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