Q-exponential Distribution

[Keesha Ondieki]

Asthe Lomax distribution is a shifted version of the Pareto distribution, the q-exponential is a shifted reparameterized generalization of the Pareto. When q > 1, the q-exponential is equivalent to the Pareto shifted to have support starting at zero. Specifically, if

Being a power transform, it is a usual technique in statistics for stabilizing the variance, making the data more normal distribution-like and improving the validity of measures of association such as the Pearson correlation between variables. It has been found to be an accurate model for train delays.[3]It is also found in atomic physics and quantum optics, for example processes of molecular condensate creation via transition through the Feshbach resonance.[4]

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Abstract: The Gibbs distribution of statistical physics is an exponential family of probability distributions, which has a mathematical basis of duality in the form of the Legendre transformation. Recent studies of complex systems have found lots of distributions obeying the power law rather than the standard Gibbs type distributions. The Tsallis q-entropy is a typical example capturing such phenomena. We treat the q-Gibbs distribution or the q-exponential family by generalizing the exponential function to the q-family of power functions, which is useful for studying various complex or non-standard physical phenomena. We give a new mathematical structure to the q-exponential family different from those previously given. It has a dually flat geometrical structure derived from the Legendre transformation and the conformal geometry is useful for understanding it. The q-version of the maximum entropy theorem is naturally induced from the q-Pythagorean theorem. We also show that the maximizer of the q-escort distribution is a Bayesian MAP (Maximum A posteriori Probability) estimator. Keywords: q-exponential family; q-entropy; information geometry; q-Pythagorean theorem; q-Max-Ent theorem; conformal transformation

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The q-exponential distributions, which are generalizations of the Zipf-Mandelbrot power-law distribution, are frequently encountered in complex systems at their stationary states. From the viewpoint of the principle of maximum entropy, they can apparently be derived from three different generalized entropies: the Rnyi entropy, the Tsallis entropy, and the normalized Tsallis entropy. Accordingly, mere fittings of observed data by the q-exponential distributions do not lead to identification of the correct physical entropy. Here, stabilities of these entropies, i.e., their behaviors under arbitrary small deformation of a distribution, are examined. It is shown that, among the three, the Tsallis entropy is stable and can provide an entropic basis for the q-exponential distributions, whereas the others are unstable and cannot represent any experimentally observable quantities.

The critical fraction fc for malicious attacks as a function of q for λ=0.1 (green triangles), λ=1 (red squares), and λ=100 (black circles). The results are obtained for networks of size N=500000 by averaging over 2000 samples. The fraction from which the largest cluster does not obey Molloy-Reed's criterion is the critical fraction fc. The q-exponential networks with smaller values of λ are clearly more robust, as a larger fraction of nodes needs to be removed to attain the critical point. The continuous lines represent guides to the eye.

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Following you can see a part of data, for which I am trying to fit a q-exponential distribution functin as I explained above. I am uiung nls function in R to obtain estimates (q-exponential) for the given data.

If you just evaluate model() at the initial parameters you can see that some of the derivatives with respect to q are NaN (it is always worthwhile to try evaluating the objective function and gradient at the starting values to make sure they make sense). In addition to this, if you look at the values of the objective function you can see that they're nowhere close to the data (they increase from 1 to 42, whereas the data decrease from 200000 to 3000 ...)

I am running a simulation task which is supposed to sample data from an exponential distribution, but the results are always larger than correct ones. I converted my julia code to matlab and it produced correct results (without any logic and algorithm changes).

When running this line of code repeatedly in the REPL, I found that it displayed significantly 2 different histograms, and the incorrect one was clearer when I combined them into one plot (see below).

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. In particular, the role of q-distributions in the interdisciplinary field of complex systems has been expanding. Here, we make a brief review of q-exponential, q-Gaussian and q-Weibull distributions focusing some of their basic properties and recent applications. The richness of systems analyzed may indicate future directions in this field.

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. In particular, the role of q-distributions in the interdisciplinary field of complex systems has been expanding. Here, we make a brief review of q-exponential, q-Gaussian and q-Weibull distributions focusing some of their basic properties and recent applications. The richness of systems analyzed may indicate future directions in this field.

Common characteristics of complex systems include long-range correlations, multifractality and non-Gaussian distributions with asymptotic power law behavior. Typically, such systems are not well described by approaches based on the usual statistical mechanics. In this scenario, a new formalism capable of providing a better description of complex systems is welcome. This is the case of the generalized (nonextensive) statistical mechanics proposed by Tsallis -nowadays, an intense and growing research field[1-4].

Concepts related with nonextensive statistical mechanics have found applications in a variety of disciplines including physics, chemistry, biology, mathematics, geography, economics, medicine, informatics, linguistics among others[5- 7]. Probability distributions which emerge from the nonextensive formalism -also called q-distributions -have been applied to an impressive variety of problems indiverse research areas including the interdisciplinary field of complex systems.

In the present work we focus on q-exponential, q-Gaussian and q-Weibull distributions. We summarized some of their basic properties and provide useful references of recent applications. The richness of systems analyzed may indicate future directions in this research line.

The q-exponential function given by eq. (1) has been employed in a growing number of theoretical and empirical works on a large variety of themes. Examples include scale-free networks[10-14], dynamical systems[15-27], algebraic structures[28-31] among other topics in statistical physcics[32-36].

As specific examples of q-exponential distributions in complex systems, let us consider results on population of cities[37] and circulation of magazines[38]. Figure 2 shows the cumulative distribution of the population of cities in the USA and Brazil. Figure 3 shows the cumulative distribution of circulation of magazines in the USA and UK. In both cases - population of cities and circulation of magazines - the empirical data are consistent with a q-exponential distribution, with q 1.4.

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