# Kaniadakis Functions beyond Statistical Mechanics: Weakest-Link Scaling, Power-Law Tails, and Modified Lognormal Distribution

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## Abstract

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## 1. Introduction

- We formulate an autoregressive, intermittent precipitation model based on the $\kappa $-modified Box–Cox transform in Section 3.3. We show that the resulting precipitation time series has higher “peaks” than those obtained with the Box–Cox transform with the same parameter value.
- We review the $\kappa $-Weibull distribution focusing on its connection with weakest-link theory (Section 4). This demonstrates that the $\kappa $-Weibull is a physically motivated generalization of the classical Weibull distribution for the mechanical strength of brittle materials, unlike modified Weibull distributions which fail to satisfy the weakest-link principle.
- We show that for several physical quantities, including the thickness of magmatic sheet intrusions, the tensile strength of steel, earthquake waiting times, and precipitation amounts the $\kappa $-Weibull distribution provides a better fit than the Weibull according to model selection criteria.
- We introduce the $\kappa $-lognormal distribution, which provides a deformation of the lognormal with lighter tails than the latter in Section 5. The $\kappa $-lognormal can be used to model asymmetric data distributions which concentrate more probability mass around the median than the lognormal. We discuss the importance of the generalized mean (power mean) of the lognormal distribution for estimating the effective permeability of heterogeneous porous media, and we calculate the generalized mean of the $\kappa $-lognormal distribution.

## 2. Mathematical Preliminaries

#### 2.1. The $\kappa $-Exponential Function

#### 2.2. The $\kappa $-Logarithm Function

## 3. Nonlinear Transformation of Data Based on the $\mathbf{\kappa}$-Logarithm

#### 3.1. Box–Cox Transform and the Replica Trick

#### 3.2. The $\kappa $-Logarithmic Transform

#### 3.3. Application to Precipitation Modeling

## 4. The $\mathbf{\kappa}$-Weibull Distribution and Its Applications

#### 4.1. $\kappa $-Weibull Probability Functions

#### 4.2. Connection with Weakest-Link Scaling Theory

#### 4.3. $\kappa $-Weibull Plot for Graphical Testing

#### 4.4. Application to Real Data

## 5. The $\mathbf{\kappa}$-Lognormal Distribution

#### 5.1. Effective Permeability of Random Media

#### 5.2. Generalized Mean of the $\kappa $-Lognormal Distribution

## 6. Discussion

## 7. Conclusions

- The modified Box–Cox transform given by Equation (22).
- Application of the modified Box–Cox transform to an autoregressive, intermittent model of precipitation as described in Section 3.3.
- Connection between the $\kappa $-Weibull probability model with the theory of weakest-link scaling as shown in Section 4.2.
- The study of the $\kappa $-lognormal distribution which is a generalization of the lognormal model with lighter tails. The PDF of this new model is given by Equation (32).
- The calculation of the power-mean (generalized mean) of the $\kappa $-lognormal as shown in Section 5.2.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

AR | Autoregressive |

AIC | Akaike Information Criterion |

BCT | Box-Cox transform |

BIC | Bayesian Information Criterion |

CDF | Cumulative distribution function |

KLT | $\kappa $-logarithm transform |

LLM | Landau-Lifshitz-Matheron (ansatz) |

NLL | Negative log-likelihood |

Probability density function |

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**Figure 1.**Plots of the Box–Cox and $\kappa $-logarithmic transform for $\lambda =\kappa =1$ ($\lambda $ is the Box–Cox parameter and $\kappa $ is the deformation parameter of the Kaniadakis logarithm).

**Figure 2.**Plots of the Box–Cox (

**left**) and $\kappa $-logarithmic (

**right**) transform for different values of $\lambda =\kappa $ ($\lambda $ is the Box–Cox parameter and $\kappa $ is the deformation parameter of the Kaniadakis logarithm).

**Figure 3.**Realizations of six time series generated by the censored and transformed AR(1) model of Equation (24) with ${\varphi}_{1}=0.5$ and ${\sigma}_{\u03f5}=0.6$. The nonlinear transform uses BCT (blue, continuous lines) and KLT (red, broken lines) for $\kappa =\lambda \in \{0.001,0.2,0.4,0.6,0.8,0.95\}$.

**Figure 4.**Schematic illustrating how long tails can emerge if the observation window (blue square) is a nested insider a larger, interacting system (see text for explanation). Blue stars indicate events inside the observation window, while the red star refers to an event outside the window.

**Figure 5.**Survival functions for the Weibull and $\kappa $-Weibull distributions for different values of $\kappa $ and ${x}_{s}=m=1$.

**Figure 6.**Link survival function for different effective system sizes. The horizontal axis denotes the variable $z\left(x\right)={x}^{m}/{\tilde{x}}_{l}^{m}$. Larger values of ${N}_{\mathrm{eff}}$ correspond to slower decay of ${S}_{1}\left(x\right)$.

**Figure 7.**Plots of estimated ${\mathrm{\Phi}}_{\kappa}\left(x\right)$ obtained from $\kappa $-Weibull synthetically generated samples with different values of the Weibull modulus m and the deformation parameter $\kappa $.

**Figure 8.**Probability density functions resulting from the $\kappa $-logarithmic (

**left**) and Box–Cox (

**right**) transformations of the standard normal distribution, given by Equations (32) and (33) respectively. The curves correspond to different values of $\lambda =\kappa $ ($\lambda $ is the Box–Cox parameter and $\kappa $ is the deformation parameter of the Kaniadakis logarithm).

**Figure 9.**Parametric plots (versus x) of the $\kappa $-lognormal PDF defined in Equation (32) (

**top**) and the ratio function ${R}_{f}(x;\kappa )$ defined in Equation (34); the latter compares the tails of the $\kappa $-lognormal relative to the lognormal distribution for different values of the deformation parameter $\kappa $ (

**bottom**).

**Figure 10.**Parametric plots of the generalized mean versus $\kappa $ for different values of the averaging exponent $\alpha $ (

**left**) and the generalized mean versus $\alpha $ for different values of the deformation parameter $\kappa $ (

**right**).

**Table 1.**Results of maximum likelihood estimated fits to the Weibull and $\kappa $-Weibull distribution. 1. Tensile strength of carbon fibers. 2. Daily averaged wind speeds from 1 January 2009 to 4 October 2009 for Cairo, Egypt. 3–6. Thickness of magmatic sheet intrusions for different tectonic settings. 7. Tensile strength of low-alloy steels. 8. Recurrence times of aftershocks (A.R.T.) from 25 October 2018 until 31 May 2019, following the major M

_{w}6.9 Zakynthos earthquake (Greece). 9. Recurrence times of foreshocks (F.R.T.) preceding the Zakynthos earthquake (from 1 January 2014 until 25 October 2018). For more information regarding the data see the relevant sources. N, sample length; ${x}_{s}$, scale parameter; m, shape parameter; $\kappa $, Weibull deformation parameter. Values are rounded off to the second decimal digit. NLL, Negative log-likelihood.

Weibull | $\mathit{\kappa}$-Weibull | |||||||
---|---|---|---|---|---|---|---|---|

Data | $\mathit{N}$ | ${\mathit{x}}_{\mathit{s}}$ | $\mathit{m}$ | NLL | ${\mathit{x}}_{\mathit{s}}$ | $\mathit{m}$ | $\mathit{\kappa}$ | NLL |

1. C fibers (GPa) [69] | 100 | 2.94 | 2.79 | 141.53 | 2.90 | 2.98 | 0.285 | 141.23 |

2. Wind (mph) [70] | 100 | 8.05 | 2.78 | 240.21 | 7.63 | 3.28 | 0.56 | 239.32 |

3. Dyrfjöll (m) [71] | 487 | 0.90 | 1.26 | 378.05 | 0.84 | 1.46 | 0.42 | 368.40 |

4. Geitafell (m) [71] | 546 | 0.57 | 1.02 | 233.88 | 0.52 | 1.17 | 0.43 | 225.62 |

5. Tenerife (m) [71] | 550 | 1.83 | 1.02 | 875.18 | 1.65 | 1.18 | 0.45 | 867.38 |

6. La Palma (m) [71] | 2093 | 0.43 | 1.14 | 206.51 | 0.37 | 1.53 | 0.66 | 83.98 |

7. Steel (MPa) [72] | 915 | 548.58 | 1.98 | 6194.22 | 524.26 | 4.81 | 0.52 | 5753.13 |

8. A.R.T. (days) [74] | 7822 | 0.027 | 0.94 | −20207 | 0.024 | 1.19 | 0.49 | −20374 |

9. F.R.T. (days) [74] | 4731 | 0.28 | 0.68 | −692.60 | 0.27 | 0.70 | 0.17 | −698.37 |

**Table 2.**Measures of fit to the Weibull and $\kappa $-Weibull distributions for the datasets listed in Table 1. NLL, negative log-likelihood; AIC’, value of Akaike information criterion value per sample point, i.e., AIC’ = AIC/N = $2(k+\mathrm{NLL})/N$; BIC’, value of Bayesian information criterion per sample point, i.e., BIC’ = BIC/N = $(klogN+2\phantom{\rule{0.166667em}{0ex}}\mathrm{NLL})/N$.

Weibull | $\mathit{\kappa}$-Weibull | ||||||
---|---|---|---|---|---|---|---|

Data | $\mathit{N}$ | NLL | AIC’ | BIC’ | NLL | AIC’ | BIC’ |

1. C fibers (GPa) | 100 | 141.53 | 2.8706 | 2.9227 | 141.23 | 2.8846 | 2.9628 |

2. Wind (mph) | 100 | 240.21 | 4.8842 | 4.8963 | 239.32 | 4.8464 | 4.9246 |

3. Dyrfjöll (m) | 487 | 378.05 | 1.5608 | 1.5780 | 368.40 | 1.5253 | 1.5511 |

4. Geitafell (m) | 546 | 233.88 | 0.8640 | 0.8798 | 225.62 | 0.8374 | 0.8611 |

5. Tenerife (m) | 550 | 875.18 | 3.1897 | 3.2054 | 867.38 | 3.1650 | 3.1885 |

6. La Palma (m) | 2093 | 206.51 | 0.1992 | 0.2046 | 83.98 | 0.0831 | 0.0912 |

7. Steel (MPa) | 915 | 6194.22 | 13.5437 | 13.5542 | 5753.13 | 12.5817 | 12.5975 |

8. A.R.T. (days) | 7822 | −20207 | −5.1662 | −5.1644 | −20374 | −5.2086 | −5.2060 |

9. F.R.T. (days) | 4731 | −692.60 | −0.2919 | −0.2892 | −692.60 | −0.2940 | −0.2899 |

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Hristopulos, D.T.; Baxevani, A.
Kaniadakis Functions beyond Statistical Mechanics: Weakest-Link Scaling, Power-Law Tails, and Modified Lognormal Distribution. *Entropy* **2022**, *24*, 1362.
https://doi.org/10.3390/e24101362

**AMA Style**

Hristopulos DT, Baxevani A.
Kaniadakis Functions beyond Statistical Mechanics: Weakest-Link Scaling, Power-Law Tails, and Modified Lognormal Distribution. *Entropy*. 2022; 24(10):1362.
https://doi.org/10.3390/e24101362

**Chicago/Turabian Style**

Hristopulos, Dionissios T., and Anastassia Baxevani.
2022. "Kaniadakis Functions beyond Statistical Mechanics: Weakest-Link Scaling, Power-Law Tails, and Modified Lognormal Distribution" *Entropy* 24, no. 10: 1362.
https://doi.org/10.3390/e24101362