# Multifractal Dimensional Dependence Assessment Based on Tsallis Mutual Information

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Entropy, Divergence and Mutual Information

#### 2.1.1. Entropy

#### 2.1.2. Divergence

#### 2.1.3. Mutual Information

#### 2.2. Scaling Behavior and Multifractality

#### 2.2.1. Generalized Rényi Dimensions

- “Capacity” dimension, ${D}_{0}$: This shows how the points of a multifractal pattern fill the domain under study. The larger the value of this dimension, the better the space is covered.
- “Entropy” dimension, ${D}_{1}$: This is a measure of order-disorder of the points in the domain under study. Larger values indicate higher disorder.
- “Correlation” dimension, ${D}_{2}$: This quantifies the degree of clustering/inhibition. Lower values correspond to a higher level of clustering.
- “Multifractal step”, ${D}_{-\infty}-{D}_{\infty}$: This indicates the degree of multifractality. Larger values correspond to a stronger multifractal behavior. The multifractal step is zero for monofractal behavior.

#### 2.2.2. Generalized Tsallis Dimensions

## 3. Dependence Analysis

#### 3.1. Generalized Dependence Coefficients

#### 3.1.1. A Formal Justification and Discussion

- (i)
- ${\rho}_{q}(X,Y)={\rho}_{q}(Y,X)$,
- (ii)
- $0\le {\rho}_{q}(X,Y)\le 1$,
- (iii)
- ${\rho}_{q}(X,Y)=0$ if and only if X and Y are independent and $q=1$,
- (iv)
- ${\rho}_{q}(X,Y)=1$ if and only if $X\equiv Y$.

#### 3.2. Generalized Dependence Coefficients in the Multifractal Domain

## 4. Application to a Real Seismic Series

**Figure 3.**Temporal dynamics of the values of multifractal dependence coefficients for $S\leftrightarrow M$ based on sliding windows (size 120 events, overlapping 10 percent).

**Figure 4.**Temporal dynamics of the values of multifractal dependence coefficients for $S\leftrightarrow M$ based on sliding windows (size 240 events, overlapping 10 percent).

**Figure 5.**Temporal dynamics of multifractal dependence coefficients for $S\leftrightarrow T$ based on sliding windows (size 120 events, overlapping 10 percent).

**Figure 6.**Temporal dynamics of multifractal dependence coefficients for $S\leftrightarrow T$ based on sliding windows (size 240 events, overlapping 10 percent).

**Figure 7.**Values of multifractal dependence coefficients for $S\leftrightarrow T$ for q varying from 1.0001–5, based on data segments of size 120 (left plots) and 240 (right plots) events, in the two periods immediately before and after the catastrophic event and in a period of stability of the system (from top to bottom).

## 5. Conclusion

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Beckerman, L.P. Application of Complex Systems Science to Systems Engineering. Syst. Eng.
**2000**, 3, 96–102. [Google Scholar] [CrossRef] - Newth, D.; Finnigan, J. Emergence and Self-organization in Chemistry and Biology. Aust. J. Chem.
**2007**, 59, 841–848. [Google Scholar] [CrossRef] - Prokopenko, M.; Boschetti, F.; Ryan, A.J. An Information-Theoretic Primer on Complexity, Self-organization, and Emergence. Complexity
**2009**, 15, 11–28. [Google Scholar] [CrossRef] - Baranger, M. Chaos, Complexity, and Entropy. Available online: http://necsi.edu/projects/baranger/cce.pdf (accessed on 21 July 2015).
- Samet, R.H. Long-Range Futures Research: An Application of Complexity Science; Book Surge Publishing: North Charleston, SC, USA, 2009. [Google Scholar]
- Biswas, A.; Zeleke, T.B.; Si, B.C. Multifractal Detrended Fluctuation Analysis in Examining Scaling Properties of the Spatial Patterns of Soil Water Storage. Nonlin. Proc. Geophys.
**2012**, 19, 227–238. [Google Scholar] [CrossRef] - Harte, D. Multifractals: Theory and Applications; CRC Press: Boca Raton, FL, USA, 2001. [Google Scholar]
- Kantelhardt, J.W.; Zschiegner, S.A.; Koscielny-Bunde, E.; Havlin, S.; Bunde, A.; Stanley, H.E. Multifractal Detrended Fluctuation Analysis of Nonstationary Time Series. Physica A
**2002**, 316, 87–114. [Google Scholar] [CrossRef] - Lin, A.; Ma, H.; Shang, P. The Scaling Properties of Stock Markets Based on Modified Multiscale Multifractal Detrended Fluctuation Analysis. Physica A
**2015**, 436, 525–537. [Google Scholar] [CrossRef] - Stanley, H.E.; Meakin, P. Multifractal Phenomena in Physics and Chemistry. Nature
**1988**, 335, 405–409. [Google Scholar] - Bouchet, F.; Gupta, S.; Mukamel, D. Thermodynamics and Dynamics of Systems with Long-Range Interactions. Physica A
**2010**, 389, 4389–4405. [Google Scholar] [CrossRef] - Michas, G.; Vallianatos, F.; Sammonds, P. Non-extensivity and Long-Range Correlations in the Earthquake Activity at the West Corinth Rift (Greece). Nonlinear Proc. Geoph.
**2013**, 20, 713–724. [Google Scholar] [CrossRef] - Prehl, J.; Essex, C.; Hoffmann, K.H. Tsallis Relative Entropy and Anomalous Diffusion. Entropy
**2012**, 14, 701–716. [Google Scholar] [CrossRef] - Tsallis, C. Possible Generalization of Boltzmann-Gibbs Statistics. J. Stat. Phys.
**1988**, 52, 479–487. [Google Scholar] [CrossRef] - Yamano, T. Information Theory Based on Non-additive Information Content. Phys. Rev. E
**2001**, 63, 046105. [Google Scholar] [CrossRef] - Yamano, T. A Possible Extension of Shannon's Information Theory. Entropy
**2001**, 3, 280–292. [Google Scholar] [CrossRef] - Furuichi, S. Information Theoretical Properties of Tsallis Entropies. J. Math. Phys.
**2006**, 47, 023302. [Google Scholar] [CrossRef] - Angulo, J.M.; Esquivel, F.J. Structural Complexity in Space-Time Seismic Event Data. Stoch. Env. Res. Risk. A
**2014**, 28, 1187–1206. [Google Scholar] [CrossRef] - Esquivel, F.J.; Angulo, J.M. Non-extensive Analysis of the Seismic Activity Involving the 2011 Volcanic Eruption in El Hierro. Spat. Stat.
**2015**. (accepted). [Google Scholar] - Bak, P.; Christensen, K.; Danon, L.; Scanlon, T. Unified Scaling Law for Earthquakes. Phys. Rev. Lett.
**2002**, 88, 178501. [Google Scholar] [CrossRef] - Lennartz, S.; Bunde, A.; Turcotte, D.L. Modelling Seismic Catalogues by Cascade Models: Do We Need Long-Term Magnitude Correlations? Geophys. J. Int.
**2011**, 184, 1214–1222. [Google Scholar] [CrossRef] - Main, I. Statistical Physics, Seismogenesis, and Seismic Hazard. Rev. Geophys.
**1996**, 34, 433–462. [Google Scholar] [CrossRef] - Rundle, J.B.; Turcotte, D.L.; Shcherbakov, R.; Klein, W.; Sammis, C. Statistical Physics Approach to Understanding the Multiscale Dynamics of Earthquake Fault Systems. Rev. Geophys.
**2003**, 41, 1019. [Google Scholar] - Saleur, H.; Sammis, C.G.; Sornette, D. Discrete Scale Invariance, Complex Fractal Dimensions and Log-Periodic Fluctuations in Seismicity. J. Geophys. Res.
**1996**, 101, 17661–17677. [Google Scholar] [CrossRef] - Turcotte, D.L. Fractals and Chaos in Geology and Geophysics, 2nd ed.; Cambridge University Press: Cambridge, UK, 1997. [Google Scholar]
- Varotsos, P.A.; Sarlis, N.V.; Tanaka, H.K.; Skordas, E.S. Similarity of Fluctuations in Correlated Systems: The Case of Seismicity. Phys. Rev. E
**2005**, 72, 041103. [Google Scholar] [CrossRef] - Vere-Jones, D. Seismology—A Statistical Vignette. J. Am. Stat. Assoc.
**2000**, 95, 975–978. [Google Scholar] - Vere-Jones, D. The Marriage of Statistics and Seismology. J. Appl. Probab.
**2001**, 38, 1–5. [Google Scholar] [CrossRef] - Vere-Jones, D. Foundations of Statistical Seismology. Pure Appl. Geophys.
**2010**, 167, 645–653. [Google Scholar] [CrossRef] - Sotolongo-Costa, O.; Rodriguez, A.H.; Rodgers, G.J. Tsallis Entropy and the Transition to Scaling in Fragmentation. Entropy
**2000**, 2, 172–177. [Google Scholar] [CrossRef] - Telesca, L. A Non-extensive Approach in Investigating the Seismicity of L'Aquila Area (Central Italy), Struck by the 6 April 2009 Earthquake (ML = 5.8). Terra Nova
**2010**, 22, 87–93. [Google Scholar] [CrossRef] - Telesca, L. Analysis of Italian Seismicity by Using a Nonextensive Approach. Tectonophysics
**2010**, 494, 155–162. [Google Scholar] [CrossRef] - Telesca, L. Tsallis-Based Nonextensive Analysis of the Southern California Seismicity. Entropy
**2011**, 13, 1267–1280. [Google Scholar] [CrossRef] - Telesca, L. Maximum Likelihood Estimation of the Nonextensive Parameters of the Earthquake Cumulative Magnitude Distribution. Bull. Seismol. Soc. Am.
**2012**, 102, 886–891. [Google Scholar] [CrossRef] - Vallianatos, F.; Michas, G.; Papadakis, G.; Tzanis, A. Evidence of Non-extensivity in the Seismicity Observed during the 2011–2012 Unrest at the Santorini Volcanic Complex, Greece. Nat. Hazards Earth Syst. Sci.
**2013**, 13, 177–185. [Google Scholar] [CrossRef] - Shannon, C.E. A Mathematical Theory of Communication. Bell Syst. Tech. J.
**1948**, 27, 379–423. [Google Scholar] [CrossRef] - Rényi, A. On Measures of Entropy and Information. In Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Berkeley, CA, USA, June 20–July 30 1960; Neyman, J., Ed.; University of California Press: Berkeley, CA, USA, 1961; Volume 1, pp. 547–561. [Google Scholar]
- Hanel, R.; Thurner, S. Generalized (c,d)-Entropy and Aging Random Walks. Entropy
**2013**, 15, 5324–5337. [Google Scholar] [CrossRef] - Kullback, S.; Leibler, R.A. On Information and Sufficiency. Ann. Math. Stat.
**1951**, 22, 79–86. [Google Scholar] [CrossRef] - Tsallis, C. Generalized Entropy-Based Criterion for Consistent Testing. Phys. Rev. E
**1998**, 58, 1442–1445. [Google Scholar] [CrossRef] - Swingle, B. Rényi Entropy, Mutual Information, and Fluctuation Properties of Fermi Liquids. Phys. Rev. B
**2012**, 86, 045109. [Google Scholar] [CrossRef] - Cvejic, N.; Canagarajah, C.N.; Bull, D.R. Image Fusion Metric Based on Mutual Information and Tsallis Entropy. Electron. Lett.
**2006**, 42, 626–627. [Google Scholar] [CrossRef] - Sun, S.; Zhang, L.; Guo, C.P. Medical Image Registration by Minimizing Divergence Measure Based on Tsallis Entropy. Int. J. Biol. Sci.
**2008**, 13, 809–814. [Google Scholar] - Vila, M.; Bardera, A.; Feixas, M.; Sbert, M. Tsallis Mutual Information for Document Classification. Entropy
**2011**, 13, 1694–1707. [Google Scholar] [CrossRef] - Halsey, T.C.; Jensen, M.H.; Kadanoff, L.P.; Procaccia, I.; Shraiman, B.I. Fractal Measures and Their Singularities: The Characterization of Strange Sets. Nucl. Phys. B
**1986**, 33, 1141–1151. [Google Scholar] [CrossRef] - Hentschel, H.; Procaccia, I. The Infinite Number of Generalized Dimensions of Fractals and Strange Attractors. Physica D
**1983**, 8, 435–444. [Google Scholar] [CrossRef] - Morales, J.; Cantavella, J.V.; Mancilla, F.; Lozano, L.; Stich, D.; Herraiz, E.; Matín, J.B.; Lopez-Comino, J.A.; Martinez-Solares, J.M. The 2011 Lorca Seismic Series: Temporal Evolution, Faulting Parameters and Hypocentral Relocation. Bull. Earthquake Eng.
**2014**, 12, 1871–1888. [Google Scholar] [CrossRef]

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**MDPI and ACS Style**

Angulo, J.M.; Esquivel, F.J.
Multifractal Dimensional Dependence Assessment Based on Tsallis Mutual Information. *Entropy* **2015**, *17*, 5382-5401.
https://doi.org/10.3390/e17085382

**AMA Style**

Angulo JM, Esquivel FJ.
Multifractal Dimensional Dependence Assessment Based on Tsallis Mutual Information. *Entropy*. 2015; 17(8):5382-5401.
https://doi.org/10.3390/e17085382

**Chicago/Turabian Style**

Angulo, José M., and Francisco J. Esquivel.
2015. "Multifractal Dimensional Dependence Assessment Based on Tsallis Mutual Information" *Entropy* 17, no. 8: 5382-5401.
https://doi.org/10.3390/e17085382