Power-law Probability Distribution Research Articles

Fractional calculus ties the microscopic and macroscopic scales of complex network dynamics

A two-state, master equation-based decision-making model has been shown to generate phase transitions, to be topologically complex, and to manifest temporal complexity through an inverse power-law probability distribution function in the switching times between the two critical states of consensus. These properties are entailed by the fundamental assumption that the network elements in the decision-making model imperfectly imitate one another. The process of subordination establishes that a single network element can be described by a fractional master equation whose analytic solution yields the observed inverse power-law probability distribution obtained by numerical integration of the two-state master equation to a high degree of accuracy.

Stochastic generation of synthetic minutely irradiance time series derived from mean hourly weather observation data

Synthetic minutely irradiance time series are utilised in non-spatial solar energy system research simulations. It is necessary that they accurately capture irradiance fluctuations and variability inherent in the solar resource. This article describes a methodology to generate a synthetic minutely irradiance time series from widely available hourly weather observation data. The weather observation data are used to produce a set of Markov chains taking into account seasonal, diurnal, and pressure influences on transition probabilities of cloud cover. Cloud dynamics are based on a power-law probability distribution, from which cloud length and duration are derived. Atmospheric transmission losses are simulated with minutely variability, using atmospheric profiles from meteorological reanalysis data and cloud attenuation derived real-world observations. Both direct and diffuse irradiance are calculated, from which total irradiance is determined on an arbitrary plane. The method is applied to the city of Leeds, UK, and validated using independent hourly radiation measurements from the same site. Variability and ramp rate are validated using 1-min resolution irradiance data from the town of Cambourne, Cornwall, UK. The hourly irradiance frequency distribution correlates with R2=0.996 whilst the mean hourly irradiance correlates with R2=0.971, the daily variability indices cumulative probability distribution function (CDF), 1-min irradiance ramp rate CDF and 1-min irradiance frequency CDF are also shown to correlate with R2=0.9903,1.000, and 0.9994 respectively. Kolmogorov–Smirnov tests on 1-min data for each day show that the ramp rate frequency of occurrence is captured with a high significance level of 99.99%, whilst the irradiance frequency distribution and minutely variability indices are captured at significances of 99% and 97.5% respectively. The use of multiple Markov chains and detailed consideration of the atmospheric losses are shown to be essential elements for the generation of realistic minutely irradiance time series over a typical meteorological year. A freely downloadable example of the model is made available and may be configured to the particular requirements of users or incorporated into other models.

Molecular clouds have power-law probability distribution functions

In this Letter we investigate the shape of the probability distribution of column densities (PDF) in molecular clouds. Through the use of low-noise, extinction-calibrated \textit{Herschel}/\textit{Planck} emission data for eight molecular clouds, we demonstrate that, contrary to common belief, the PDFs of molecular clouds are not described well by log-normal functions, but are instead power laws with exponents close to two and with breaks between $A_K \simeq 0.1$ and $0.2\,\mathrm{mag}$, so close to the CO self-shielding limit and not far from the transition between molecular and atomic gas. Additionally, we argue that the intrinsic functional form of the PDF cannot be securely determined below $A_K \simeq 0.1\,\mathrm{mag}$, limiting our ability to investigate more complex models for the shape of the cloud PDF.

The Burr XII Distribution Family and the Maximum Entropy Principle: Power-Law Phenomena Are Not Necessarily “Nonextensive”

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-Bingh-Maddala (burr12) double power law probability distribution function and its approximations (Pareto, loglogistic ..) and extension 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.

Automated detection of solar eruptions

Observation of the solar atmosphere reveals a wide range of motions, from small scale jets and spicules to global-scale coronal mass ejections. Identifying and characterizing these motions are essential to advancing our understanding the drivers of space weather. Both automated and visual identifications are currently used in identifying CMEs. To date, eruptions near the solar surface (which may be precursors to CMEs) have been identified primarily by visual inspection. Here we report on EruptionPatrol (EP): a software module that is designed to automatically identify eruptions from data collected by SDO/AIA. We describe the method underlying the module and compare its results to previous identifications found in the Heliophysics Event Knowledgebase. EP identifies eruptions events that are consistent with those found by human annotations, but in a significantly more consistent and quantitative manner. Eruptions are found to be distributed within 15Mm of the solar surface. They possess peak speeds ranging from 4 to 100 km/sec and display a power-law probability distribution over that range. These characteristics are consistent with previous observations of prominences.

A numerical study of Lévy random walks: Mean square displacement and power-law propagators

Non-diffusive transport, for which the particle mean free path grows nonlinearly in time, is envisaged for many space and laboratory plasmas. In particular, superdiffusion, i.e. 〈Δx2〉 ∝tαwith α > 1, can be described in terms of a Lévy random walk, in which case the probability of free-path lengths has power-law tails. Here, we develop a direct numerical simulation to reproduce the Lévy random walk, as distinct from the Lévy flights. This implies that in the free-path probability distributionΨ(x, t) there is a space-time coupling, that is, the free-path length is proportional to the free-path duration. A power-law probability distribution for the free-path duration is assumed, so that the numerical model depends on the power-law slope μ and on the scale distancex0. The numerical model is able to reproduce the expected mean square deviation, which grows in a superdiffusive way, and the expected propagatorP(x, t), which exhibits power-law tails, too. The difference in the power-law slope between the Lévy flights propagator and the Lévy walks propagator is also estimated.

Beyond Gibbs-Boltzmann-Shannon: general entropies—the Gibbs-Lorentzian example

We propose a generalisation of Gibbs' statistical mechanics into the domain of non-negligible phase space correlations. Derived are the probability distribution and entropy as a generalised ensemble average, replacing Gibbs-Boltzmann-Shannon's entropy definition enabling construction of new forms of statistical mechanics. The general entropy may also be of importance in information theory and data analysis. Application to generalised Lorentzian phase space elements yields the Gibbs-Lorentzian power law probability distribution and statistical mechanics. Details can be found in arXiv:1406.6639

Estimating individual tree mid- and understory rank-size distributions from airborne laser scanning in semi-arid forests

Limitations inherent to airborne laser scanning (ALS) technology and the complex sorting and packing relationships of forests complicate accurate remote sensing of mid- and understory trees, especially in denser forest stands. Self-similarities in rank-sized individual tree distributions (ITD), e.g. bole diameter or height, are a well-understood property of natural, non-plantation, forests undergoing density dependent self-thinning and thus offer an approach to solving this problem. Alternately, semi-arid conifer forests of the southwestern USA that experience episodic wildfires and herbivory tend to exist as open stands compared to forests where disturbances are less common. We found the ITD for semi-arid forest plots with ALS-estimated canopy cover<50% had a low rate of omission error for mid- and understory ALS trees making distribution fitting of the mid- and understory ITD unnecessary. In dense semi-arid forest plots (>50% canopy cover) the ITD correlated significantly with a tapered Pareto distribution, a power law probability distribution that is not heavy right-tailed. Two-sample Kolmogorov–Smirnov tests confirmed that observed vs. ALS-estimated overstory ITD parameters were not significantly different regardless of canopy cover. Therefore an overstory ITD derived from ALS is sufficient for fitting a continuous distribution function to estimate the ITD of the forest understory when the scale parameter is established a priori. Foresters and ecologists interested in measuring and modeling stand dynamics from ALS can use this approach to correct for stand density effects when developing ALS-derived single-tree inventories. Canopy cover can be used as a proxy for stand density when developing a combined ITD with area-based approaches for estimating understory in semi-arid forests.

Scale criticality in estimating ecosystem carbon dynamics

Scaling is central to ecology and Earth system sciences. However, the importance of scale (i.e. resolution and extent) for understanding carbon dynamics across scales is poorly understood and quantified. We simulated carbon dynamics under a wide range of combinations of resolution (nine spatial resolutions of 250 m, 500 m, 1 km, 2 km, 5 km, 10 km, 20 km, 50 km, and 100 km) and extent (57 geospatial extents ranging from 108 to 1 247 034 km(2) ) in the southeastern United States to explore the existence of scale dependence of the simulated regional carbon balance. Results clearly show the existence of a critical threshold resolution for estimating carbon sequestration within a given extent and an error limit. Furthermore, an invariant power law scaling relationship was found between the critical resolution and the spatial extent as the critical resolution is proportional to A(n) (n is a constant, and A is the extent). Scale criticality and the power law relationship might be driven by the power law probability distributions of land surface and ecological quantities including disturbances at landscape to regional scales. The current overwhelming practices without considering scale criticality might have largely contributed to difficulties in balancing carbon budgets at regional and global scales.

Greedy routing in small-world networks with power-law degrees

In this paper we study decentralized routing in small-world networks that combine a wide variation in node degrees with a notion of spatial embedding. Specifically, we consider a variant of J. Kleinberg’s grid-based small-world model in which (1) the number of long-range edges of each node is not fixed, but is drawn from a power-law probability distribution with exponent parameter \(\alpha \ge 0\) and constant mean, and (2) the long-range edges are considered to be bidirectional for the purposes of routing. This model is motivated by empirical observations indicating that several real networks have degrees that follow a power-law distribution. The measured power-law exponent \(\alpha \) for these networks is often in the range between 2 and 3. For the small-world model we consider, we show that when \(2 < \alpha < 3\) the standard greedy routing algorithm, in which a node forwards the message to its neighbor that is closest to the target in the grid, finishes in an expected number of \(O(\log ^{\alpha -1} n\cdot \log \log n)\) steps, for any source–target pair. This is asymptotically smaller than the \(O(\log ^2 n)\) steps needed in Kleinberg’s original model with the same average degree, and approaches \(O(\log n)\) as \(\alpha \) approaches 2. Further, we show that when \(0\le \alpha < 2\) or \(\alpha \ge 3\) the expected number of steps is \(O(\log ^2 n)\), while for \(\alpha = 2\) it is \(O(\log ^{4/3} n)\). We complement these results with lower bounds that match the upper bounds within at most a \(\log \log n\) factor.

Binary-coded extremal optimization for the design of PID controllers

Design of an effective and efficient PID controller to obtain high-quality performances such as high stability and satisfied transient response is of great theoretical and practical significance. This paper presents a novel design method for PID controllers based on the binary-coded extremal optimization algorithm (BCEO). The basic idea behind the proposed method is encoding the PID parameters into a binary string, evaluating the control performance by a more reasonable index than the integral of absolute error (IAE) and the integral of time weighted absolute error (ITAE), updating the solution by the selection based on power-law probability distribution and binary mutation for the selected bad elements. The experimental results on some benchmark instances have shown that the proposed BCEO-based PID design method is simpler, more efficient and effective than the existing popular evolutionary algorithms, such as the adaptive genetic algorithm (AGA), the self-organizing genetic algorithm (SOGA) and probability based binary particle swarm optimization (PBPSO) for single-variable plants. Moreover, the superiority of the BCEO method to AGA and PBPSO is demonstrated by the experimental results on the multivariable benchmark plant.

An Improved Real-Coded Population-Based Extremal Optimization Method for Continuous Unconstrained Optimization Problems

As a novel evolutionary optimization method, extremal optimization (EO) has been successfully applied to a variety of combinatorial optimization problems. However, the applications of EO in continuous optimization problems are relatively rare. This paper proposes an improved real-coded population-based EO method (IRPEO) for continuous unconstrained optimization problems. The key operations of IRPEO include generation of real-coded random initial population, evaluation of individual and population fitness, selection of bad elements according to power-law probability distribution, generation of new population based on uniform random mutation, and updating the population by accepting the new population unconditionally. The experimental results on 10 benchmark test functions with the dimensionN=30have shown that IRPEO is competitive or even better than the recently reported various genetic algorithm (GA) versions with different mutation operations in terms of simplicity, effectiveness, and efficiency. Furthermore, the superiority of IRPEO to other evolutionary algorithms such as original population-based EO, particle swarm optimization (PSO), and the hybrid PSO-EO is also demonstrated by the experimental results on some benchmark functions.

Hydroperiod regime controls the organization of plant species in wetlands

With urban, agricultural, and industrial needs growing throughout the past decades, wetland ecosystems have experienced profound changes. Most critically, the biodiversity of wetlands is intimately linked to its hydrologic dynamics, which in turn are being drastically altered by ongoing climate changes. Hydroperiod regimes, e.g., percentage of time a site is inundated, exert critical control in the creation of niches for different plant species in wetlands. However, the spatial signatures of the organization of plant species in wetlands and how the different drivers interact to yield such signatures are unknown. Focusing on Everglades National Park (ENP) in Florida, we show here that cluster sizes of each species follow a power law probability distribution and that such clusters have well-defined fractal characteristics. Moreover, we individuate and model those signatures via the interplay between global forcings arising from the hydroperiod regime and local controls exerted by neighboring vegetation. With power law clustering often associated with systems near critical transitions, our findings are highly relevant for the management of wetland ecosystems. In addition, our results show that changes in climate and land management have a quantifiable predictable impact on the type of vegetation and its spatial organization in wetlands.

Velocity Inversion In Cylindrical Couette Gas Flows

We investigate a power-law probability distribution function to describe the mean free path of rarefied gas molecules in non-planar geometries. A new curvature-dependent model is derived by taking into account the boundary-limiting effects on the molecular mean free path for surfaces with both convex and concave curvatures. In comparison to a planar wall, we find that the mean free path for a convex surface is higher at the wall and exhibits a sharper gradient within the Knudsen layer. In contrast, a concave wall exhibits a lower mean free path near the surface and the gradients in the Knudsen layer are shallower. The Navier-Stokes constitutive relations and velocity-slip boundary conditions are modified based on a power-law scaling to describe the mean free path, in accordance with the kinetic theory of gases, i.e. transport properties can be described in terms of the mean free path.Velocity profiles for isothermal cylindrical Couette flow are obtained using the power-law model. We demonstrate that our model is more accurate than the classical slip solution, especially in the transition regime, and we are able to capture important non-linear trends associated with the non-equilibrium physics of the Knudsen layer. In addition, we establish a new criterion for the critical accommodation coefficient that leads to the non-intuitive phenomena of velocity-inversion. Our results are compared with conventional hydrodynamic models and direct simulation Monte Carlo data. The power-law model predicts that the critical accommodation coefficient is significantly lower than that calculated using the classical slip solution and is in good agreement with available DSMC data. Our proposed constitutive scaling for non-planar surfaces is based on simple physical arguments and can be readily implemented in conventional fluid dynamics codes for arbitrary geometric configurations.

Parameter Estimation for Exponentially Tempered Power Law Distributions

Tail estimates are developed for power law probability distributions with exponential tempering, using a conditional maximum likelihood approach based on the upper-order statistics. Tempered power law distributions are intermediate between heavy power-law tails and Laplace or exponential tails, and are sometimes called “semi-heavy” tailed distributions. The estimation method is demonstrated on simulated data from a tempered stable distribution, and for several data sets from geophysics and finance that show a power law probability tail with some tempering.

Beyond power laws: A new approach for analyzing single molecule photoluminescence intermittency

The photoluminescence intermittency (PI) exhibited by single emitters has been studied for over a decade. To date, the vast majority of PI analyses involve parsing the data into emissive and non-emissive events, constructing histograms of event durations, and fitting these histograms to either exponential or power law probability distributions functions (PDFs). Here, a new method for analyzing PI data is presented where the data are used directly to construct a cumulative distribution function (CDF), and maximum-likelihood estimation techniques are used to determine the best fit of a model PDF to the CDF. Statistical tests are then employed to quantitatively evaluate the hypothesis that the CDF (data) is represented by the model PDF. The analysis method is outlined and applied to PI exhibited by single CdSe∕CdS core-shell nanocrystals and the organic chromophore violamine R isolated in single crystals of potassium-acid phthalate. Contrary to previous studies, the analysis presented here demonstrates that the PI exhibited by these systems is not described by a power law. The analysis developed here is also used to quantify heterogeneity within PI data obtained from a collection of CdSe/CdS nanocrytals, and for the determination of statistically significant changes in PI accompanying perturbation of the emitter. In summary, the analysis methodology presented here provides a more statistically robust approach for analyzing PI data.

Benford’s Law in Time Series Analysis of Seismic Clusters

Benford’s analysis is applied to the recurrence times of approximately 17,000 seismic events in different geological contexts of Italy over the last 6 years, including the Mt. Etna volcanic area and the seismic series associated with the destructive Mw 6.3, 2009 L’Aquila earthquake. A close conformity to Benford’s law and a power-law probability distribution for the recurrence times of consecutive events is found, as typical of random multiplicative processes. The application of Benford’s law to the recurrence event times in seismic series of specific seismogenic regions represents a novel approach, which enlarges the occurrence and relevance of Benford-like asymmetries, with implications on the physics of natural systems approaching a power law behaviour. Moreover, we propose that the shift from a close conformity of Benford’s law to Brownian dynamics, observed for time separations among non-consecutive events in the study seismic series, may be ruled by a periodical noise factor, such as the effects of Earth tides on seismicity tuning.

Evolutionary Trade-Offs, Pareto Optimality, and the Geometry of Phenotype Space

Biological systems that perform multiple tasks face a fundamental trade-off: A given phenotype cannot be optimal at all tasks. Here we ask how trade-offs affect the range of phenotypes found in nature. Using the Pareto front concept from economics and engineering, we find that best-trade-off phenotypes are weighted averages of archetypes--phenotypes specialized for single tasks. For two tasks, phenotypes fall on the line connecting the two archetypes, which could explain linear trait correlations, allometric relationships, as well as bacterial gene-expression patterns. For three tasks, phenotypes fall within a triangle in phenotype space, whose vertices are the archetypes, as evident in morphological studies, including on Darwin's finches. Tasks can be inferred from measured phenotypes based on the behavior of organisms nearest the archetypes.

Simulation of landscape evolution using a global flow path search method

The choice of a flow path extraction method is a key issue in numerical simulation of fluvial landscape evolution. Most whole landscape evolution models use the deterministic eight-neighbor flow direction retrieval method (D8), which generates angular uncertainty in determined flow paths. In the modeling of landscape evolution, the uncertainty generated at each local flow direction accumulates over both space and time. Recently, a new method which searches for flow paths over global scale, called global D8 or GD8, was proposed as an alternative to D8. GD8 relaxes uncertainty generated at a local level over an entire flow path while still defining specific flow paths without artificial dispersion. On the basis of these advantages of GD8 demonstrated for static landscapes, this paper presents the first landscape evolution model that uses GD8, i.e., a new Landscape Evolution model using Global Search (LEGS). Using LEGS, the difference between D8 and GD8 simulations is investigated with focus on both evolution rate and resulting topography. Theoretical landscapes simulated by LEGS are evaluated using typical characteristics of natural river networks such as concave profiles, Hack's law, the power-law exceedance probability distribution of contributing areas, and the power-law variogram.

Wind Velocity Vertical Extrapolation by Extended Power Law

Wind energy gains more attention day by day as one of the clean renewable energy resources. We predicted wind speed vertical extrapolation by using extended power law. In this study, an extended vertical wind velocity extrapolation formulation is derived on the basis of perturbation theory by considering power law and Weibull wind speed probability distribution function. In the proposed methodology not only the mean values of the wind speeds at different elevations but also their standard deviations and the cross-correlation coefficient between different elevations are taken into consideration. The application of the presented methodology is performed for wind speed measurements at Karaburun/Istanbul, Turkey. At this location, hourly wind speed measurements are available for three different heights above the earth surface.