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Journal of Computational and Nonlinear Dynamics | Volume 11 | Issue 3 | research-article
Research Papers

Entropy Analysis of Industrial Accident Data Series

[+] Author and Article Information
António M. Lopes

Institute of Mechanical Engineering,
Faculty of Engineering,
University of Porto,
Rua Dr. Roberto Frias,
Porto 4200-465, Portugal
e-mail: aml@fe.up.pt

J. A. Tenreiro Machado

Institute of Engineering,
Polytechnic of Porto,
Rua Dr. António Bernardino de Almeida, 431,
Porto 4200-072, Portugal
e-mail: jtm@isep.ipp.pt

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received December 21, 2014; final manuscript received July 28, 2015; published online October 23, 2015. Assoc. Editor: Hiroshi Yabuno.

J. Comput. Nonlinear Dynam 11(3), 031006 (Oct 23, 2015) (7 pages) Paper No: CND-14-1327; doi: 10.1115/1.4031195 History: Received December 21, 2014; Revised July 28, 2015
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Complex industrial plants exhibit multiple interactions among smaller parts and with human operators. Failure in one part can propagate across subsystem boundaries causing a serious disaster. This paper analyzes the industrial accident data series in the perspective of dynamical systems. First, we process real world data and show that the statistics of the number of fatalities reveal features that are well described by power law (PL) distributions. For early years, the data reveal double PL behavior, while, for more recent time periods, a single PL fits better into the experimental data. Second, we analyze the entropy of the data series statistics over time. Third, we use the Kullback–Leibler divergence to compare the empirical data and multidimensional scaling (MDS) techniques for data analysis and visualization. Entropy-based analysis is adopted to assess complexity, having the advantage of yielding a single parameter to express relationships between the data. The classical and the generalized (fractional) entropy and Kullback–Leibler divergence are used. The generalized measures allow a clear identification of patterns embedded in the data.
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References

1
Bhattacharya, P. , Chakrabarti, B. K. , and Kamal , 2011, “ A Fractal Model of Earthquake Occurrence: Theory, Simulations and Comparisons With the Aftershock Data,” J. Phys.: Conf. Ser., 319(1), p. 012004. [CrossRef]
2
Turcotte, D. L. , and Malamud, B. D. , 2002, “ 14 Earthquakes as a Complex System,” Int. Geophys., 81, pp. 209–227–IV.
3
Makowiec, D. , Dudkowska, A. , Gałaska, R. , and Rynkiewicz, A. , 2009, “ Multifractal Estimates of Monofractality in RR-Heart Series in Power Spectrum Ranges,” Phys. A, 388(17), pp. 3486–3502. [CrossRef]
4
Machado, J. T. , 2012, “ Accessing Complexity From Genome Information,” Commun. Nonlinear Sci. Numer. Simul., 17(6), pp. 2237–2243. [CrossRef]
5
Johnson, N. F. , Jefferies, P. , and Hui, P. M. , 2003, Financial Market Complexity, Oxford University Press, Oxford.
6
Huang, W.-Q. , Zhuang, X.-T. , and Yao, S. , 2009, “ A Network Analysis of the Chinese Stock Market,” Phys. A, 388(14), pp. 2956–2964. [CrossRef]
7
Amaral, L. A. N. , Scala, A. , Barthélémy, M. , and Stanley, H. E. , 2000, “ Classes of Small-World Networks,” Proc. Natl. Acad. Sci. U. S. A., 97(21), pp. 11149–11152. [CrossRef] [PubMed]
8
Sen, P. , Dasgupta, S. , Chatterjee, A. , Sreeram, P. , Mukherjee, G. , and Manna, S. , 2003, “ Small-World Properties of the Indian Railway Network,” Phys. Rev. E, 67(3), p. 036106. [CrossRef]
9
M'Chirgui, Z. , 2012, “ Small-World or Scale-Free Phenomena in Internet: What Implications for the Next-Generation Networks?,” Rev. Eur. Stud., 4(1), pp. 85–93.
10
Zhang, W.-B. , 2002, “ Theory of Complex Systems and Economic Dynamics,” Nonlinear Dyn., Psychol., Life Sci., 6(2), pp. 83–101. [CrossRef]
11
Foxon, T. J. , Köhler, J. , Michie, J. , and Oughton, C. , 2013, “ Towards a New Complexity Economics for Sustainability,” Cambridge J. Econ., 37(1), pp. 187–208. [CrossRef]
12
Barabási, A.-L. , Jeong, H. , Néda, Z. , Ravasz, E. , Schubert, A. , and Vicsek, T. , 2002, “ Evolution of the Social Network of Scientific Collaborations,” Phys. A, 311(3), pp. 590–614. [CrossRef]
13
Li, W. , Zhang, X. , and Hu, G. , 2007, “ How Scale-Free Networks and Large-Scale Collective Cooperation Emerge in Complex Homogeneous Social Systems,” Phys. Rev. E, 76(4), p. 045102. [CrossRef]
14
Huberman, B. A. , Pirolli, P. L. , Pitkow, J. E. , and Lukose, R. M. , 1998, “ Strong Regularities in World Wide Web Surfing,” Science, 280(5360), pp. 95–97. [CrossRef] [PubMed]
15
Adamic, L. A. , and Huberman, B. A. , 2000, “ Power-Law Distribution of the World Wide Web,” Science, 287(5461), p. 2115. [CrossRef]
16
Kostić, S. , Vasović, N. , Franović, I. , and Todorović, K. , 2014, “ Complex Dynamics of Spring-Block Earthquake Model Under Periodic Parameter Perturbations,” ASME J. Comput. Nonlinear Dyn., 9(3), p. 031019. [CrossRef]
17
Lopes, A. M. , and Machado, J. T. , 2012, “ Dynamical Behaviour of Multi-Particle Large-Scale Systems,” Nonlinear Dyn., 69(3), pp. 913–925. [CrossRef]
18
Strogatz, S. H. , 2001, “ Exploring Complex Networks,” Nature, 410(6825), pp. 268–276. [CrossRef] [PubMed]
19
Haken, 2006, Information and Self-Organization: A Macroscopic Approach to Complex Systems, Springer, Berlin, Heidelberg.
20
Pinto, C. , Lopes, A. M. , and Machado, J. , 2012, “ A Review of Power Laws in Real Life Phenomena,” Commun. Nonlinear Sci. Numer. Simul., 17(9), pp. 3558–3578. [CrossRef]
21
Newman, M. E. , 2005, “ Power Laws, Pareto Distributions and Zipf's Law,” Contemp. Phys., 46(5), pp. 323–351. [CrossRef]
22
Baleanu, D. , 2008, “ Fractional Constrained Systems and Caputo Derivatives,” ASME J. Comput. Nonlinear Dyn., 3(2), p. 021102. [CrossRef]
23
Guzzetti, F. , Malamud, B. D. , Turcotte, D. L. , and Reichenbach, P. , 2002, “ Power-Law Correlations of Landslide Areas in Central Italy,” Earth Planet. Sci. Lett., 195(3), pp. 169–183. [CrossRef]
24
Balasis, G. , Daglis, I. A. , Papadimitriou, C. , Anastasiadis, A. , Sandberg, I. , and Eftaxias, K. , 2011, “ Quantifying Dynamical Complexity of Magnetic Storms and Solar Flares Via Nonextensive Tsallis Entropy,” Entropy, 13(10), pp. 1865–1881. [CrossRef]
25
Levada, A. , 2014, “ Learning From Complex Systems: On the Roles of Entropy and Fisher Information in Pairwise Isotropic Gaussian Markov Random Fields,” Entropy, 16(2), pp. 1002–1036. [CrossRef]
26
Seely, A. J. , Newman, K. D. , and Herry, C. L. , 2014, “ Fractal Structure and Entropy Production Within the Central Nervous System,” Entropy, 16(8), pp. 4497–4520. [CrossRef]
27
Feldman, D. P. , and Crutchfield, J. P. , 1998, “ Measures of Statistical Complexity: Why?,” Phys. Lett. A, 238(4), pp. 244–252. [CrossRef]
28
Kwapień, J. , and Drożdż, S. , 2012, “ Physical Approach to Complex Systems,” Phys. Rep., 515(3), pp. 115–226. [CrossRef]
29
Machado, J. A. T. , and Lopes, A. M. , 2013, “ Analysis and Visualization of Seismic Data Using Mutual Information,” Entropy, 15(9), pp. 3892–3909. [CrossRef]
30
Pidgeon, N. , and O'Leary, M. , 2000, “ Man-Made Disasters: Why Technology and Organizations (Sometimes) Fail,” Saf. Sci., 34(1), pp. 15–30. [CrossRef]
31
Liu, T. , Zhong, M. , and Xing, J. , 2005, “ Industrial Accidents: Challenges for China's Economic and Social Development,” Saf. Sci., 43(8), pp. 503–522. [CrossRef]
32
Costa, M. , Goldberger, A. L. , and Peng, C.-K. , 2002, “ Multiscale Entropy Analysis of Complex Physiologic Time Series,” Phys. Rev. Lett., 89(6), p. 068102. [CrossRef] [PubMed]
33
Baranger, M. , 2000, Chaos, Complexity, and Entropy, New England Complex Systems Institute, Cambridge, MA.
34
Solé, R. V. , and Valverde, S. , 2004, “ Information Theory of Complex Networks: On Evolution and Architectural Constraints,” Complex Networks, Springer, Berlin, Heidelberg, pp. 189–207.
35
Svítek, M. , 2015, “ Towards Complex System Theory,” Neural Network World, 25(1), pp. 5–33. [CrossRef]
36
Cai, Y. , Qi, L. , and Wang, C. , 2013, “ A Data Mining Model of Complex System Based on Improved Cluster Analysis Model and Rough Set Theory,” Int. J. Appl. Math. Stat., 43(13), pp. 45–51.
37
Sacchi, L. , Dagliati, A. , and Bellazzi, R. , 2015, “ Analyzing Complex Patients? Temporal Histories: New Frontiers in Temporal Data Mining,” Data Mining in Clinical Medicine, Springer, New York, pp. 89–105.
38
Sugihara, G. , May, R. , Ye, H. , Hsieh, C.-H. , Deyle, E. , Fogarty, M. , and Munch, S. , 2012, “ Detecting Causality in Complex Ecosystems,” Science, 338(6106), pp. 496–500. [CrossRef] [PubMed]
39
Shao, Y.-H. , Gu, G.-F. , Jiang, Z.-Q. , Zhou, W.-X. , and Sornette, D. , 2012, “ Comparing the Performance of FA, DFA and DMA Using Different Synthetic Long-Range Correlated Time Series,” Sci. Rep., 2, pp. 1–5. [CrossRef]
40
Martinez, G. J. , Adamatzky, A. , and Alonso-Sanz, R. , 2012, “ Complex Dynamics of Elementary Cellular Automata Emerging From Chaotic Rules,” Int. J. Bifurcation Chaos, 22(2), p. 1250023. [CrossRef]
41
Cervelle, J. , Dennunzio, A. , and Formenti, E. , 2012, “ Chaotic Behavior of Cellular Automata,” Computational Complexity: Theory, Techniques, and Applications, Springer, New York, pp. 479–489.
42
Holcombe, M. , Adra, S. , Bicak, M. , Chin, S. , Coakley, S. , Graham, A. I. , Green, J. , Greenough, C. , Jackson, D. , Kiran, M. , MacNeil, S. , Maleki-Dizaji, A. , McMinn, P. , Pogson, M. , Poole, R. , Qwarnstrom, E. , Ratnieks, F. , Rolfe, M. D. , Smallwood, R. , Sun, T. , and Worth, D. , 2012, “ Modeling Complex Biological Systems Using an Agent-Based Approach,” Integr. Biol., 4(1), pp. 53–64. [CrossRef]
43
Niazi, M. A. , and Hussain, A. , 2012, Cognitive Agent-Based Computing-I: A Unified Framework for Modeling Complex Adaptive Systems Using Agent-Based and Complex Network-Based Methods, Vol. 1, Springer Science & Business Media, Dordrecht, Heidelberg.
44
Albert, R. , and Barabási, A.-L. , 2002, “ Statistical Mechanics of Complex Networks,” Rev. Mod. Phys., 74(1), pp. 47–97. [CrossRef]
45
Newman, M. E. , 2003, “ The Structure and Function of Complex Networks,” SIAM Rev., 45(2), pp. 167–256. [CrossRef]
46
Ravasz, E. , and Barabási, A.-L. , 2003, “ Hierarchical Organization in Complex Networks,” Phys. Rev. E, 67(2), p. 026112. [CrossRef]
47
“Emergency Events Database EM-DAT,” 2013, http://www.emdat.be/
48
Clauset, A. , Shalizi, C. R. , and Newman, M. E. , 2009, “ Power-Law Distributions in Empirical Data,” SIAM Rev., 51(4), pp. 661–703. [CrossRef]
49
Csányi, G. , and Szendrői, B. , 2004, “ Structure of a Large Social Network,” Phys. Rev. E, 69(3), p. 036131. [CrossRef]
50
Pinto, C. , Lopes, A. M. , and Machado, J. T. , 2014, “ Double Power Laws, Fractals and Self-Similarity,” Appl. Math. Modell., 38(15–16), pp. 4019–4026. [CrossRef]
51
Jóhannesson, G. , Björnsson, G. , and Gudmundsson, E. H. , 2006, “ Afterglow Light Curves and Broken Power Laws: A Statistical Study,” Astrophys. J., Lett., 640(1), pp. L5–L8. [CrossRef]
52
Khinchin, A. I. , 1957, Mathematical Foundations of Information Theory, Vol. 434, Courier Dover Publications, New York.
53
Machado, J. T. , 2014, “ Fractional Order Generalized Information,” Entropy, 16(4), pp. 2350–2361. [CrossRef]
54
Valério, D. , Trujillo, J. J. , Rivero, M. , Machado, J. T. , and Baleanu, D. , 2013, “ Fractional Calculus: A Survey of Useful Formulas,” Eur. Phys. J.: Spec. Top., 222(8), pp. 1827–1846. [CrossRef]
55
Kullback, S. , and Leibler, R. A. , 1951, “ On Information and Sufficiency,” Annals Math. Stat., 22(1), pp. 79–86. [CrossRef]
56
Cox, T. F. , and Cox, M. A. , 2000, Multidimensional Scaling, CRC Press, Boca Raton, FL.
57
Machado, J. T. , 2013, “ Visualizing Non-Linear Control System Performance by Means of Multidimensional Scaling,” ASME J. Comput. Nonlinear Dyn., 8(4), p. 041017. [CrossRef]
58
Kantz, H. , and Schreiber, T. , 2004, Nonlinear Time Series Analysis, Vol. 7, Cambridge University, Cambridge, UK.
59
Marwan, N. , Romano, M. C. , Thiel, M. , and Kurths, J. , 2007, “ Recurrence Plots for the Analysis of Complex Systems,” Phys. Rep., 438(5), pp. 237–329. [CrossRef]
60
Perc, M. , Green, A. K. , Dixon, C. J. , and Marhl, M. , 2008, “ Establishing the Stochastic Nature of Intracellular Calcium Oscillations From Experimental Data,” Biophys. Chem., 132(1), pp. 33–38. [CrossRef] [PubMed]
61
Kostić, S. , Vasović, N. , Perc, M. , Toljić, M. , and Nikolić, D. , 2013, “ Stochastic Nature of Earthquake Ground Motion,” Phys. A, 392(18), pp. 4134–4145. [CrossRef]
62
Takens, F. , 1981, Detecting Strange Attractors in Turbulence, Springer, Berlin, Heidelberg.
63
Kaplan, D. T. , and Glass, L. , 1992, “ Direct Test for Determinism in a Time Series,” Phys. Rev. Lett., 68(4), pp. 427–430. [CrossRef] [PubMed]
64
Fraser, A. M. , and Swinney, H. L. , 1986, “ Independent Coordinates for Strange Attractors From Mutual Information,” Phys. Rev. A, 33(2), pp. 1134–1140. [CrossRef]
65
Kennel, M. B. , Brown, R. , and Abarbanel, H. D. , 1992, “ Determining Embedding Dimension for Phase-Space Reconstruction Using a Geometrical Construction,” Phys. Rev. A, 45(6), pp. 3403–3411. [CrossRef] [PubMed]
66
Theiler, J. , Eubank, S. , Longtin, A. , Galdrikian, B. , and Farmer, J. D. , 1992, “ Testing for Nonlinearity in Time Series: The Method of Surrogate Data,” Phys. D, 58(1), pp. 77–94. [CrossRef]
67
Machado, J. T. , 2012, “ Fractional Order Modeling of Fractional-Order Holds,” Nonlinear Dyn., 70(1), pp. 789–796. [CrossRef]
68
Borg, I. , and Groenen, P. J. , 2005, “ Modeling Asymmetric Data,” Modern Multidimensional Scaling: Theory and Applications, Springer, New York, pp. 495–518.

Figures

Grahic Jump Location
Fig. 1

Annual evolution of the number of occurrences and human casualties: (a) total number of accidents and (b) total number of deaths. The time period of analysis is years 1903–2012.

+Annual evolution of the number of occurrences and human casualties: (a) total number of accidents and (b) total number of deaths. The time period of analysis is years 1903–2012.
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Annual evolution of the number of occurrences and human casualties: (a) total number of accidents and (b) total number of deaths. The time period of analysis is years 1903–2012.
Grahic Jump Location
Fig. 2

Parameters for the determinism test of human casualties time-series: (a) MI versus time-delay and (b) fnn versus embedding dimension

+Parameters for the determinism test of human casualties time-series: (a) MI versus time-delay and (b) fnn versus embedding dimension
View Large  |  Save Figure  |  Download Slide (.ppt)
Parameters for the determinism test of human casualties time-series: (a) MI versus time-delay and (b) fnn versus embedding dimension
Grahic Jump Location
Fig. 3

Complementary cumulative distributions corresponding to the 13 time windows analyzed. The minimum severity of events is 20 and the time period of analysis is 1903–2012.

+Complementary cumulative distributions corresponding to the 13 time windows analyzed. The minimum severity of events is 20 and the time period of analysis is 1903–2012.
View Large  |  Save Figure  |  Download Slide (.ppt)
Complementary cumulative distributions corresponding to the 13 time windows analyzed. The minimum severity of events is 20 and the time period of analysis is 1903–2012.
Grahic Jump Location
Fig. 4

Complementary cumulative distributions and PL approximations for the number of deadly victims occurred in industrial accidents: (a) time window w5 and (b) time window w13

+Complementary cumulative distributions and PL approximations for the number of deadly victims occurred in industrial accidents: (a) time window w5 and (b) time window w13
View Large  |  Save Figure  |  Download Slide (.ppt)
Complementary cumulative distributions and PL approximations for the number of deadly victims occurred in industrial accidents: (a) time window w5 and (b) time window w13
Grahic Jump Location
Fig. 5

Locus of the PL parameters in (C *, α*) plane

+Locus of the PL parameters in (C *, α*) plane
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Locus of the PL parameters in (C *, α*) plane
Grahic Jump Location
Fig. 6

Generalized entropy versus time windows, wi (i = 1, … , 13)

+Generalized entropy versus time windows, wi (i = 1, … , 13)
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Generalized entropy versus time windows, wi (i = 1, … , 13)
Grahic Jump Location
Fig. 7

MDS two- and three-dimensional maps comparing the time windows (wi, wj), (i, j) = 1, … , 13, distributions, based on Kullback–Leibler divergence, for α = 0

+MDS two- and three-dimensional maps comparing the time windows (wi, wj), (i, j) = 1, … , 13, distributions, based on Kullback–Leibler divergence, for α = 0
View Large  |  Save Figure  |  Download Slide (.ppt)
MDS two- and three-dimensional maps comparing the time windows (wi, wj), (i, j) = 1, … , 13, distributions, based on Kullback–Leibler divergence, for α = 0
Grahic Jump Location
Fig. 8

Sheppard plots corresponding to the MDS two- and three-dimensional maps comparing the time windows (wi, wj), (i, j) = 1, … , 13, distributions, based on Kullback–Leibler divergence, for α = 0

+Sheppard plots corresponding to the MDS two- and three-dimensional maps comparing the time windows (wi, wj), (i, j) = 1, … , 13, distributions, based on Kullback–Leibler divergence, for α = 0
View Large  |  Save Figure  |  Download Slide (.ppt)
Sheppard plots corresponding to the MDS two- and three-dimensional maps comparing the time windows (wi, wj), (i, j) = 1, … , 13, distributions, based on Kullback–Leibler divergence, for α = 0
Grahic Jump Location
Fig. 9

Stress plot for the MDS comparing the time windows(wi, wj), (i, j) = 1, … , 13, distributions, based on Kullback–Leibler divergence, for α = 0

+Stress plot for the MDS comparing the time windows(wi, wj), (i, j) = 1, … , 13, distributions, based on Kullback–Leibler divergence, for α = 0
View Large  |  Save Figure  |  Download Slide (.ppt)
Stress plot for the MDS comparing the time windows(wi, wj), (i, j) = 1, … , 13, distributions, based on Kullback–Leibler divergence, for α = 0
Grahic Jump Location
Fig. 10

MDS two- and three-dimensional maps comparing the time windows (wi, wj), (i, j) = 1, … , 13, distributions, based on Kullback–Leibler divergence, for α = 0.6

+MDS two- and three-dimensional maps comparing the time windows (wi, wj), (i, j) = 1, … , 13, distributions, based on Kullback–Leibler divergence, for α = 0.6
View Large  |  Save Figure  |  Download Slide (.ppt)
MDS two- and three-dimensional maps comparing the time windows (wi, wj), (i, j) = 1, … , 13, distributions, based on Kullback–Leibler divergence, for α = 0.6
Grahic Jump Location
Fig. 11

Sheppard plots corresponding to the MDS two- and three-dimensional maps comparing the time windows (wi, wj), (i, j) = 1, … , 13, distributions, based on Kullback–Leibler divergence, for α = 0.6

+Sheppard plots corresponding to the MDS two- and three-dimensional maps comparing the time windows (wi, wj), (i, j) = 1, … , 13, distributions, based on Kullback–Leibler divergence, for α = 0.6
View Large  |  Save Figure  |  Download Slide (.ppt)
Sheppard plots corresponding to the MDS two- and three-dimensional maps comparing the time windows (wi, wj), (i, j) = 1, … , 13, distributions, based on Kullback–Leibler divergence, for α = 0.6
Grahic Jump Location
Fig. 12

Stress plot for the MDS comparing the time windows(wi, wj), (i, j) = 1, … , 13, distributions, based on Kullback–Leibler divergence, for α = 0.6

+Stress plot for the MDS comparing the time windows(wi, wj), (i, j) = 1, … , 13, distributions, based on Kullback–Leibler divergence, for α = 0.6
View Large  |  Save Figure  |  Download Slide (.ppt)
Stress plot for the MDS comparing the time windows(wi, wj), (i, j) = 1, … , 13, distributions, based on Kullback–Leibler divergence, for α = 0.6

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