# Kaniadakis’s Information Geometry of Compositional Data

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

#### 1.1. Why a Geometric Methodology

#### 1.2. CoDa

#### 1.3. CoDa and Systemic Financial Risk

#### 1.4. Data and Methods

#### 1.5. Kaniadakis’ Logarithm

#### 1.6. Kaniadakis’ Exponential Form of a Positive Probability Function

#### 1.7. Properties of the Cumulant Function ${K}_{p}$

#### 1.8. Bibliographical Notes

## 2. Affine Space

#### 2.1. Statistical Bundle

#### 2.2. Velocity and Auto-Parallel Curves

#### 2.3. Surfaces of Constant Divergence

#### 2.4. Displacement

#### 2.5. Barycenter and Deviation

## 3. Data Analysis

#### 3.1. Kaniadakis Divergence

#### 3.2. Mixture Displacement

#### 3.3. Exponential Displacement

## 4. Conclusions and Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CoDa | Compositional Data |

SRISK | Systemic Risk |

IG | Information Geometry |

## References

- Kaniadakis, G. Non-linear kinetics underlying generalized statistics. Phys. A
**2001**, 296, 405–425. [Google Scholar] [CrossRef] [Green Version] - Kaniadakis, G. H-theorem and generalized entropies within the framework of nonlinear kinetics. Phys. Lett. A
**2001**, 288, 283–291. [Google Scholar] [CrossRef] [Green Version] - Tsallis, C. Possible generalization of Boltzmann-Gibbs statistics. J. Statist. Phys.
**1988**, 52, 479–487. [Google Scholar] [CrossRef] - Tsallis, C. Introduction to nonextensive statistical mechanics. In Approaching a Complex World; Springer: New York, NY, USA, 2009; p. xvii+382. [Google Scholar]
- Amari, S.; Nagaoka, H. Methods of Information Geometry; American Mathematical Society: Providence, RI, USA, 2000; p. x+206, (Translated from the 1993 Japanese original by Daishi Harada). [Google Scholar]
- Pistone, G. κ-exponential models from the geometrical viewpoint. Eur. Phys. J. B Condens. Matter Phys.
**2009**, 71, 29–37. [Google Scholar] [CrossRef] [Green Version] - Chirco, G.; Pistone, G. Dually affine Information Geometry modeled on a Banach space. arXiv
**2022**, arXiv:2204.00917. [Google Scholar] - Pawlowsky-Glahn, V.; Egozcue, J.J.; Tolosana-Delgado, R. Modelling and Analysis of Compositional Data; John Wiley & Sons, Ltd.: Hoboken, NJ, USA, 2015. [Google Scholar] [CrossRef]
- Pawlowsky-Glahn, V.; Egozcue, J.J. Compositional data and their analysis: An introduction. Geol. Soc. Lond. Spec. Publ.
**2006**, 264, 1–10. [Google Scholar] [CrossRef] [Green Version] - Egozcue, J.J.; Pawlowsky-Glahn, V. Compositional data: The sample space and its structure. Test
**2019**, 28, 599–638. [Google Scholar] [CrossRef] - Coenders, G.; Ferrer-Rosell, B. Compositional data analysis in tourism: Review and future directions. Tour. Anal.
**2020**, 25, 153–168. [Google Scholar] [CrossRef] - Fiori, A.M.; Porro, F. A compositional analysis of systemic risk in European financial institutions. Ann. Financ.
**2023**, 19, 1–30. [Google Scholar] [CrossRef] - Boonen, T.J.; Guillen, M.; Santolino, M. Forecasting compositional risk allocations. Insur. Math. Econ.
**2019**, 84, 79–86. [Google Scholar] [CrossRef] [Green Version] - Grifoll, M.; Ortego, M.; Egozcue, J. Compositional data techniques for the analysis of the container traffic share in a multi-port region. Eur. Transp. Res. Rev.
**2019**, 11, 1–15. [Google Scholar] [CrossRef] - Linares-Mustarós, S.; Coenders, G.; Vives-Mestres, M. Financial performance and distress profiles. From classification according to financial ratios to compositional classification. Adv. Account.
**2018**, 40, 1–10. [Google Scholar] [CrossRef] - Acharya, V.; Engle, R.; Richardson, M. Capital shortfall: A new approach to ranking and regulating systemic risks. Am. Econ. Rev.
**2012**, 102, 59–64. [Google Scholar] [CrossRef] [Green Version] - Acharya, V.V.; Richardson, M.P. (Eds.) Restoring Financial Stability: How to Repair a Failed System; John Wiley & Sons, Ltd.: Hoboken, NJ, USA, 2009; Volume 542. [Google Scholar]
- Engle, R. Systemic risk 10 years later. Annu. Rev. Financ. Econ.
**2018**, 10, 125–152. [Google Scholar] [CrossRef] - Stolbov, M.; Shchepeleva, M. Systemic risk in Europe: Deciphering leading measures, common patterns and real effects. Ann. Financ.
**2018**, 14, 49–91. [Google Scholar] [CrossRef] - Engle, R.; Jondeau, E.; Rockinger, M. Systemic risk in Europe. Rev. Financ.
**2015**, 19, 145–190. [Google Scholar] [CrossRef] - Aitchison, J. The Statistical Analysis of Compositional Data; Monographs on Statistics and Applied Probability; Chapman & Hall: London, UK, 1986; p. xvi+416. [Google Scholar] [CrossRef]
- Naudts, J. Generalised exponential families and associated entropy functions. Entropy
**2008**, 10, 131–149. [Google Scholar] [CrossRef] [Green Version] - Pistone, G.; Riccomagno, E.; Wynn, H.P. Algebraic Statistics: Computational Commutative Algebra in Statistics; Monographs on Statistics and Applied Probability; Chapman & Hall/CRC: Boca Raton, FL, USA, 2001; Volume 89, p. xvii+160. [Google Scholar]
- Montrucchio, L.; Pistone, G. A Class of Non-parametric Deformed Exponential Statistical Models. In Geometric Structures of Information; Springer International Publishing: Berlin/Heidelberg, Germany, 2018; pp. 15–35. [Google Scholar] [CrossRef] [Green Version]
- Pistone, G. Information Geometry of the Probability Simplex: A Short Course. Nonlinear Phenom. Complex Syst.
**2020**, 23, 221–242. [Google Scholar] [CrossRef] - Kaniadakis, G. Statistical mechanics in the context of special relativity. II. Phys. Rev. E
**2005**, 72, 036108. [Google Scholar] [CrossRef] [Green Version] - Kaniadakis, G. Statistical mechanics in the context of special relativity. Phys. Rev. E
**2002**, 66, 056125. [Google Scholar] [CrossRef] [Green Version] - Montrucchio, L.; Pistone, G. Deformed Exponential Bundle: The Linear Growth Case. In Lecture Notes in Computer Science; Springer International Publishing: Berlin/Heidelberg, Germany, 2017; pp. 239–246. [Google Scholar] [CrossRef]
- Weyl, H. Space—Time—Matter; Dover: New York, NY, USA, 1952; (Translation of the 1921 RAUM ZEIT MATERIE). [Google Scholar]
- Berger, M. Geometry I; Universitext; Springer: Berlin, Germany, 1994; p. xiv+427, (Translated from the 1977 French original by M. Cole and S. Levy, Corrected reprint of the 1987 translation). [Google Scholar]
- Pistone, G. Lagrangian Function on the Finite State Space Statistical Bundle. Entropy
**2018**, 20, 139. [Google Scholar] [CrossRef] [PubMed] - Chirco, G.; Malagò, L.; Pistone, G. Lagrangian and Hamiltonian dynamics for probabilities on the statistical bundle. Int. J. Geom. Methods Mod. Phys.
**2022**, 19, 2250214. [Google Scholar] [CrossRef] - Eguchi, S. Second order efficiency of minimum contrast estimators in a curved exponential family. Ann. Statist.
**1983**, 11, 793–803. [Google Scholar] [CrossRef]

**Figure 2.**(

**A**) Mixture displacement on compositional data by taking 2008 as a reference and (

**B**) mixture displacement on compositional data by taking mean as reference.

**Figure 3.**(

**A**) Exponential displacement on compositional data by taking 2008 as a reference and (

**B**) exponential displacement on compositional data by taking mean as a reference.

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**MDPI and ACS Style**

Pistone, G.; Shoaib, M.
Kaniadakis’s Information Geometry of Compositional Data. *Entropy* **2023**, *25*, 1107.
https://doi.org/10.3390/e25071107

**AMA Style**

Pistone G, Shoaib M.
Kaniadakis’s Information Geometry of Compositional Data. *Entropy*. 2023; 25(7):1107.
https://doi.org/10.3390/e25071107

**Chicago/Turabian Style**

Pistone, Giovanni, and Muhammad Shoaib.
2023. "Kaniadakis’s Information Geometry of Compositional Data" *Entropy* 25, no. 7: 1107.
https://doi.org/10.3390/e25071107