Nonextensive statistics and incomplete information

Wang, Qiuping A.

doi:10.1140/epjb/e20020100

Cited by 36 publications

(51 citation statements)

References 21 publications

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“…In other words, the statistic range of Shannon entropy is restricted to additive signals, which like B-G entropy only fits into extensive systems in thermodynamics [13]. It is well known that Boltzmann-Gibbs statistics (BGS) are inadequate for treating some complex systems [14].…”

Section: The Theoretical Defects Of Sweementioning

confidence: 99%

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Tsallis Wavelet Entropy and Its Application in Power Signal Analysis

Chen

2014

Entropy

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As a novel data mining approach, a wavelet entropy algorithm is used to perform entropy statistics on wavelet coefficients (or reconstructed signals) at various wavelet scales on the basis of wavelet decomposition and entropy statistic theory. Shannon wavelet energy entropy, one kind of wavelet entropy algorithm, has been taken into consideration and utilized in many areas since it came into being. However, as there is wavelet aliasing after the wavelet decomposition, and the information set of different-scale wavelet decomposition coefficients (or reconstructed signals) is non-additive to a certain extent, Shannon entropy, which is more adaptable to extensive systems, couldn't do accurate uncertainty statistics on the wavelet decomposition results. Therefore, the transient signal features are extracted incorrectly by using Shannon wavelet energy entropy. From the two aspects, the theoretical limitations and negative effects of wavelet aliasing on extraction accuracy, the problems which exist in the feature extraction process of transient signals by Shannon wavelet energy entropy, are discussed in depth. Considering the defects of Shannon wavelet energy entropy, a novel wavelet entropy named Tsallis wavelet energy entropy is proposed by using Tsallis entropy instead of Shannon entropy, and it is applied to the feature extraction of transient signals in power systems. Theoretical derivation and experimental result prove that compared with Shannon wavelet energy entropy, Tsallis wavelet energy entropy could reduce the negative effects of wavelet aliasing on accuracy of feature extraction and extract transient signal feature of power system accurately.

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Section: The Theoretical Defects Of Sweementioning

confidence: 99%

“…Taking a three-level system as the analysis object, according to Equation (13), the statistical results are calculated, and the corresponding relation between Tsallis entropy with q and probability distribution are shown in Figure 7. …”

Section: The Inhibition On Wavelet Aliasing Using Tweementioning

confidence: 99%

Tsallis Wavelet Entropy and Its Application in Power Signal Analysis

Chen

2014

Entropy

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“…Sometimes this assumption may fail [16][17][18][19] and the sum of the probability of events becomes less than unity, n a=1 p a < 1, which means that we do not know the set of events completely and some events may possibly be remained unknown. For example earthquakes with strength of greater than ten Richter have not been recorded yet but they would be possible.…”

Section: Incomplete Information Theorymentioning

confidence: 99%

“…Wang, in his works assumed that, a real parameter q exists so that p q a = 1. He also changed the last two axioms of the information theory to include the incomplete information condition [16,17],…”

Section: Incomplete Information Theorymentioning

confidence: 99%

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Tsallis Entropy, Escort Probability and the Incomplete Information Theory

Darooneh

Naeimi

Mehri

et al. 2010

Entropy

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Non-extensive statistical mechanics appears as a powerful way to describe complex systems. Tsallis entropy, the main core of this theory has been remained as an unproven assumption. Many people have tried to derive the Tsallis entropy axiomatically. Here we follow the work of Wang (EPJB, 2002) and use the incomplete information theory to retrieve the Tsallis entropy. We change the incomplete information axioms to consider the escort probability and obtain a correct form of Tsallis entropy in comparison with Wang's work.

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Thermoequilibrium statistics for a finite system with energy nonextensivity

Zheng

2011

Chin. Sci. Bull.

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A first-principles derivation is presented of canonical distributions for a finite thermostat taking into account nonextensive energy.Parameterizing this energy by , we derive an explicit form for the distribution functions by regulating , and then explore the nontrivial relationship between these functions and energy nonextensivity, as well other system parameters such as system size. A variational entropy function is also derived from these distribution functions.. finite system, energy nonextensivity, first-principles method, canonical distribution function Citation:Zheng L, Li W. Thermoequilibrium statistics for a finite system with energy nonextensivity. Chinese Sci Bull, 2011Bull, , 56: 3666-3670, doi: 10.1007 Studies into the family of small or finite-size systems reveal that these systems show complex statistical characteristics very different from large systems, and expose failings in our understanding of thermostatistics for finite systems and their nonequilibrium dynamics [1][2][3][4]. One of these characteristics is nonextensivity (refer to [2] and references therein for examples), which means that macroscopic quantities of such systems may not be proportional to system size. A thermodynamic quantity of a nonextensive system is called nonadditive if this quantity is not the sum of that of its subsystems. Nonextensivity might arise from surface effects or interactions between subsystems. Thus, extensive energy and entropy may become inappropriate in treating finite systems. There has already been much discussion on the properties of finite systems that has raised many questions and controversies [5][6][7]. To describe the behaviors of these systems, one point worth noting is that the Boltzmann-Gibbs statistical mechanics, which is based on the concept of extensive entropy, is not applicable [1,8,9]. Consequently, it is questionable to assume an exponential distribution for finite systems. Therefore, one of the first things to work out for a statistical description of finite systems is to find the appropriate probability distribution. Some published results obtained by first principles (see [8] for example) have shown that a small system, in equilibrium with a finite reservoir, may follow a q-exponential distribution of nonextensive statistical mechanics (NSM), as proposed by Tsallis [9]. This NSM has been used widely in many areas, and considered effective in solving many physical problems [10][11][12][13]. Results from mathematical proofs are able to demonstrate a connection between the system's finiteness and nonextensivity of the theory. However, most proofs have relied on the additivity of energy. Clearly, for large systems, this assumption is acceptable and provides a helpful approximation in obtaining the statistics in the thermodynamic limit. The assumption is, however, questionable when establishing statistics for finite systems. Related problems arising from this additive energy assumption can be found in [14][15][16][17].Work in [18] represents a first attempt to build statistics for fin...

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Nonextensive statistics and incomplete information

Cited by 36 publications

References 21 publications

Tsallis Wavelet Entropy and Its Application in Power Signal Analysis

Tsallis Wavelet Entropy and Its Application in Power Signal Analysis

Tsallis Entropy, Escort Probability and the Incomplete Information Theory

Thermoequilibrium statistics for a finite system with energy nonextensivity

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