentally, these three distributions are fitted to histogram data from nine optical coherence tomography scans of various samples and biological tissues, in vivo and ex vivo. The distributions are also compared with classical models such as the Rayleigh, K, and gamma distributions. The results indicate that across OCT datasets of various tissue types, the proposed power-law distributions are more appropriate models yielding novel parameters for characterizing the physics of scattering from biological tissue. Thus, the overall framework brings to the field new biomarkers from OCT measures of speckle in tissues, grounded in basic biophysics and with wide applications to diagnostic imaging in clinical use.

Entropy variation in a fractal phase space

In this paper we recall for physicists how it is possible, using the principle of maximization of the Boltzmann-Shannon entropy, to derive the Burr-Singh-Maddala (BurrXII) double power law probability distribution function and its approximations (Pareto, loglogistic ..) and extension (GB2…) first used in econometrics. This is possible using a deformation of the power function, as this has been done in complex systems for the exponential function. We give to that distribution a deep stochastic interpretation using the theory of Weron et al. Applied to thermodynamics the entropy nonextensivity can be accounted for by assuming that the asymptotic exponents are scale dependent. Therefore functions which describe phenomena presenting power-law asymptotic behaviour can be obtained without introducing exotic forms of the entropy.

GUJARATI POETRY CLASSIFICATION BASED ON EMOTIONS USING DEEP LEARNING

Poetries are the way to express the word that represents emotions and thoughts with the use of any language in the world and the computational linguistic study to the emotion recognition from poetry is an overwhelming and complex task too. Ultimate goal of this study is to disclose emotions through Gujarati poetries with the use of variety of characteristic there within Gujarati poems. Study presents a novel perspective in sentiment capture as of Gujarati Poems. ‘Kavan’ Gujarati poems collection was by hand interpreted upon the Indian idea mentioned in ‘Navarasa’. In the collection, 300+ poems were categorized into nine feeling mentioned in ‘Navarasa’, respectively ‘?????’, ‘???????’, ‘??????’, ‘? ???????’, ‘???????’, ‘????????’, ‘????????’, ‘???????’, and ‘???????’. The Zipfian allotment is part a family of related discrete power law probability distributions, which is used to identify the probability of one ‘rasa’ from the given poem.. Deep Learning has colossal set of pattern recognition tools that can be used for natural language processing and have incredible potential to locate a solution for challenging machine learning problem. Result accuracy was found up to ~87.62% of emotion classification task from Gujarati poetry corpus.

Log-Weighted Pareto Distribution And Its Statistical Properties

The Pareto distribution is a power law probability distribution that is used to describe social scientific, geophysical, actuarial, and many other types of observable phenomena. A new weighted Pareto distribution is proposed using a logarithmic weight function. Several statistical properties of the weighted Pareto distribution are studied and derived including cumulative distribution function, location measures such as mode, median and mean, reliability measures such as reliability function, hazard and reversed hazard functions and the mean residual life, moments, shape indices such as skewness and kurtosis coefficients and order statistics. A parametric estimation is performed to obtain estimators for the distribution parameters using three different estimation methods the maximum likelihood method, the L-moments method and the method of moments. Numerical simulation is carried out to validate the robustness of the proposed distribution. The distribution is fitted to a real data set to show its importance in real life applications.

Altered Weibull Degree Distribution in Resting-State Functional Brain Networks Is Associated With Cognitive Decline in Mild Cognitive Impairment.

The topological organization of human brain networks can be mathematically characterized by the connectivity degree distribution of network nodes. However, there is no clear consensus on whether the topological structure of brain networks follows a power law or other probability distributions, and whether it is altered in Alzheimer's disease (AD). Here we employed resting-state functional MRI and graph theory approaches to investigate the fitting of degree distributions of the whole-brain functional networks and seven subnetworks in healthy subjects and individuals with amnestic mild cognitive impairment (aMCI), i.e., the prodromal stage of AD, and whether they are altered and correlated with cognitive performance in patients. Forty-one elderly cognitively healthy controls and 30 aMCI subjects were included. We constructed functional connectivity matrices among brain voxels and examined nodal degree distributions that were fitted by maximum likelihood estimation. In the whole-brain networks and all functional subnetworks, the connectivity degree distributions were fitted better by the Weibull distribution [f(x)~x(β−1)e(−λxβ)] than power law or power law with exponential cutoff. Compared with the healthy control group, the aMCI group showed lower Weibull β parameters (shape factor) in both the whole-brain networks and all seven subnetworks (false-discovery rate-corrected, p < 0.05). These decreases of the Weibull β parameters in the whole-brain networks and all subnetworks except for ventral attention were associated with reduced cognitive performance in individuals with aMCI. Thus, we provided a short-tailed model to capture intrinsic connectivity structure of the human brain functional networks in health and disease.

Financial and Actuarial Properties of the Beta-Pareto as a Long-Tail Distribution

Undoubtedly, the single parameter Pareto distribution is one of the most attractive distribution in statistics; a power-law probability distribution that is found in a large number of real-world situations inside and outside the field of economics. Furthermore, it is usually used as a basis for excess of loss quotations as it gives a pretty good description of the random behaviour of large losses. In this paper, we provide properties of the Beta-Pareto distribution which can be useful in Economics, and in Financial and Actuarial fields, mainly related to the analysis of the tail of the distribution that makes it a candidate model for fitting actuarial data with extreme observations. As empirical applications two well-known data sources considered in general insurance are used to account for the suitability of the model.

Shannon entropy as a robust estimator of Zipf's Law in animal vocal communication repertoires

Abstract Information complexity in animals is an indicator of advanced communication and an intricate socio‐ecology. Zipf's Law of least effort has been used to assess the potential information content of animal repertoires, including whether or not a particular animal communication could be ‘language‐like’. As all human languages follow Zipf's law, with a power law coefficient (PLC) close to −1, animal signals with similar probability distributions are postulated to possess similar information characteristics to language. However, estimation of the PLC from limited empirical datasets (e.g. most animal communication studies) is problematic because of biases from small sample sizes. The traditional approach to estimating Zipf's law PLC is to find the slope of a log–log rank‐frequency plot. Our alternative option uses the underlying equivalence between Shannon entropy (i.e. whether successive elements of a sequence are unpredictable, or repetitive) and PLC. Here, we test whether an entropy approach yields more robust estimates of Zipf's law PLC than the traditional approach. We examined the efficacy of the entropy approach in two ways. First, we estimated the PLC from synthetic datasets generated with a priori known power law probability distributions. This revealed that the estimated PLC using the traditional method is particularly inaccurate for highly stereotyped sequences, even at modest repertoire sizes. Estimation via Shannon entropy is accurate with modest sample sizes even for repertoires with thousands of distinct elements. Second, we applied these approaches to empirical data taken from 11 animal species. Shannon entropy produced a more robust estimate of PLC with lower variance than the traditional method, even when the true PLC is unknown. Our approach for the first time reveals Zipf's law operating in the vocal systems of multiple lineages: songbirds, hyraxes and cetaceans. As different methods of estimating the PLC can lead to misleading results in real data, estimating the balance of a communication system between simplicity and complexity is best performed using the entropy approach. This provides a more robust way to investigate the evolutionary constraints and processes that have acted on animal communication systems, and the parallels between these processes and the evolution of language.

Altered Weibull degree distribution in resting‐state functional brain networks in Alzheimer’s disease

AbstractBackgroundThe topological organization of human brain networks can be mathematically characterized by the connectivity degree distribution of network nodes. However, there is no clear consensus on whether the topological structure of brain networks follows a power law or other probability distributions, and whether it is disrupted in Alzheimer’s disease(AD) and its prodromal stage, mild cognitive impairment(MCI). We investigated on the fitting of nodal degree distribution of the whole‐brain functional networks and 7 subnetworks in healthy controls(HC) and individuals with MCI and AD.MethodWe included 40 HCs, 34 MCI and 34 early‐stage AD subjects who were recruited from a memory clinic at the Peking University Sixth Hospital. Resting‐state fMRI scans underwent standard preprocessing and were spatially normalized to the standard space. For each subject, we built whole‐brain FC matrices by computing Pearson’s correlations among brain voxels, followed by a threshold process(r from 0.4 to 0.6). Seven pre‐defined subnetworks were obtained from previous studies, including visual, sensorimotor, dorsal attention, ventral attention, limbic, frontoparietal and default mode (DMN). For a given subnetwork, degree distributions were calculated within and between subnetworks, respectively. Maximum likelihood estimation was used to determine model parameters. The goodness of fit of each fitting was evaluated by bootstrapping and the Kolmogorov‐Smirnov test. Group differences in the estimated parameters were tested by ANCOVAs, controlled for age, gender and education.ResultIn the whole‐brain networks and all subnetworks, the connectivity degree distributions were fitted better by a Weibull distribution(f(x)∼x^(β‐1)e^(‐λx^β)) than power law or alternative models. Compared with the HC group, the MCI group showed lower Weibull β parameter (shape factor) in both the whole‐brain and subnetworks, especially in visual, somatomotor and DMN across all threshold(ps&lt;0.05), some of which were distributed in within‐network(somatomotor), between‐networks(dorsal attention and limbic), or both(visual, ventral attention, frontoparietal and DMN). In the AD group, the β parameters were lower in the frontalparietal(p&lt;0.05) and DMN(p&lt;0.05, except for r=0.6) than those in the HC.ConclusionWe show that the Weibull model fits human brain functional networks better than a power law model, and this network structure is disrupted in AD and MCI. Such a short‐tailed model may capture intrinsic network structure of the human brain in health and disease.

Non-Markovian Pitch-angle Scattering as the Origin of Particle Superdiffusion Parallel to the Magnetic Field

Abstract We develop a theoretical model for particle superdiffusive transport parallel to the average magnetic field, due to the pitch-angle scattering times having a non-Markovian, power-law probability distribution. We show that a non-Markovian Fokker–Planck equation can be derived, where the traditional time derivative is changed for a fractional time derivative. By solving the fractional Fokker–Planck equation, with the time-dependent part having solutions that are expressed by the Mittag-Leffler functions, it is found that an initial pitch-angle distribution slowly decays toward isotropy. This leads to a parallel velocity autocorrelation function that also has a slow power-law decay in time, thus implying superdiffusive transport in the direction parallel to the background magnetic field. In this framework, we derive for the first time the anomalous diffusion coefficient as a function of physical parameters like the background magnetic field, the resonant turbulence level, and the particle speed.