Kaniadakis distribution
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In statistics, a Kaniadakis distribution (also known as κ-distribution) is a statistical distribution that emerges from the Kaniadakis statistics.[1] There are several families of Kaniadakis distributions related to different constraints used in the maximization of the Kaniadakis entropy, such as the κ-Exponential distribution, κ-Gaussian distribution, Kaniadakis κ-Gamma distribution and κ-Weibull distribution. The κ-distributions have been applied for modeling a vast phenomenology of experimental statistical distributions in natural or artificial complex systems, such as, in epidemiology,[2] quantum statistics,[3][4][5] in astrophysics and cosmology,[6][7][8] in geophysics,[9][10][11] in economy,[12][13][14] in machine learning.[15]
The κ-distributions are written as function of the κ-deformed exponential, taking the form
enables the power-law description of complex systems following the consistent κ-generalized statistical theory.,[16][17] where is the Kaniadakis κ-exponential function.
The κ-distribution becomes the common Boltzmann distribution at low energies, while it has a power-law tail at high energies, the feature of high interest of many researchers.
List of κ-statistical distributions
[edit]Supported on the whole real line
[edit]![](http://upload.wikimedia.org/wikipedia/commons/thumb/2/2b/Kaniadakis_Gaussian_Distribution_Type_II_pdf.png/337px-Kaniadakis_Gaussian_Distribution_Type_II_pdf.png)
- The Kaniadakis Gaussian distribution, also called the κ-Gaussian distribution. The normal distribution is a particular case when
- The Kaniadakis double exponential distribution, as known as Kaniadakis κ-double exponential distribution or κ-Laplace distribution. The Laplace distribution is a particular case when [18]
Supported on semi-infinite intervals, usually [0,∞)
[edit]![](http://upload.wikimedia.org/wikipedia/commons/thumb/9/91/Kaniadakis_Gamma_Distribution_pdf.png/340px-Kaniadakis_Gamma_Distribution_pdf.png)
- The Kaniadakis Exponential distribution, also called the κ-Exponential distribution. The exponential distribution is a particular case when
- The Kaniadakis Gamma distribution, also called the κ-Gamma distribution, which is a four-parameter () deformation of the generalized Gamma distribution.
- The κ-Gamma distribution becomes a ...
- κ-Exponential distribution of Type I when .
- κ-Erlang distribution when and positive integer.
- κ-Half-Normal distribution, when and .
- Generalized Gamma distribution, when ;
- In the limit , the κ-Gamma distribution becomes a ...
- Erlang distribution, when and positive integer;
- Chi-Squared distribution, when and half integer;
- Nakagami distribution, when and ;
- Rayleigh distribution, when and ;
- Chi distribution, when and half integer;
- Maxwell distribution, when and ;
- Half-Normal distribution, when and ;
- Weibull distribution, when and ;
- Stretched Exponential distribution, when and ;
- The κ-Gamma distribution becomes a ...
Common Kaniadakis distributions
[edit]κ-Exponential distribution
[edit]κ-Gaussian distribution
[edit]κ-Gamma distribution
[edit]κ-Weibull distribution
[edit]κ-Logistic distribution
[edit]κ-Erlang distribution
[edit]κ-Distribution Type IV
[edit] Probability density function ![]() Plot of the κ-Distribution Type IV for typical κ-values, and . | |||
Cumulative distribution function ![]() | |||
Parameters | shape (real) rate (real) | ||
---|---|---|---|
Support | |||
CDF | |||
Method of moments |
The Kaniadakis distribution of Type IV (or κ-Distribution Type IV) is a three-parameter family of continuous statistical distributions.[1]
The κ-Distribution Type IV distribution has the following probability density function:
valid for , where is the entropic index associated with the Kaniadakis entropy, is the scale parameter, and is the shape parameter.
The cumulative distribution function of κ-Distribution Type IV assumes the form:
The κ-Distribution Type IV does not admit a classical version, since the probability function and its cumulative reduces to zero in the classical limit .
Its moment of order given by
The moment of order of the κ-Distribution Type IV is finite for .
See also
[edit]- Giorgio Kaniadakis
- Kaniadakis statistics
- Kaniadakis κ-Exponential distribution
- Kaniadakis κ-Gaussian distribution
- Kaniadakis κ-Gamma distribution
- Kaniadakis κ-Weibull distribution
- Kaniadakis κ-Logistic distribution
- Kaniadakis κ-Erlang distribution
References
[edit]- ^ a b Kaniadakis, G. (2021-01-01). "New power-law tailed distributions emerging in κ-statistics (a)". Europhysics Letters. 133 (1): 10002. arXiv:2203.01743. Bibcode:2021EL....13310002K. doi:10.1209/0295-5075/133/10002. ISSN 0295-5075. S2CID 234144356.
- ^ Kaniadakis, Giorgio; Baldi, Mauro M.; Deisboeck, Thomas S.; Grisolia, Giulia; Hristopulos, Dionissios T.; Scarfone, Antonio M.; Sparavigna, Amelia; Wada, Tatsuaki; Lucia, Umberto (2020). "The κ-statistics approach to epidemiology". Scientific Reports. 10 (1): 19949. arXiv:2012.00629. Bibcode:2020NatSR..1019949K. doi:10.1038/s41598-020-76673-3. ISSN 2045-2322. PMC 7673996. PMID 33203913.
- ^ Santos, A.P.; Silva, R.; Alcaniz, J.S.; Anselmo, D.H.A.L. (2011). "Generalized quantum entropies". Physics Letters A. 375 (35): 3119–3123. Bibcode:2011PhLA..375.3119S. doi:10.1016/j.physleta.2011.07.001.
- ^ Ourabah, Kamel; Tribeche, Mouloud (2014-06-24). "Planck radiation law and Einstein coefficients reexamined in Kaniadakis κ statistics". Physical Review E. 89 (6): 062130. Bibcode:2014PhRvE..89f2130O. doi:10.1103/PhysRevE.89.062130. ISSN 1539-3755. PMID 25019747.
- ^ Lourek, Imene; Tribeche, Mouloud (2017). "Thermodynamic properties of the blackbody radiation: A Kaniadakis approach". Physics Letters A. 381 (5): 452–456. Bibcode:2017PhLA..381..452L. doi:10.1016/j.physleta.2016.12.019.
- ^ Carvalho, J. C.; do Nascimento, J. D.; Silva, R.; De Medeiros, J. R. (2009-05-01). "Non-Gaussian Statistics and Stellar Rotational Velocities of Main-Sequence Field Stars". The Astrophysical Journal. 696 (1): L48–L51. arXiv:0903.0868. Bibcode:2009ApJ...696L..48C. doi:10.1088/0004-637X/696/1/L48. ISSN 0004-637X. S2CID 17161421.
- ^ Abreu, Everton M.C.; Ananias Neto, Jorge; Mendes, Albert C.R.; de Paula, Rodrigo M. (2019). "Loop quantum gravity Immirzi parameter and the Kaniadakis statistics". Chaos, Solitons & Fractals. 118: 307–310. arXiv:1808.01891. Bibcode:2019CSF...118..307A. doi:10.1016/j.chaos.2018.11.033. S2CID 119207713.
- ^ Soares, Bráulio B.; Barboza, Edésio M.; Abreu, Everton M.C.; Neto, Jorge Ananias (2019). "Non-Gaussian thermostatistical considerations upon the Saha equation". Physica A: Statistical Mechanics and Its Applications. 532: 121590. arXiv:1901.01839. Bibcode:2019PhyA..53221590S. doi:10.1016/j.physa.2019.121590. S2CID 119539402.
- ^ Hristopulos, Dionissios T.; Petrakis, Manolis P.; Kaniadakis, Giorgio (2014-05-28). "Finite-size effects on return interval distributions for weakest-link-scaling systems". Physical Review E. 89 (5): 052142. arXiv:1308.1881. Bibcode:2014PhRvE..89e2142H. doi:10.1103/PhysRevE.89.052142. ISSN 1539-3755. PMID 25353774. S2CID 22310350.
- ^ da Silva, Sérgio Luiz E.F. (2021). "κ -generalised Gutenberg–Richter law and the self-similarity of earthquakes". Chaos, Solitons & Fractals. 143: 110622. Bibcode:2021CSF...14310622D. doi:10.1016/j.chaos.2020.110622. S2CID 234063959.
- ^ da Silva, Sérgio Luiz E. F.; Carvalho, Pedro Tiago C.; de Araújo, João M.; Corso, Gilberto (2020-05-27). "Full-waveform inversion based on Kaniadakis statistics". Physical Review E. 101 (5): 053311. Bibcode:2020PhRvE.101e3311D. doi:10.1103/PhysRevE.101.053311. ISSN 2470-0045. PMID 32575242. S2CID 219746493.
- ^ Clementi, Fabio; Gallegati, Mauro; Kaniadakis, Giorgio; Landini, Simone (2016). "κ-generalized models of income and wealth distributions: A survey". The European Physical Journal Special Topics. 225 (10): 1959–1984. arXiv:1610.08676. Bibcode:2016EPJST.225.1959C. doi:10.1140/epjst/e2016-60014-2. ISSN 1951-6355. S2CID 125503224.
- ^ Clementi, Fabio; Gallegati, Mauro; Kaniadakis, Giorgio (2012). "A new model of income distribution: the κ-generalized distribution". Journal of Economics. 105 (1): 63–91. doi:10.1007/s00712-011-0221-0. hdl:11393/73598. ISSN 0931-8658. S2CID 155080665.
- ^ Trivellato, Barbara (2013-09-02). "Deformed Exponentials and Applications to Finance" (PDF). Entropy. 15 (12): 3471–3489. Bibcode:2013Entrp..15.3471T. doi:10.3390/e15093471. ISSN 1099-4300.
- ^ Passos, Leandro Aparecido; Cleison Santana, Marcos; Moreira, Thierry; Papa, Joao Paulo (2019). "κ-Entropy Based Restricted Boltzmann Machines". 2019 International Joint Conference on Neural Networks (IJCNN). Budapest, Hungary: IEEE. pp. 1–8. doi:10.1109/IJCNN.2019.8851714. ISBN 978-1-7281-1985-4. S2CID 203605811.
- ^ Kaniadakis, Giorgio (2013-09-25). "Theoretical Foundations and Mathematical Formalism of the Power-Law Tailed Statistical Distributions". Entropy. 15 (12): 3983–4010. arXiv:1309.6536. Bibcode:2013Entrp..15.3983K. doi:10.3390/e15103983. ISSN 1099-4300.
- ^ Kaniadakis, G. (2001). "Non-linear kinetics underlying generalized statistics". Physica A: Statistical Mechanics and Its Applications. 296 (3–4): 405–425. arXiv:cond-mat/0103467. Bibcode:2001PhyA..296..405K. doi:10.1016/S0378-4371(01)00184-4. S2CID 44275064.
- ^ da Silva, Sérgio Luiz E. F.; dos Santos Lima, Gustavo Z.; Volpe, Ernani V.; de Araújo, João M.; Corso, Gilberto (2021). "Robust approaches for inverse problems based on Tsallis and Kaniadakis generalised statistics". The European Physical Journal Plus. 136 (5): 518. Bibcode:2021EPJP..136..518D. doi:10.1140/epjp/s13360-021-01521-w. ISSN 2190-5444. S2CID 236575441.
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