# Possible Generalization Of Boltzmann Gibbs Statistics Pdf And Cdf

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*The flow velocity distribution in partially-filled circular pipe was investigated in this paper. The velocity profile is different from full-filled pipe flow, since the flow is driven by gravity, not by pressure.*

- Forecasting Using Information and Entropy Based on Belief Functions
- The κ-statistics approach to epidemiology
- Copula–entropy theory for multivariate stochastic modeling in water engineering

*Wind-speed statistics are generally modelled using the Weibull distribution. However, the Weibull distribution is based on empirical rather than physical justification and might display strong limitations for its applications. Here, we derive wind-speed distributions analytically with different assumptions on the wind components to model wind anisotropy, wind extremes and multiple wind regimes.*

## Forecasting Using Information and Entropy Based on Belief Functions

Wind-speed statistics are generally modelled using the Weibull distribution. However, the Weibull distribution is based on empirical rather than physical justification and might display strong limitations for its applications. Here, we derive wind-speed distributions analytically with different assumptions on the wind components to model wind anisotropy, wind extremes and multiple wind regimes.

We quantitatively confront these distributions with an extensive set of meteorological data 89 stations covering various sub-climatic regions in France to identify distributions that perform best and the reasons for this, and we analyze the sensitivity of the proposed distributions to the diurnal to seasonal variability.

We find that local topography, unsteady wind fluctuations as well as persistent wind regimes are determinants for the performances of these distributions, as they induce anisotropy or non-Gaussian fluctuations of the wind components. A Rayleigh—Rice distribution is proposed to model the combination of weak isotropic wind and persistent wind regimes. It outperforms all other tested distributions Weibull, elliptical and non-Gaussian and is the only proposed distribution able to catch accurately the diurnal and seasonal variability.

Understanding and modelling wind-speed statistics is key to a better understanding of atmospheric turbulence and diffusion, and at stake in practical applications such as air quality and pollution transport modelling, estimation of wind loads on buildings, prediction of atmospheric or space probe and missile trajectory, and wind-power analysis. This is done using the distribution of wind speed M. The Weibull distribution is an extremely commonly used paradigm used to model wind-speed statistics e.

In wind-energy research the Weibull distribution is used to obtain additional flexibility in order to fit an observed wind-speed histogram. To produce e. The method of moments is a method of estimation of distribution parameters introduced by e.

Pearson , a , b , , and one begins with deriving equations that relate the moments of the distribution i. Then a sample is drawn and the moments are estimated from the sample, with the equations then solved for the parameters, using the sample moments. This results in estimates of those parameters. The method of moments ensures the best estimation of wind-energy potential but does not ensure the maximum likelihood with the observed histograms.

This can lead to large errors when considering only a fraction of the wind distribution, between the cut-in and cut-out wind speeds of a specific wind-turbine power curve for instance. For some design applications including wind loads and structural safety, it is also necessary to have information on the distribution of the complete population of wind speed at a site. Estimation of fatigue damage must account for damage accumulation over a range of extreme winds, the distribution of which is usually fitted with a distribution of the Weibull type Davenport In chemistry-transport modelling, the Weibull distribution is used to represent the subgrid-scale variability of the wind speed.

This allows improvement of the simulation of aerosol saltation at the surface and emission fluxes into the atmosphere that are triggered by a threshold in the wind speed Menut In such contexts, maximizing the likelihood of fitted distributions to observed wind-speed histograms is a major issue. However, it has been long known that the Weibull distribution is only an approximation and may fit poorly the wind-speed statistics, especially in the case of non-circular i.

The wide use of the Weibull distribution is purely empirical and there is a lack of physical background justifying the use of the Weibull distribution to model wind statistics. Many previous studies have considered the limitations of the Weibull distribution for modelling wind speeds e.

In situ and modelled oceanic surface wind speeds from extratropical latitudes are reasonably well simulated by the Weibull distribution Bauer Conversely, the remotely sensed wind speeds agree poorly with the corresponding empirical distributions. Possible better suited surface wind-speed frequency distributions have been investigated e. The Weibull distribution parameters present a series of advantages with respect to other distributions e.

Therefore, its generalized use cannot be justified. Other distributions based on an expansion of orthogonal polynomials Morrissey and Greene or the maximum entropy principle Li and Li can produce more accurate estimates of the wind-speed distribution than the Weibull function, and can represent a wider range of data types as well.

In general, the other tested wind-speed distributions displayed lower performance. In this study, we propose to derive wind-speed distributions based on the use of the bivariate distribution with the two wind components. Indeed, by contrast with the wind-speed modulus, wind components obey the momentum conservation equations e.

As stated above, several authors proposed the use of the bivariate normal distribution with wind speed and direction as variables.

In Weber , no restrictions are imposed on the standard deviations of the longitudinal and lateral fluctuations. Here, we go one step further, and use wind records at 89 locations in France to:.

Finally Sect. This database has long time records, but we only use measurements made between January 1 and December 31 because the accuracy is better over these recent years: the wind speed M is binned with intervals of 1 knot 0. The min averaged wind speeds and directions are recorded every hour. The calm conditions represent 5.

When wind components are used instead of wind speed, the zonal west—east and meridional south—north wind components u and v are derived from the wind speeds and directions.

The wind components are often correlated so we use the procedure described by Crutcher and Baer to reduce component correlation to zero. The procedure relies on the property that any Gaussian joint probability distribution is characterized by a covariance matrix, which is positive and definite.

Considering correlation free components for the wind field allows the manipulation of simpler expressions for the distributions.

The topography of France is presented in Fig. The stations located in the northern and western parts of France are in rather flat terrain. In the northern and western regions, all wind directions are experienced, even though this part of France is located in the storm track so that strong winds are often from the west, from the Atlantic Ocean Vautard ; Plaut and Vautard ; Simonnet and Plaut Indeed, over the north-west Atlantic cyclones originate, travel eastwards and affect the European continent.

In the southern region, frequent channelled flow can persist for several days. The mistral shares its occurrence with a northerly land breeze and southerly sea breeze e.

In such a region, accounting for such persistent wind systems for modelling the wind-speed statistics is thus mandatory. We introduce several wind-speed distributions and analyze how they fit the observational data.

It must be noted that the Weibull distribution leads to an analytic expression for the CDF, but not so for the other distributions that will be derived hereafter. Because of the coarse resolution of the observations 1 knot , this empirical CDF is a step function so we add a small random noise to the wind-speed data in order to smooth the CDF.

Adding this noise does not affect the fitting of the distributions and it is not necessary to compute the distance scores. So smoothing the empirical CDF enables us to better quantify the performances of the distributions and to make comparison between different stations easier. The goodness-of-fit scores used herein are Cramer—von Mises and modified Anderson—Darling statistics.

A general equation for these scores is. Here, the centre of the distribution is not one single point but the region around the median, mean or maximum of the distribution where the PDF is above, say, 0.

As in any goodness-of-fit test, there are thresholds for rejection of the null hypothesis i. But these thresholds depend on the distribution F , on the autocorrelation in the data and could only be estimated by simulations e. Therefore the question of limit values for the metrics is very complex and beyond the scope of the present work. Rather than defining thresholds we will simply compare the scores of different fits at each station. As a reference before introducing other distributions, we consider the Weibull distribution for the wind speed M Eq.

First we benchmark four different methods for fitting the Weibull distribution because there are several possible fitting procedures. Popular methods include the method of moments, often used in a wind atlas, or by the usual maximum likelihood estimate MLE. Examples of the Weibull fits from the four fitting methods are given in Fig. While good fits have a similar shape regardless of the fitting methods in Nantes Fig. ADR minimization is a good compromise favouring the tail but not overly so, and also yields results similar to the MLE.

For these reasons we adopt the ADR minimization for all fits throughout the study. Black observed distributions. The y -axis is divided into a linear axis and a logarithmic axis to better resolve the tail.

In the flat terrain of northern France, as at Nantes Fig. In wind energy, the tail of the wind distribution is not important for estimating the wind resource but it is of high importance when addressing wind loading and damage fatigue, pollutant transport or the impact of wind storms.

In more complex terrain, the fit to the Weibull distribution is less accurate. So we cannot compare one station to another but we can compare the scores of several fits at a unique station see Sect. Nevertheless, it is important to provide an estimate of the fit quality.

This is consistent with that we observed at the three stations where only the fit at the centre of the distribution at Nantes is excellent.

In southern France, the measured wind-speed histograms deviate significantly from the Weibull distribution. Note that the systematic deviation from the Weibull distribution in the southern region might be related to the complex topography. In the following, we investigate alternative distributions, where topography-induced effects such as wind anisotropy and the existence of persistent wind regimes are captured.

Weibull distribution fit by minimizing the ADR score. For a deeper insight into the differences between observed wind-speed distributions and the commonly used Weibull distribution, we now consider bivariate distributions of the two wind components to take into account the wind-field anisotropy.

At many stations in the southern regions, where the Weibull distribution does not model the wind statistics well, the wind field is very anisotropic. Indeed, we see in Fig. Plain curves observed distributions for u red and v blue. Dashed curves their fits using a Gaussian distribution. This particular bivariate normal distribution will be called elliptical hereafter. With the previous elliptical distribution, we assumed the wind components to follow a Gaussian distribution.

However, this assumption is not always valid, since Gaussian curves sometimes fail to describe the histograms—see Fig. We can evaluate the departure of each component u and v from a Gaussian shape by computing the ADR score for the two components, i. Theoretically, the strict Gaussianity is reached when the sum equals zero, however, from visual inspection, a value around 20 can still be considered as reasonably Gaussian. In the flat terrain of north-western France, the scores are not too high, indicating a good fit to a Gaussian.

This is however not the case in southern and eastern France. In the following, we use super-statistics defined by Beck and Cohen to address such a deviation from Gaussianity. This approach consists in representing the long-term stationary state by a superposition of different states that are weighted with a certain probability density.

For the sake of brevity, we focus on one component, u , throughout the following.

## The κ-statistics approach to epidemiology

Thank you for visiting nature. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser or turn off compatibility mode in Internet Explorer. In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript. A great variety of complex physical, natural and artificial systems are governed by statistical distributions, which often follow a standard exponential function in the bulk, while their tail obeys the Pareto power law.

Picoli Jr. Mendes; L. Malacarne; R. The nonextensive statistical mechanics proposed by Tsallis is today an intense and growing research field. Probability distributions which emerges from the nonextensive formalism q -distributions have been applied to an impressive variety of problems.

This paper introduces an entropy-based belief function to the forecasting problem. While the likelihood-based belief function needs to know the distribution of the objective function for the prediction, the entropy-based belief function does not. This is because the observed data likelihood is somewhat complex in practice. We, thus, replace the likelihood function with the entropy. That is, we propose an approach in which a belief function is built from the entropy function. As an illustration, the proposed method is compared to the likelihood-based belief function in the simulation and empirical studies.

The Tsallis entropy is widely used in physics to study the distribution characterizing the motion of cold Possible generalization of Boltzmann-Gibbs statistics.

## Copula–entropy theory for multivariate stochastic modeling in water engineering

Metrics details. The copula—entropy theory combines the entropy theory and the copula theory. The entropy theory has been extensively applied to derive the most probable univariate distribution subject to specified constraints by applying the principle of maximum entropy.

Distributional properties and stochastic comparisons with other known weighted distributions are given. Furthermore, an upper bound for the k -order moment of the random variables associated with the new family and a characterization result are obtained. This is a preview of subscription content, access via your institution.

*The q -exponential distribution is a probability distribution arising from the maximization of the Tsallis entropy under appropriate constraints, including constraining the domain to be positive. It is one example of a Tsallis distribution. The q -exponential is a generalization of the exponential distribution in the same way that Tsallis entropy is a generalization of standard Boltzmann—Gibbs entropy or Shannon entropy.*

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FradinmigavKeywords: Topp-Leone distribution, beta-generated, generalized exponential, parame- the probability density function (pdf) and describes empirical data the possible states by maximizing the Boltzmann-Gibbs entropy S.

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