A family of nonlinear ordinary differential equations with arbitrary order is obtained by using nonextensive concepts related to the Tsallis entropy. Applications of these equations are given here. In particular, a connection between Tsallis entropy and the one-dimensional correlated anomalous diffusion equation is established. It is also developed explicitly a WKB-like method for second order ...

Kolmogorov complexity and Shannon entropy are conceptually different measures. However, for any recursive probability distribution, the expected value of Kolmogorov complexity equals its Shannon entropy, up to a constant. We study if a similar relationship holds for Rényi and Tsallis entropies of order α, showing that it only holds for α = 1. Regarding a time-bounded analogue relationship, we s...

In this paper a novel feature extraction method for image processing via PCNN and Tsallis entropy is presented. We describe the mathematical model of the PCNN and the basic concept of Tsallis entropy in order to find a recognition method for isolated objects. Experiments show that the novel feature is translation and scale independent, while rotation independence is a bit weak at diagonal angle...

We discuss the generalized von Neumann (Tsallis) entropy and the generalized Fisher information (GFI) in nonextensive quantum systems, by using the interpolation approximation (IA) which has been shown to yield good results for the quantal distributions within O(q − 1) and in highand lowtemperature limits, q being the entropic index [H. Hasegawa, Phys. Rev. E 80 (2009) 011126]. Three types of G...

A two-parameter family of statistical measures of complexity are introduced based on the Tsallis-type nonadditive entropies. This provides a unified framework for the study of the recently proposed various measures of complexity as well as for the discussion of a whole new class of measures. As a special case, a generalization of the measure proposed by Landsberg and his co-workers based on the...

Landsberg’s notion of disorder, entropy normalized to maximum entropy, was originally proposed for the Shannon information-theoretic entropy to overcome deficiencies of entropy as a measure of disorder due to extensivity. We generalize Landsberg’s concept to three classes of extended entropies: Rényi, Tsallis and Landsberg-Vedral. Three examples are treated, including one based on the logistic ...

Tsallis’ ‘statistical thermodynamic’ formulation of the nonadditive entropy of degree-α is neither correct nor self-consistent. It is well known that the maximum entropy formalism [1], the minimum discrimination information [2], and Gauss’ principle [3, 4] all lead to the same results when a certain condition on the prior probability distribution is imposed [5]. All these methods lead to the sa...

E.T. Jaynes, originator of the maximum entropy interpretation of statistical mechanics, emphasized that there is an inevitable trade-off between the conflicting requirements of robustness and accuracy for any inferencing algorithm. This is because robustness requires discarding of information in order to reduce the sensitivity to outliers. The principal of nonlinear statistical coupling, which ...

The Shannon entropy is a widely used summary statistic, for example, network traffic measurement, anomaly detection, neural computations, spike trains, etc. This study focuses on estimating Shannon entropy of data streams. It is known that Shannon entropy can be approximated by Rényi entropy or Tsallis entropy, which are both functions of the αth frequency moments and approach Shannon entropy a...

In this paper, we show that 1) additive energy is not appropriate for discussing the validity of Tsallis or Rényi statistics for nonextensive systems at equilibrium; 2) equilibrium N-body systems with nonad-ditive energy or entropy should be described by generalized statistics whose nature is prescribed by the existence of thermodynamic equilibrium. The nonextensive statistics may be Tsallis or...