Kaniadakis distribution

{{Distinguish|text=the K-distribution of probability distributions}}

In statistics, a Kaniadakis distribution (also known as κ-distribution) is a statistical distribution that emerges from the Kaniadakis statistics.{{Cite journal |last=Kaniadakis |first=G. |date=2021-01-01 |title=New power-law tailed distributions emerging in κ-statistics (a) |url=https://iopscience.iop.org/article/10.1209/0295-5075/133/10002 |journal=Europhysics Letters |volume=133 |issue=1 |pages=10002 |doi=10.1209/0295-5075/133/10002 |arxiv=2203.01743 |bibcode=2021EL....13310002K |s2cid=234144356 |issn=0295-5075}} 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,{{Cite journal |last1=Kaniadakis |first1=Giorgio |last2=Baldi |first2=Mauro M. |last3=Deisboeck |first3=Thomas S. |last4=Grisolia |first4=Giulia |last5=Hristopulos |first5=Dionissios T. |last6=Scarfone |first6=Antonio M. |last7=Sparavigna |first7=Amelia |last8=Wada |first8=Tatsuaki |last9=Lucia |first9=Umberto |date=2020 |title=The κ-statistics approach to epidemiology |journal=Scientific Reports |language=en |volume=10 |issue=1 |pages=19949 |doi=10.1038/s41598-020-76673-3 |issn=2045-2322 |pmc=7673996 |pmid=33203913|arxiv=2012.00629 |bibcode=2020NatSR..1019949K }} quantum statistics,{{Cite journal |last1=Santos |first1=A.P. |last2=Silva |first2=R. |last3=Alcaniz |first3=J.S. |last4=Anselmo |first4=D.H.A.L. |date=2011 |title=Generalized quantum entropies |journal=Physics Letters A |language=en |volume=375 |issue=35 |pages=3119–3123 |doi=10.1016/j.physleta.2011.07.001|bibcode=2011PhLA..375.3119S |url=https://repositorio.ufrn.br/jspui/handle/123456789/29118 }}{{Cite journal |last1=Ourabah |first1=Kamel |last2=Tribeche |first2=Mouloud |date=2014-06-24 |title=Planck radiation law and Einstein coefficients reexamined in Kaniadakis κ statistics |url=https://link.aps.org/doi/10.1103/PhysRevE.89.062130 |journal=Physical Review E |language=en |volume=89 |issue=6 |pages=062130 |doi=10.1103/PhysRevE.89.062130 |pmid=25019747 |bibcode=2014PhRvE..89f2130O |issn=1539-3755|url-access=subscription }}{{Cite journal |last1=Lourek |first1=Imene |last2=Tribeche |first2=Mouloud |date=2017 |title=Thermodynamic properties of the blackbody radiation: A Kaniadakis approach |journal=Physics Letters A |language=en |volume=381 |issue=5 |pages=452–456 |doi=10.1016/j.physleta.2016.12.019|bibcode=2017PhLA..381..452L }} in astrophysics and cosmology,{{Cite journal |last1=Carvalho |first1=J. C. |last2=do Nascimento |first2=J. D. |last3=Silva |first3=R. |last4=De Medeiros |first4=J. R. |title=Non-Gaussian Statistics and Stellar Rotational Velocities of Main-Sequence Field Stars |date=2009-05-01 |url=https://iopscience.iop.org/article/10.1088/0004-637X/696/1/L48 |journal=The Astrophysical Journal |volume=696 |issue=1 |pages=L48–L51 |doi=10.1088/0004-637X/696/1/L48 |s2cid=17161421 |issn=0004-637X|arxiv=0903.0868 |bibcode=2009ApJ...696L..48C }}{{Cite journal |last1=Abreu |first1=Everton M.C. |last2=Ananias Neto |first2=Jorge |last3=Mendes |first3=Albert C.R. |last4=de Paula |first4=Rodrigo M. |date=2019 |title=Loop quantum gravity Immirzi parameter and the Kaniadakis statistics |journal=Chaos, Solitons & Fractals |language=en |volume=118 |pages=307–310 |doi=10.1016/j.chaos.2018.11.033|arxiv=1808.01891 |bibcode=2019CSF...118..307A |s2cid=119207713 }}{{Cite journal |last1=Soares |first1=Bráulio B. |last2=Barboza |first2=Edésio M. |last3=Abreu |first3=Everton M.C. |last4=Neto |first4=Jorge Ananias |date=2019 |title=Non-Gaussian thermostatistical considerations upon the Saha equation |journal=Physica A: Statistical Mechanics and Its Applications |language=en |volume=532 |pages=121590 |doi=10.1016/j.physa.2019.121590|arxiv=1901.01839 |bibcode=2019PhyA..53221590S |s2cid=119539402 }} in geophysics,{{Cite journal |last1=Hristopulos |first1=Dionissios T. |last2=Petrakis |first2=Manolis P. |last3=Kaniadakis |first3=Giorgio |date=2014-05-28 |title=Finite-size effects on return interval distributions for weakest-link-scaling systems |url=https://link.aps.org/doi/10.1103/PhysRevE.89.052142 |journal=Physical Review E |language=en |volume=89 |issue=5 |pages=052142 |doi=10.1103/PhysRevE.89.052142 |pmid=25353774 |arxiv=1308.1881 |bibcode=2014PhRvE..89e2142H |s2cid=22310350 |issn=1539-3755}}{{Cite journal |last=da Silva |first=Sérgio Luiz E.F. |date=2021 |title=κ -generalised Gutenberg–Richter law and the self-similarity of earthquakes |journal=Chaos, Solitons & Fractals |language=en |volume=143 |pages=110622 |doi=10.1016/j.chaos.2020.110622|bibcode=2021CSF...14310622D |s2cid=234063959 }}{{Cite journal |last1=da Silva |first1=Sérgio Luiz E. F. |last2=Carvalho |first2=Pedro Tiago C. |last3=de Araújo |first3=João M. |last4=Corso |first4=Gilberto |date=2020-05-27 |title=Full-waveform inversion based on Kaniadakis statistics |url=https://link.aps.org/doi/10.1103/PhysRevE.101.053311 |journal=Physical Review E |language=en |volume=101 |issue=5 |pages=053311 |doi=10.1103/PhysRevE.101.053311 |pmid=32575242 |bibcode=2020PhRvE.101e3311D |s2cid=219746493 |issn=2470-0045|url-access=subscription }} in economy,{{Cite journal |last1=Clementi |first1=Fabio |last2=Gallegati |first2=Mauro |last3=Kaniadakis |first3=Giorgio |last4=Landini |first4=Simone |date=2016 |title=κ-generalized models of income and wealth distributions: A survey |url=http://link.springer.com/10.1140/epjst/e2016-60014-2 |journal=The European Physical Journal Special Topics |language=en |volume=225 |issue=10 |pages=1959–1984 |doi=10.1140/epjst/e2016-60014-2 |arxiv=1610.08676 |bibcode=2016EPJST.225.1959C |s2cid=125503224 |issn=1951-6355}}{{Cite journal |last1=Clementi |first1=Fabio |last2=Gallegati |first2=Mauro |last3=Kaniadakis |first3=Giorgio |date=2012 |title=A new model of income distribution: the κ-generalized distribution |url=http://link.springer.com/10.1007/s00712-011-0221-0 |journal=Journal of Economics |language=en |volume=105 |issue=1 |pages=63–91 |doi=10.1007/s00712-011-0221-0 |hdl=11393/73598 |s2cid=155080665 |issn=0931-8658|hdl-access=free }}{{Cite journal |last=Trivellato |first=Barbara |date=2013-09-02 |title=Deformed Exponentials and Applications to Finance |journal=Entropy |language=en |volume=15 |issue=12 |pages=3471–3489 |doi=10.3390/e15093471 |bibcode=2013Entrp..15.3471T |issn=1099-4300|doi-access=free |url=http://pdfs.semanticscholar.org/18ab/6d30fe656eaac3f9510d9d58b7f70fe6b76f.pdf }} in machine learning.{{Cite book |last1=Passos |first1=Leandro Aparecido |last2=Cleison Santana |first2=Marcos |last3=Moreira |first3=Thierry |last4=Papa |first4=Joao Paulo |title=2019 International Joint Conference on Neural Networks (IJCNN) |chapter=κ-Entropy Based Restricted Boltzmann Machines |date=2019 |chapter-url=https://ieeexplore.ieee.org/document/8851714 |location=Budapest, Hungary |publisher=IEEE |pages=1–8 |doi=10.1109/IJCNN.2019.8851714 |isbn=978-1-7281-1985-4|s2cid=203605811 }}

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.,{{Cite journal |last=Kaniadakis |first=Giorgio |date=2013-09-25 |title=Theoretical Foundations and Mathematical Formalism of the Power-Law Tailed Statistical Distributions |journal=Entropy |language=en |volume=15 |issue=12 |pages=3983–4010 |doi=10.3390/e15103983 |arxiv=1309.6536 |bibcode=2013Entrp..15.3983K |issn=1099-4300|doi-access=free }}{{Cite journal |last=Kaniadakis |first=G. |date=2001 |title=Non-linear kinetics underlying generalized statistics |journal=Physica A: Statistical Mechanics and Its Applications |language=en |volume=296 |issue=3–4 |pages=405–425 |doi=10.1016/S0378-4371(01)00184-4|arxiv=cond-mat/0103467 |bibcode=2001PhyA..296..405K |s2cid=44275064 }} 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.

List of κ-statistical distributions

= Supported on the whole real line =

File: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 $\kappa \rightarrow 0.$
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 $\kappa \rightarrow 0.$ {{Cite journal |last1=da Silva |first1=Sérgio Luiz E. F. |last2=dos Santos Lima |first2=Gustavo Z. |last3=Volpe |first3=Ernani V. |last4=de Araújo |first4=João M. |last5=Corso |first5=Gilberto |date=2021 |title=Robust approaches for inverse problems based on Tsallis and Kaniadakis generalised statistics |url=https://link.springer.com/10.1140/epjp/s13360-021-01521-w |journal=The European Physical Journal Plus |language=en |volume=136 |issue=5 |pages=518 |doi=10.1140/epjp/s13360-021-01521-w |bibcode=2021EPJP..136..518D |s2cid=236575441 |issn=2190-5444|url-access=subscription }}

= Supported on semi-infinite intervals, usually [0,∞) =

File:Kaniadakis Gamma Distribution pdf.png

The Kaniadakis Exponential distribution, also called the κ-Exponential distribution. The exponential distribution is a particular case when $\kappa \rightarrow 0.$
The Kaniadakis Gamma distribution, also called the κ-Gamma distribution, which is a four-parameter ( $\kappa, \alpha, \beta, \nu$ ) deformation of the generalized Gamma distribution.
The κ-Gamma distribution becomes a ...
κ-Exponential distribution of Type I when $\alpha = \nu = 1$ .
κ-Erlang distribution when $\alpha = 1$ and $\nu = n =$ positive integer.
κ-Half-Normal distribution, when $\alpha = 2$ and $\nu = 1/2$ .
Generalized Gamma distribution, when $\alpha = 1$ ;
In the limit $\kappa \rightarrow 0$ , the κ-Gamma distribution becomes a ...
Erlang distribution, when $\alpha = 1$ and $\nu = n =$ positive integer;
Chi-Squared distribution, when $\alpha = 1$ and $\nu =$ half integer;
Nakagami distribution, when $\alpha = 2$ and $\nu > 0$ ;
Rayleigh distribution, when $\alpha = 2$ and $\nu = 1$ ;
Chi distribution, when $\alpha = 2$ and $\nu =$ half integer;
Maxwell distribution, when $\alpha = 2$ and $\nu = 3/2$ ;
Half-Normal distribution, when $\alpha = 2$ and $\nu = 1/2$ ;
Weibull distribution, when $\alpha > 0$ and $\nu = 1$ ;
Stretched Exponential distribution, when $\alpha > 0$ and $\nu = 1/\alpha$ ;

Common Kaniadakis distributions

= κ-Exponential distribution =

= κ-Gaussian distribution =

= κ-Gamma distribution =

= κ-Weibull distribution =

= κ-Logistic distribution =

= κ-Erlang distribution =

= κ-Distribution Type IV =

{{Infobox probability distribution

| name = κ-Distribution Type IV

| type = density

| parameters = $0 \leq \kappa < 1$
$\alpha > 0$ shape (real)
$\beta> 0$ rate (real)

| support = $x \in [0, +\infty)$

| pdf = $\frac{\alpha}{\kappa} (2\kappa \beta )^{1/\kappa } \left(1 - \frac{\kappa \beta x^\alpha}{\sqrt{1+\kappa^2\beta^2x^{2\alpha} } } \right) x^{ -1 + \alpha / \kappa} \exp_\kappa(-\beta x^\alpha)$

| cdf = $(2\kappa \beta )^{1/\kappa} x^{\alpha / \kappa} \exp_\kappa(-\beta x^\alpha)$

|moments= $\frac{(2 \kappa \beta)^{-m/\alpha} }{ 1 + \kappa \frac{ m }{ 2\alpha } } \frac{\Gamma\Big(\frac{1}{\kappa} + \frac{m}{\alpha}\Big) \Gamma\Big(1 - \frac{m}{2\alpha}\Big)}{\Gamma\Big(\frac{1}{\kappa} + \frac{m}{2\alpha}\Big)}$ |pdf_image=FILE:Kaniadakis typeIV Distribution pdf.png|pdf_caption=Plot of the κ-Distribution Type IV for typical κ-values, and $\alpha = \beta = 1$ .|cdf_image=FILE:Kaniadakis typeIV Distribution cdf.png}}

The Kaniadakis distribution of Type IV (or κ-Distribution Type IV) is a three-parameter family of continuous statistical distributions.

The κ-Distribution Type IV distribution has the following probability density function:

: $f_{_{\kappa}}(x) = \frac{\alpha}{\kappa} (2\kappa \beta )^{1/\kappa} \left(1 - \frac{\kappa \beta x^\alpha}{\sqrt{1+\kappa^2\beta^2x^{2\alpha} } } \right) x^{ -1 + \alpha / \kappa} \exp_\kappa(-\beta x^\alpha)$

valid for $x \geq 0$ , where $0 \leq |\kappa| < 1$ is the entropic index associated with the Kaniadakis entropy, $\beta > 0$ is the scale parameter, and $\alpha > 0$ is the shape parameter.

The cumulative distribution function of κ-Distribution Type IV assumes the form:

: $F_\kappa(x) = (2\kappa \beta )^{1/\kappa} x^{\alpha / \kappa} \exp_\kappa(-\beta x^\alpha)$

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 $\kappa \rightarrow 0$ .

Its moment of order $m$ given by

: $\operatorname{E}[X^m] = \frac{(2 \kappa \beta)^{-m/\alpha} }{ 1 + \kappa \frac{ m }{ 2\alpha } } \frac{\Gamma\Big(\frac{1}{\kappa} + \frac{m}{\alpha}\Big) \Gamma\Big(1 - \frac{m}{2\alpha}\Big)}{\Gamma\Big(\frac{1}{\kappa} + \frac{m}{2\alpha}\Big)}$

The moment of order $m$ of the κ-Distribution Type IV is finite for $m < 2\alpha$ .

References

External links

[https://scholar.google.it/citations?user=pFlYesUAAAAJ&hl=en Giorgio Kaniadakis Google Scholar page]
[https://arxiv.org/search/?query=kaniadakis+statistics&searchtype=all&abstracts=show&order=-announced_date_first&size=200 Kaniadakis Statistics on arXiv.org]

Category:Probability distributions

Category:Statistical mechanics