Kaniadakis distribution information

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

f_{i}=\exp _{\kappa }(-\beta E_{i}+\beta \mu )

enables the power-law description of complex systems following the consistent κ-generalized statistical theory.,^[16]^[17] where $\exp _{\kappa }(x)=({\sqrt {1+\kappa ^{2}x^{2}}}+\kappa x)^{1/\kappa }$ 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.

^ 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.

[:0-1] 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.

[2] 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.

[3] 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.

[4] 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.

[5] 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.

[6] 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.

[7] 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.

[8] 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.

[9] 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.

[10] 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.

[11] 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.

[12] 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.

[13] 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.

[14] 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.

[15] 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.

[16] 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.

[17] 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.

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