{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,19]],"date-time":"2025-10-19T15:46:59Z","timestamp":1760888819507,"version":"build-2065373602"},"reference-count":63,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2013,9,16]],"date-time":"2013-09-16T00:00:00Z","timestamp":1379289600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Seismic data is difficult to analyze and classical mathematical tools reveal strong limitations in exposing hidden relationships between earthquakes. In this paper, we study earthquake phenomena in the perspective of complex systems. Global seismic data, covering the period from 1962 up to 2011 is analyzed. The events, characterized by their magnitude, geographic location and time of occurrence, are divided into groups, either according to the Flinn-Engdahl (F-E) seismic regions of Earth or using a rectangular grid based in latitude and longitude coordinates. Two methods of analysis are considered and compared in this study. In a first method, the distributions of magnitudes are approximated by Gutenberg-Richter (G-R) distributions and the parameters used to reveal the relationships among regions. In the second method, the mutual information is calculated and adopted as a measure of similarity between regions. In both cases, using clustering analysis, visualization maps are generated, providing an intuitive and useful representation of the complex relationships that are present among seismic data. Such relationships might not be perceived on classical geographic maps. Therefore, the generated charts are a valid alternative to other visualization tools, for understanding the global behavior of earthquakes.<\/jats:p>","DOI":"10.3390\/e15093892","type":"journal-article","created":{"date-parts":[[2013,9,16]],"date-time":"2013-09-16T11:20:11Z","timestamp":1379330411000},"page":"3892-3909","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":40,"title":["Analysis and Visualization of Seismic Data Using Mutual Information"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4274-4879","authenticated-orcid":false,"given":"Jos\u00e9","family":"Machado","sequence":"first","affiliation":[{"name":"Institute of Engineering, Polytechnic of Porto, Rua Dr. Ant\u00f3nio Bernardino de Almeida, 431,  Porto 4200-072, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7359-4370","authenticated-orcid":false,"given":"Ant\u00f3nio","family":"Lopes","sequence":"additional","affiliation":[{"name":"Institute of Mechanical Engineering, Faculty of Engineering, University of Porto,  Rua Dr. Roberto Frias, Porto 4200-465, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2013,9,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"312","DOI":"10.1016\/j.engstruct.2005.09.028","article-title":"The impact of the 26 December 2004 earthquake and tsunami on structures and infrastructure","volume":"28","author":"Ghobarah","year":"2006","journal-title":"Eng. Struct."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"319","DOI":"10.1007\/s11069-009-9372-5","article-title":"Global earthquake casualties due to secondary effects: a quantitative analysis for improving rapid loss analyses","volume":"52","author":"Marano","year":"2010","journal-title":"Nat. Hazards"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"933","DOI":"10.1193\/1.2977493","article-title":"Fire following earthquake\u2014Reviewing the state-of-the-art modelling","volume":"24","author":"Lee","year":"2008","journal-title":"Earthq. Spectra"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1016\/j.enggeo.2004.05.006","article-title":"Earthquake losses due to ground failure","volume":"75","author":"Bird","year":"2004","journal-title":"Eng. Geol."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"298","DOI":"10.1111\/j.1468-0297.2010.02378.x","article-title":"Estimating the direct economic damages of the earthquake in Haiti","volume":"120","author":"Cavallo","year":"2010","journal-title":"Econ. J."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1055","DOI":"10.1007\/s11069-011-9889-2","article-title":"Natural disaster management mechanisms for probabilistic earthquake loss","volume":"60","author":"Tseng","year":"2012","journal-title":"Nat. Hazards"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"161","DOI":"10.1007\/s12665-011-1078-9","article-title":"Regional indirect economic impact evaluation of the 2008 Wenchuan Earthquake","volume":"65","author":"Wu","year":"2012","journal-title":"Environ. Earth Sci."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1530","DOI":"10.1016\/j.worlddev.2011.02.010","article-title":"Earthquake propensity and the politics of mortality prevention","volume":"39","author":"Keefer","year":"2011","journal-title":"World Dev."},{"key":"ref_9","first-page":"39","article-title":"Multifractal analysis of the spatial distribution of earthquake epicenters in the Zagros and Alborz-Kopeh Dagh regions of Iran","volume":"A1","author":"Zamani","year":"2011","journal-title":"Iran J. Sci. Technol."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Sornette, D., and Pisarenko, V. (2003). Fractal plate tectonics. Geophys. Res. Lett.","DOI":"10.1029\/2002GL015043"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"012004","DOI":"10.1088\/1742-6596\/319\/1\/012004","article-title":"A fractal model of earthquake occurrence: Theory, simulations and comparisons with the aftershock data","volume":"319","author":"Bhattacharya","year":"2011","journal-title":"J. Phys. Conf. Ser."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"2599","DOI":"10.1103\/PhysRevLett.76.2599","article-title":"Self-affine asperity model for earthquakes","volume":"76","author":"Hallgass","year":"1996","journal-title":"Phys. Rev. Lett."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1346","DOI":"10.1103\/PhysRevE.56.1346","article-title":"Earthquake statistics and fractal faults","volume":"56","author":"Hallgass","year":"1997","journal-title":"Phys. Rev. E"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"023123","DOI":"10.1063\/1.4711374","article-title":"Natural time analysis of the centennial earthquake catalog","volume":"22","author":"Sarlis","year":"2012","journal-title":"Chaos"},{"key":"ref_15","unstructured":"Jennings, P., Kanamori, H., and Lee, W. (2002). International Handbook of Earthquake and Engineering Seismology, Academic Press."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1429","DOI":"10.1088\/0034-4885\/67\/8\/R03","article-title":"The physics of earthquakes","volume":"67","author":"Kanamori","year":"2004","journal-title":"Rep. Prog. Phys."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"551","DOI":"10.1785\/gssrl.80.4.551","article-title":"Mid-continent earthquakes as a complex system","volume":"80","author":"Stein","year":"2009","journal-title":"Seismol. Res. Lett."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"69001","DOI":"10.1209\/0295-5075\/81\/69001","article-title":"Long-term memory in earthquakes and the distribution of interoccurrence times","volume":"81","author":"Lennartz","year":"2008","journal-title":"Europhys. Lett."},{"key":"ref_19","first-page":"207","article-title":"On a fractional model for earthquakes","volume":"178","author":"Ahmed","year":"2006","journal-title":"Appl. Math. Comput."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"647","DOI":"10.1016\/j.camwa.2013.02.003","article-title":"Fractional dynamics and MDS visualization of earthquake phenomena","volume":"66","author":"Lopes","year":"2013","journal-title":"Comput. Math. Appl."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1209\/0295-5075\/9\/3\/002","article-title":"Self-organized criticality and earthquakes","volume":"9","author":"Sornette","year":"1989","journal-title":"Europhys. Lett."},{"key":"ref_22","unstructured":"Shahin, A.M., Ahmed, E., Elgazzar, A.S., and Omar, Y.A. (2009)."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"535","DOI":"10.1016\/S0378-4371(98)00550-0","article-title":"Fractional calculus and the evolution of fractal phenomena","volume":"265","author":"Rocco","year":"1999","journal-title":"Physica A"},{"key":"ref_24","unstructured":"Samko, S., Kilbas, A., and Marichev, O. (1993). Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers."},{"key":"ref_25","unstructured":"Podlubny, I. (1999). Fractional Differential Equations, Academic Press."},{"key":"ref_26","unstructured":"Kilbas, A., Srivastava, H.M., and Trujillo, J. (2006). Theory and Applications of Fractional Differential Equations, Elsevier."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Baleanu, D., Diethelm, K., Scalas, E., and Trujillo, J. (2012). Fractional Calculus: Models and Numerical Methods, World Scientific Publishing.","DOI":"10.1142\/9789814355216"},{"key":"ref_28","unstructured":"International Seismological Centre (2010) On-line Bulletin, Internatl. Seis. Cent., Thatcham, UK. Available online: http:\/\/www.isc.ac.uk."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"185","DOI":"10.1785\/BSSA0340040185","article-title":"Frequency of earthquakes in California","volume":"34","author":"Gutenberg","year":"1944","journal-title":"Bull. Seismol. Soc. Am."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"8729","DOI":"10.1029\/92JB00427","article-title":"Variation of the Gutenberg-Richter b values and nontrivial temporal correlations in a spring-block model for earthquakes","volume":"97","author":"Christensen","year":"1992","journal-title":"J. Geophys. Res."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"727","DOI":"10.1111\/j.1365-246X.1993.tb04663.x","article-title":"Analysis of temporal and spatial heterogeneity of magnitude frequency distribution inferred from earthquake catalogues","volume":"113","author":"Ogata","year":"1993","journal-title":"Geophys. J. Int."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1002\/j.1538-7305.1948.tb01338.x","article-title":"A mathematical theory of communication","volume":"27","author":"Shannon","year":"1948","journal-title":"Bell Syst. Tech. J."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"269","DOI":"10.1016\/S0031-9201(00)00181-3","article-title":"Spatio-temporal seismicity patterns using mutual information application to southern Iberian peninsula (Spain) earthquakes","volume":"122","author":"Posadas","year":"2000","journal-title":"Phys. Earth Planet. Inter."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1267","DOI":"10.3390\/e13071267","article-title":"Tsallis-based nonextensive analysis of the southern California seismicity","volume":"13","author":"Telesca","year":"2011","journal-title":"Entropy"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"800","DOI":"10.3390\/e14040800","article-title":"Entropies and scaling exponents of street and fracture networks","volume":"14","author":"Mohajeri","year":"2012","journal-title":"Entropy"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"3096","DOI":"10.1103\/PhysRevE.62.3096","article-title":"Physical nature of higher-order mutual information: Intrinsic correlations and frustration","volume":"62","author":"Matsuda","year":"2000","journal-title":"Phys. Rev. E"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1134","DOI":"10.1103\/PhysRevA.33.1134","article-title":"Independent coordinates for strange attractors from mutual information","volume":"33","author":"Fraser","year":"1986","journal-title":"Phys. Rev. A"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"1773","DOI":"10.1103\/PhysRevLett.60.1773","article-title":"Information transport in spatiotemporal systems","volume":"60","author":"Vastano","year":"1988","journal-title":"Phys. Rev. Lett."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"391","DOI":"10.1016\/0167-2789(89)90263-7","article-title":"Reconstructing attractors from scalar time series: A Comparison of singular system and redundancy criteria","volume":"34","author":"Fraser","year":"1989","journal-title":"Phys. D"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1016\/0960-0779(94)90020-5","article-title":"Finite sample effects in sequence analysis","volume":"4","author":"Herzel","year":"1994","journal-title":"Chaos Soliton Fractals"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"1612","DOI":"10.1016\/j.camwa.2011.03.005","article-title":"Entropy analysis of DNA code dynamics in human chromosomes","volume":"62","author":"Costa","year":"2011","journal-title":"Comput. Math. Appl."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"3135","DOI":"10.1016\/j.nonrwa.2011.05.013","article-title":"Shannon, R\u00e9nyie and Tsallis entropy analysis of DNA using phase plane","volume":"12","author":"Costa","year":"2011","journal-title":"Nonlinear Anal. Real World Appl."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"839","DOI":"10.1007\/BF02330576","article-title":"Mutual information of Ising systems","volume":"35","author":"Matsuda","year":"1996","journal-title":"Int. J. Theor. Phys."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"275","DOI":"10.1016\/S0167-2789(97)00302-3","article-title":"Edge of chaos in rule-changing cellular automata","volume":"116","author":"Mori","year":"1998","journal-title":"Phys. D"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"244","DOI":"10.1016\/S0375-9601(97)00855-4","article-title":"Measures of statistical complexity: Why?","volume":"238","author":"Feldman","year":"1998","journal-title":"Phys. Lett. A"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"051125","DOI":"10.1103\/PhysRevE.75.051125","article-title":"Mutual information as a tool for identifying phase transitions in dynamical complex systems with limited data","volume":"75","author":"Wicks","year":"2007","journal-title":"Phys. Rev. E"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1016\/j.physrep.2012.01.007","article-title":"Physical approach to complex systems","volume":"515","year":"2012","journal-title":"Phys. Rep."},{"key":"ref_48","unstructured":"Bagozzi, R.P. (1994). Advanced Methods in Marketing Research, Blackwell."},{"key":"ref_49","unstructured":"Bishop, C.M. (2006). Pattern Recognition and Machine Learning, Springer-Verlag."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"406","DOI":"10.1109\/TIFS.2010.2049842","article-title":"Face matching and retrieval using soft biometrics","volume":"5","author":"Park","year":"2010","journal-title":"IEEE Trans. Inf. Forensics Secur."},{"key":"ref_51","unstructured":"Cover, T.M., and Thomas, J.A. (1991). Elements of Information Theory, John Wiley & Sons."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"651","DOI":"10.1016\/j.patrec.2009.09.011","article-title":"Data clustering: 50 years beyond K-means","volume":"31","author":"Jain","year":"2010","journal-title":"Pattern Recognit. Lett."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"881","DOI":"10.1109\/TPAMI.2002.1017616","article-title":"An efficient K-means clustering algorithm: Analysis and implementation","volume":"24","author":"Kanungo","year":"2002","journal-title":"IEEE Trans. Pattern Anal."},{"key":"ref_54","unstructured":"Jain, A.K., and Dubes, R. (1988). Algorithms for Clustering Data, Prentice-Hall."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"241","DOI":"10.1007\/BF02289588","article-title":"Hierarchical clustering schemes","volume":"32","author":"Johnson","year":"1966","journal-title":"Psychometrika"},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"54","DOI":"10.2307\/2346439","article-title":"Minimum spanning trees and single linkage cluster analysis","volume":"18","author":"Gower","year":"1969","journal-title":"Appl. Stat."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"236","DOI":"10.1080\/01621459.1963.10500845","article-title":"Hierarchical grouping to optimize an objective function","volume":"58","author":"Ward","year":"1963","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"771","DOI":"10.1785\/BSSA0643-20771","article-title":"Seismic and geographical regionalization","volume":"64","author":"Flinn","year":"1974","journal-title":"Bull. Seismol. Soc. Am."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1029\/RG003i001p00123","article-title":"A proposed basis for geographical and seismic regionalization","volume":"3","author":"Flinn","year":"1965","journal-title":"Rev. Geophys."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"063010","DOI":"10.1088\/1367-2630\/12\/6\/063010","article-title":"A non-universal aspect in the temporal occurrence of earthquakes","volume":"12","author":"Zhao","year":"2010","journal-title":"New J. Phys."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1016\/0377-0427(87)90125-7","article-title":"Silhouettes: A graphical aid to the interpretation and validation of cluster analysis","volume":"20","author":"Rousseeuw","year":"1987","journal-title":"J. Comput. Appl. Math."},{"key":"ref_62","doi-asserted-by":"crossref","unstructured":"Omi, T., Kanter, I., and Shinomoto, S. (2011). Optimal observation time window for forecasting the next earthquake. Phys. Rev. E, 83.","DOI":"10.1103\/PhysRevE.83.026101"},{"key":"ref_63","unstructured":"Felsenstein, J. (2005). PHYLIP, Department of Genome Sciences, University of Washington. version 3.6; free phylogeny inference package; Distributed by the author."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/15\/9\/3892\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T21:49:20Z","timestamp":1760219360000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/15\/9\/3892"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,9,16]]},"references-count":63,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2013,9]]}},"alternative-id":["e15093892"],"URL":"https:\/\/doi.org\/10.3390\/e15093892","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2013,9,16]]}}}