Kaniadakis distribution
Not to be confused with 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 .[ 1] 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] astrophysics and cosmology ,[ 6] [ 7] [ 8] geophysics ,[ 9] [ 10] [ 11] economy ,[ 12] [ 13] [ 14] machine learning .[ 15]
The κ-distributions are written as function of the κ-deformed exponential, taking the form
f
i
=
exp
κ κ -->
-->
(
− − -->
β β -->
E
i
+
β β -->
μ μ -->
)
{\displaystyle 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]
exp
κ κ -->
-->
(
x
)
=
(
1
+
κ κ -->
2
x
2
+
κ κ -->
x
)
1
/
κ κ -->
{\displaystyle \exp _{\kappa }(x)=({\sqrt {1+\kappa ^{2}x^{2}}}+\kappa x)^{1/\kappa }}
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.
Supported on the whole real line
Plot of the κ-Gaussian distribution for typical κ-values. The case κ=0 corresponds to the normal distribution. The Kaniadakis Gaussian distribution , also called the κ-Gaussian distribution. The normal distribution is a particular case when
κ κ -->
→ → -->
0.
{\displaystyle \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
κ κ -->
→ → -->
0.
{\displaystyle \kappa \rightarrow 0.}
[ 18]
Plot of the κ-Gamma distribution for typical κ-values. The Kaniadakis Exponential distribution , also called the κ-Exponential distribution. The exponential distribution is a particular case when
κ κ -->
→ → -->
0.
{\displaystyle \kappa \rightarrow 0.}
The Kaniadakis Gamma distribution , also called the κ-Gamma distribution, which is a four-parameter (
κ κ -->
,
α α -->
,
β β -->
,
ν ν -->
{\displaystyle \kappa ,\alpha ,\beta ,\nu }
generalized Gamma distribution .
The κ-Gamma distribution becomes a ...
κ-Exponential distribution of Type I when
α α -->
=
ν ν -->
=
1
{\displaystyle \alpha =\nu =1}
κ-Erlang distribution when
α α -->
=
1
{\displaystyle \alpha =1}
ν ν -->
=
n
=
{\displaystyle \nu =n=}
κ -Half-Normal distribution
α α -->
=
2
{\displaystyle \alpha =2}
ν ν -->
=
1
/
2
{\displaystyle \nu =1/2}
Generalized Gamma distribution , when
α α -->
=
1
{\displaystyle \alpha =1}
In the limit
κ κ -->
→ → -->
0
{\displaystyle \kappa \rightarrow 0}
κ-Gamma distribution becomes a ...
Erlang distribution , when
α α -->
=
1
{\displaystyle \alpha =1}
ν ν -->
=
n
=
{\displaystyle \nu =n=}
Chi-Squared distribution , when
α α -->
=
1
{\displaystyle \alpha =1}
ν ν -->
=
{\displaystyle \nu =}
Nakagami distribution , when
α α -->
=
2
{\displaystyle \alpha =2}
ν ν -->
>
0
{\displaystyle \nu >0}
Rayleigh distribution , when
α α -->
=
2
{\displaystyle \alpha =2}
ν ν -->
=
1
{\displaystyle \nu =1}
Chi distribution , when
α α -->
=
2
{\displaystyle \alpha =2}
ν ν -->
=
{\displaystyle \nu =}
Maxwell distribution, when
α α -->
=
2
{\displaystyle \alpha =2}
ν ν -->
=
3
/
2
{\displaystyle \nu =3/2}
Half-Normal distribution , when
α α -->
=
2
{\displaystyle \alpha =2}
ν ν -->
=
1
/
2
{\displaystyle \nu =1/2}
Weibull distribution , when
α α -->
>
0
{\displaystyle \alpha >0}
ν ν -->
=
1
{\displaystyle \nu =1}
Stretched Exponential distribution , when
α α -->
>
0
{\displaystyle \alpha >0}
ν ν -->
=
1
/
α α -->
{\displaystyle \nu =1/\alpha }
Common Kaniadakis distributions
Continuous probability distribution
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 :
f
κ κ -->
(
x
)
=
α α -->
κ κ -->
(
2
κ κ -->
β β -->
)
1
/
κ κ -->
(
1
− − -->
κ κ -->
β β -->
x
α α -->
1
+
κ κ -->
2
β β -->
2
x
2
α α -->
)
x
− − -->
1
+
α α -->
/
κ κ -->
exp
κ κ -->
-->
(
− − -->
β β -->
x
α α -->
)
{\displaystyle f_{_{\kappa }}(x)={\frac {\alpha }{\kappa }}(2\kappa \beta )^{1/\kappa }\left(1-{\frac {\kappa \beta x^{\alpha }}{\sqrt {1+\kappa ^{2}\beta ^{2}x^{2\alpha }}}}\right)x^{-1+\alpha /\kappa }\exp _{\kappa }(-\beta x^{\alpha })}
valid for
x
≥ ≥ -->
0
{\displaystyle x\geq 0}
0
≤ ≤ -->
|
κ κ -->
|
<
1
{\displaystyle 0\leq |\kappa |<1}
Kaniadakis entropy ,
β β -->
>
0
{\displaystyle \beta >0}
α α -->
>
0
{\displaystyle \alpha >0}
The cumulative distribution function of κ-Distribution Type IV assumes the form:
F
κ κ -->
(
x
)
=
(
2
κ κ -->
β β -->
)
1
/
κ κ -->
x
α α -->
/
κ κ -->
exp
κ κ -->
-->
(
− − -->
β β -->
x
α α -->
)
{\displaystyle 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
κ κ -->
→ → -->
0
{\displaystyle \kappa \rightarrow 0}
Its moment of order
m
{\displaystyle m}
E
-->
[
X
m
]
=
(
2
κ κ -->
β β -->
)
− − -->
m
/
α α -->
1
+
κ κ -->
m
2
α α -->
Γ Γ -->
(
1
κ κ -->
+
m
α α -->
)
Γ Γ -->
(
1
− − -->
m
2
α α -->
)
Γ Γ -->
(
1
κ κ -->
+
m
2
α α -->
)
{\displaystyle \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
{\displaystyle m}
m
<
2
α α -->
{\displaystyle m<2\alpha }
See also
References
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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". 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External links
![{\displaystyle \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 )}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7023dfc05ad42a3bb1ce9794ffe50e8e6acfbc5b)
