Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 562320 10 pageshttpdxdoiorg1011552013562320

Review ArticlePower Law and Entropy Analysis of Catastrophic Phenomena

J A Tenreiro Machado1 Carla M A Pinto2 and A Mendes Lopes3

1 Department of Electrical Engineering Institute of Engineering Polytechnic of Porto (ISEP)Rua Dr Antonio Bernardino de Almeida 431 4200-072 Porto Portugal

2 Institute of Engineering Polytechnic of Porto (ISEP) and Center of Mathematics of the University of PortoRua Dr Antonio Bernardino de Almeida 431 4200-072 Porto Portugal

3 UISPA IDMEC-Polo FEUP Faculty of Engineering University of Porto Rua Dr Roberto Frias4200-465 Porto Portugal

Correspondence should be addressed to J A Tenreiro Machado jtmisepipppt

Received 17 September 2012 Accepted 13 December 2012

Academic Editor Jozsef Kazmer Tar

Copyright copy 2013 J A Tenreiro Machado et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Catastrophic events such as wars and terrorist attacks tornadoes and hurricanes earthquakes tsunamis floods and landslidesare always accompanied by a large number of casualties The size distribution of these casualties has separately been shown tofollow approximate power law (PL) distributions In this paper we analyze the statistical distributions of the number of victims ofcatastrophic phenomena in particular terrorism and finddouble PL behaviorThismeans that the data sets are better approximatedby two PLs instead of a single one We plot the PL parameters corresponding to several events and observe an interestingpattern in the charts where the lines that connect each pair of points defining the double PLs are almost parallel to each otherA complementary data analysis is performed by means of the computation of the entropy The results reveal relationships hiddenin the data that may trigger a future comprehensive explanation of this type of phenomena

1 Introduction

Power laws (PLs) have been widely reported in the model-ing of distinct phenomena and have been associated withlong memory behavior self-similarity fractal structures andfractional calculus In [1] for example PLs are interpretedas a manifestation of the long memory property of systemswith fractional dynamics and in [2] several complex systemsexhibiting PL behavior are analysed in the perspective offractional dynamics Fractional calculus and PLs are usedin [3 4] to model the dynamics of financial markets In [5]the complexity of the human body is characterized throughfractal measures and its dynamics described by means offractional calculus

Catastrophic events are characterized by a huge severityusually defined by a large number of casualties By catastroph-ic events we mean wars terrorist attacks tornadoes earth-quakes floods and landslidesThedistribution of the numberof casualties in these events is proved to be a PL [6ndash12]

PL distributions were first mentioned in 1896 whenPareto described the distribution of income [13] Paretoproved that the relative number of individuals with an annualincome larger than a certain value 119909 was proportional toa power of 119909 This has been known by Pareto distributionAfter this work Auerbach [14] demonstrated an analogousresult for city size distributions Ranking cities from 1 to 119899with the city with bigger population ranked as 1 Auerbachdemonstrated that the product of cities populations by theirranks was approximately constant for a given territoryEstoup [15] and Zipf [16 17] applied PLs to words frequenciesin texts They found that there are words that are used moreoften than others and the distribution of word frequenciesfollows a PL Zipf [17] described the distribution of city sizesby a Pareto distribution

Often to show that a certain data set follows a PL dis-tribution researchers depict a plot of the size versus fre-quency of the event studied In logarithmic scales theyobtain a straight line with negative slope In the case of

2 Mathematical Problems in Engineering

the Pareto distribution the behavior is exactly linear and isgiven by

ln (119875 [119883 ge 119909]) = ln119862 minus ln minus ln 119909 (1)

where 119883 is a random variable following a PL distribution gt 0 119862 = 119862 gt 0 In these distributions the tail fallsasymptotically according to the value of translating inheavy tails comparatively to other distributions Zipf rsquos lawis a special case of the Paretorsquos law with coefficient = 1Relevant reviews on PL distributions can be found in [18ndash20]

In many cases a single PL holds for the entire range of therandom variable that represents the system In other casesthe statistical distribution is better described by multiple PLs[21] In such cases different PLs characterized by distinctPL parameters fit more adequately the real data Double PLbehaviors have been pointed out by others in different phe-nomena For example in [22] many instances are shown oftwo PLs expressed bymeans of a generalized beta distributionfunction and in [23] the double PL behavior is explicitlystudied in the frequency of words in texts Moreover beyondthe ranking problem two PLs are manifested in other type ofproblems such as in turbulence earth magnetic pole fluctua-tions paleolake sedimentation density subject to volcanismand avalanche distributions to mention a few [24]

In this paper we analyze the statistical distributions ofthe number of victims caused by catastrophic phenomenaand find double PL behavior Moreover we plot the PLparameters corresponding to several events and observe aninteresting pattern in the charts where the lines that connecteach pair of points defining the double PLs are roughlyparallel to each other Then a complementary data analysisis performed by means of the computation of the Shannonentropy The results reveal relationships hidden in the datathat may trigger a future comprehensive explanation of thistype of phenomena

Bearing these ideas in mind this paper is organizedas follows In Section 2 the results found in the literatureconcerning PLs and casualties in natural and human-madedisasters are summarized Section 3 analyses the PL behaviorof catastrophic phenomena using data from real disasters InSection 4 several entropy measures are used to characterizethe data Finally in Section 5 the main results and conclu-sions of this paper are discussed

2 Brief Review of PLs inCatastrophic Occurrences

Patterns seen in wars terrorist attacks tornadoes earth-quakes landslides floods and other severe occurrences havebeen at close attention by various researchers [6ndash12 25ndash28]Many attentive explanations have arisen in the literatureNevertheless a complete understanding of these patternsis a complex task Important and intricate political geo-graphical historical and even cultural factors oppose to

a better understanding Predicting the number of casualtiesin natural or human-made disasters is extremely important indeveloping predisaster strategies Aspects like rationalizationof medical supplies and food gathering emergency teamsorganize shelter spaces amongst others have to be dealt within order to minimize the damage

A PL behavior is indicative of a particular property ofa system it indicates that the size of an event is inverselyproportional to its frequency In this sense large casualtiesare associated with low frequency phenomena and morefrequent events are less harmful in terms of preserving humanlives [7 25] Examples of phenomena with low probabilityand huge casualties are the twoworldwars (WWs) highmag-nitude earthquakes strong tornadoes huge tsunamis andamongst others

In 1948 Richardson [7] analyzed domestic and inter-national cases of violence in the period from 1820 to 1945He distributed the cases according to casualties measuredin powers of 10 into five categories The two WWs wereclassified in the highest category In a later work [25] thesame author showed that if the frequency of an occurrencedecreased by a factor close to three then the number of casu-alties increased by a power of 10

Guzzetti [26] considers landslide events in specific peri-ods in different countries such as Italy Canada Alps HongKong Japan and China He shows that the plot of thecumulative distribution function of the number of landslideevents versus the number of casualties is well approximatedby a straight line This result suggests a PL distribution of thedata

Cederman [11] followed Richardsonrsquos work [7 25] Heused data from the Correlates of War (COW) Project [29]focusing on interstate wars He computed the cumulativerelative frequency of war size and showed that it obeyed aPL The author proposed a self-organized critical dynamicalsystem that replicated the PL behavior seen in real data Itsmodel allowed conflict to spread and diffuse potentially overlong periods of time due to the quasi-parallel execution

In 2005 Jonkman [27] studied the distribution of killingsin global events focusing on the number of human deathscaused by three types of floods (river floods flash floods anddrainage issues) between January 1975 and June 2002 Theauthor plotted the global frequency of events with119873 or moredeaths versus119873 He observed a PL behavior for earthquakesbut not for flood data Becerra et al [30] use the same dataset as Jonkman [27] but consider all disasters combinedboth globally and disaggregated by continent They obtainedstraight-line log-log plots for all disasters combined Theslopes of the casualties PL distributions were smaller thanthose for modern wars and terrorism The explanation forthis remained an open question Another unsolved issue wasthe existence of PL behavior in combined disasters and not inindividual disasters such as floods Here it is worth mention-ing that casualties in earthquakes verified a PL distribution[6 27 30]

Johnson et al [28] suggested a microscopic theory toexplain similarity in patterns of violence such as war andglobal terrorism The similarity was observed regardless of

Mathematical Problems in Engineering 3

underlying ideologies motivations and the terrain in whichevents occurred The authors introduced a model where theinsurgent force behaved as a self-organizing system whichevolved dynamically through the continual coalescence andfragmentation of its constituent groups They analyzed casu-altiesrsquo patterns arising within a given war unlike previousstudies that focused on the total casualty figure for oneparticular war [7 11 25 31] A PL behavior fitted well the datanot only from Iraq Colombia and non-G7 terrorism butalso with data obtained from the war in Afghanistan The PLparameter for Iraq Colombia andAfghanistan was (close to) = 25 This value of the coefficient equalized the coefficientvalue characterizing non-G7 terrorism In the literature thePL parameter value was = 251 for non-G7 countries [32]and = 1713 for G7 countries This result suggested that PLpatterns would emerge within any modern asymmetric warfought by loosely-organized insurgent groups

In 2006 Bogen and Jones [33] treated the severity of ter-rorist attacks in terms of deaths and injuredThey applied a PLdistribution to victimevent rates and used the PL to predictmortality due to terrorism through the year 2080 Authorsclaimed that these PL models could be used to improvestrategies ldquoto assess prevent and manage terror-related risksand consequencesrdquo

Clauset et al [34] studied the frequency and the numberof casualties (deaths and injuries) of terrorist attacks since1968 They observed a scale-invariance behavior with thefrequency being an inverse power of the casualties Thisbehavior was independent of the type of weapon economicdevelopment and distinct time scales The authors presenteda new model to fit the frequency of severe terrorist attackssince previous models in the literature failed to produce theheavy tail in the PL distribution Their model assumed thatthe severity of an occurrence was a function of the executionplan and that selection tools were better suited to modelcompetition between states and nonstate actors Finally re-searchers claimed that periodicity was a common feature inglobal terrorism with period close to roughly 13 years

Bohorquez et al [12] studied the quantitative relationbetween human insurgency global terrorism and ecologyThey introduced a newmodel to explain the size distributionof casualties or the timing ofwithin-conflict eventsThey con-sidered insurgent populations as self-organized groups thatdynamically evolved through decision-making processesThe main assumptions of the model were (i) being consistentwith work on human group dynamics in everyday environ-ments (ii) having a new perception of modern insurgenciesas fragmented transient and evolving and (iii) using adecision-making process about when to attack based oncompetition for media attention Authors applied a PL distri-bution to Iraq andColombiawars with parameter value closeto = 25 A coefficient value of = 25 was in concordancewith the coefficient value of = 248 plusmn 007 obtained byClauset et al [34] on global terrorism A PL fit to Spanish andAmerican Civil wars revealed a PL parameter value smaller(around = 17) Authors claimed that theirmodel suggesteda remarkable link between violent and nonviolent humanactions due to its similarity to financial market models

Rank

max

rank

Sizemax size10minus3 10minus2 10minus1 10010minus2

10minus1

100

ln(119910) = minus085503 ln(119909) minus 474389

Figure 1 Rankfrequency log-log plot corresponding to the distri-bution of casualties caused by industrial accidents in CentralSouthAmerica over the period 1900ndash2011 (min size = 10 max size = 2700max rank = 66)

3 Power Law Behavior inCatastrophic Phenomena

In this section we investigate the statistical distributionsof random variables that represent the number of humancasualties in several human-made and natural hazards

Data from the EM-DAT International Disaster Database(httpwwwemdatbe) and the Global Terrorism Database(GTD) (httpwwwstartumdedugtd) are analyzed TheEM-DAT database contains information on over than 18000worldwide natural and technological disasters from 1900to present The EM-DAT is maintained by the Centre forResearch on the Epidemiology of Disasters (CRED) at theSchool of Public Health of the Universite Catholique deLouvain located inBrussels Belgium [35]TheGTDdatabaseis an open-source database that includes information onmorethan 98000 worldwide terrorist attacks from 1970 up to 2010[31]

PLs are observed in several natural and man-madesystems Examples of single and double PLs in real data aregiven in Figures 1 and 2 respectively The former representsthe complementary cumulative distribution of the severityof industrial accidents in CentralSouth America over theperiod 1900ndash2011 The adopted measure to quantify theseverity of an event is the total number of fatalities Thedepicted graph corresponds to a rankfrequency log-log plotTo construct the graph we first sort the data (ie theaccidents) in decreasing order according to their severity andnumber them consecutively starting from one [36] Then anormalization of the values is carried out meaning that thenumber of fatalities (119909-axis) is divided by the correspondinghighest value and the rank (119910-axis) is divided by the rank ofthe smallest event Finally PLs are adjusted to the data using aleast squares algorithm All the log-log plots presented in thispaper are made following this procedure

Figure 2 corresponds to the distribution of casual-ties caused by earthquakes in CentralSouth America in

4 Mathematical Problems in EngineeringRa

nkm

ax ra

nk

Sizemax size

ln(119910) = minus0246959 ln(119909) minus 299476

ln(119910) = minus0499454 ln(119909) minus 492208

10minus4 10minus310minus6 10minus5 10minus2 10minus1 10010minus3

10minus2

10minus1

100

Figure 2 Rankfrequency log-log plot corresponding to the dis-tribution of casualties caused by earthquakes in CentralSouthAmerica over the period 1900ndash2011 (min size = 1 max size = 222570max rank = 179)

the period 1900ndash2011 representing one event that can beapproximated by a double PL

As can be seen in Figure 1 a single PL (SPL) with para-meters (119862 ) = (00087 08550) fits to the data The dis-tribution depicted in Figure 2 is better approximated by adouble PL (DPL) with parameters (119862

1 1) = (00500 02470)

and (1198622 2) = (00073 04995) The change in the behavior

occurs at the relative value of 119909 = 0000539 approximatelyWe analyzed the data available at the EM-DAT database

in terms of disaster type (DT119895) and disaster location (DL

119896)

119895 = 1 11 and 119896 = 1 6 categories respectivelyDT119895= Drought Earthquake Epidemic Extreme tempera-

ture Flood Industrial accident Mass movement wet StormTransport accident VolcanoWildfire DL

119896= Africa North

America Central amp South America Europe (including Rus-sia) Asia (not including SE Asia) Oceania (including SEAsia) The period of analysis was 1900ndash2011 for every caseThe total number of combinations (locationtype) is 11 times6 Nevertheless for 14 cases there is insufficient data tocompute reliable statistical distributions For all cases takingthe number of casualties as the variable of interest we obtainstatistical distributions that can be approximated by either aSPL (16 cases) or aDPL (36 cases) similar to the ones depictedin Figures 1 and 2

In Figure 3 we depict the locus of the parameters(119862119894 119894) 119894 = 1 2 corresponding to the analyzed cases As can

be seen an interesting pattern emerges where the linesthat connect the pairs of points that characterize the DPLshave identical orientation This geometrical pattern reflects arelationship between the two parts of the DPL distributions(DPL1mdashpart closer to the head and DPL2mdashpart closer to thetail) Besides the observation that

2gt 1 in all cases further

investigation on the reason for this behavior is neededWe pursued our study with the analysis of the GTD

database First the events associated to human casualtieswere grouped by year (119884

119903) starting in 1980 up to 2010

0

02

04

06

08

1

12

14

10minus4 10minus3 10minus2 10minus1 100

119862

Figure 3 Locus of the parameters (119862119894 119894) 119894 = 1 2 that characterize

the PLs corresponding to the number of casualties in certaincombinations of disaster typelocation DT

119895DL119896

(except 1993 because there is no data available) 119884119903

=

1980 2010 1993 119903 = 1 30 We found that allthe statistical distributions can be approximated by DPLs InFigures 4 and 5 the time evolution of the parameters of theDPLs (119862

119894 119894) 119894 = 1 2 is shown Regarding the parameters

119862119894 it can be seen that they have identical behavior although

1198622varies more than 119862

1and is always smaller than it

With respect to 119894 we have a similar evolution but in this

case the parameter 2is always greater than

1 As severe

terrorist attacks correspond to points closer to the tail of thedistribution DPL2 which is characterized by a larger thismeans that those events are more similar between each otherthan the smaller events (that correspond to DPL1)

To complement the analysis with respect to the date of theoccurrences the parameters (119862

119894 119894) of the PLs correspond-

ing to 119884119903 were plotted (Figure 6) As can be seen a pattern

similar to the described previously (Figure 3) is observedWe have also studied the distributions of the casualties in

terrorist attacks occurred in the period 1970ndash2010 but withrespect to other criteria namely the type of used weapon(119882119894) region where the event took place (119877

119895) target (119879

119896)

and type of attack (119860119897) Each criterion was then divided

into 119894 = 1 6 119895 = 1 13 119896 = 1 19 and 119897 =

1 8 categories respectively119882119894= Chemical Explosives

Firearms Incendiary Melee Vehicle 119877119895= Australasia amp

Oceania Central America amp Caribbean Central Asia EastAsia Eastern Europe Middle East amp North Africa NorthAmerica South America South Asia Southeast Asia Sub-Saharan Africa USSR amp Newly Independent States (NIS)Western Europe 119879

119896= Airports amp Airlines Business

Educational Institution Food or Water Supply Govern-ment (Diplomatic) Government (General) Journalists ampMedia Maritime Military NGO Police Private Citizensamp Property Religious FiguresInstitutions Telecommuni-cation Terrorists Tourists Transportation Utilities Vio-lent Political Party 119860

119897= Armed Assault Assassination

Mathematical Problems in Engineering 5

1980 1990 2000 2010Year

11986211198622

10minus4

10minus3

10minus2

10minus1

119862

Figure 4 Time evolution of parameters 119862119894 119894 = 1 2 of the DPLs

corresponding to terrorist attacks over the period 1980ndash2010 119884119903

1980 1990 2000 2010Year

0

05

1

15

2

25

3

12057211205722

Figure 5 Time evolution of parameters 119894 119894 = 1 2 of the DPLs

corresponding to terrorist attacks over the period 1980ndash2010 119884119903

BombingExplosion FacilityInfrastructure Attack Hijack-ing Hostage Taking (Barricade Incident) Hostage Taking(Kidnapping) and Unarmed Assault

Most cases are characterized by DPLs However in afew situations a SPL fits better to the data The main resultsare summarized in Table 1 Moreover we observed that theparameters corresponding to all distributions characterizedby DPLs display a pattern similar to the ones mentionedpreviously (Figures 3 and 6) where the lines connecting theslopes and intercepts of DPL1 (

1 1198621) to its companionDPL2

(2 1198622) for the same data set have identical orientation in

the (119862 ) Cartesian space

4 Entropy of Catastrophic Phenomena

In this section we analyse the entropy of data collected fromthe GTD database that is data related to terrorism To

1980 1981 1982 1983 1984 19851986 1987 1988 1989 1990 19911992 1994 1995 1996 1997 19981999 2000 2001 2002 2003 20042005 2006 2007 2008 2009 2010

0

05

1

15

2

25

3

10minus4 10minus3 10minus2 10minus1

119862

Figure 6 Locus of the parameters (119862119894 ) 119894 = 1 2 that characterize

the distributions of terrorist attacks over the period 1980ndash2010 119884119903

calculate the entropies we construct histograms of relativefrequencies using bins of width one (one casualty) andapproximate the probabilities 119901

119894by the relative frequencies

We present results obtained for terrorist events grouped byyear (119884

119903) as defined in the previous section Nevertheless it

should be noticed that similar results are obtained for all otherhuman-made and natural hazards

Clausius [37] and Boltzmann [38] were the first authorsto define entropy in the field of thermodynamics Later onShannon [39] and Jaynes [40] applied their results to infor-mation theory [41]

The most celebrated entropy is the so-called Shannonentropy 119878 defined by

119878 = minus

119882

sum

119894=1

119901119894ln119901119894 (2)

The Shannon entropy represents the expected value of theinformation minus ln119901

119894 Therefore for the uniform probability

distribution we have 119901119894= 119882minus1 and the Shannon entropy

takes its maximum value 119878 = ln119882 yielding the Boltzmannrsquosfamous formula up to a multiplicative factor 119896 denoting theBoltzmann constant Thus in thermodynamic equilibriumthe Shannon entropy can be identified as the ldquophysicalentropyrdquo of the system

Renyi and Tsallis entropies are generalizations of Shan-nonrsquos entropy and are given by respectively

119878(119877)

119902=

1

1 minus 119902

ln(119882

sum

119894=1

119901119902

119894) 119902 gt 0

119878(119879)

119902=

1

119902 minus 1

(1 minus

119882

sum

119894=1

119901119902

119894)

(3)

6 Mathematical Problems in Engineering

Table 1 PL fit to the distributions of casualties in terrorist attacks

Criterion Category SPL DPL1 DPL2(119862 ) (119862

1 1) (119862

2 2)

Weapon (119882119894)

1198821

00391 058491198822

00082 07514 00002 171981198823

00017 08660 00000 196241198824

00236 06877 00021 148901198825

00057 06909 00003 135791198826

00249 07801

Region (119877119894)

1198771

00438 101471198772

00214 06412 00009 169051198773

00174 119051198774

00065 090731198775

00012 121411198776

00091 08051 00006 170881198777

00009 114151198778

00193 07814 00008 206751198779

00036 08742 00001 2055211987710

00128 08561 00012 1657811987711

00091 06454 00004 1313211987712

00023 0995611987713

00001 15523

Target (119879119894)

1198791

00438 05144 00092 124881198792

00057 08884 00003 171801198793

00002 13589 00012 084581198794

00452 079231198795

00015 112591198796

00006 11152 00001 151621198797

00014 161061198798

00576 04694 00094 090701198799

00353 05669 00009 1629111987910

00090 09645 00031 1257311987911

00047 08991 00001 1835411987912

00036 07778 00000 1866711987913

00158 07353 00018 1610611987914

00462 0977411987915

00168 08610 00026 1682411987916

00234 0763511987917

00340 05892 00014 1517811987918

00225 07389 00074 1070811987919

00108 07918 00025 11650

Type (119860119894)

1198601

00049 07346 00000 179821198602

00014 15201 00001 265551198603

00059 08116 00001 206211198604

00261 06339 00016 148801198605

00305 071861198606

00116 073161198607

00072 09689 00014 162141198608

00471 07006 00034 20186

Mathematical Problems in Engineering 7

1980 1985 1990 1995 2000 2005 201005

05506

06507

07508

08509

0951

Year

Nor

mal

ized

entro

py

119878(119879)05

119878(119877)05

119878(119880)05

119878(119879)2

119878(119877)2

119878(119880)2

119878

Figure 7 Total Shannon Tsallis Renyi and Ubriaco normalizedentropies as a function of the year of the events 119884

119903 119902 = 05 and

119902 = 2

Tsallis entropy reduces to Renyi entropy when 119902 rarr 1 Tsallisentropy was applied to diffusion equations [42] and Fokker-Planck systems [43] Renyi entropy has an inverse power lawequilibrium distribution [44] and satisfies the zeroth law ofthermodynamics [45] The two parameters Sharma-Mittalentropy [46] is accepted as a generalization of Tsallis Renyiand Boltzmann-Gibbs entropies for limiting cases of theparameters [47]

Recently more general entropy measures have been pro-posed in the literature where the additivity axiom has beenrelaxed For instance Ubriaco [48] proposed the followingformula for the fractional entropy

119878(119880)

119902=

119882

sum

119894=1

(minus ln119901119894)119902

119901119894

(4)

that has the same properties as the Shannon entropy exceptadditivity

Applications of entropy in distinct complex systems canbe found in [49ndash57]

In Figure 7 the total Shannon normalized entropy (119878) isdepicted as well as Tsallisrsquo (119878(119879)

119902) Renyirsquos (119878(119877)

119902) and Ubriacorsquos

(119878(119880)

119902) for 119902 = 05 and 119902 = 2 as a function of the year of

the events 119884119903 Figures 9 10 and 11 show 119878

(119879)

119902 119878(119877)119902

and 119878(119880)119902

normalized entropies as a function of the year 119884119903 and

entropy parameter 01 le 119902 le 10In Figure 7 we observe two types of behavior namely

short- and long-term phenomena In what concerns shorttime behavior we verify peaks during 1983ndash1985 1997-19982004ndash2007 and minima at 1980 1995 and 2010 In whatconcerns long-time relationships a smooth decreasing isobserved for 119878(119879)

05 119878(119877)05

and 119878(119880)2

Removing the maxima andminima we get a time series for years 1981 1982 1986ndash1992 1994ndash1996 1999 2000 2002 2003 2008 and 2009(Figure 8) Largersmaller entropies correspond to chartscloserafar uniform distributions therefore seemingly wehave lessmore organized terrorist events in global terms

1980 1985 1990 1995 2000 2005 2010075

08

085

09

095

1

Year

Nor

mal

ized

entro

py

119878(119879)05

119878(119877)05

119878(119880)2

Lin Reg 119878(119879)05Lin Reg 119878(119877)05Lin Reg 119878(119880)2

Figure 8 Total Tsallis and Renyi normalized entropies (119902 = 05)

and Ubriacorsquos (119902 = 2) as a function of the events in years 1981 19821986ndash1992 1994ndash1996 1999 2000 2002 2003 2008 2009

1980 1985 1990 1995 2000 2005 2010

01234567

8910040506070809

1

Year

119902

119878(119879)119902

Figure 9 Total Tsallis normalized entropy 119878(119879)119902 as a function of the

year of the events 119884119903 and parameter 119902

These conclusions remain invariant for Figures 9 to 11 wherewe vary both the entropy definition and the parameter tuningTherefore we conclude that such results are robust againstsuch type of variations

As discussed in the previous section the statistical dis-tributions of real data can be approximated by either singleor double PLs In the latter case we study not only the totalentropy but also the entropy associated to each part of thedistributions Therefore we compute the entropy associatedto DPL1 and DPL2 that approximate the first and secondpart of the distributions respectively When adopting thisprocedure we are restricted to the Shannon and Ubriacoentropies as Tsallisrsquo and Renyirsquos do not admit the associativitydescribed above

In the sequel we present several results of the analysistaking into account the grouping criteria 119884

119903 119882119894 119877119895 119879119896 and

119860119897 Figure 12 depicts the Shannon entropy versus parameter

for all statistical distributions The black squares (denotedldquoSPLrdquo) correspond to the plot of versus the total entropyof the respective distributions 119878 The black circles (denotedldquoDPL1rdquo) are the plot of

1versus the entropy associated to the

first parts of the distributions 1198781 The white circles (denoted

ldquoDPL2rdquo) represent the plot of 2versus the entropy associated

8 Mathematical Problems in Engineering

1980 1985 1990 1995 2000 2005 2010

012345678910040608

1

Year

119902

119878(119877)119902

Figure 10 Total Renyi normalized entropy 119878(119877)119902 as a function of the

year of the events 119884119903 and parameter 119902

1980 1985 1990 1995 2000 2005 2010

012345678910

0402

0608

1

Year

119902

119878(119880)119902

Figure 11 Total Ubriaco normalized entropy 119878(119880)119902

as a function ofthe year of the events 119884

119903 and parameter 119902

to the second parts of the distributions 1198782 As can be seen for

the distributions that behave as single PLs higher entropiescorrespond to the lower values of the parameter and the twoparameters are linearly related A similar pattern is observedfor the parameters corresponding to DPL1 For DPL2 theparameter

2increases with entropy but the almost linear

relation between both parameters remainsFigure 13 shows identical results for the Ubriaco entropy

Theplot corresponds to 119902 = 05 nevertheless identical resultsare obtained for other values

For all DPLs related to terrorist events 119899 (119899 = 1 60)we find that the parameters (119862

119894 119894) 119894 = 1 2 obey the follow-

ing relation

1119899

10038161003816100381610038161003816log1198621119899|119901= 2119899

10038161003816100381610038161003816log1198622119899|119901+ 120598119899 (5)

where 119901 = minus1612 The mean value of 120598119899is 120598 = 0002 and the

corresponding standard deviation is 120590120598= 0036 Moreover

for the analyzed data we find 119894119899| log119862

119894119899|119901= 119896119894119899 Parameters

119896119894119899are approximately constant with mean value 119896 = 0277

and standard deviation 120590119896= 006

It is worth noticing that (5) is similar to Poissonrsquos law ofan adiabatic reversible process involving ideal gases given by

1198751119881120574

1= 1198752119881120574

2 (6)

where variables 119875119894and 119881

119894represent pressure and specific

volume respectively Equation (6) implies that 119875119881120574 = constParameter 120574 is called Poissonrsquos coefficient taking values 120574 =

53 ≃ 167 and 120574 = 75 = 140 for monoatomic and diatomicgases respectively Additionally it should be noticed that theabsolute value of the exponent 119901 is very similar to the oneobserved for an ideal gas undergoing a reversible adiabaticprocess

0

05

1

15

2

25

3

0 05 1 15 2 25Shannon entropy

SPLDPL1

DPL2

Figure 12 Parameters versus Shannon entropies for the dis-tributions corresponding to the criteria 119884

119903 119882119894 119877119895 119879119896 and 119860

119897

ldquoSPLrdquo corresponds to (119878 ) ldquoDPL1rdquo corresponds to (1198781 1) ldquoDPL2rdquo

corresponds to (1198782 2)

0

05

1

15

2

25

3

0 02 04 06 08 1 12 14 16

SPL DPL1DPL2

Ubriaco entropy (119902 = 05)

Figure 13 Parameters versus Ubriaco entropies for the distri-butions corresponding to the criteria 119884

119903 119882119894 119877119895 119879119896 and 119860

119897 ldquoSPLrdquo

corresponds to (119878(119880)119902 ) ldquoDPL1rdquo corresponds to (119878(119880)

1199021 1) ldquoDPL2rdquo

corresponds to (119878(119880)1199022 2) 119902 = 05

5 Conclusions

PLs have been widely reported in the modeling of distinctphenomena and have been associated with long memorybehavior self-similarity fractal structures and fractionalcalculus

In this paper we reviewed interesting and importantresults on PLs distributions and their applications to the

Mathematical Problems in Engineering 9

modeling of the number of victims in catastrophic eventsWe found double PL behavior in real data of catastrophicoccurrences in particular terrorismWe have plotted the twoPLs parameters (119862

119894 119894) 119894 = 1 2 corresponding to certain

events and observed an interesting pattern in the chartwhere the lines that connect each pair of points defining thedouble PLs are almost aligned to each other We have alsocomputed the entropy of the data sets This complementaryanalysis of the numerical data revealed extra relationships butthe fact is that these phenomena have a dense and rich volumeof characteristics and further research efforts are needed to adeeper understanding

References

[1] J Tenreiro Machado F B Duarte and G M Duarte ldquoPowerlaw analysis of financial index dynamicsrdquo Discrete Dynamics inNature and Society vol 2012 Article ID 120518 12 pages 2012

[2] J Tenreiro Machado ldquoAnd I say to myself lsquoWhat a fractionalworlrsquordquo Fractional Calculus and Applied Analysis vol 14 no 4pp 635ndash654 2011

[3] E Scalas R Gorenflo and FMainardi ldquoFractional calculus andcontinuous-time financerdquo Physica A vol 284 no 1ndash4 pp 376ndash384 2000

[4] F Mainardi M Raberto R Gorenflo and E Scalas ldquoFractionalcalculus and continuous-time finance II the waiting-time dis-tributionrdquo Physica A vol 287 no 3-4 pp 468ndash481 2000

[5] B J West ldquoFractal physiology and the fractional calculus aperspectiverdquo Frontiers in Physiology vol 1 p 12 2010

[6] B Gutenberg and R F Richter ldquoFrequency of earthquakes incaliforniardquo Bulletin of the Seismological Society of America vol34 pp 185ndash188 1944

[7] L F Richardson ldquoVariation of the Frequency of Fatal QuarrelswithMagnituderdquo Journal of the American Statistical Associationvol 43 pp 523ndash546 1948

[8] J M Carlson and J S Langer ldquoMechanical model of an earth-quake faultrdquo Physical Review A vol 40 no 11 pp 6470ndash64841989

[9] D C Roberts and D L Turcotte ldquoFractality and self-organizedcriticality of warsrdquo Fractals vol 6 no 4 pp 351ndash357 1998

[10] D R Davis and D E Weinstein ldquoBones bombs and breakpoints the geography of economic activityrdquo The AmericanEconomic Review vol 92 no 5 pp 1269ndash1289 2002

[11] L E Cederman ldquoModeling the size of wars from billiard ballsto sandpilesrdquo American Political Science Review vol 97 no 1pp 135ndash150 2003

[12] J C Bohorquez S Gourley A R Dixon M Spagat and N FJohnson ldquoCommon ecology quantifies human insurgencyrdquoNature vol 462 no 7275 pp 911ndash914 2009

[13] V Pareto Cours drsquoEconomie Politique Droz Geneva Switzer-land 1896

[14] F Auerbach ldquoDas Gesetz der BelvolkerungskoncentrationrdquoPetermanns GeographischeMitteilungen vol 59 pp 74ndash76 1913

[15] J B Estoup Gammes Stenographiques Institut de France 1916[16] G Zipf Selective Studies and the Principle of Relative Frequency

in Language Harvard University Press Cambridge Mass USA1932

[17] G Zipf Human Behavior and the Priciple of Least EffortAddison-Wesley Cambridge Mass USA 1949

[18] W Li ldquoReferences on Zipf rsquos lawrdquo httpwwwsorinsolomonnetsimsorinccszipfreferences20on20zipf27s20lawhtm

[19] D Sornette Critical Phenomena in Natural Sciences chapter 14Springer Heidelberg Germany 2nd edition 2003

[20] C M A Pinto A M Lopes and J A Tenreiro Machado ldquoAreview of power laws in real life phenomenardquo Communicationsin Nonlinear Science and Numerical Simulations vol 17 no 9pp 3558ndash3578 2012

[21] J Tuldava ldquoThe frequency spectrum of text and vocabularyrdquoJournal of Quantitative Linguistics vol 3 no 1 pp 38ndash50 1996

[22] GMartınez-Mekler R AlvarezMartınezM Beltran del Rıo RMansilla P Miramontes and G Cocho ldquoUniversality of rank-ordering distributions in the arts and sciencesrdquo PLoS One vol4 no 3 Article ID e4791 2009

[23] R F I Cancho and R V Sole ldquoTwo regimes in the frequency ofwords and the origins of complex lexicons Zipf rsquos law revisitedrdquoJournal of Quantitative Linguistics vol 8 no 3 pp 165ndash1732001

[24] R Alvarez-Martinez G Martinez-Mekler and G CochoaldquoOrder-disorder transition in conflicting dynamics leading torank-frequency generalized beta distributionsrdquo Physica A vol390 no 1 pp 120ndash130 2011

[25] L F Richardson Statistics of Deadly Quarrels QuadrangleBooks Chicago Ill USA 1960

[26] F Guzzetti ldquoLandslide fatalities and the evaluation of landsliderisk in Italyrdquo Engineering Geology vol 58 no 2 pp 89ndash1072000

[27] S N Jonkman ldquoGlobal perspectives on loss of human lifecaused by floodsrdquo Natural Hazards vol 34 no 2 pp 151ndash1752005

[28] N F Johnson M Spagat J A Restrepo et al ldquoUniversal pat-terns underlying ongoingwars and terrorismrdquo httparxivorgabsphysics0605035

[29] D S Geller and J David Singer Nations at War A ScientificStudy of International Conflict Cambridge Studies in Interna-tional Relations Cambridge University Press Cambridge UK1998

[30] O Becerra N Johnson P Meier J Restrepo and M Spa-gat ldquoNatural disasters casualties and power laws a com-parative analysis with armed conflictrdquo in Proceedings of theAnnual Meeting of the American Political Science AssociationLoews Philadelphia and the Pennsylvania Convention Cen-ter Philadelphia Pa USA 2006 httpwwwallacademiccommetap151714 indexhtml

[31] M E J Newman ldquoPower laws Pareto distributions and Zipf rsquoslawrdquo Contemporary Physics vol 46 pp 323ndash351 2005

[32] A Clauset andM Young ldquoScale invariance in global terrorismrdquohttparxivorgabsphysics0502014

[33] K T Bogen and E D Jones ldquoRisks of mortality and morbidityfrom worldwide terrorism 1968ndash2004rdquo Risk Analysis vol 26no 1 pp 45ndash59 2006

[34] A Clauset M Young and K S Gleditsch ldquoOn the frequency ofsevere terrorist eventsrdquo Journal of Conflict Resolution vol 51 no1 pp 58ndash87 2007

[35] EM-DAT ldquoTheOFDACRED International Disaster DatabaserdquoUniversite catholique de Louvain Brussels Belgium httpwwwemdatnet

[36] National Consortium for the Study of Terrorism and Responsesto Terrorism (START) ldquoGlobal Terrorism Databaserdquo 2011httpwwwstartumdedugtd

[37] R ClausiusTheMechanicalTheory of Heat with Its Applicationsto the Steam Engine and to Physical Properties of Bodies Johnvan Voorst London UK 1865

[38] L Boltzmann Vorlesungen ber Gastheorie J A Barth LeipzigGermany 1896

10 Mathematical Problems in Engineering

[39] C E Shannon ldquoAmathematical theory of communicationrdquoTheBell System Technical Journal vol 27 pp 379ndash423 623ndash6561948

[40] E T Jaynes ldquoInformation theory and statistical mechanicsrdquo vol106 pp 620ndash630 1957

[41] A I Khinchin Mathematical Foundations of Information The-ory Dover New York NY USA 1957

[42] A R Plastino M Casas and A Plastino ldquoA nonextensive max-imum entropy approach to a family of nonlinear reaction-diffusion equationsrdquo Physica A vol 280 no 3 pp 289ndash3032000

[43] T D Frank and A Daffertshofer ldquo119867-theorem for nonlinearFokker-Planck equations related to generalized thermostatis-ticsrdquo Physica A vol 295 no 3-4 pp 455ndash474 2001

[44] E K Lenzi R SMendes and L R da Silva ldquoStatisticalmechan-ics based onRenyi entropyrdquo Physica A vol 280 no 3-4 pp 337ndash345 2000

[45] A S Parvan andT S Biro ldquoExtensive Renyi statistics fromnon-extensive entropyrdquo Physics Letters A vol 340 no 5-6 pp 375ndash387 2005

[46] B D Sharma and D P Mittal ldquoNew nonadditive measures ofentropy for discrete probability distributionsrdquo Journal of Math-ematical Sciences vol 10 pp 28ndash40 1975

[47] T D Frank and A Daffertshofer ldquoExact time-dependentsolutions of the Renyi Fokker-Planck equation and the Fokker-Planck equations related to the entropies proposed by Sharmaand Mittalrdquo Physica A vol 285 no 3 pp 351ndash366 2000

[48] M R Ubriaco ldquoEntropies based on fractional calculusrdquo PhysicsLetters A vol 373 no 30 pp 2516ndash2519 2009

[49] P T Landsberg and V Vedral ldquoDistributions and channelcapacities in generalized statistical mechanicsrdquo Physics LettersA vol 247 no 3 pp 211ndash217 1998

[50] A Plastino and A R Plastino ldquoTsallis Entropy and Jaynesrsquoinformation theory formalismrdquo Brazilian Journal of Physics vol29 no 1 pp 50ndash60 1999

[51] X Li C Essex M Davison K H Hoffmann and C SchulzkyldquoFractional diffusion irreversibility and entropyrdquo Journal ofNon-Equilibrium Thermodynamics vol 28 no 3 pp 279ndash2912003

[52] H J Haubold A M Mathai and R K Saxena ldquoBoltzmann-Gibbs entropy versus Tsallis entropy recent contributions toresolving the argument of Einstein concerning ldquoneither HerrBoltzmann nor Herr Planck has given a definition ofWrdquo Essayreviewrdquo Astrophysics and Space Science vol 290 no 3-4 pp241ndash245 2004

[53] P N Rathie and S da Silva ldquoShannon Levy and Tsallis a noterdquoApplied Mathematical Sciences vol 2 no 28 pp 1359ndash13632008

[54] R M Gray Entropy and Information Theory Springer NewYork NY USA 2009

[55] J A Tenreiro Machado A C Costa and M F M Lima ldquoDy-namical analysis of compositionsrdquoNonlinear Dynamics vol 65no 4 pp 339ndash412 2011

[56] J A T Machado A C Costa and M D Quelhas ldquoEntropyanalysis of the DNA code dynamics in human chromosomesrdquoComputers amp Mathematics with Applications vol 62 no 3 pp1612ndash1617 2011

[57] J A TenreiroMachado A C Costa andM D Quelhas ldquoShan-non Renyie and Tsallis entropy analysis of DNA using phaseplanerdquo Nonlinear Analysis Real World Applications vol 12 no6 pp 3135ndash3144 2011

2 Mathematical Problems in Engineering

the Pareto distribution the behavior is exactly linear and isgiven by

ln (119875 [119883 ge 119909]) = ln119862 minus ln minus ln 119909 (1)

where 119883 is a random variable following a PL distribution gt 0 119862 = 119862 gt 0 In these distributions the tail fallsasymptotically according to the value of translating inheavy tails comparatively to other distributions Zipf rsquos lawis a special case of the Paretorsquos law with coefficient = 1Relevant reviews on PL distributions can be found in [18ndash20]

In many cases a single PL holds for the entire range of therandom variable that represents the system In other casesthe statistical distribution is better described by multiple PLs[21] In such cases different PLs characterized by distinctPL parameters fit more adequately the real data Double PLbehaviors have been pointed out by others in different phe-nomena For example in [22] many instances are shown oftwo PLs expressed bymeans of a generalized beta distributionfunction and in [23] the double PL behavior is explicitlystudied in the frequency of words in texts Moreover beyondthe ranking problem two PLs are manifested in other type ofproblems such as in turbulence earth magnetic pole fluctua-tions paleolake sedimentation density subject to volcanismand avalanche distributions to mention a few [24]

In this paper we analyze the statistical distributions ofthe number of victims caused by catastrophic phenomenaand find double PL behavior Moreover we plot the PLparameters corresponding to several events and observe aninteresting pattern in the charts where the lines that connecteach pair of points defining the double PLs are roughlyparallel to each other Then a complementary data analysisis performed by means of the computation of the Shannonentropy The results reveal relationships hidden in the datathat may trigger a future comprehensive explanation of thistype of phenomena

Bearing these ideas in mind this paper is organizedas follows In Section 2 the results found in the literatureconcerning PLs and casualties in natural and human-madedisasters are summarized Section 3 analyses the PL behaviorof catastrophic phenomena using data from real disasters InSection 4 several entropy measures are used to characterizethe data Finally in Section 5 the main results and conclu-sions of this paper are discussed

2 Brief Review of PLs inCatastrophic Occurrences

Patterns seen in wars terrorist attacks tornadoes earth-quakes landslides floods and other severe occurrences havebeen at close attention by various researchers [6ndash12 25ndash28]Many attentive explanations have arisen in the literatureNevertheless a complete understanding of these patternsis a complex task Important and intricate political geo-graphical historical and even cultural factors oppose to

a better understanding Predicting the number of casualtiesin natural or human-made disasters is extremely important indeveloping predisaster strategies Aspects like rationalizationof medical supplies and food gathering emergency teamsorganize shelter spaces amongst others have to be dealt within order to minimize the damage

A PL behavior is indicative of a particular property ofa system it indicates that the size of an event is inverselyproportional to its frequency In this sense large casualtiesare associated with low frequency phenomena and morefrequent events are less harmful in terms of preserving humanlives [7 25] Examples of phenomena with low probabilityand huge casualties are the twoworldwars (WWs) highmag-nitude earthquakes strong tornadoes huge tsunamis andamongst others

In 1948 Richardson [7] analyzed domestic and inter-national cases of violence in the period from 1820 to 1945He distributed the cases according to casualties measuredin powers of 10 into five categories The two WWs wereclassified in the highest category In a later work [25] thesame author showed that if the frequency of an occurrencedecreased by a factor close to three then the number of casu-alties increased by a power of 10

Guzzetti [26] considers landslide events in specific peri-ods in different countries such as Italy Canada Alps HongKong Japan and China He shows that the plot of thecumulative distribution function of the number of landslideevents versus the number of casualties is well approximatedby a straight line This result suggests a PL distribution of thedata

Cederman [11] followed Richardsonrsquos work [7 25] Heused data from the Correlates of War (COW) Project [29]focusing on interstate wars He computed the cumulativerelative frequency of war size and showed that it obeyed aPL The author proposed a self-organized critical dynamicalsystem that replicated the PL behavior seen in real data Itsmodel allowed conflict to spread and diffuse potentially overlong periods of time due to the quasi-parallel execution

In 2005 Jonkman [27] studied the distribution of killingsin global events focusing on the number of human deathscaused by three types of floods (river floods flash floods anddrainage issues) between January 1975 and June 2002 Theauthor plotted the global frequency of events with119873 or moredeaths versus119873 He observed a PL behavior for earthquakesbut not for flood data Becerra et al [30] use the same dataset as Jonkman [27] but consider all disasters combinedboth globally and disaggregated by continent They obtainedstraight-line log-log plots for all disasters combined Theslopes of the casualties PL distributions were smaller thanthose for modern wars and terrorism The explanation forthis remained an open question Another unsolved issue wasthe existence of PL behavior in combined disasters and not inindividual disasters such as floods Here it is worth mention-ing that casualties in earthquakes verified a PL distribution[6 27 30]

Johnson et al [28] suggested a microscopic theory toexplain similarity in patterns of violence such as war andglobal terrorism The similarity was observed regardless of

Mathematical Problems in Engineering 3

underlying ideologies motivations and the terrain in whichevents occurred The authors introduced a model where theinsurgent force behaved as a self-organizing system whichevolved dynamically through the continual coalescence andfragmentation of its constituent groups They analyzed casu-altiesrsquo patterns arising within a given war unlike previousstudies that focused on the total casualty figure for oneparticular war [7 11 25 31] A PL behavior fitted well the datanot only from Iraq Colombia and non-G7 terrorism butalso with data obtained from the war in Afghanistan The PLparameter for Iraq Colombia andAfghanistan was (close to) = 25 This value of the coefficient equalized the coefficientvalue characterizing non-G7 terrorism In the literature thePL parameter value was = 251 for non-G7 countries [32]and = 1713 for G7 countries This result suggested that PLpatterns would emerge within any modern asymmetric warfought by loosely-organized insurgent groups

In 2006 Bogen and Jones [33] treated the severity of ter-rorist attacks in terms of deaths and injuredThey applied a PLdistribution to victimevent rates and used the PL to predictmortality due to terrorism through the year 2080 Authorsclaimed that these PL models could be used to improvestrategies ldquoto assess prevent and manage terror-related risksand consequencesrdquo

Clauset et al [34] studied the frequency and the numberof casualties (deaths and injuries) of terrorist attacks since1968 They observed a scale-invariance behavior with thefrequency being an inverse power of the casualties Thisbehavior was independent of the type of weapon economicdevelopment and distinct time scales The authors presenteda new model to fit the frequency of severe terrorist attackssince previous models in the literature failed to produce theheavy tail in the PL distribution Their model assumed thatthe severity of an occurrence was a function of the executionplan and that selection tools were better suited to modelcompetition between states and nonstate actors Finally re-searchers claimed that periodicity was a common feature inglobal terrorism with period close to roughly 13 years

Bohorquez et al [12] studied the quantitative relationbetween human insurgency global terrorism and ecologyThey introduced a newmodel to explain the size distributionof casualties or the timing ofwithin-conflict eventsThey con-sidered insurgent populations as self-organized groups thatdynamically evolved through decision-making processesThe main assumptions of the model were (i) being consistentwith work on human group dynamics in everyday environ-ments (ii) having a new perception of modern insurgenciesas fragmented transient and evolving and (iii) using adecision-making process about when to attack based oncompetition for media attention Authors applied a PL distri-bution to Iraq andColombiawars with parameter value closeto = 25 A coefficient value of = 25 was in concordancewith the coefficient value of = 248 plusmn 007 obtained byClauset et al [34] on global terrorism A PL fit to Spanish andAmerican Civil wars revealed a PL parameter value smaller(around = 17) Authors claimed that theirmodel suggesteda remarkable link between violent and nonviolent humanactions due to its similarity to financial market models

Rank

max

rank

Sizemax size10minus3 10minus2 10minus1 10010minus2

10minus1

100

ln(119910) = minus085503 ln(119909) minus 474389

Figure 1 Rankfrequency log-log plot corresponding to the distri-bution of casualties caused by industrial accidents in CentralSouthAmerica over the period 1900ndash2011 (min size = 10 max size = 2700max rank = 66)

3 Power Law Behavior inCatastrophic Phenomena

In this section we investigate the statistical distributionsof random variables that represent the number of humancasualties in several human-made and natural hazards

Data from the EM-DAT International Disaster Database(httpwwwemdatbe) and the Global Terrorism Database(GTD) (httpwwwstartumdedugtd) are analyzed TheEM-DAT database contains information on over than 18000worldwide natural and technological disasters from 1900to present The EM-DAT is maintained by the Centre forResearch on the Epidemiology of Disasters (CRED) at theSchool of Public Health of the Universite Catholique deLouvain located inBrussels Belgium [35]TheGTDdatabaseis an open-source database that includes information onmorethan 98000 worldwide terrorist attacks from 1970 up to 2010[31]

PLs are observed in several natural and man-madesystems Examples of single and double PLs in real data aregiven in Figures 1 and 2 respectively The former representsthe complementary cumulative distribution of the severityof industrial accidents in CentralSouth America over theperiod 1900ndash2011 The adopted measure to quantify theseverity of an event is the total number of fatalities Thedepicted graph corresponds to a rankfrequency log-log plotTo construct the graph we first sort the data (ie theaccidents) in decreasing order according to their severity andnumber them consecutively starting from one [36] Then anormalization of the values is carried out meaning that thenumber of fatalities (119909-axis) is divided by the correspondinghighest value and the rank (119910-axis) is divided by the rank ofthe smallest event Finally PLs are adjusted to the data using aleast squares algorithm All the log-log plots presented in thispaper are made following this procedure

Figure 2 corresponds to the distribution of casual-ties caused by earthquakes in CentralSouth America in

4 Mathematical Problems in EngineeringRa

nkm

ax ra

nk

Sizemax size

ln(119910) = minus0246959 ln(119909) minus 299476

ln(119910) = minus0499454 ln(119909) minus 492208

10minus4 10minus310minus6 10minus5 10minus2 10minus1 10010minus3

10minus2

10minus1

100

Figure 2 Rankfrequency log-log plot corresponding to the dis-tribution of casualties caused by earthquakes in CentralSouthAmerica over the period 1900ndash2011 (min size = 1 max size = 222570max rank = 179)

the period 1900ndash2011 representing one event that can beapproximated by a double PL

As can be seen in Figure 1 a single PL (SPL) with para-meters (119862 ) = (00087 08550) fits to the data The dis-tribution depicted in Figure 2 is better approximated by adouble PL (DPL) with parameters (119862

1 1) = (00500 02470)

and (1198622 2) = (00073 04995) The change in the behavior

occurs at the relative value of 119909 = 0000539 approximatelyWe analyzed the data available at the EM-DAT database

in terms of disaster type (DT119895) and disaster location (DL

119896)

119895 = 1 11 and 119896 = 1 6 categories respectivelyDT119895= Drought Earthquake Epidemic Extreme tempera-

ture Flood Industrial accident Mass movement wet StormTransport accident VolcanoWildfire DL

119896= Africa North

America Central amp South America Europe (including Rus-sia) Asia (not including SE Asia) Oceania (including SEAsia) The period of analysis was 1900ndash2011 for every caseThe total number of combinations (locationtype) is 11 times6 Nevertheless for 14 cases there is insufficient data tocompute reliable statistical distributions For all cases takingthe number of casualties as the variable of interest we obtainstatistical distributions that can be approximated by either aSPL (16 cases) or aDPL (36 cases) similar to the ones depictedin Figures 1 and 2

In Figure 3 we depict the locus of the parameters(119862119894 119894) 119894 = 1 2 corresponding to the analyzed cases As can

be seen an interesting pattern emerges where the linesthat connect the pairs of points that characterize the DPLshave identical orientation This geometrical pattern reflects arelationship between the two parts of the DPL distributions(DPL1mdashpart closer to the head and DPL2mdashpart closer to thetail) Besides the observation that

2gt 1 in all cases further

investigation on the reason for this behavior is neededWe pursued our study with the analysis of the GTD

database First the events associated to human casualtieswere grouped by year (119884

119903) starting in 1980 up to 2010

0

02

04

06

08

1

12

14

10minus4 10minus3 10minus2 10minus1 100

119862

Figure 3 Locus of the parameters (119862119894 119894) 119894 = 1 2 that characterize

the PLs corresponding to the number of casualties in certaincombinations of disaster typelocation DT

119895DL119896

(except 1993 because there is no data available) 119884119903

=

1980 2010 1993 119903 = 1 30 We found that allthe statistical distributions can be approximated by DPLs InFigures 4 and 5 the time evolution of the parameters of theDPLs (119862

119894 119894) 119894 = 1 2 is shown Regarding the parameters

119862119894 it can be seen that they have identical behavior although

1198622varies more than 119862

1and is always smaller than it

With respect to 119894 we have a similar evolution but in this

case the parameter 2is always greater than

1 As severe

terrorist attacks correspond to points closer to the tail of thedistribution DPL2 which is characterized by a larger thismeans that those events are more similar between each otherthan the smaller events (that correspond to DPL1)

To complement the analysis with respect to the date of theoccurrences the parameters (119862

119894 119894) of the PLs correspond-

ing to 119884119903 were plotted (Figure 6) As can be seen a pattern

similar to the described previously (Figure 3) is observedWe have also studied the distributions of the casualties in

terrorist attacks occurred in the period 1970ndash2010 but withrespect to other criteria namely the type of used weapon(119882119894) region where the event took place (119877

119895) target (119879

119896)

and type of attack (119860119897) Each criterion was then divided

into 119894 = 1 6 119895 = 1 13 119896 = 1 19 and 119897 =

1 8 categories respectively119882119894= Chemical Explosives

Firearms Incendiary Melee Vehicle 119877119895= Australasia amp

Oceania Central America amp Caribbean Central Asia EastAsia Eastern Europe Middle East amp North Africa NorthAmerica South America South Asia Southeast Asia Sub-Saharan Africa USSR amp Newly Independent States (NIS)Western Europe 119879

119896= Airports amp Airlines Business

Educational Institution Food or Water Supply Govern-ment (Diplomatic) Government (General) Journalists ampMedia Maritime Military NGO Police Private Citizensamp Property Religious FiguresInstitutions Telecommuni-cation Terrorists Tourists Transportation Utilities Vio-lent Political Party 119860

119897= Armed Assault Assassination

Mathematical Problems in Engineering 5

1980 1990 2000 2010Year

11986211198622

10minus4

10minus3

10minus2

10minus1

119862

Figure 4 Time evolution of parameters 119862119894 119894 = 1 2 of the DPLs

corresponding to terrorist attacks over the period 1980ndash2010 119884119903

1980 1990 2000 2010Year

0

05

1

15

2

25

3

12057211205722

Figure 5 Time evolution of parameters 119894 119894 = 1 2 of the DPLs

corresponding to terrorist attacks over the period 1980ndash2010 119884119903

BombingExplosion FacilityInfrastructure Attack Hijack-ing Hostage Taking (Barricade Incident) Hostage Taking(Kidnapping) and Unarmed Assault

Most cases are characterized by DPLs However in afew situations a SPL fits better to the data The main resultsare summarized in Table 1 Moreover we observed that theparameters corresponding to all distributions characterizedby DPLs display a pattern similar to the ones mentionedpreviously (Figures 3 and 6) where the lines connecting theslopes and intercepts of DPL1 (

1 1198621) to its companionDPL2

(2 1198622) for the same data set have identical orientation in

the (119862 ) Cartesian space

4 Entropy of Catastrophic Phenomena

In this section we analyse the entropy of data collected fromthe GTD database that is data related to terrorism To

1980 1981 1982 1983 1984 19851986 1987 1988 1989 1990 19911992 1994 1995 1996 1997 19981999 2000 2001 2002 2003 20042005 2006 2007 2008 2009 2010

0

05

1

15

2

25

3

10minus4 10minus3 10minus2 10minus1

119862

Figure 6 Locus of the parameters (119862119894 ) 119894 = 1 2 that characterize

the distributions of terrorist attacks over the period 1980ndash2010 119884119903

calculate the entropies we construct histograms of relativefrequencies using bins of width one (one casualty) andapproximate the probabilities 119901

119894by the relative frequencies

We present results obtained for terrorist events grouped byyear (119884

119903) as defined in the previous section Nevertheless it

should be noticed that similar results are obtained for all otherhuman-made and natural hazards

Clausius [37] and Boltzmann [38] were the first authorsto define entropy in the field of thermodynamics Later onShannon [39] and Jaynes [40] applied their results to infor-mation theory [41]

The most celebrated entropy is the so-called Shannonentropy 119878 defined by

119878 = minus

119882

sum

119894=1

119901119894ln119901119894 (2)

The Shannon entropy represents the expected value of theinformation minus ln119901

119894 Therefore for the uniform probability

distribution we have 119901119894= 119882minus1 and the Shannon entropy

takes its maximum value 119878 = ln119882 yielding the Boltzmannrsquosfamous formula up to a multiplicative factor 119896 denoting theBoltzmann constant Thus in thermodynamic equilibriumthe Shannon entropy can be identified as the ldquophysicalentropyrdquo of the system

Renyi and Tsallis entropies are generalizations of Shan-nonrsquos entropy and are given by respectively

119878(119877)

119902=

1

1 minus 119902

ln(119882

sum

119894=1

119901119902

119894) 119902 gt 0

119878(119879)

119902=

1

119902 minus 1

(1 minus

119882

sum

119894=1

119901119902

119894)

(3)

6 Mathematical Problems in Engineering

Table 1 PL fit to the distributions of casualties in terrorist attacks

Criterion Category SPL DPL1 DPL2(119862 ) (119862

1 1) (119862

2 2)

Weapon (119882119894)

1198821

00391 058491198822

00082 07514 00002 171981198823

00017 08660 00000 196241198824

00236 06877 00021 148901198825

00057 06909 00003 135791198826

00249 07801

Region (119877119894)

1198771

00438 101471198772

00214 06412 00009 169051198773

00174 119051198774

00065 090731198775

00012 121411198776

00091 08051 00006 170881198777

00009 114151198778

00193 07814 00008 206751198779

00036 08742 00001 2055211987710

00128 08561 00012 1657811987711

00091 06454 00004 1313211987712

00023 0995611987713

00001 15523

Target (119879119894)

1198791

00438 05144 00092 124881198792

00057 08884 00003 171801198793

00002 13589 00012 084581198794

00452 079231198795

00015 112591198796

00006 11152 00001 151621198797

00014 161061198798

00576 04694 00094 090701198799

00353 05669 00009 1629111987910

00090 09645 00031 1257311987911

00047 08991 00001 1835411987912

00036 07778 00000 1866711987913

00158 07353 00018 1610611987914

00462 0977411987915

00168 08610 00026 1682411987916

00234 0763511987917

00340 05892 00014 1517811987918

00225 07389 00074 1070811987919

00108 07918 00025 11650

Type (119860119894)

1198601

00049 07346 00000 179821198602

00014 15201 00001 265551198603

00059 08116 00001 206211198604

00261 06339 00016 148801198605

00305 071861198606

00116 073161198607

00072 09689 00014 162141198608

00471 07006 00034 20186

Mathematical Problems in Engineering 7

1980 1985 1990 1995 2000 2005 201005

05506

06507

07508

08509

0951

Year

Nor

mal

ized

entro

py

119878(119879)05

119878(119877)05

119878(119880)05

119878(119879)2

119878(119877)2

119878(119880)2

119878

Figure 7 Total Shannon Tsallis Renyi and Ubriaco normalizedentropies as a function of the year of the events 119884

119903 119902 = 05 and

119902 = 2

Tsallis entropy reduces to Renyi entropy when 119902 rarr 1 Tsallisentropy was applied to diffusion equations [42] and Fokker-Planck systems [43] Renyi entropy has an inverse power lawequilibrium distribution [44] and satisfies the zeroth law ofthermodynamics [45] The two parameters Sharma-Mittalentropy [46] is accepted as a generalization of Tsallis Renyiand Boltzmann-Gibbs entropies for limiting cases of theparameters [47]

Recently more general entropy measures have been pro-posed in the literature where the additivity axiom has beenrelaxed For instance Ubriaco [48] proposed the followingformula for the fractional entropy

119878(119880)

119902=

119882

sum

119894=1

(minus ln119901119894)119902

119901119894

(4)

that has the same properties as the Shannon entropy exceptadditivity

Applications of entropy in distinct complex systems canbe found in [49ndash57]

In Figure 7 the total Shannon normalized entropy (119878) isdepicted as well as Tsallisrsquo (119878(119879)

119902) Renyirsquos (119878(119877)

119902) and Ubriacorsquos

(119878(119880)

119902) for 119902 = 05 and 119902 = 2 as a function of the year of

the events 119884119903 Figures 9 10 and 11 show 119878

(119879)

119902 119878(119877)119902

and 119878(119880)119902

normalized entropies as a function of the year 119884119903 and

entropy parameter 01 le 119902 le 10In Figure 7 we observe two types of behavior namely

short- and long-term phenomena In what concerns shorttime behavior we verify peaks during 1983ndash1985 1997-19982004ndash2007 and minima at 1980 1995 and 2010 In whatconcerns long-time relationships a smooth decreasing isobserved for 119878(119879)

05 119878(119877)05

and 119878(119880)2

Removing the maxima andminima we get a time series for years 1981 1982 1986ndash1992 1994ndash1996 1999 2000 2002 2003 2008 and 2009(Figure 8) Largersmaller entropies correspond to chartscloserafar uniform distributions therefore seemingly wehave lessmore organized terrorist events in global terms

1980 1985 1990 1995 2000 2005 2010075

08

085

09

095

1

Year

Nor

mal

ized

entro

py

119878(119879)05

119878(119877)05

119878(119880)2

Lin Reg 119878(119879)05Lin Reg 119878(119877)05Lin Reg 119878(119880)2

Figure 8 Total Tsallis and Renyi normalized entropies (119902 = 05)

and Ubriacorsquos (119902 = 2) as a function of the events in years 1981 19821986ndash1992 1994ndash1996 1999 2000 2002 2003 2008 2009

1980 1985 1990 1995 2000 2005 2010

01234567

8910040506070809

1

Year

119902

119878(119879)119902

Figure 9 Total Tsallis normalized entropy 119878(119879)119902 as a function of the

year of the events 119884119903 and parameter 119902

These conclusions remain invariant for Figures 9 to 11 wherewe vary both the entropy definition and the parameter tuningTherefore we conclude that such results are robust againstsuch type of variations

As discussed in the previous section the statistical dis-tributions of real data can be approximated by either singleor double PLs In the latter case we study not only the totalentropy but also the entropy associated to each part of thedistributions Therefore we compute the entropy associatedto DPL1 and DPL2 that approximate the first and secondpart of the distributions respectively When adopting thisprocedure we are restricted to the Shannon and Ubriacoentropies as Tsallisrsquo and Renyirsquos do not admit the associativitydescribed above

In the sequel we present several results of the analysistaking into account the grouping criteria 119884

119903 119882119894 119877119895 119879119896 and

119860119897 Figure 12 depicts the Shannon entropy versus parameter

for all statistical distributions The black squares (denotedldquoSPLrdquo) correspond to the plot of versus the total entropyof the respective distributions 119878 The black circles (denotedldquoDPL1rdquo) are the plot of

1versus the entropy associated to the

first parts of the distributions 1198781 The white circles (denoted

ldquoDPL2rdquo) represent the plot of 2versus the entropy associated

8 Mathematical Problems in Engineering

1980 1985 1990 1995 2000 2005 2010

012345678910040608

1

Year

119902

119878(119877)119902

Figure 10 Total Renyi normalized entropy 119878(119877)119902 as a function of the

year of the events 119884119903 and parameter 119902

1980 1985 1990 1995 2000 2005 2010

012345678910

0402

0608

1

Year

119902

119878(119880)119902

Figure 11 Total Ubriaco normalized entropy 119878(119880)119902

as a function ofthe year of the events 119884

119903 and parameter 119902

to the second parts of the distributions 1198782 As can be seen for

the distributions that behave as single PLs higher entropiescorrespond to the lower values of the parameter and the twoparameters are linearly related A similar pattern is observedfor the parameters corresponding to DPL1 For DPL2 theparameter

2increases with entropy but the almost linear

relation between both parameters remainsFigure 13 shows identical results for the Ubriaco entropy

Theplot corresponds to 119902 = 05 nevertheless identical resultsare obtained for other values

For all DPLs related to terrorist events 119899 (119899 = 1 60)we find that the parameters (119862

119894 119894) 119894 = 1 2 obey the follow-

ing relation

1119899

10038161003816100381610038161003816log1198621119899|119901= 2119899

10038161003816100381610038161003816log1198622119899|119901+ 120598119899 (5)

where 119901 = minus1612 The mean value of 120598119899is 120598 = 0002 and the

corresponding standard deviation is 120590120598= 0036 Moreover

for the analyzed data we find 119894119899| log119862

119894119899|119901= 119896119894119899 Parameters

119896119894119899are approximately constant with mean value 119896 = 0277

and standard deviation 120590119896= 006

It is worth noticing that (5) is similar to Poissonrsquos law ofan adiabatic reversible process involving ideal gases given by

1198751119881120574

1= 1198752119881120574

2 (6)

where variables 119875119894and 119881

119894represent pressure and specific

volume respectively Equation (6) implies that 119875119881120574 = constParameter 120574 is called Poissonrsquos coefficient taking values 120574 =

53 ≃ 167 and 120574 = 75 = 140 for monoatomic and diatomicgases respectively Additionally it should be noticed that theabsolute value of the exponent 119901 is very similar to the oneobserved for an ideal gas undergoing a reversible adiabaticprocess

0

05

1

15

2

25

3

0 05 1 15 2 25Shannon entropy

SPLDPL1

DPL2

Figure 12 Parameters versus Shannon entropies for the dis-tributions corresponding to the criteria 119884

119903 119882119894 119877119895 119879119896 and 119860

119897

ldquoSPLrdquo corresponds to (119878 ) ldquoDPL1rdquo corresponds to (1198781 1) ldquoDPL2rdquo

corresponds to (1198782 2)

0

05

1

15

2

25

3

0 02 04 06 08 1 12 14 16

SPL DPL1DPL2

Ubriaco entropy (119902 = 05)

Figure 13 Parameters versus Ubriaco entropies for the distri-butions corresponding to the criteria 119884

119903 119882119894 119877119895 119879119896 and 119860

119897 ldquoSPLrdquo

corresponds to (119878(119880)119902 ) ldquoDPL1rdquo corresponds to (119878(119880)

1199021 1) ldquoDPL2rdquo

corresponds to (119878(119880)1199022 2) 119902 = 05

5 Conclusions

PLs have been widely reported in the modeling of distinctphenomena and have been associated with long memorybehavior self-similarity fractal structures and fractionalcalculus

In this paper we reviewed interesting and importantresults on PLs distributions and their applications to the

Mathematical Problems in Engineering 9

modeling of the number of victims in catastrophic eventsWe found double PL behavior in real data of catastrophicoccurrences in particular terrorismWe have plotted the twoPLs parameters (119862

119894 119894) 119894 = 1 2 corresponding to certain

events and observed an interesting pattern in the chartwhere the lines that connect each pair of points defining thedouble PLs are almost aligned to each other We have alsocomputed the entropy of the data sets This complementaryanalysis of the numerical data revealed extra relationships butthe fact is that these phenomena have a dense and rich volumeof characteristics and further research efforts are needed to adeeper understanding

References

[1] J Tenreiro Machado F B Duarte and G M Duarte ldquoPowerlaw analysis of financial index dynamicsrdquo Discrete Dynamics inNature and Society vol 2012 Article ID 120518 12 pages 2012

[2] J Tenreiro Machado ldquoAnd I say to myself lsquoWhat a fractionalworlrsquordquo Fractional Calculus and Applied Analysis vol 14 no 4pp 635ndash654 2011

[3] E Scalas R Gorenflo and FMainardi ldquoFractional calculus andcontinuous-time financerdquo Physica A vol 284 no 1ndash4 pp 376ndash384 2000

[4] F Mainardi M Raberto R Gorenflo and E Scalas ldquoFractionalcalculus and continuous-time finance II the waiting-time dis-tributionrdquo Physica A vol 287 no 3-4 pp 468ndash481 2000

[5] B J West ldquoFractal physiology and the fractional calculus aperspectiverdquo Frontiers in Physiology vol 1 p 12 2010

[6] B Gutenberg and R F Richter ldquoFrequency of earthquakes incaliforniardquo Bulletin of the Seismological Society of America vol34 pp 185ndash188 1944

[7] L F Richardson ldquoVariation of the Frequency of Fatal QuarrelswithMagnituderdquo Journal of the American Statistical Associationvol 43 pp 523ndash546 1948

[8] J M Carlson and J S Langer ldquoMechanical model of an earth-quake faultrdquo Physical Review A vol 40 no 11 pp 6470ndash64841989

[9] D C Roberts and D L Turcotte ldquoFractality and self-organizedcriticality of warsrdquo Fractals vol 6 no 4 pp 351ndash357 1998

[10] D R Davis and D E Weinstein ldquoBones bombs and breakpoints the geography of economic activityrdquo The AmericanEconomic Review vol 92 no 5 pp 1269ndash1289 2002

[11] L E Cederman ldquoModeling the size of wars from billiard ballsto sandpilesrdquo American Political Science Review vol 97 no 1pp 135ndash150 2003

[12] J C Bohorquez S Gourley A R Dixon M Spagat and N FJohnson ldquoCommon ecology quantifies human insurgencyrdquoNature vol 462 no 7275 pp 911ndash914 2009

[13] V Pareto Cours drsquoEconomie Politique Droz Geneva Switzer-land 1896

[14] F Auerbach ldquoDas Gesetz der BelvolkerungskoncentrationrdquoPetermanns GeographischeMitteilungen vol 59 pp 74ndash76 1913

[15] J B Estoup Gammes Stenographiques Institut de France 1916[16] G Zipf Selective Studies and the Principle of Relative Frequency

in Language Harvard University Press Cambridge Mass USA1932

[17] G Zipf Human Behavior and the Priciple of Least EffortAddison-Wesley Cambridge Mass USA 1949

[18] W Li ldquoReferences on Zipf rsquos lawrdquo httpwwwsorinsolomonnetsimsorinccszipfreferences20on20zipf27s20lawhtm

[19] D Sornette Critical Phenomena in Natural Sciences chapter 14Springer Heidelberg Germany 2nd edition 2003

[20] C M A Pinto A M Lopes and J A Tenreiro Machado ldquoAreview of power laws in real life phenomenardquo Communicationsin Nonlinear Science and Numerical Simulations vol 17 no 9pp 3558ndash3578 2012

[21] J Tuldava ldquoThe frequency spectrum of text and vocabularyrdquoJournal of Quantitative Linguistics vol 3 no 1 pp 38ndash50 1996

[22] GMartınez-Mekler R AlvarezMartınezM Beltran del Rıo RMansilla P Miramontes and G Cocho ldquoUniversality of rank-ordering distributions in the arts and sciencesrdquo PLoS One vol4 no 3 Article ID e4791 2009

[23] R F I Cancho and R V Sole ldquoTwo regimes in the frequency ofwords and the origins of complex lexicons Zipf rsquos law revisitedrdquoJournal of Quantitative Linguistics vol 8 no 3 pp 165ndash1732001

[24] R Alvarez-Martinez G Martinez-Mekler and G CochoaldquoOrder-disorder transition in conflicting dynamics leading torank-frequency generalized beta distributionsrdquo Physica A vol390 no 1 pp 120ndash130 2011

[25] L F Richardson Statistics of Deadly Quarrels QuadrangleBooks Chicago Ill USA 1960

[26] F Guzzetti ldquoLandslide fatalities and the evaluation of landsliderisk in Italyrdquo Engineering Geology vol 58 no 2 pp 89ndash1072000

[27] S N Jonkman ldquoGlobal perspectives on loss of human lifecaused by floodsrdquo Natural Hazards vol 34 no 2 pp 151ndash1752005

[28] N F Johnson M Spagat J A Restrepo et al ldquoUniversal pat-terns underlying ongoingwars and terrorismrdquo httparxivorgabsphysics0605035

[29] D S Geller and J David Singer Nations at War A ScientificStudy of International Conflict Cambridge Studies in Interna-tional Relations Cambridge University Press Cambridge UK1998

[30] O Becerra N Johnson P Meier J Restrepo and M Spa-gat ldquoNatural disasters casualties and power laws a com-parative analysis with armed conflictrdquo in Proceedings of theAnnual Meeting of the American Political Science AssociationLoews Philadelphia and the Pennsylvania Convention Cen-ter Philadelphia Pa USA 2006 httpwwwallacademiccommetap151714 indexhtml

[31] M E J Newman ldquoPower laws Pareto distributions and Zipf rsquoslawrdquo Contemporary Physics vol 46 pp 323ndash351 2005

[32] A Clauset andM Young ldquoScale invariance in global terrorismrdquohttparxivorgabsphysics0502014

[33] K T Bogen and E D Jones ldquoRisks of mortality and morbidityfrom worldwide terrorism 1968ndash2004rdquo Risk Analysis vol 26no 1 pp 45ndash59 2006

[34] A Clauset M Young and K S Gleditsch ldquoOn the frequency ofsevere terrorist eventsrdquo Journal of Conflict Resolution vol 51 no1 pp 58ndash87 2007

[35] EM-DAT ldquoTheOFDACRED International Disaster DatabaserdquoUniversite catholique de Louvain Brussels Belgium httpwwwemdatnet

[36] National Consortium for the Study of Terrorism and Responsesto Terrorism (START) ldquoGlobal Terrorism Databaserdquo 2011httpwwwstartumdedugtd

[37] R ClausiusTheMechanicalTheory of Heat with Its Applicationsto the Steam Engine and to Physical Properties of Bodies Johnvan Voorst London UK 1865

[38] L Boltzmann Vorlesungen ber Gastheorie J A Barth LeipzigGermany 1896

10 Mathematical Problems in Engineering

[39] C E Shannon ldquoAmathematical theory of communicationrdquoTheBell System Technical Journal vol 27 pp 379ndash423 623ndash6561948

[40] E T Jaynes ldquoInformation theory and statistical mechanicsrdquo vol106 pp 620ndash630 1957

[41] A I Khinchin Mathematical Foundations of Information The-ory Dover New York NY USA 1957

[42] A R Plastino M Casas and A Plastino ldquoA nonextensive max-imum entropy approach to a family of nonlinear reaction-diffusion equationsrdquo Physica A vol 280 no 3 pp 289ndash3032000

[43] T D Frank and A Daffertshofer ldquo119867-theorem for nonlinearFokker-Planck equations related to generalized thermostatis-ticsrdquo Physica A vol 295 no 3-4 pp 455ndash474 2001

[44] E K Lenzi R SMendes and L R da Silva ldquoStatisticalmechan-ics based onRenyi entropyrdquo Physica A vol 280 no 3-4 pp 337ndash345 2000

[45] A S Parvan andT S Biro ldquoExtensive Renyi statistics fromnon-extensive entropyrdquo Physics Letters A vol 340 no 5-6 pp 375ndash387 2005

[46] B D Sharma and D P Mittal ldquoNew nonadditive measures ofentropy for discrete probability distributionsrdquo Journal of Math-ematical Sciences vol 10 pp 28ndash40 1975

[47] T D Frank and A Daffertshofer ldquoExact time-dependentsolutions of the Renyi Fokker-Planck equation and the Fokker-Planck equations related to the entropies proposed by Sharmaand Mittalrdquo Physica A vol 285 no 3 pp 351ndash366 2000

[48] M R Ubriaco ldquoEntropies based on fractional calculusrdquo PhysicsLetters A vol 373 no 30 pp 2516ndash2519 2009

[49] P T Landsberg and V Vedral ldquoDistributions and channelcapacities in generalized statistical mechanicsrdquo Physics LettersA vol 247 no 3 pp 211ndash217 1998

[50] A Plastino and A R Plastino ldquoTsallis Entropy and Jaynesrsquoinformation theory formalismrdquo Brazilian Journal of Physics vol29 no 1 pp 50ndash60 1999

[51] X Li C Essex M Davison K H Hoffmann and C SchulzkyldquoFractional diffusion irreversibility and entropyrdquo Journal ofNon-Equilibrium Thermodynamics vol 28 no 3 pp 279ndash2912003

[52] H J Haubold A M Mathai and R K Saxena ldquoBoltzmann-Gibbs entropy versus Tsallis entropy recent contributions toresolving the argument of Einstein concerning ldquoneither HerrBoltzmann nor Herr Planck has given a definition ofWrdquo Essayreviewrdquo Astrophysics and Space Science vol 290 no 3-4 pp241ndash245 2004

[53] P N Rathie and S da Silva ldquoShannon Levy and Tsallis a noterdquoApplied Mathematical Sciences vol 2 no 28 pp 1359ndash13632008

[54] R M Gray Entropy and Information Theory Springer NewYork NY USA 2009

[55] J A Tenreiro Machado A C Costa and M F M Lima ldquoDy-namical analysis of compositionsrdquoNonlinear Dynamics vol 65no 4 pp 339ndash412 2011

[56] J A T Machado A C Costa and M D Quelhas ldquoEntropyanalysis of the DNA code dynamics in human chromosomesrdquoComputers amp Mathematics with Applications vol 62 no 3 pp1612ndash1617 2011

[57] J A TenreiroMachado A C Costa andM D Quelhas ldquoShan-non Renyie and Tsallis entropy analysis of DNA using phaseplanerdquo Nonlinear Analysis Real World Applications vol 12 no6 pp 3135ndash3144 2011

Mathematical Problems in Engineering 3

underlying ideologies motivations and the terrain in whichevents occurred The authors introduced a model where theinsurgent force behaved as a self-organizing system whichevolved dynamically through the continual coalescence andfragmentation of its constituent groups They analyzed casu-altiesrsquo patterns arising within a given war unlike previousstudies that focused on the total casualty figure for oneparticular war [7 11 25 31] A PL behavior fitted well the datanot only from Iraq Colombia and non-G7 terrorism butalso with data obtained from the war in Afghanistan The PLparameter for Iraq Colombia andAfghanistan was (close to) = 25 This value of the coefficient equalized the coefficientvalue characterizing non-G7 terrorism In the literature thePL parameter value was = 251 for non-G7 countries [32]and = 1713 for G7 countries This result suggested that PLpatterns would emerge within any modern asymmetric warfought by loosely-organized insurgent groups

In 2006 Bogen and Jones [33] treated the severity of ter-rorist attacks in terms of deaths and injuredThey applied a PLdistribution to victimevent rates and used the PL to predictmortality due to terrorism through the year 2080 Authorsclaimed that these PL models could be used to improvestrategies ldquoto assess prevent and manage terror-related risksand consequencesrdquo

Clauset et al [34] studied the frequency and the numberof casualties (deaths and injuries) of terrorist attacks since1968 They observed a scale-invariance behavior with thefrequency being an inverse power of the casualties Thisbehavior was independent of the type of weapon economicdevelopment and distinct time scales The authors presenteda new model to fit the frequency of severe terrorist attackssince previous models in the literature failed to produce theheavy tail in the PL distribution Their model assumed thatthe severity of an occurrence was a function of the executionplan and that selection tools were better suited to modelcompetition between states and nonstate actors Finally re-searchers claimed that periodicity was a common feature inglobal terrorism with period close to roughly 13 years

Bohorquez et al [12] studied the quantitative relationbetween human insurgency global terrorism and ecologyThey introduced a newmodel to explain the size distributionof casualties or the timing ofwithin-conflict eventsThey con-sidered insurgent populations as self-organized groups thatdynamically evolved through decision-making processesThe main assumptions of the model were (i) being consistentwith work on human group dynamics in everyday environ-ments (ii) having a new perception of modern insurgenciesas fragmented transient and evolving and (iii) using adecision-making process about when to attack based oncompetition for media attention Authors applied a PL distri-bution to Iraq andColombiawars with parameter value closeto = 25 A coefficient value of = 25 was in concordancewith the coefficient value of = 248 plusmn 007 obtained byClauset et al [34] on global terrorism A PL fit to Spanish andAmerican Civil wars revealed a PL parameter value smaller(around = 17) Authors claimed that theirmodel suggesteda remarkable link between violent and nonviolent humanactions due to its similarity to financial market models

Rank

max

rank

Sizemax size10minus3 10minus2 10minus1 10010minus2

10minus1

100

ln(119910) = minus085503 ln(119909) minus 474389

Figure 1 Rankfrequency log-log plot corresponding to the distri-bution of casualties caused by industrial accidents in CentralSouthAmerica over the period 1900ndash2011 (min size = 10 max size = 2700max rank = 66)

3 Power Law Behavior inCatastrophic Phenomena

In this section we investigate the statistical distributionsof random variables that represent the number of humancasualties in several human-made and natural hazards

Data from the EM-DAT International Disaster Database(httpwwwemdatbe) and the Global Terrorism Database(GTD) (httpwwwstartumdedugtd) are analyzed TheEM-DAT database contains information on over than 18000worldwide natural and technological disasters from 1900to present The EM-DAT is maintained by the Centre forResearch on the Epidemiology of Disasters (CRED) at theSchool of Public Health of the Universite Catholique deLouvain located inBrussels Belgium [35]TheGTDdatabaseis an open-source database that includes information onmorethan 98000 worldwide terrorist attacks from 1970 up to 2010[31]

PLs are observed in several natural and man-madesystems Examples of single and double PLs in real data aregiven in Figures 1 and 2 respectively The former representsthe complementary cumulative distribution of the severityof industrial accidents in CentralSouth America over theperiod 1900ndash2011 The adopted measure to quantify theseverity of an event is the total number of fatalities Thedepicted graph corresponds to a rankfrequency log-log plotTo construct the graph we first sort the data (ie theaccidents) in decreasing order according to their severity andnumber them consecutively starting from one [36] Then anormalization of the values is carried out meaning that thenumber of fatalities (119909-axis) is divided by the correspondinghighest value and the rank (119910-axis) is divided by the rank ofthe smallest event Finally PLs are adjusted to the data using aleast squares algorithm All the log-log plots presented in thispaper are made following this procedure

Figure 2 corresponds to the distribution of casual-ties caused by earthquakes in CentralSouth America in

4 Mathematical Problems in EngineeringRa

nkm

ax ra

nk

Sizemax size

ln(119910) = minus0246959 ln(119909) minus 299476

ln(119910) = minus0499454 ln(119909) minus 492208

10minus4 10minus310minus6 10minus5 10minus2 10minus1 10010minus3

10minus2

10minus1

100

Figure 2 Rankfrequency log-log plot corresponding to the dis-tribution of casualties caused by earthquakes in CentralSouthAmerica over the period 1900ndash2011 (min size = 1 max size = 222570max rank = 179)

the period 1900ndash2011 representing one event that can beapproximated by a double PL

As can be seen in Figure 1 a single PL (SPL) with para-meters (119862 ) = (00087 08550) fits to the data The dis-tribution depicted in Figure 2 is better approximated by adouble PL (DPL) with parameters (119862

1 1) = (00500 02470)

and (1198622 2) = (00073 04995) The change in the behavior

occurs at the relative value of 119909 = 0000539 approximatelyWe analyzed the data available at the EM-DAT database

in terms of disaster type (DT119895) and disaster location (DL

119896)

119895 = 1 11 and 119896 = 1 6 categories respectivelyDT119895= Drought Earthquake Epidemic Extreme tempera-

ture Flood Industrial accident Mass movement wet StormTransport accident VolcanoWildfire DL

119896= Africa North

America Central amp South America Europe (including Rus-sia) Asia (not including SE Asia) Oceania (including SEAsia) The period of analysis was 1900ndash2011 for every caseThe total number of combinations (locationtype) is 11 times6 Nevertheless for 14 cases there is insufficient data tocompute reliable statistical distributions For all cases takingthe number of casualties as the variable of interest we obtainstatistical distributions that can be approximated by either aSPL (16 cases) or aDPL (36 cases) similar to the ones depictedin Figures 1 and 2

In Figure 3 we depict the locus of the parameters(119862119894 119894) 119894 = 1 2 corresponding to the analyzed cases As can

be seen an interesting pattern emerges where the linesthat connect the pairs of points that characterize the DPLshave identical orientation This geometrical pattern reflects arelationship between the two parts of the DPL distributions(DPL1mdashpart closer to the head and DPL2mdashpart closer to thetail) Besides the observation that

2gt 1 in all cases further

investigation on the reason for this behavior is neededWe pursued our study with the analysis of the GTD

database First the events associated to human casualtieswere grouped by year (119884

119903) starting in 1980 up to 2010

0

02

04

06

08

1

12

14

10minus4 10minus3 10minus2 10minus1 100

119862

Figure 3 Locus of the parameters (119862119894 119894) 119894 = 1 2 that characterize

the PLs corresponding to the number of casualties in certaincombinations of disaster typelocation DT

119895DL119896

(except 1993 because there is no data available) 119884119903

=

1980 2010 1993 119903 = 1 30 We found that allthe statistical distributions can be approximated by DPLs InFigures 4 and 5 the time evolution of the parameters of theDPLs (119862

119894 119894) 119894 = 1 2 is shown Regarding the parameters

119862119894 it can be seen that they have identical behavior although

1198622varies more than 119862

1and is always smaller than it

With respect to 119894 we have a similar evolution but in this

case the parameter 2is always greater than

1 As severe

terrorist attacks correspond to points closer to the tail of thedistribution DPL2 which is characterized by a larger thismeans that those events are more similar between each otherthan the smaller events (that correspond to DPL1)

To complement the analysis with respect to the date of theoccurrences the parameters (119862

119894 119894) of the PLs correspond-

ing to 119884119903 were plotted (Figure 6) As can be seen a pattern

similar to the described previously (Figure 3) is observedWe have also studied the distributions of the casualties in

terrorist attacks occurred in the period 1970ndash2010 but withrespect to other criteria namely the type of used weapon(119882119894) region where the event took place (119877

119895) target (119879

119896)

and type of attack (119860119897) Each criterion was then divided

into 119894 = 1 6 119895 = 1 13 119896 = 1 19 and 119897 =

1 8 categories respectively119882119894= Chemical Explosives

Firearms Incendiary Melee Vehicle 119877119895= Australasia amp

Oceania Central America amp Caribbean Central Asia EastAsia Eastern Europe Middle East amp North Africa NorthAmerica South America South Asia Southeast Asia Sub-Saharan Africa USSR amp Newly Independent States (NIS)Western Europe 119879

119896= Airports amp Airlines Business

Educational Institution Food or Water Supply Govern-ment (Diplomatic) Government (General) Journalists ampMedia Maritime Military NGO Police Private Citizensamp Property Religious FiguresInstitutions Telecommuni-cation Terrorists Tourists Transportation Utilities Vio-lent Political Party 119860

119897= Armed Assault Assassination

Mathematical Problems in Engineering 5

1980 1990 2000 2010Year

11986211198622

10minus4

10minus3

10minus2

10minus1

119862

Figure 4 Time evolution of parameters 119862119894 119894 = 1 2 of the DPLs

corresponding to terrorist attacks over the period 1980ndash2010 119884119903

1980 1990 2000 2010Year

0

05

1

15

2

25

3

12057211205722

Figure 5 Time evolution of parameters 119894 119894 = 1 2 of the DPLs

corresponding to terrorist attacks over the period 1980ndash2010 119884119903

BombingExplosion FacilityInfrastructure Attack Hijack-ing Hostage Taking (Barricade Incident) Hostage Taking(Kidnapping) and Unarmed Assault

Most cases are characterized by DPLs However in afew situations a SPL fits better to the data The main resultsare summarized in Table 1 Moreover we observed that theparameters corresponding to all distributions characterizedby DPLs display a pattern similar to the ones mentionedpreviously (Figures 3 and 6) where the lines connecting theslopes and intercepts of DPL1 (

1 1198621) to its companionDPL2

(2 1198622) for the same data set have identical orientation in

the (119862 ) Cartesian space

4 Entropy of Catastrophic Phenomena

In this section we analyse the entropy of data collected fromthe GTD database that is data related to terrorism To

1980 1981 1982 1983 1984 19851986 1987 1988 1989 1990 19911992 1994 1995 1996 1997 19981999 2000 2001 2002 2003 20042005 2006 2007 2008 2009 2010

0

05

1

15

2

25

3

10minus4 10minus3 10minus2 10minus1

119862

Figure 6 Locus of the parameters (119862119894 ) 119894 = 1 2 that characterize

the distributions of terrorist attacks over the period 1980ndash2010 119884119903

calculate the entropies we construct histograms of relativefrequencies using bins of width one (one casualty) andapproximate the probabilities 119901

119894by the relative frequencies

We present results obtained for terrorist events grouped byyear (119884

119903) as defined in the previous section Nevertheless it

should be noticed that similar results are obtained for all otherhuman-made and natural hazards

Clausius [37] and Boltzmann [38] were the first authorsto define entropy in the field of thermodynamics Later onShannon [39] and Jaynes [40] applied their results to infor-mation theory [41]

The most celebrated entropy is the so-called Shannonentropy 119878 defined by

119878 = minus

119882

sum

119894=1

119901119894ln119901119894 (2)

The Shannon entropy represents the expected value of theinformation minus ln119901

119894 Therefore for the uniform probability

distribution we have 119901119894= 119882minus1 and the Shannon entropy

takes its maximum value 119878 = ln119882 yielding the Boltzmannrsquosfamous formula up to a multiplicative factor 119896 denoting theBoltzmann constant Thus in thermodynamic equilibriumthe Shannon entropy can be identified as the ldquophysicalentropyrdquo of the system

Renyi and Tsallis entropies are generalizations of Shan-nonrsquos entropy and are given by respectively

119878(119877)

119902=

1

1 minus 119902

ln(119882

sum

119894=1

119901119902

119894) 119902 gt 0

119878(119879)

119902=

1

119902 minus 1

(1 minus

119882

sum

119894=1

119901119902

119894)

(3)

6 Mathematical Problems in Engineering

Table 1 PL fit to the distributions of casualties in terrorist attacks

Criterion Category SPL DPL1 DPL2(119862 ) (119862

1 1) (119862

2 2)

Weapon (119882119894)

1198821

00391 058491198822

00082 07514 00002 171981198823

00017 08660 00000 196241198824

00236 06877 00021 148901198825

00057 06909 00003 135791198826

00249 07801

Region (119877119894)

1198771

00438 101471198772

00214 06412 00009 169051198773

00174 119051198774

00065 090731198775

00012 121411198776

00091 08051 00006 170881198777

00009 114151198778

00193 07814 00008 206751198779

00036 08742 00001 2055211987710

00128 08561 00012 1657811987711

00091 06454 00004 1313211987712

00023 0995611987713

00001 15523

Target (119879119894)

1198791

00438 05144 00092 124881198792

00057 08884 00003 171801198793

00002 13589 00012 084581198794

00452 079231198795

00015 112591198796

00006 11152 00001 151621198797

00014 161061198798

00576 04694 00094 090701198799

00353 05669 00009 1629111987910

00090 09645 00031 1257311987911

00047 08991 00001 1835411987912

00036 07778 00000 1866711987913

00158 07353 00018 1610611987914

00462 0977411987915

00168 08610 00026 1682411987916

00234 0763511987917

00340 05892 00014 1517811987918

00225 07389 00074 1070811987919

00108 07918 00025 11650

Type (119860119894)

1198601

00049 07346 00000 179821198602

00014 15201 00001 265551198603

00059 08116 00001 206211198604

00261 06339 00016 148801198605

00305 071861198606

00116 073161198607

00072 09689 00014 162141198608

00471 07006 00034 20186

Mathematical Problems in Engineering 7

1980 1985 1990 1995 2000 2005 201005

05506

06507

07508

08509

0951

Year

Nor

mal

ized

entro

py

119878(119879)05

119878(119877)05

119878(119880)05

119878(119879)2

119878(119877)2

119878(119880)2

119878

Figure 7 Total Shannon Tsallis Renyi and Ubriaco normalizedentropies as a function of the year of the events 119884

119903 119902 = 05 and

119902 = 2

Tsallis entropy reduces to Renyi entropy when 119902 rarr 1 Tsallisentropy was applied to diffusion equations [42] and Fokker-Planck systems [43] Renyi entropy has an inverse power lawequilibrium distribution [44] and satisfies the zeroth law ofthermodynamics [45] The two parameters Sharma-Mittalentropy [46] is accepted as a generalization of Tsallis Renyiand Boltzmann-Gibbs entropies for limiting cases of theparameters [47]

Recently more general entropy measures have been pro-posed in the literature where the additivity axiom has beenrelaxed For instance Ubriaco [48] proposed the followingformula for the fractional entropy

119878(119880)

119902=

119882

sum

119894=1

(minus ln119901119894)119902

119901119894

(4)

that has the same properties as the Shannon entropy exceptadditivity

Applications of entropy in distinct complex systems canbe found in [49ndash57]

In Figure 7 the total Shannon normalized entropy (119878) isdepicted as well as Tsallisrsquo (119878(119879)

119902) Renyirsquos (119878(119877)

119902) and Ubriacorsquos

(119878(119880)

119902) for 119902 = 05 and 119902 = 2 as a function of the year of

the events 119884119903 Figures 9 10 and 11 show 119878

(119879)

119902 119878(119877)119902

and 119878(119880)119902

normalized entropies as a function of the year 119884119903 and

entropy parameter 01 le 119902 le 10In Figure 7 we observe two types of behavior namely

short- and long-term phenomena In what concerns shorttime behavior we verify peaks during 1983ndash1985 1997-19982004ndash2007 and minima at 1980 1995 and 2010 In whatconcerns long-time relationships a smooth decreasing isobserved for 119878(119879)

05 119878(119877)05

and 119878(119880)2

Removing the maxima andminima we get a time series for years 1981 1982 1986ndash1992 1994ndash1996 1999 2000 2002 2003 2008 and 2009(Figure 8) Largersmaller entropies correspond to chartscloserafar uniform distributions therefore seemingly wehave lessmore organized terrorist events in global terms

1980 1985 1990 1995 2000 2005 2010075

08

085

09

095

1

Year

Nor

mal

ized

entro

py

119878(119879)05

119878(119877)05

119878(119880)2

Lin Reg 119878(119879)05Lin Reg 119878(119877)05Lin Reg 119878(119880)2

Figure 8 Total Tsallis and Renyi normalized entropies (119902 = 05)

and Ubriacorsquos (119902 = 2) as a function of the events in years 1981 19821986ndash1992 1994ndash1996 1999 2000 2002 2003 2008 2009

1980 1985 1990 1995 2000 2005 2010

01234567

8910040506070809

1

Year

119902

119878(119879)119902

Figure 9 Total Tsallis normalized entropy 119878(119879)119902 as a function of the

year of the events 119884119903 and parameter 119902

These conclusions remain invariant for Figures 9 to 11 wherewe vary both the entropy definition and the parameter tuningTherefore we conclude that such results are robust againstsuch type of variations

As discussed in the previous section the statistical dis-tributions of real data can be approximated by either singleor double PLs In the latter case we study not only the totalentropy but also the entropy associated to each part of thedistributions Therefore we compute the entropy associatedto DPL1 and DPL2 that approximate the first and secondpart of the distributions respectively When adopting thisprocedure we are restricted to the Shannon and Ubriacoentropies as Tsallisrsquo and Renyirsquos do not admit the associativitydescribed above

In the sequel we present several results of the analysistaking into account the grouping criteria 119884

119903 119882119894 119877119895 119879119896 and

119860119897 Figure 12 depicts the Shannon entropy versus parameter

for all statistical distributions The black squares (denotedldquoSPLrdquo) correspond to the plot of versus the total entropyof the respective distributions 119878 The black circles (denotedldquoDPL1rdquo) are the plot of

1versus the entropy associated to the

first parts of the distributions 1198781 The white circles (denoted

ldquoDPL2rdquo) represent the plot of 2versus the entropy associated

8 Mathematical Problems in Engineering

1980 1985 1990 1995 2000 2005 2010

012345678910040608

1

Year

119902

119878(119877)119902

Figure 10 Total Renyi normalized entropy 119878(119877)119902 as a function of the

year of the events 119884119903 and parameter 119902

1980 1985 1990 1995 2000 2005 2010

012345678910

0402

0608

1

Year

119902

119878(119880)119902

Figure 11 Total Ubriaco normalized entropy 119878(119880)119902

as a function ofthe year of the events 119884

119903 and parameter 119902

to the second parts of the distributions 1198782 As can be seen for

the distributions that behave as single PLs higher entropiescorrespond to the lower values of the parameter and the twoparameters are linearly related A similar pattern is observedfor the parameters corresponding to DPL1 For DPL2 theparameter

2increases with entropy but the almost linear

relation between both parameters remainsFigure 13 shows identical results for the Ubriaco entropy

Theplot corresponds to 119902 = 05 nevertheless identical resultsare obtained for other values

For all DPLs related to terrorist events 119899 (119899 = 1 60)we find that the parameters (119862

119894 119894) 119894 = 1 2 obey the follow-

ing relation

1119899

10038161003816100381610038161003816log1198621119899|119901= 2119899

10038161003816100381610038161003816log1198622119899|119901+ 120598119899 (5)

where 119901 = minus1612 The mean value of 120598119899is 120598 = 0002 and the

corresponding standard deviation is 120590120598= 0036 Moreover

for the analyzed data we find 119894119899| log119862

119894119899|119901= 119896119894119899 Parameters

119896119894119899are approximately constant with mean value 119896 = 0277

and standard deviation 120590119896= 006

It is worth noticing that (5) is similar to Poissonrsquos law ofan adiabatic reversible process involving ideal gases given by

1198751119881120574

1= 1198752119881120574

2 (6)

where variables 119875119894and 119881

119894represent pressure and specific

volume respectively Equation (6) implies that 119875119881120574 = constParameter 120574 is called Poissonrsquos coefficient taking values 120574 =

53 ≃ 167 and 120574 = 75 = 140 for monoatomic and diatomicgases respectively Additionally it should be noticed that theabsolute value of the exponent 119901 is very similar to the oneobserved for an ideal gas undergoing a reversible adiabaticprocess

0

05

1

15

2

25

3

0 05 1 15 2 25Shannon entropy

SPLDPL1

DPL2

Figure 12 Parameters versus Shannon entropies for the dis-tributions corresponding to the criteria 119884

119903 119882119894 119877119895 119879119896 and 119860

119897

ldquoSPLrdquo corresponds to (119878 ) ldquoDPL1rdquo corresponds to (1198781 1) ldquoDPL2rdquo

corresponds to (1198782 2)

0

05

1

15

2

25

3

0 02 04 06 08 1 12 14 16

SPL DPL1DPL2

Ubriaco entropy (119902 = 05)

Figure 13 Parameters versus Ubriaco entropies for the distri-butions corresponding to the criteria 119884

119903 119882119894 119877119895 119879119896 and 119860

119897 ldquoSPLrdquo

corresponds to (119878(119880)119902 ) ldquoDPL1rdquo corresponds to (119878(119880)

1199021 1) ldquoDPL2rdquo

corresponds to (119878(119880)1199022 2) 119902 = 05

5 Conclusions

PLs have been widely reported in the modeling of distinctphenomena and have been associated with long memorybehavior self-similarity fractal structures and fractionalcalculus

In this paper we reviewed interesting and importantresults on PLs distributions and their applications to the

Mathematical Problems in Engineering 9

modeling of the number of victims in catastrophic eventsWe found double PL behavior in real data of catastrophicoccurrences in particular terrorismWe have plotted the twoPLs parameters (119862

119894 119894) 119894 = 1 2 corresponding to certain

events and observed an interesting pattern in the chartwhere the lines that connect each pair of points defining thedouble PLs are almost aligned to each other We have alsocomputed the entropy of the data sets This complementaryanalysis of the numerical data revealed extra relationships butthe fact is that these phenomena have a dense and rich volumeof characteristics and further research efforts are needed to adeeper understanding

References

[1] J Tenreiro Machado F B Duarte and G M Duarte ldquoPowerlaw analysis of financial index dynamicsrdquo Discrete Dynamics inNature and Society vol 2012 Article ID 120518 12 pages 2012

[2] J Tenreiro Machado ldquoAnd I say to myself lsquoWhat a fractionalworlrsquordquo Fractional Calculus and Applied Analysis vol 14 no 4pp 635ndash654 2011

[3] E Scalas R Gorenflo and FMainardi ldquoFractional calculus andcontinuous-time financerdquo Physica A vol 284 no 1ndash4 pp 376ndash384 2000

[4] F Mainardi M Raberto R Gorenflo and E Scalas ldquoFractionalcalculus and continuous-time finance II the waiting-time dis-tributionrdquo Physica A vol 287 no 3-4 pp 468ndash481 2000

[5] B J West ldquoFractal physiology and the fractional calculus aperspectiverdquo Frontiers in Physiology vol 1 p 12 2010

[6] B Gutenberg and R F Richter ldquoFrequency of earthquakes incaliforniardquo Bulletin of the Seismological Society of America vol34 pp 185ndash188 1944

[7] L F Richardson ldquoVariation of the Frequency of Fatal QuarrelswithMagnituderdquo Journal of the American Statistical Associationvol 43 pp 523ndash546 1948

[8] J M Carlson and J S Langer ldquoMechanical model of an earth-quake faultrdquo Physical Review A vol 40 no 11 pp 6470ndash64841989

[9] D C Roberts and D L Turcotte ldquoFractality and self-organizedcriticality of warsrdquo Fractals vol 6 no 4 pp 351ndash357 1998

[10] D R Davis and D E Weinstein ldquoBones bombs and breakpoints the geography of economic activityrdquo The AmericanEconomic Review vol 92 no 5 pp 1269ndash1289 2002

[11] L E Cederman ldquoModeling the size of wars from billiard ballsto sandpilesrdquo American Political Science Review vol 97 no 1pp 135ndash150 2003

[12] J C Bohorquez S Gourley A R Dixon M Spagat and N FJohnson ldquoCommon ecology quantifies human insurgencyrdquoNature vol 462 no 7275 pp 911ndash914 2009

[13] V Pareto Cours drsquoEconomie Politique Droz Geneva Switzer-land 1896

[14] F Auerbach ldquoDas Gesetz der BelvolkerungskoncentrationrdquoPetermanns GeographischeMitteilungen vol 59 pp 74ndash76 1913

[15] J B Estoup Gammes Stenographiques Institut de France 1916[16] G Zipf Selective Studies and the Principle of Relative Frequency

in Language Harvard University Press Cambridge Mass USA1932

[17] G Zipf Human Behavior and the Priciple of Least EffortAddison-Wesley Cambridge Mass USA 1949

[18] W Li ldquoReferences on Zipf rsquos lawrdquo httpwwwsorinsolomonnetsimsorinccszipfreferences20on20zipf27s20lawhtm

[19] D Sornette Critical Phenomena in Natural Sciences chapter 14Springer Heidelberg Germany 2nd edition 2003

[20] C M A Pinto A M Lopes and J A Tenreiro Machado ldquoAreview of power laws in real life phenomenardquo Communicationsin Nonlinear Science and Numerical Simulations vol 17 no 9pp 3558ndash3578 2012

[21] J Tuldava ldquoThe frequency spectrum of text and vocabularyrdquoJournal of Quantitative Linguistics vol 3 no 1 pp 38ndash50 1996

[22] GMartınez-Mekler R AlvarezMartınezM Beltran del Rıo RMansilla P Miramontes and G Cocho ldquoUniversality of rank-ordering distributions in the arts and sciencesrdquo PLoS One vol4 no 3 Article ID e4791 2009

[23] R F I Cancho and R V Sole ldquoTwo regimes in the frequency ofwords and the origins of complex lexicons Zipf rsquos law revisitedrdquoJournal of Quantitative Linguistics vol 8 no 3 pp 165ndash1732001

[24] R Alvarez-Martinez G Martinez-Mekler and G CochoaldquoOrder-disorder transition in conflicting dynamics leading torank-frequency generalized beta distributionsrdquo Physica A vol390 no 1 pp 120ndash130 2011

[25] L F Richardson Statistics of Deadly Quarrels QuadrangleBooks Chicago Ill USA 1960

[26] F Guzzetti ldquoLandslide fatalities and the evaluation of landsliderisk in Italyrdquo Engineering Geology vol 58 no 2 pp 89ndash1072000

[27] S N Jonkman ldquoGlobal perspectives on loss of human lifecaused by floodsrdquo Natural Hazards vol 34 no 2 pp 151ndash1752005

[28] N F Johnson M Spagat J A Restrepo et al ldquoUniversal pat-terns underlying ongoingwars and terrorismrdquo httparxivorgabsphysics0605035

[29] D S Geller and J David Singer Nations at War A ScientificStudy of International Conflict Cambridge Studies in Interna-tional Relations Cambridge University Press Cambridge UK1998

[30] O Becerra N Johnson P Meier J Restrepo and M Spa-gat ldquoNatural disasters casualties and power laws a com-parative analysis with armed conflictrdquo in Proceedings of theAnnual Meeting of the American Political Science AssociationLoews Philadelphia and the Pennsylvania Convention Cen-ter Philadelphia Pa USA 2006 httpwwwallacademiccommetap151714 indexhtml

[31] M E J Newman ldquoPower laws Pareto distributions and Zipf rsquoslawrdquo Contemporary Physics vol 46 pp 323ndash351 2005

[32] A Clauset andM Young ldquoScale invariance in global terrorismrdquohttparxivorgabsphysics0502014

[33] K T Bogen and E D Jones ldquoRisks of mortality and morbidityfrom worldwide terrorism 1968ndash2004rdquo Risk Analysis vol 26no 1 pp 45ndash59 2006

[34] A Clauset M Young and K S Gleditsch ldquoOn the frequency ofsevere terrorist eventsrdquo Journal of Conflict Resolution vol 51 no1 pp 58ndash87 2007

[35] EM-DAT ldquoTheOFDACRED International Disaster DatabaserdquoUniversite catholique de Louvain Brussels Belgium httpwwwemdatnet

[36] National Consortium for the Study of Terrorism and Responsesto Terrorism (START) ldquoGlobal Terrorism Databaserdquo 2011httpwwwstartumdedugtd

[37] R ClausiusTheMechanicalTheory of Heat with Its Applicationsto the Steam Engine and to Physical Properties of Bodies Johnvan Voorst London UK 1865

[38] L Boltzmann Vorlesungen ber Gastheorie J A Barth LeipzigGermany 1896

10 Mathematical Problems in Engineering

[39] C E Shannon ldquoAmathematical theory of communicationrdquoTheBell System Technical Journal vol 27 pp 379ndash423 623ndash6561948

[40] E T Jaynes ldquoInformation theory and statistical mechanicsrdquo vol106 pp 620ndash630 1957

[41] A I Khinchin Mathematical Foundations of Information The-ory Dover New York NY USA 1957

[42] A R Plastino M Casas and A Plastino ldquoA nonextensive max-imum entropy approach to a family of nonlinear reaction-diffusion equationsrdquo Physica A vol 280 no 3 pp 289ndash3032000

[43] T D Frank and A Daffertshofer ldquo119867-theorem for nonlinearFokker-Planck equations related to generalized thermostatis-ticsrdquo Physica A vol 295 no 3-4 pp 455ndash474 2001

[44] E K Lenzi R SMendes and L R da Silva ldquoStatisticalmechan-ics based onRenyi entropyrdquo Physica A vol 280 no 3-4 pp 337ndash345 2000

[45] A S Parvan andT S Biro ldquoExtensive Renyi statistics fromnon-extensive entropyrdquo Physics Letters A vol 340 no 5-6 pp 375ndash387 2005

[46] B D Sharma and D P Mittal ldquoNew nonadditive measures ofentropy for discrete probability distributionsrdquo Journal of Math-ematical Sciences vol 10 pp 28ndash40 1975

[47] T D Frank and A Daffertshofer ldquoExact time-dependentsolutions of the Renyi Fokker-Planck equation and the Fokker-Planck equations related to the entropies proposed by Sharmaand Mittalrdquo Physica A vol 285 no 3 pp 351ndash366 2000

[48] M R Ubriaco ldquoEntropies based on fractional calculusrdquo PhysicsLetters A vol 373 no 30 pp 2516ndash2519 2009

[49] P T Landsberg and V Vedral ldquoDistributions and channelcapacities in generalized statistical mechanicsrdquo Physics LettersA vol 247 no 3 pp 211ndash217 1998

[50] A Plastino and A R Plastino ldquoTsallis Entropy and Jaynesrsquoinformation theory formalismrdquo Brazilian Journal of Physics vol29 no 1 pp 50ndash60 1999

[51] X Li C Essex M Davison K H Hoffmann and C SchulzkyldquoFractional diffusion irreversibility and entropyrdquo Journal ofNon-Equilibrium Thermodynamics vol 28 no 3 pp 279ndash2912003

[52] H J Haubold A M Mathai and R K Saxena ldquoBoltzmann-Gibbs entropy versus Tsallis entropy recent contributions toresolving the argument of Einstein concerning ldquoneither HerrBoltzmann nor Herr Planck has given a definition ofWrdquo Essayreviewrdquo Astrophysics and Space Science vol 290 no 3-4 pp241ndash245 2004

[53] P N Rathie and S da Silva ldquoShannon Levy and Tsallis a noterdquoApplied Mathematical Sciences vol 2 no 28 pp 1359ndash13632008

[54] R M Gray Entropy and Information Theory Springer NewYork NY USA 2009

[55] J A Tenreiro Machado A C Costa and M F M Lima ldquoDy-namical analysis of compositionsrdquoNonlinear Dynamics vol 65no 4 pp 339ndash412 2011

[56] J A T Machado A C Costa and M D Quelhas ldquoEntropyanalysis of the DNA code dynamics in human chromosomesrdquoComputers amp Mathematics with Applications vol 62 no 3 pp1612ndash1617 2011

[57] J A TenreiroMachado A C Costa andM D Quelhas ldquoShan-non Renyie and Tsallis entropy analysis of DNA using phaseplanerdquo Nonlinear Analysis Real World Applications vol 12 no6 pp 3135ndash3144 2011

4 Mathematical Problems in EngineeringRa

nkm

ax ra

nk

Sizemax size

ln(119910) = minus0246959 ln(119909) minus 299476

ln(119910) = minus0499454 ln(119909) minus 492208

10minus4 10minus310minus6 10minus5 10minus2 10minus1 10010minus3

10minus2

10minus1

100

Figure 2 Rankfrequency log-log plot corresponding to the dis-tribution of casualties caused by earthquakes in CentralSouthAmerica over the period 1900ndash2011 (min size = 1 max size = 222570max rank = 179)

the period 1900ndash2011 representing one event that can beapproximated by a double PL

As can be seen in Figure 1 a single PL (SPL) with para-meters (119862 ) = (00087 08550) fits to the data The dis-tribution depicted in Figure 2 is better approximated by adouble PL (DPL) with parameters (119862

1 1) = (00500 02470)

and (1198622 2) = (00073 04995) The change in the behavior

occurs at the relative value of 119909 = 0000539 approximatelyWe analyzed the data available at the EM-DAT database

in terms of disaster type (DT119895) and disaster location (DL

119896)

119895 = 1 11 and 119896 = 1 6 categories respectivelyDT119895= Drought Earthquake Epidemic Extreme tempera-

ture Flood Industrial accident Mass movement wet StormTransport accident VolcanoWildfire DL

119896= Africa North

America Central amp South America Europe (including Rus-sia) Asia (not including SE Asia) Oceania (including SEAsia) The period of analysis was 1900ndash2011 for every caseThe total number of combinations (locationtype) is 11 times6 Nevertheless for 14 cases there is insufficient data tocompute reliable statistical distributions For all cases takingthe number of casualties as the variable of interest we obtainstatistical distributions that can be approximated by either aSPL (16 cases) or aDPL (36 cases) similar to the ones depictedin Figures 1 and 2

In Figure 3 we depict the locus of the parameters(119862119894 119894) 119894 = 1 2 corresponding to the analyzed cases As can

be seen an interesting pattern emerges where the linesthat connect the pairs of points that characterize the DPLshave identical orientation This geometrical pattern reflects arelationship between the two parts of the DPL distributions(DPL1mdashpart closer to the head and DPL2mdashpart closer to thetail) Besides the observation that

2gt 1 in all cases further

investigation on the reason for this behavior is neededWe pursued our study with the analysis of the GTD

database First the events associated to human casualtieswere grouped by year (119884

119903) starting in 1980 up to 2010

0

02

04

06

08

1

12

14

10minus4 10minus3 10minus2 10minus1 100

119862

Figure 3 Locus of the parameters (119862119894 119894) 119894 = 1 2 that characterize

the PLs corresponding to the number of casualties in certaincombinations of disaster typelocation DT

119895DL119896

(except 1993 because there is no data available) 119884119903

=

1980 2010 1993 119903 = 1 30 We found that allthe statistical distributions can be approximated by DPLs InFigures 4 and 5 the time evolution of the parameters of theDPLs (119862

119894 119894) 119894 = 1 2 is shown Regarding the parameters

119862119894 it can be seen that they have identical behavior although

1198622varies more than 119862

1and is always smaller than it

With respect to 119894 we have a similar evolution but in this

case the parameter 2is always greater than

1 As severe

terrorist attacks correspond to points closer to the tail of thedistribution DPL2 which is characterized by a larger thismeans that those events are more similar between each otherthan the smaller events (that correspond to DPL1)

To complement the analysis with respect to the date of theoccurrences the parameters (119862

119894 119894) of the PLs correspond-

ing to 119884119903 were plotted (Figure 6) As can be seen a pattern

similar to the described previously (Figure 3) is observedWe have also studied the distributions of the casualties in

terrorist attacks occurred in the period 1970ndash2010 but withrespect to other criteria namely the type of used weapon(119882119894) region where the event took place (119877

119895) target (119879

119896)

and type of attack (119860119897) Each criterion was then divided

into 119894 = 1 6 119895 = 1 13 119896 = 1 19 and 119897 =

1 8 categories respectively119882119894= Chemical Explosives

Firearms Incendiary Melee Vehicle 119877119895= Australasia amp

Oceania Central America amp Caribbean Central Asia EastAsia Eastern Europe Middle East amp North Africa NorthAmerica South America South Asia Southeast Asia Sub-Saharan Africa USSR amp Newly Independent States (NIS)Western Europe 119879

119896= Airports amp Airlines Business

Educational Institution Food or Water Supply Govern-ment (Diplomatic) Government (General) Journalists ampMedia Maritime Military NGO Police Private Citizensamp Property Religious FiguresInstitutions Telecommuni-cation Terrorists Tourists Transportation Utilities Vio-lent Political Party 119860

119897= Armed Assault Assassination

Mathematical Problems in Engineering 5

1980 1990 2000 2010Year

11986211198622

10minus4

10minus3

10minus2

10minus1

119862

Figure 4 Time evolution of parameters 119862119894 119894 = 1 2 of the DPLs

corresponding to terrorist attacks over the period 1980ndash2010 119884119903

1980 1990 2000 2010Year

0

05

1

15

2

25

3

12057211205722

Figure 5 Time evolution of parameters 119894 119894 = 1 2 of the DPLs

corresponding to terrorist attacks over the period 1980ndash2010 119884119903

BombingExplosion FacilityInfrastructure Attack Hijack-ing Hostage Taking (Barricade Incident) Hostage Taking(Kidnapping) and Unarmed Assault

Most cases are characterized by DPLs However in afew situations a SPL fits better to the data The main resultsare summarized in Table 1 Moreover we observed that theparameters corresponding to all distributions characterizedby DPLs display a pattern similar to the ones mentionedpreviously (Figures 3 and 6) where the lines connecting theslopes and intercepts of DPL1 (

1 1198621) to its companionDPL2

(2 1198622) for the same data set have identical orientation in

the (119862 ) Cartesian space

4 Entropy of Catastrophic Phenomena

In this section we analyse the entropy of data collected fromthe GTD database that is data related to terrorism To

1980 1981 1982 1983 1984 19851986 1987 1988 1989 1990 19911992 1994 1995 1996 1997 19981999 2000 2001 2002 2003 20042005 2006 2007 2008 2009 2010

0

05

1

15

2

25

3

10minus4 10minus3 10minus2 10minus1

119862

Figure 6 Locus of the parameters (119862119894 ) 119894 = 1 2 that characterize

the distributions of terrorist attacks over the period 1980ndash2010 119884119903

calculate the entropies we construct histograms of relativefrequencies using bins of width one (one casualty) andapproximate the probabilities 119901

119894by the relative frequencies

We present results obtained for terrorist events grouped byyear (119884

119903) as defined in the previous section Nevertheless it

should be noticed that similar results are obtained for all otherhuman-made and natural hazards

Clausius [37] and Boltzmann [38] were the first authorsto define entropy in the field of thermodynamics Later onShannon [39] and Jaynes [40] applied their results to infor-mation theory [41]

The most celebrated entropy is the so-called Shannonentropy 119878 defined by

119878 = minus

119882

sum

119894=1

119901119894ln119901119894 (2)

The Shannon entropy represents the expected value of theinformation minus ln119901

119894 Therefore for the uniform probability

distribution we have 119901119894= 119882minus1 and the Shannon entropy

takes its maximum value 119878 = ln119882 yielding the Boltzmannrsquosfamous formula up to a multiplicative factor 119896 denoting theBoltzmann constant Thus in thermodynamic equilibriumthe Shannon entropy can be identified as the ldquophysicalentropyrdquo of the system

Renyi and Tsallis entropies are generalizations of Shan-nonrsquos entropy and are given by respectively

119878(119877)

119902=

1

1 minus 119902

ln(119882

sum

119894=1

119901119902

119894) 119902 gt 0

119878(119879)

119902=

1

119902 minus 1

(1 minus

119882

sum

119894=1

119901119902

119894)

(3)

6 Mathematical Problems in Engineering

Table 1 PL fit to the distributions of casualties in terrorist attacks

Criterion Category SPL DPL1 DPL2(119862 ) (119862

1 1) (119862

2 2)

Weapon (119882119894)

1198821

00391 058491198822

00082 07514 00002 171981198823

00017 08660 00000 196241198824

00236 06877 00021 148901198825

00057 06909 00003 135791198826

00249 07801

Region (119877119894)

1198771

00438 101471198772

00214 06412 00009 169051198773

00174 119051198774

00065 090731198775

00012 121411198776

00091 08051 00006 170881198777

00009 114151198778

00193 07814 00008 206751198779

00036 08742 00001 2055211987710

00128 08561 00012 1657811987711

00091 06454 00004 1313211987712

00023 0995611987713

00001 15523

Target (119879119894)

1198791

00438 05144 00092 124881198792

00057 08884 00003 171801198793

00002 13589 00012 084581198794

00452 079231198795

00015 112591198796

00006 11152 00001 151621198797

00014 161061198798

00576 04694 00094 090701198799

00353 05669 00009 1629111987910

00090 09645 00031 1257311987911

00047 08991 00001 1835411987912

00036 07778 00000 1866711987913

00158 07353 00018 1610611987914

00462 0977411987915

00168 08610 00026 1682411987916

00234 0763511987917

00340 05892 00014 1517811987918

00225 07389 00074 1070811987919

00108 07918 00025 11650

Type (119860119894)

1198601

00049 07346 00000 179821198602

00014 15201 00001 265551198603

00059 08116 00001 206211198604

00261 06339 00016 148801198605

00305 071861198606

00116 073161198607

00072 09689 00014 162141198608

00471 07006 00034 20186

Mathematical Problems in Engineering 7

1980 1985 1990 1995 2000 2005 201005

05506

06507

07508

08509

0951

Year

Nor

mal

ized

entro

py

119878(119879)05

119878(119877)05

119878(119880)05

119878(119879)2

119878(119877)2

119878(119880)2

119878

Figure 7 Total Shannon Tsallis Renyi and Ubriaco normalizedentropies as a function of the year of the events 119884

119903 119902 = 05 and

119902 = 2

Tsallis entropy reduces to Renyi entropy when 119902 rarr 1 Tsallisentropy was applied to diffusion equations [42] and Fokker-Planck systems [43] Renyi entropy has an inverse power lawequilibrium distribution [44] and satisfies the zeroth law ofthermodynamics [45] The two parameters Sharma-Mittalentropy [46] is accepted as a generalization of Tsallis Renyiand Boltzmann-Gibbs entropies for limiting cases of theparameters [47]

Recently more general entropy measures have been pro-posed in the literature where the additivity axiom has beenrelaxed For instance Ubriaco [48] proposed the followingformula for the fractional entropy

119878(119880)

119902=

119882

sum

119894=1

(minus ln119901119894)119902

119901119894

(4)

that has the same properties as the Shannon entropy exceptadditivity

Applications of entropy in distinct complex systems canbe found in [49ndash57]

In Figure 7 the total Shannon normalized entropy (119878) isdepicted as well as Tsallisrsquo (119878(119879)

119902) Renyirsquos (119878(119877)

119902) and Ubriacorsquos

(119878(119880)

119902) for 119902 = 05 and 119902 = 2 as a function of the year of

the events 119884119903 Figures 9 10 and 11 show 119878

(119879)

119902 119878(119877)119902

and 119878(119880)119902

normalized entropies as a function of the year 119884119903 and

entropy parameter 01 le 119902 le 10In Figure 7 we observe two types of behavior namely

short- and long-term phenomena In what concerns shorttime behavior we verify peaks during 1983ndash1985 1997-19982004ndash2007 and minima at 1980 1995 and 2010 In whatconcerns long-time relationships a smooth decreasing isobserved for 119878(119879)

05 119878(119877)05

and 119878(119880)2

Removing the maxima andminima we get a time series for years 1981 1982 1986ndash1992 1994ndash1996 1999 2000 2002 2003 2008 and 2009(Figure 8) Largersmaller entropies correspond to chartscloserafar uniform distributions therefore seemingly wehave lessmore organized terrorist events in global terms

1980 1985 1990 1995 2000 2005 2010075

08

085

09

095

1

Year

Nor

mal

ized

entro

py

119878(119879)05

119878(119877)05

119878(119880)2

Lin Reg 119878(119879)05Lin Reg 119878(119877)05Lin Reg 119878(119880)2

Figure 8 Total Tsallis and Renyi normalized entropies (119902 = 05)

and Ubriacorsquos (119902 = 2) as a function of the events in years 1981 19821986ndash1992 1994ndash1996 1999 2000 2002 2003 2008 2009

1980 1985 1990 1995 2000 2005 2010

01234567

8910040506070809

1

Year

119902

119878(119879)119902

Figure 9 Total Tsallis normalized entropy 119878(119879)119902 as a function of the

year of the events 119884119903 and parameter 119902

These conclusions remain invariant for Figures 9 to 11 wherewe vary both the entropy definition and the parameter tuningTherefore we conclude that such results are robust againstsuch type of variations

As discussed in the previous section the statistical dis-tributions of real data can be approximated by either singleor double PLs In the latter case we study not only the totalentropy but also the entropy associated to each part of thedistributions Therefore we compute the entropy associatedto DPL1 and DPL2 that approximate the first and secondpart of the distributions respectively When adopting thisprocedure we are restricted to the Shannon and Ubriacoentropies as Tsallisrsquo and Renyirsquos do not admit the associativitydescribed above

In the sequel we present several results of the analysistaking into account the grouping criteria 119884

119903 119882119894 119877119895 119879119896 and

119860119897 Figure 12 depicts the Shannon entropy versus parameter

for all statistical distributions The black squares (denotedldquoSPLrdquo) correspond to the plot of versus the total entropyof the respective distributions 119878 The black circles (denotedldquoDPL1rdquo) are the plot of

1versus the entropy associated to the

first parts of the distributions 1198781 The white circles (denoted

ldquoDPL2rdquo) represent the plot of 2versus the entropy associated

8 Mathematical Problems in Engineering

1980 1985 1990 1995 2000 2005 2010

012345678910040608

1

Year

119902

119878(119877)119902

Figure 10 Total Renyi normalized entropy 119878(119877)119902 as a function of the

year of the events 119884119903 and parameter 119902

1980 1985 1990 1995 2000 2005 2010

012345678910

0402

0608

1

Year

119902

119878(119880)119902

Figure 11 Total Ubriaco normalized entropy 119878(119880)119902

as a function ofthe year of the events 119884

119903 and parameter 119902

to the second parts of the distributions 1198782 As can be seen for

the distributions that behave as single PLs higher entropiescorrespond to the lower values of the parameter and the twoparameters are linearly related A similar pattern is observedfor the parameters corresponding to DPL1 For DPL2 theparameter

2increases with entropy but the almost linear

relation between both parameters remainsFigure 13 shows identical results for the Ubriaco entropy

Theplot corresponds to 119902 = 05 nevertheless identical resultsare obtained for other values

For all DPLs related to terrorist events 119899 (119899 = 1 60)we find that the parameters (119862

119894 119894) 119894 = 1 2 obey the follow-

ing relation

1119899

10038161003816100381610038161003816log1198621119899|119901= 2119899

10038161003816100381610038161003816log1198622119899|119901+ 120598119899 (5)

where 119901 = minus1612 The mean value of 120598119899is 120598 = 0002 and the

corresponding standard deviation is 120590120598= 0036 Moreover

for the analyzed data we find 119894119899| log119862

119894119899|119901= 119896119894119899 Parameters

119896119894119899are approximately constant with mean value 119896 = 0277

and standard deviation 120590119896= 006

It is worth noticing that (5) is similar to Poissonrsquos law ofan adiabatic reversible process involving ideal gases given by

1198751119881120574

1= 1198752119881120574

2 (6)

where variables 119875119894and 119881

119894represent pressure and specific

volume respectively Equation (6) implies that 119875119881120574 = constParameter 120574 is called Poissonrsquos coefficient taking values 120574 =

53 ≃ 167 and 120574 = 75 = 140 for monoatomic and diatomicgases respectively Additionally it should be noticed that theabsolute value of the exponent 119901 is very similar to the oneobserved for an ideal gas undergoing a reversible adiabaticprocess

0

05

1

15

2

25

3

0 05 1 15 2 25Shannon entropy

SPLDPL1

DPL2

Figure 12 Parameters versus Shannon entropies for the dis-tributions corresponding to the criteria 119884

119903 119882119894 119877119895 119879119896 and 119860

119897

ldquoSPLrdquo corresponds to (119878 ) ldquoDPL1rdquo corresponds to (1198781 1) ldquoDPL2rdquo

corresponds to (1198782 2)

0

05

1

15

2

25

3

0 02 04 06 08 1 12 14 16

SPL DPL1DPL2

Ubriaco entropy (119902 = 05)

Figure 13 Parameters versus Ubriaco entropies for the distri-butions corresponding to the criteria 119884

119903 119882119894 119877119895 119879119896 and 119860

119897 ldquoSPLrdquo

corresponds to (119878(119880)119902 ) ldquoDPL1rdquo corresponds to (119878(119880)

1199021 1) ldquoDPL2rdquo

corresponds to (119878(119880)1199022 2) 119902 = 05

5 Conclusions

PLs have been widely reported in the modeling of distinctphenomena and have been associated with long memorybehavior self-similarity fractal structures and fractionalcalculus

In this paper we reviewed interesting and importantresults on PLs distributions and their applications to the

Mathematical Problems in Engineering 9

modeling of the number of victims in catastrophic eventsWe found double PL behavior in real data of catastrophicoccurrences in particular terrorismWe have plotted the twoPLs parameters (119862

119894 119894) 119894 = 1 2 corresponding to certain

events and observed an interesting pattern in the chartwhere the lines that connect each pair of points defining thedouble PLs are almost aligned to each other We have alsocomputed the entropy of the data sets This complementaryanalysis of the numerical data revealed extra relationships butthe fact is that these phenomena have a dense and rich volumeof characteristics and further research efforts are needed to adeeper understanding

References

[1] J Tenreiro Machado F B Duarte and G M Duarte ldquoPowerlaw analysis of financial index dynamicsrdquo Discrete Dynamics inNature and Society vol 2012 Article ID 120518 12 pages 2012

[2] J Tenreiro Machado ldquoAnd I say to myself lsquoWhat a fractionalworlrsquordquo Fractional Calculus and Applied Analysis vol 14 no 4pp 635ndash654 2011

[3] E Scalas R Gorenflo and FMainardi ldquoFractional calculus andcontinuous-time financerdquo Physica A vol 284 no 1ndash4 pp 376ndash384 2000

[4] F Mainardi M Raberto R Gorenflo and E Scalas ldquoFractionalcalculus and continuous-time finance II the waiting-time dis-tributionrdquo Physica A vol 287 no 3-4 pp 468ndash481 2000

[5] B J West ldquoFractal physiology and the fractional calculus aperspectiverdquo Frontiers in Physiology vol 1 p 12 2010

[6] B Gutenberg and R F Richter ldquoFrequency of earthquakes incaliforniardquo Bulletin of the Seismological Society of America vol34 pp 185ndash188 1944

[7] L F Richardson ldquoVariation of the Frequency of Fatal QuarrelswithMagnituderdquo Journal of the American Statistical Associationvol 43 pp 523ndash546 1948

[8] J M Carlson and J S Langer ldquoMechanical model of an earth-quake faultrdquo Physical Review A vol 40 no 11 pp 6470ndash64841989

[9] D C Roberts and D L Turcotte ldquoFractality and self-organizedcriticality of warsrdquo Fractals vol 6 no 4 pp 351ndash357 1998

[10] D R Davis and D E Weinstein ldquoBones bombs and breakpoints the geography of economic activityrdquo The AmericanEconomic Review vol 92 no 5 pp 1269ndash1289 2002

[11] L E Cederman ldquoModeling the size of wars from billiard ballsto sandpilesrdquo American Political Science Review vol 97 no 1pp 135ndash150 2003

[12] J C Bohorquez S Gourley A R Dixon M Spagat and N FJohnson ldquoCommon ecology quantifies human insurgencyrdquoNature vol 462 no 7275 pp 911ndash914 2009

[13] V Pareto Cours drsquoEconomie Politique Droz Geneva Switzer-land 1896

[14] F Auerbach ldquoDas Gesetz der BelvolkerungskoncentrationrdquoPetermanns GeographischeMitteilungen vol 59 pp 74ndash76 1913

[15] J B Estoup Gammes Stenographiques Institut de France 1916[16] G Zipf Selective Studies and the Principle of Relative Frequency

in Language Harvard University Press Cambridge Mass USA1932

[17] G Zipf Human Behavior and the Priciple of Least EffortAddison-Wesley Cambridge Mass USA 1949

[18] W Li ldquoReferences on Zipf rsquos lawrdquo httpwwwsorinsolomonnetsimsorinccszipfreferences20on20zipf27s20lawhtm

[19] D Sornette Critical Phenomena in Natural Sciences chapter 14Springer Heidelberg Germany 2nd edition 2003

[20] C M A Pinto A M Lopes and J A Tenreiro Machado ldquoAreview of power laws in real life phenomenardquo Communicationsin Nonlinear Science and Numerical Simulations vol 17 no 9pp 3558ndash3578 2012

[21] J Tuldava ldquoThe frequency spectrum of text and vocabularyrdquoJournal of Quantitative Linguistics vol 3 no 1 pp 38ndash50 1996

[22] GMartınez-Mekler R AlvarezMartınezM Beltran del Rıo RMansilla P Miramontes and G Cocho ldquoUniversality of rank-ordering distributions in the arts and sciencesrdquo PLoS One vol4 no 3 Article ID e4791 2009

[23] R F I Cancho and R V Sole ldquoTwo regimes in the frequency ofwords and the origins of complex lexicons Zipf rsquos law revisitedrdquoJournal of Quantitative Linguistics vol 8 no 3 pp 165ndash1732001

[24] R Alvarez-Martinez G Martinez-Mekler and G CochoaldquoOrder-disorder transition in conflicting dynamics leading torank-frequency generalized beta distributionsrdquo Physica A vol390 no 1 pp 120ndash130 2011

[25] L F Richardson Statistics of Deadly Quarrels QuadrangleBooks Chicago Ill USA 1960

[26] F Guzzetti ldquoLandslide fatalities and the evaluation of landsliderisk in Italyrdquo Engineering Geology vol 58 no 2 pp 89ndash1072000

[27] S N Jonkman ldquoGlobal perspectives on loss of human lifecaused by floodsrdquo Natural Hazards vol 34 no 2 pp 151ndash1752005

[28] N F Johnson M Spagat J A Restrepo et al ldquoUniversal pat-terns underlying ongoingwars and terrorismrdquo httparxivorgabsphysics0605035

[29] D S Geller and J David Singer Nations at War A ScientificStudy of International Conflict Cambridge Studies in Interna-tional Relations Cambridge University Press Cambridge UK1998

[30] O Becerra N Johnson P Meier J Restrepo and M Spa-gat ldquoNatural disasters casualties and power laws a com-parative analysis with armed conflictrdquo in Proceedings of theAnnual Meeting of the American Political Science AssociationLoews Philadelphia and the Pennsylvania Convention Cen-ter Philadelphia Pa USA 2006 httpwwwallacademiccommetap151714 indexhtml

[31] M E J Newman ldquoPower laws Pareto distributions and Zipf rsquoslawrdquo Contemporary Physics vol 46 pp 323ndash351 2005

[32] A Clauset andM Young ldquoScale invariance in global terrorismrdquohttparxivorgabsphysics0502014

[33] K T Bogen and E D Jones ldquoRisks of mortality and morbidityfrom worldwide terrorism 1968ndash2004rdquo Risk Analysis vol 26no 1 pp 45ndash59 2006

[34] A Clauset M Young and K S Gleditsch ldquoOn the frequency ofsevere terrorist eventsrdquo Journal of Conflict Resolution vol 51 no1 pp 58ndash87 2007

[35] EM-DAT ldquoTheOFDACRED International Disaster DatabaserdquoUniversite catholique de Louvain Brussels Belgium httpwwwemdatnet

[36] National Consortium for the Study of Terrorism and Responsesto Terrorism (START) ldquoGlobal Terrorism Databaserdquo 2011httpwwwstartumdedugtd

[37] R ClausiusTheMechanicalTheory of Heat with Its Applicationsto the Steam Engine and to Physical Properties of Bodies Johnvan Voorst London UK 1865

[38] L Boltzmann Vorlesungen ber Gastheorie J A Barth LeipzigGermany 1896

10 Mathematical Problems in Engineering

[39] C E Shannon ldquoAmathematical theory of communicationrdquoTheBell System Technical Journal vol 27 pp 379ndash423 623ndash6561948

[40] E T Jaynes ldquoInformation theory and statistical mechanicsrdquo vol106 pp 620ndash630 1957

[41] A I Khinchin Mathematical Foundations of Information The-ory Dover New York NY USA 1957

[42] A R Plastino M Casas and A Plastino ldquoA nonextensive max-imum entropy approach to a family of nonlinear reaction-diffusion equationsrdquo Physica A vol 280 no 3 pp 289ndash3032000

[43] T D Frank and A Daffertshofer ldquo119867-theorem for nonlinearFokker-Planck equations related to generalized thermostatis-ticsrdquo Physica A vol 295 no 3-4 pp 455ndash474 2001

[44] E K Lenzi R SMendes and L R da Silva ldquoStatisticalmechan-ics based onRenyi entropyrdquo Physica A vol 280 no 3-4 pp 337ndash345 2000

[45] A S Parvan andT S Biro ldquoExtensive Renyi statistics fromnon-extensive entropyrdquo Physics Letters A vol 340 no 5-6 pp 375ndash387 2005

[46] B D Sharma and D P Mittal ldquoNew nonadditive measures ofentropy for discrete probability distributionsrdquo Journal of Math-ematical Sciences vol 10 pp 28ndash40 1975

[47] T D Frank and A Daffertshofer ldquoExact time-dependentsolutions of the Renyi Fokker-Planck equation and the Fokker-Planck equations related to the entropies proposed by Sharmaand Mittalrdquo Physica A vol 285 no 3 pp 351ndash366 2000

[48] M R Ubriaco ldquoEntropies based on fractional calculusrdquo PhysicsLetters A vol 373 no 30 pp 2516ndash2519 2009

[49] P T Landsberg and V Vedral ldquoDistributions and channelcapacities in generalized statistical mechanicsrdquo Physics LettersA vol 247 no 3 pp 211ndash217 1998

[50] A Plastino and A R Plastino ldquoTsallis Entropy and Jaynesrsquoinformation theory formalismrdquo Brazilian Journal of Physics vol29 no 1 pp 50ndash60 1999

[51] X Li C Essex M Davison K H Hoffmann and C SchulzkyldquoFractional diffusion irreversibility and entropyrdquo Journal ofNon-Equilibrium Thermodynamics vol 28 no 3 pp 279ndash2912003

[52] H J Haubold A M Mathai and R K Saxena ldquoBoltzmann-Gibbs entropy versus Tsallis entropy recent contributions toresolving the argument of Einstein concerning ldquoneither HerrBoltzmann nor Herr Planck has given a definition ofWrdquo Essayreviewrdquo Astrophysics and Space Science vol 290 no 3-4 pp241ndash245 2004

[53] P N Rathie and S da Silva ldquoShannon Levy and Tsallis a noterdquoApplied Mathematical Sciences vol 2 no 28 pp 1359ndash13632008

[54] R M Gray Entropy and Information Theory Springer NewYork NY USA 2009

[55] J A Tenreiro Machado A C Costa and M F M Lima ldquoDy-namical analysis of compositionsrdquoNonlinear Dynamics vol 65no 4 pp 339ndash412 2011

[56] J A T Machado A C Costa and M D Quelhas ldquoEntropyanalysis of the DNA code dynamics in human chromosomesrdquoComputers amp Mathematics with Applications vol 62 no 3 pp1612ndash1617 2011

[57] J A TenreiroMachado A C Costa andM D Quelhas ldquoShan-non Renyie and Tsallis entropy analysis of DNA using phaseplanerdquo Nonlinear Analysis Real World Applications vol 12 no6 pp 3135ndash3144 2011

Mathematical Problems in Engineering 5

1980 1990 2000 2010Year

11986211198622

10minus4

10minus3

10minus2

10minus1

119862

Figure 4 Time evolution of parameters 119862119894 119894 = 1 2 of the DPLs

corresponding to terrorist attacks over the period 1980ndash2010 119884119903

1980 1990 2000 2010Year

0

05

1

15

2

25

3

12057211205722

Figure 5 Time evolution of parameters 119894 119894 = 1 2 of the DPLs

corresponding to terrorist attacks over the period 1980ndash2010 119884119903

BombingExplosion FacilityInfrastructure Attack Hijack-ing Hostage Taking (Barricade Incident) Hostage Taking(Kidnapping) and Unarmed Assault

Most cases are characterized by DPLs However in afew situations a SPL fits better to the data The main resultsare summarized in Table 1 Moreover we observed that theparameters corresponding to all distributions characterizedby DPLs display a pattern similar to the ones mentionedpreviously (Figures 3 and 6) where the lines connecting theslopes and intercepts of DPL1 (

1 1198621) to its companionDPL2

(2 1198622) for the same data set have identical orientation in

the (119862 ) Cartesian space

4 Entropy of Catastrophic Phenomena

In this section we analyse the entropy of data collected fromthe GTD database that is data related to terrorism To

1980 1981 1982 1983 1984 19851986 1987 1988 1989 1990 19911992 1994 1995 1996 1997 19981999 2000 2001 2002 2003 20042005 2006 2007 2008 2009 2010

0

05

1

15

2

25

3

10minus4 10minus3 10minus2 10minus1

119862

Figure 6 Locus of the parameters (119862119894 ) 119894 = 1 2 that characterize

the distributions of terrorist attacks over the period 1980ndash2010 119884119903

calculate the entropies we construct histograms of relativefrequencies using bins of width one (one casualty) andapproximate the probabilities 119901

119894by the relative frequencies

We present results obtained for terrorist events grouped byyear (119884

119903) as defined in the previous section Nevertheless it

should be noticed that similar results are obtained for all otherhuman-made and natural hazards

Clausius [37] and Boltzmann [38] were the first authorsto define entropy in the field of thermodynamics Later onShannon [39] and Jaynes [40] applied their results to infor-mation theory [41]

The most celebrated entropy is the so-called Shannonentropy 119878 defined by

119878 = minus

119882

sum

119894=1

119901119894ln119901119894 (2)

The Shannon entropy represents the expected value of theinformation minus ln119901

119894 Therefore for the uniform probability

distribution we have 119901119894= 119882minus1 and the Shannon entropy

takes its maximum value 119878 = ln119882 yielding the Boltzmannrsquosfamous formula up to a multiplicative factor 119896 denoting theBoltzmann constant Thus in thermodynamic equilibriumthe Shannon entropy can be identified as the ldquophysicalentropyrdquo of the system

Renyi and Tsallis entropies are generalizations of Shan-nonrsquos entropy and are given by respectively

119878(119877)

119902=

1

1 minus 119902

ln(119882

sum

119894=1

119901119902

119894) 119902 gt 0

119878(119879)

119902=

1

119902 minus 1

(1 minus

119882

sum

119894=1

119901119902

119894)

(3)

6 Mathematical Problems in Engineering

Table 1 PL fit to the distributions of casualties in terrorist attacks

Criterion Category SPL DPL1 DPL2(119862 ) (119862

1 1) (119862

2 2)

Weapon (119882119894)

1198821

00391 058491198822

00082 07514 00002 171981198823

00017 08660 00000 196241198824

00236 06877 00021 148901198825

00057 06909 00003 135791198826

00249 07801

Region (119877119894)

1198771

00438 101471198772

00214 06412 00009 169051198773

00174 119051198774

00065 090731198775

00012 121411198776

00091 08051 00006 170881198777

00009 114151198778

00193 07814 00008 206751198779

00036 08742 00001 2055211987710

00128 08561 00012 1657811987711

00091 06454 00004 1313211987712

00023 0995611987713

00001 15523

Target (119879119894)

1198791

00438 05144 00092 124881198792

00057 08884 00003 171801198793

00002 13589 00012 084581198794

00452 079231198795

00015 112591198796

00006 11152 00001 151621198797

00014 161061198798

00576 04694 00094 090701198799

00353 05669 00009 1629111987910

00090 09645 00031 1257311987911

00047 08991 00001 1835411987912

00036 07778 00000 1866711987913

00158 07353 00018 1610611987914

00462 0977411987915

00168 08610 00026 1682411987916

00234 0763511987917

00340 05892 00014 1517811987918

00225 07389 00074 1070811987919

00108 07918 00025 11650

Type (119860119894)

1198601

00049 07346 00000 179821198602

00014 15201 00001 265551198603

00059 08116 00001 206211198604

00261 06339 00016 148801198605

00305 071861198606

00116 073161198607

00072 09689 00014 162141198608

00471 07006 00034 20186

Mathematical Problems in Engineering 7

1980 1985 1990 1995 2000 2005 201005

05506

06507

07508

08509

0951

Year

Nor

mal

ized

entro

py

119878(119879)05

119878(119877)05

119878(119880)05

119878(119879)2

119878(119877)2

119878(119880)2

119878

Figure 7 Total Shannon Tsallis Renyi and Ubriaco normalizedentropies as a function of the year of the events 119884

119903 119902 = 05 and

119902 = 2

Tsallis entropy reduces to Renyi entropy when 119902 rarr 1 Tsallisentropy was applied to diffusion equations [42] and Fokker-Planck systems [43] Renyi entropy has an inverse power lawequilibrium distribution [44] and satisfies the zeroth law ofthermodynamics [45] The two parameters Sharma-Mittalentropy [46] is accepted as a generalization of Tsallis Renyiand Boltzmann-Gibbs entropies for limiting cases of theparameters [47]

Recently more general entropy measures have been pro-posed in the literature where the additivity axiom has beenrelaxed For instance Ubriaco [48] proposed the followingformula for the fractional entropy

119878(119880)

119902=

119882

sum

119894=1

(minus ln119901119894)119902

119901119894

(4)

that has the same properties as the Shannon entropy exceptadditivity

Applications of entropy in distinct complex systems canbe found in [49ndash57]

In Figure 7 the total Shannon normalized entropy (119878) isdepicted as well as Tsallisrsquo (119878(119879)

119902) Renyirsquos (119878(119877)

119902) and Ubriacorsquos

(119878(119880)

119902) for 119902 = 05 and 119902 = 2 as a function of the year of

the events 119884119903 Figures 9 10 and 11 show 119878

(119879)

119902 119878(119877)119902

and 119878(119880)119902

normalized entropies as a function of the year 119884119903 and

entropy parameter 01 le 119902 le 10In Figure 7 we observe two types of behavior namely

short- and long-term phenomena In what concerns shorttime behavior we verify peaks during 1983ndash1985 1997-19982004ndash2007 and minima at 1980 1995 and 2010 In whatconcerns long-time relationships a smooth decreasing isobserved for 119878(119879)

05 119878(119877)05

and 119878(119880)2

Removing the maxima andminima we get a time series for years 1981 1982 1986ndash1992 1994ndash1996 1999 2000 2002 2003 2008 and 2009(Figure 8) Largersmaller entropies correspond to chartscloserafar uniform distributions therefore seemingly wehave lessmore organized terrorist events in global terms

1980 1985 1990 1995 2000 2005 2010075

08

085

09

095

1

Year

Nor

mal

ized

entro

py

119878(119879)05

119878(119877)05

119878(119880)2

Lin Reg 119878(119879)05Lin Reg 119878(119877)05Lin Reg 119878(119880)2

Figure 8 Total Tsallis and Renyi normalized entropies (119902 = 05)

and Ubriacorsquos (119902 = 2) as a function of the events in years 1981 19821986ndash1992 1994ndash1996 1999 2000 2002 2003 2008 2009

1980 1985 1990 1995 2000 2005 2010

01234567

8910040506070809

1

Year

119902

119878(119879)119902

Figure 9 Total Tsallis normalized entropy 119878(119879)119902 as a function of the

year of the events 119884119903 and parameter 119902

These conclusions remain invariant for Figures 9 to 11 wherewe vary both the entropy definition and the parameter tuningTherefore we conclude that such results are robust againstsuch type of variations

As discussed in the previous section the statistical dis-tributions of real data can be approximated by either singleor double PLs In the latter case we study not only the totalentropy but also the entropy associated to each part of thedistributions Therefore we compute the entropy associatedto DPL1 and DPL2 that approximate the first and secondpart of the distributions respectively When adopting thisprocedure we are restricted to the Shannon and Ubriacoentropies as Tsallisrsquo and Renyirsquos do not admit the associativitydescribed above

In the sequel we present several results of the analysistaking into account the grouping criteria 119884

119903 119882119894 119877119895 119879119896 and

119860119897 Figure 12 depicts the Shannon entropy versus parameter

for all statistical distributions The black squares (denotedldquoSPLrdquo) correspond to the plot of versus the total entropyof the respective distributions 119878 The black circles (denotedldquoDPL1rdquo) are the plot of

1versus the entropy associated to the

first parts of the distributions 1198781 The white circles (denoted

ldquoDPL2rdquo) represent the plot of 2versus the entropy associated

8 Mathematical Problems in Engineering

1980 1985 1990 1995 2000 2005 2010

012345678910040608

1

Year

119902

119878(119877)119902

Figure 10 Total Renyi normalized entropy 119878(119877)119902 as a function of the

year of the events 119884119903 and parameter 119902

1980 1985 1990 1995 2000 2005 2010

012345678910

0402

0608

1

Year

119902

119878(119880)119902

Figure 11 Total Ubriaco normalized entropy 119878(119880)119902

as a function ofthe year of the events 119884

119903 and parameter 119902

to the second parts of the distributions 1198782 As can be seen for

the distributions that behave as single PLs higher entropiescorrespond to the lower values of the parameter and the twoparameters are linearly related A similar pattern is observedfor the parameters corresponding to DPL1 For DPL2 theparameter

2increases with entropy but the almost linear

relation between both parameters remainsFigure 13 shows identical results for the Ubriaco entropy

Theplot corresponds to 119902 = 05 nevertheless identical resultsare obtained for other values

For all DPLs related to terrorist events 119899 (119899 = 1 60)we find that the parameters (119862

119894 119894) 119894 = 1 2 obey the follow-

ing relation

1119899

10038161003816100381610038161003816log1198621119899|119901= 2119899

10038161003816100381610038161003816log1198622119899|119901+ 120598119899 (5)

where 119901 = minus1612 The mean value of 120598119899is 120598 = 0002 and the

corresponding standard deviation is 120590120598= 0036 Moreover

for the analyzed data we find 119894119899| log119862

119894119899|119901= 119896119894119899 Parameters

119896119894119899are approximately constant with mean value 119896 = 0277

and standard deviation 120590119896= 006

It is worth noticing that (5) is similar to Poissonrsquos law ofan adiabatic reversible process involving ideal gases given by

1198751119881120574

1= 1198752119881120574

2 (6)

where variables 119875119894and 119881

119894represent pressure and specific

volume respectively Equation (6) implies that 119875119881120574 = constParameter 120574 is called Poissonrsquos coefficient taking values 120574 =

53 ≃ 167 and 120574 = 75 = 140 for monoatomic and diatomicgases respectively Additionally it should be noticed that theabsolute value of the exponent 119901 is very similar to the oneobserved for an ideal gas undergoing a reversible adiabaticprocess

0

05

1

15

2

25

3

0 05 1 15 2 25Shannon entropy

SPLDPL1

DPL2

Figure 12 Parameters versus Shannon entropies for the dis-tributions corresponding to the criteria 119884

119903 119882119894 119877119895 119879119896 and 119860

119897

ldquoSPLrdquo corresponds to (119878 ) ldquoDPL1rdquo corresponds to (1198781 1) ldquoDPL2rdquo

corresponds to (1198782 2)

0

05

1

15

2

25

3

0 02 04 06 08 1 12 14 16

SPL DPL1DPL2

Ubriaco entropy (119902 = 05)

Figure 13 Parameters versus Ubriaco entropies for the distri-butions corresponding to the criteria 119884

119903 119882119894 119877119895 119879119896 and 119860

119897 ldquoSPLrdquo

corresponds to (119878(119880)119902 ) ldquoDPL1rdquo corresponds to (119878(119880)

1199021 1) ldquoDPL2rdquo

corresponds to (119878(119880)1199022 2) 119902 = 05

5 Conclusions

PLs have been widely reported in the modeling of distinctphenomena and have been associated with long memorybehavior self-similarity fractal structures and fractionalcalculus

In this paper we reviewed interesting and importantresults on PLs distributions and their applications to the

Mathematical Problems in Engineering 9

modeling of the number of victims in catastrophic eventsWe found double PL behavior in real data of catastrophicoccurrences in particular terrorismWe have plotted the twoPLs parameters (119862

119894 119894) 119894 = 1 2 corresponding to certain

events and observed an interesting pattern in the chartwhere the lines that connect each pair of points defining thedouble PLs are almost aligned to each other We have alsocomputed the entropy of the data sets This complementaryanalysis of the numerical data revealed extra relationships butthe fact is that these phenomena have a dense and rich volumeof characteristics and further research efforts are needed to adeeper understanding

References

[1] J Tenreiro Machado F B Duarte and G M Duarte ldquoPowerlaw analysis of financial index dynamicsrdquo Discrete Dynamics inNature and Society vol 2012 Article ID 120518 12 pages 2012

[2] J Tenreiro Machado ldquoAnd I say to myself lsquoWhat a fractionalworlrsquordquo Fractional Calculus and Applied Analysis vol 14 no 4pp 635ndash654 2011

[3] E Scalas R Gorenflo and FMainardi ldquoFractional calculus andcontinuous-time financerdquo Physica A vol 284 no 1ndash4 pp 376ndash384 2000

[4] F Mainardi M Raberto R Gorenflo and E Scalas ldquoFractionalcalculus and continuous-time finance II the waiting-time dis-tributionrdquo Physica A vol 287 no 3-4 pp 468ndash481 2000

[5] B J West ldquoFractal physiology and the fractional calculus aperspectiverdquo Frontiers in Physiology vol 1 p 12 2010

[6] B Gutenberg and R F Richter ldquoFrequency of earthquakes incaliforniardquo Bulletin of the Seismological Society of America vol34 pp 185ndash188 1944

[7] L F Richardson ldquoVariation of the Frequency of Fatal QuarrelswithMagnituderdquo Journal of the American Statistical Associationvol 43 pp 523ndash546 1948

[8] J M Carlson and J S Langer ldquoMechanical model of an earth-quake faultrdquo Physical Review A vol 40 no 11 pp 6470ndash64841989

[9] D C Roberts and D L Turcotte ldquoFractality and self-organizedcriticality of warsrdquo Fractals vol 6 no 4 pp 351ndash357 1998

[10] D R Davis and D E Weinstein ldquoBones bombs and breakpoints the geography of economic activityrdquo The AmericanEconomic Review vol 92 no 5 pp 1269ndash1289 2002

[11] L E Cederman ldquoModeling the size of wars from billiard ballsto sandpilesrdquo American Political Science Review vol 97 no 1pp 135ndash150 2003

[12] J C Bohorquez S Gourley A R Dixon M Spagat and N FJohnson ldquoCommon ecology quantifies human insurgencyrdquoNature vol 462 no 7275 pp 911ndash914 2009

[13] V Pareto Cours drsquoEconomie Politique Droz Geneva Switzer-land 1896

[14] F Auerbach ldquoDas Gesetz der BelvolkerungskoncentrationrdquoPetermanns GeographischeMitteilungen vol 59 pp 74ndash76 1913

[15] J B Estoup Gammes Stenographiques Institut de France 1916[16] G Zipf Selective Studies and the Principle of Relative Frequency

in Language Harvard University Press Cambridge Mass USA1932

[17] G Zipf Human Behavior and the Priciple of Least EffortAddison-Wesley Cambridge Mass USA 1949

[18] W Li ldquoReferences on Zipf rsquos lawrdquo httpwwwsorinsolomonnetsimsorinccszipfreferences20on20zipf27s20lawhtm

[19] D Sornette Critical Phenomena in Natural Sciences chapter 14Springer Heidelberg Germany 2nd edition 2003

[20] C M A Pinto A M Lopes and J A Tenreiro Machado ldquoAreview of power laws in real life phenomenardquo Communicationsin Nonlinear Science and Numerical Simulations vol 17 no 9pp 3558ndash3578 2012

[21] J Tuldava ldquoThe frequency spectrum of text and vocabularyrdquoJournal of Quantitative Linguistics vol 3 no 1 pp 38ndash50 1996

[22] GMartınez-Mekler R AlvarezMartınezM Beltran del Rıo RMansilla P Miramontes and G Cocho ldquoUniversality of rank-ordering distributions in the arts and sciencesrdquo PLoS One vol4 no 3 Article ID e4791 2009

[23] R F I Cancho and R V Sole ldquoTwo regimes in the frequency ofwords and the origins of complex lexicons Zipf rsquos law revisitedrdquoJournal of Quantitative Linguistics vol 8 no 3 pp 165ndash1732001

[24] R Alvarez-Martinez G Martinez-Mekler and G CochoaldquoOrder-disorder transition in conflicting dynamics leading torank-frequency generalized beta distributionsrdquo Physica A vol390 no 1 pp 120ndash130 2011

[25] L F Richardson Statistics of Deadly Quarrels QuadrangleBooks Chicago Ill USA 1960

[26] F Guzzetti ldquoLandslide fatalities and the evaluation of landsliderisk in Italyrdquo Engineering Geology vol 58 no 2 pp 89ndash1072000

[27] S N Jonkman ldquoGlobal perspectives on loss of human lifecaused by floodsrdquo Natural Hazards vol 34 no 2 pp 151ndash1752005

[28] N F Johnson M Spagat J A Restrepo et al ldquoUniversal pat-terns underlying ongoingwars and terrorismrdquo httparxivorgabsphysics0605035

[29] D S Geller and J David Singer Nations at War A ScientificStudy of International Conflict Cambridge Studies in Interna-tional Relations Cambridge University Press Cambridge UK1998

[30] O Becerra N Johnson P Meier J Restrepo and M Spa-gat ldquoNatural disasters casualties and power laws a com-parative analysis with armed conflictrdquo in Proceedings of theAnnual Meeting of the American Political Science AssociationLoews Philadelphia and the Pennsylvania Convention Cen-ter Philadelphia Pa USA 2006 httpwwwallacademiccommetap151714 indexhtml

[31] M E J Newman ldquoPower laws Pareto distributions and Zipf rsquoslawrdquo Contemporary Physics vol 46 pp 323ndash351 2005

[32] A Clauset andM Young ldquoScale invariance in global terrorismrdquohttparxivorgabsphysics0502014

[33] K T Bogen and E D Jones ldquoRisks of mortality and morbidityfrom worldwide terrorism 1968ndash2004rdquo Risk Analysis vol 26no 1 pp 45ndash59 2006

[34] A Clauset M Young and K S Gleditsch ldquoOn the frequency ofsevere terrorist eventsrdquo Journal of Conflict Resolution vol 51 no1 pp 58ndash87 2007

[35] EM-DAT ldquoTheOFDACRED International Disaster DatabaserdquoUniversite catholique de Louvain Brussels Belgium httpwwwemdatnet

[36] National Consortium for the Study of Terrorism and Responsesto Terrorism (START) ldquoGlobal Terrorism Databaserdquo 2011httpwwwstartumdedugtd

[37] R ClausiusTheMechanicalTheory of Heat with Its Applicationsto the Steam Engine and to Physical Properties of Bodies Johnvan Voorst London UK 1865

[38] L Boltzmann Vorlesungen ber Gastheorie J A Barth LeipzigGermany 1896

10 Mathematical Problems in Engineering

[39] C E Shannon ldquoAmathematical theory of communicationrdquoTheBell System Technical Journal vol 27 pp 379ndash423 623ndash6561948

[40] E T Jaynes ldquoInformation theory and statistical mechanicsrdquo vol106 pp 620ndash630 1957

[41] A I Khinchin Mathematical Foundations of Information The-ory Dover New York NY USA 1957

[42] A R Plastino M Casas and A Plastino ldquoA nonextensive max-imum entropy approach to a family of nonlinear reaction-diffusion equationsrdquo Physica A vol 280 no 3 pp 289ndash3032000

[43] T D Frank and A Daffertshofer ldquo119867-theorem for nonlinearFokker-Planck equations related to generalized thermostatis-ticsrdquo Physica A vol 295 no 3-4 pp 455ndash474 2001

[44] E K Lenzi R SMendes and L R da Silva ldquoStatisticalmechan-ics based onRenyi entropyrdquo Physica A vol 280 no 3-4 pp 337ndash345 2000

[45] A S Parvan andT S Biro ldquoExtensive Renyi statistics fromnon-extensive entropyrdquo Physics Letters A vol 340 no 5-6 pp 375ndash387 2005

[46] B D Sharma and D P Mittal ldquoNew nonadditive measures ofentropy for discrete probability distributionsrdquo Journal of Math-ematical Sciences vol 10 pp 28ndash40 1975

[47] T D Frank and A Daffertshofer ldquoExact time-dependentsolutions of the Renyi Fokker-Planck equation and the Fokker-Planck equations related to the entropies proposed by Sharmaand Mittalrdquo Physica A vol 285 no 3 pp 351ndash366 2000

[48] M R Ubriaco ldquoEntropies based on fractional calculusrdquo PhysicsLetters A vol 373 no 30 pp 2516ndash2519 2009

[49] P T Landsberg and V Vedral ldquoDistributions and channelcapacities in generalized statistical mechanicsrdquo Physics LettersA vol 247 no 3 pp 211ndash217 1998

[50] A Plastino and A R Plastino ldquoTsallis Entropy and Jaynesrsquoinformation theory formalismrdquo Brazilian Journal of Physics vol29 no 1 pp 50ndash60 1999

[51] X Li C Essex M Davison K H Hoffmann and C SchulzkyldquoFractional diffusion irreversibility and entropyrdquo Journal ofNon-Equilibrium Thermodynamics vol 28 no 3 pp 279ndash2912003

[52] H J Haubold A M Mathai and R K Saxena ldquoBoltzmann-Gibbs entropy versus Tsallis entropy recent contributions toresolving the argument of Einstein concerning ldquoneither HerrBoltzmann nor Herr Planck has given a definition ofWrdquo Essayreviewrdquo Astrophysics and Space Science vol 290 no 3-4 pp241ndash245 2004

[53] P N Rathie and S da Silva ldquoShannon Levy and Tsallis a noterdquoApplied Mathematical Sciences vol 2 no 28 pp 1359ndash13632008

[54] R M Gray Entropy and Information Theory Springer NewYork NY USA 2009

[55] J A Tenreiro Machado A C Costa and M F M Lima ldquoDy-namical analysis of compositionsrdquoNonlinear Dynamics vol 65no 4 pp 339ndash412 2011

[56] J A T Machado A C Costa and M D Quelhas ldquoEntropyanalysis of the DNA code dynamics in human chromosomesrdquoComputers amp Mathematics with Applications vol 62 no 3 pp1612ndash1617 2011

[57] J A TenreiroMachado A C Costa andM D Quelhas ldquoShan-non Renyie and Tsallis entropy analysis of DNA using phaseplanerdquo Nonlinear Analysis Real World Applications vol 12 no6 pp 3135ndash3144 2011

6 Mathematical Problems in Engineering

Table 1 PL fit to the distributions of casualties in terrorist attacks

Criterion Category SPL DPL1 DPL2(119862 ) (119862

1 1) (119862

2 2)

Weapon (119882119894)

1198821

00391 058491198822

00082 07514 00002 171981198823

00017 08660 00000 196241198824

00236 06877 00021 148901198825

00057 06909 00003 135791198826

00249 07801

Region (119877119894)

1198771

00438 101471198772

00214 06412 00009 169051198773

00174 119051198774

00065 090731198775

00012 121411198776

00091 08051 00006 170881198777

00009 114151198778

00193 07814 00008 206751198779

00036 08742 00001 2055211987710

00128 08561 00012 1657811987711

00091 06454 00004 1313211987712

00023 0995611987713

00001 15523

Target (119879119894)

1198791

00438 05144 00092 124881198792

00057 08884 00003 171801198793

00002 13589 00012 084581198794

00452 079231198795

00015 112591198796

00006 11152 00001 151621198797

00014 161061198798

00576 04694 00094 090701198799

00353 05669 00009 1629111987910

00090 09645 00031 1257311987911

00047 08991 00001 1835411987912

00036 07778 00000 1866711987913

00158 07353 00018 1610611987914

00462 0977411987915

00168 08610 00026 1682411987916

00234 0763511987917

00340 05892 00014 1517811987918

00225 07389 00074 1070811987919

00108 07918 00025 11650

Type (119860119894)

1198601

00049 07346 00000 179821198602

00014 15201 00001 265551198603

00059 08116 00001 206211198604

00261 06339 00016 148801198605

00305 071861198606

00116 073161198607

00072 09689 00014 162141198608

00471 07006 00034 20186

Mathematical Problems in Engineering 7

1980 1985 1990 1995 2000 2005 201005

05506

06507

07508

08509

0951

Year

Nor

mal

ized

entro

py

119878(119879)05

119878(119877)05

119878(119880)05

119878(119879)2

119878(119877)2

119878(119880)2

119878

Figure 7 Total Shannon Tsallis Renyi and Ubriaco normalizedentropies as a function of the year of the events 119884

119903 119902 = 05 and

119902 = 2

Tsallis entropy reduces to Renyi entropy when 119902 rarr 1 Tsallisentropy was applied to diffusion equations [42] and Fokker-Planck systems [43] Renyi entropy has an inverse power lawequilibrium distribution [44] and satisfies the zeroth law ofthermodynamics [45] The two parameters Sharma-Mittalentropy [46] is accepted as a generalization of Tsallis Renyiand Boltzmann-Gibbs entropies for limiting cases of theparameters [47]

Recently more general entropy measures have been pro-posed in the literature where the additivity axiom has beenrelaxed For instance Ubriaco [48] proposed the followingformula for the fractional entropy

119878(119880)

119902=

119882

sum

119894=1

(minus ln119901119894)119902

119901119894

(4)

that has the same properties as the Shannon entropy exceptadditivity

Applications of entropy in distinct complex systems canbe found in [49ndash57]

In Figure 7 the total Shannon normalized entropy (119878) isdepicted as well as Tsallisrsquo (119878(119879)

119902) Renyirsquos (119878(119877)

119902) and Ubriacorsquos

(119878(119880)

119902) for 119902 = 05 and 119902 = 2 as a function of the year of

the events 119884119903 Figures 9 10 and 11 show 119878

(119879)

119902 119878(119877)119902

and 119878(119880)119902

normalized entropies as a function of the year 119884119903 and

entropy parameter 01 le 119902 le 10In Figure 7 we observe two types of behavior namely

short- and long-term phenomena In what concerns shorttime behavior we verify peaks during 1983ndash1985 1997-19982004ndash2007 and minima at 1980 1995 and 2010 In whatconcerns long-time relationships a smooth decreasing isobserved for 119878(119879)

05 119878(119877)05

and 119878(119880)2

Removing the maxima andminima we get a time series for years 1981 1982 1986ndash1992 1994ndash1996 1999 2000 2002 2003 2008 and 2009(Figure 8) Largersmaller entropies correspond to chartscloserafar uniform distributions therefore seemingly wehave lessmore organized terrorist events in global terms

1980 1985 1990 1995 2000 2005 2010075

08

085

09

095

1

Year

Nor

mal

ized

entro

py

119878(119879)05

119878(119877)05

119878(119880)2

Lin Reg 119878(119879)05Lin Reg 119878(119877)05Lin Reg 119878(119880)2

Figure 8 Total Tsallis and Renyi normalized entropies (119902 = 05)

and Ubriacorsquos (119902 = 2) as a function of the events in years 1981 19821986ndash1992 1994ndash1996 1999 2000 2002 2003 2008 2009

1980 1985 1990 1995 2000 2005 2010

01234567

8910040506070809

1

Year

119902

119878(119879)119902

Figure 9 Total Tsallis normalized entropy 119878(119879)119902 as a function of the

year of the events 119884119903 and parameter 119902

These conclusions remain invariant for Figures 9 to 11 wherewe vary both the entropy definition and the parameter tuningTherefore we conclude that such results are robust againstsuch type of variations

As discussed in the previous section the statistical dis-tributions of real data can be approximated by either singleor double PLs In the latter case we study not only the totalentropy but also the entropy associated to each part of thedistributions Therefore we compute the entropy associatedto DPL1 and DPL2 that approximate the first and secondpart of the distributions respectively When adopting thisprocedure we are restricted to the Shannon and Ubriacoentropies as Tsallisrsquo and Renyirsquos do not admit the associativitydescribed above

In the sequel we present several results of the analysistaking into account the grouping criteria 119884

119903 119882119894 119877119895 119879119896 and

119860119897 Figure 12 depicts the Shannon entropy versus parameter

for all statistical distributions The black squares (denotedldquoSPLrdquo) correspond to the plot of versus the total entropyof the respective distributions 119878 The black circles (denotedldquoDPL1rdquo) are the plot of

1versus the entropy associated to the

first parts of the distributions 1198781 The white circles (denoted

ldquoDPL2rdquo) represent the plot of 2versus the entropy associated

8 Mathematical Problems in Engineering

1980 1985 1990 1995 2000 2005 2010

012345678910040608

1

Year

119902

119878(119877)119902

Figure 10 Total Renyi normalized entropy 119878(119877)119902 as a function of the

year of the events 119884119903 and parameter 119902

1980 1985 1990 1995 2000 2005 2010

012345678910

0402

0608

1

Year

119902

119878(119880)119902

Figure 11 Total Ubriaco normalized entropy 119878(119880)119902

as a function ofthe year of the events 119884

119903 and parameter 119902

to the second parts of the distributions 1198782 As can be seen for

the distributions that behave as single PLs higher entropiescorrespond to the lower values of the parameter and the twoparameters are linearly related A similar pattern is observedfor the parameters corresponding to DPL1 For DPL2 theparameter

2increases with entropy but the almost linear

relation between both parameters remainsFigure 13 shows identical results for the Ubriaco entropy

Theplot corresponds to 119902 = 05 nevertheless identical resultsare obtained for other values

For all DPLs related to terrorist events 119899 (119899 = 1 60)we find that the parameters (119862

119894 119894) 119894 = 1 2 obey the follow-

ing relation

1119899

10038161003816100381610038161003816log1198621119899|119901= 2119899

10038161003816100381610038161003816log1198622119899|119901+ 120598119899 (5)

where 119901 = minus1612 The mean value of 120598119899is 120598 = 0002 and the

corresponding standard deviation is 120590120598= 0036 Moreover

for the analyzed data we find 119894119899| log119862

119894119899|119901= 119896119894119899 Parameters

119896119894119899are approximately constant with mean value 119896 = 0277

and standard deviation 120590119896= 006

It is worth noticing that (5) is similar to Poissonrsquos law ofan adiabatic reversible process involving ideal gases given by

1198751119881120574

1= 1198752119881120574

2 (6)

where variables 119875119894and 119881

119894represent pressure and specific

volume respectively Equation (6) implies that 119875119881120574 = constParameter 120574 is called Poissonrsquos coefficient taking values 120574 =

53 ≃ 167 and 120574 = 75 = 140 for monoatomic and diatomicgases respectively Additionally it should be noticed that theabsolute value of the exponent 119901 is very similar to the oneobserved for an ideal gas undergoing a reversible adiabaticprocess

0

05

1

15

2

25

3

0 05 1 15 2 25Shannon entropy

SPLDPL1

DPL2

Figure 12 Parameters versus Shannon entropies for the dis-tributions corresponding to the criteria 119884

119903 119882119894 119877119895 119879119896 and 119860

119897

ldquoSPLrdquo corresponds to (119878 ) ldquoDPL1rdquo corresponds to (1198781 1) ldquoDPL2rdquo

corresponds to (1198782 2)

0

05

1

15

2

25

3

0 02 04 06 08 1 12 14 16

SPL DPL1DPL2

Ubriaco entropy (119902 = 05)

Figure 13 Parameters versus Ubriaco entropies for the distri-butions corresponding to the criteria 119884

119903 119882119894 119877119895 119879119896 and 119860

119897 ldquoSPLrdquo

corresponds to (119878(119880)119902 ) ldquoDPL1rdquo corresponds to (119878(119880)

1199021 1) ldquoDPL2rdquo

corresponds to (119878(119880)1199022 2) 119902 = 05

5 Conclusions

PLs have been widely reported in the modeling of distinctphenomena and have been associated with long memorybehavior self-similarity fractal structures and fractionalcalculus

In this paper we reviewed interesting and importantresults on PLs distributions and their applications to the

Mathematical Problems in Engineering 9

modeling of the number of victims in catastrophic eventsWe found double PL behavior in real data of catastrophicoccurrences in particular terrorismWe have plotted the twoPLs parameters (119862

119894 119894) 119894 = 1 2 corresponding to certain

events and observed an interesting pattern in the chartwhere the lines that connect each pair of points defining thedouble PLs are almost aligned to each other We have alsocomputed the entropy of the data sets This complementaryanalysis of the numerical data revealed extra relationships butthe fact is that these phenomena have a dense and rich volumeof characteristics and further research efforts are needed to adeeper understanding

References

[1] J Tenreiro Machado F B Duarte and G M Duarte ldquoPowerlaw analysis of financial index dynamicsrdquo Discrete Dynamics inNature and Society vol 2012 Article ID 120518 12 pages 2012

[2] J Tenreiro Machado ldquoAnd I say to myself lsquoWhat a fractionalworlrsquordquo Fractional Calculus and Applied Analysis vol 14 no 4pp 635ndash654 2011

[3] E Scalas R Gorenflo and FMainardi ldquoFractional calculus andcontinuous-time financerdquo Physica A vol 284 no 1ndash4 pp 376ndash384 2000

[4] F Mainardi M Raberto R Gorenflo and E Scalas ldquoFractionalcalculus and continuous-time finance II the waiting-time dis-tributionrdquo Physica A vol 287 no 3-4 pp 468ndash481 2000

[5] B J West ldquoFractal physiology and the fractional calculus aperspectiverdquo Frontiers in Physiology vol 1 p 12 2010

[6] B Gutenberg and R F Richter ldquoFrequency of earthquakes incaliforniardquo Bulletin of the Seismological Society of America vol34 pp 185ndash188 1944

[7] L F Richardson ldquoVariation of the Frequency of Fatal QuarrelswithMagnituderdquo Journal of the American Statistical Associationvol 43 pp 523ndash546 1948

[8] J M Carlson and J S Langer ldquoMechanical model of an earth-quake faultrdquo Physical Review A vol 40 no 11 pp 6470ndash64841989

[9] D C Roberts and D L Turcotte ldquoFractality and self-organizedcriticality of warsrdquo Fractals vol 6 no 4 pp 351ndash357 1998

[10] D R Davis and D E Weinstein ldquoBones bombs and breakpoints the geography of economic activityrdquo The AmericanEconomic Review vol 92 no 5 pp 1269ndash1289 2002

[11] L E Cederman ldquoModeling the size of wars from billiard ballsto sandpilesrdquo American Political Science Review vol 97 no 1pp 135ndash150 2003

[12] J C Bohorquez S Gourley A R Dixon M Spagat and N FJohnson ldquoCommon ecology quantifies human insurgencyrdquoNature vol 462 no 7275 pp 911ndash914 2009

[13] V Pareto Cours drsquoEconomie Politique Droz Geneva Switzer-land 1896

[14] F Auerbach ldquoDas Gesetz der BelvolkerungskoncentrationrdquoPetermanns GeographischeMitteilungen vol 59 pp 74ndash76 1913

[15] J B Estoup Gammes Stenographiques Institut de France 1916[16] G Zipf Selective Studies and the Principle of Relative Frequency

in Language Harvard University Press Cambridge Mass USA1932

[17] G Zipf Human Behavior and the Priciple of Least EffortAddison-Wesley Cambridge Mass USA 1949

[18] W Li ldquoReferences on Zipf rsquos lawrdquo httpwwwsorinsolomonnetsimsorinccszipfreferences20on20zipf27s20lawhtm

[19] D Sornette Critical Phenomena in Natural Sciences chapter 14Springer Heidelberg Germany 2nd edition 2003

[20] C M A Pinto A M Lopes and J A Tenreiro Machado ldquoAreview of power laws in real life phenomenardquo Communicationsin Nonlinear Science and Numerical Simulations vol 17 no 9pp 3558ndash3578 2012

[21] J Tuldava ldquoThe frequency spectrum of text and vocabularyrdquoJournal of Quantitative Linguistics vol 3 no 1 pp 38ndash50 1996

[22] GMartınez-Mekler R AlvarezMartınezM Beltran del Rıo RMansilla P Miramontes and G Cocho ldquoUniversality of rank-ordering distributions in the arts and sciencesrdquo PLoS One vol4 no 3 Article ID e4791 2009

[23] R F I Cancho and R V Sole ldquoTwo regimes in the frequency ofwords and the origins of complex lexicons Zipf rsquos law revisitedrdquoJournal of Quantitative Linguistics vol 8 no 3 pp 165ndash1732001

[24] R Alvarez-Martinez G Martinez-Mekler and G CochoaldquoOrder-disorder transition in conflicting dynamics leading torank-frequency generalized beta distributionsrdquo Physica A vol390 no 1 pp 120ndash130 2011

[25] L F Richardson Statistics of Deadly Quarrels QuadrangleBooks Chicago Ill USA 1960

[26] F Guzzetti ldquoLandslide fatalities and the evaluation of landsliderisk in Italyrdquo Engineering Geology vol 58 no 2 pp 89ndash1072000

[27] S N Jonkman ldquoGlobal perspectives on loss of human lifecaused by floodsrdquo Natural Hazards vol 34 no 2 pp 151ndash1752005

[28] N F Johnson M Spagat J A Restrepo et al ldquoUniversal pat-terns underlying ongoingwars and terrorismrdquo httparxivorgabsphysics0605035

[29] D S Geller and J David Singer Nations at War A ScientificStudy of International Conflict Cambridge Studies in Interna-tional Relations Cambridge University Press Cambridge UK1998

[30] O Becerra N Johnson P Meier J Restrepo and M Spa-gat ldquoNatural disasters casualties and power laws a com-parative analysis with armed conflictrdquo in Proceedings of theAnnual Meeting of the American Political Science AssociationLoews Philadelphia and the Pennsylvania Convention Cen-ter Philadelphia Pa USA 2006 httpwwwallacademiccommetap151714 indexhtml

[31] M E J Newman ldquoPower laws Pareto distributions and Zipf rsquoslawrdquo Contemporary Physics vol 46 pp 323ndash351 2005

[32] A Clauset andM Young ldquoScale invariance in global terrorismrdquohttparxivorgabsphysics0502014

[33] K T Bogen and E D Jones ldquoRisks of mortality and morbidityfrom worldwide terrorism 1968ndash2004rdquo Risk Analysis vol 26no 1 pp 45ndash59 2006

[34] A Clauset M Young and K S Gleditsch ldquoOn the frequency ofsevere terrorist eventsrdquo Journal of Conflict Resolution vol 51 no1 pp 58ndash87 2007

[35] EM-DAT ldquoTheOFDACRED International Disaster DatabaserdquoUniversite catholique de Louvain Brussels Belgium httpwwwemdatnet

[36] National Consortium for the Study of Terrorism and Responsesto Terrorism (START) ldquoGlobal Terrorism Databaserdquo 2011httpwwwstartumdedugtd

[37] R ClausiusTheMechanicalTheory of Heat with Its Applicationsto the Steam Engine and to Physical Properties of Bodies Johnvan Voorst London UK 1865

[38] L Boltzmann Vorlesungen ber Gastheorie J A Barth LeipzigGermany 1896

10 Mathematical Problems in Engineering

[39] C E Shannon ldquoAmathematical theory of communicationrdquoTheBell System Technical Journal vol 27 pp 379ndash423 623ndash6561948

[40] E T Jaynes ldquoInformation theory and statistical mechanicsrdquo vol106 pp 620ndash630 1957

[41] A I Khinchin Mathematical Foundations of Information The-ory Dover New York NY USA 1957

[42] A R Plastino M Casas and A Plastino ldquoA nonextensive max-imum entropy approach to a family of nonlinear reaction-diffusion equationsrdquo Physica A vol 280 no 3 pp 289ndash3032000

[43] T D Frank and A Daffertshofer ldquo119867-theorem for nonlinearFokker-Planck equations related to generalized thermostatis-ticsrdquo Physica A vol 295 no 3-4 pp 455ndash474 2001

[44] E K Lenzi R SMendes and L R da Silva ldquoStatisticalmechan-ics based onRenyi entropyrdquo Physica A vol 280 no 3-4 pp 337ndash345 2000

[45] A S Parvan andT S Biro ldquoExtensive Renyi statistics fromnon-extensive entropyrdquo Physics Letters A vol 340 no 5-6 pp 375ndash387 2005

[46] B D Sharma and D P Mittal ldquoNew nonadditive measures ofentropy for discrete probability distributionsrdquo Journal of Math-ematical Sciences vol 10 pp 28ndash40 1975

[47] T D Frank and A Daffertshofer ldquoExact time-dependentsolutions of the Renyi Fokker-Planck equation and the Fokker-Planck equations related to the entropies proposed by Sharmaand Mittalrdquo Physica A vol 285 no 3 pp 351ndash366 2000

[48] M R Ubriaco ldquoEntropies based on fractional calculusrdquo PhysicsLetters A vol 373 no 30 pp 2516ndash2519 2009

[49] P T Landsberg and V Vedral ldquoDistributions and channelcapacities in generalized statistical mechanicsrdquo Physics LettersA vol 247 no 3 pp 211ndash217 1998

[50] A Plastino and A R Plastino ldquoTsallis Entropy and Jaynesrsquoinformation theory formalismrdquo Brazilian Journal of Physics vol29 no 1 pp 50ndash60 1999

[51] X Li C Essex M Davison K H Hoffmann and C SchulzkyldquoFractional diffusion irreversibility and entropyrdquo Journal ofNon-Equilibrium Thermodynamics vol 28 no 3 pp 279ndash2912003

[52] H J Haubold A M Mathai and R K Saxena ldquoBoltzmann-Gibbs entropy versus Tsallis entropy recent contributions toresolving the argument of Einstein concerning ldquoneither HerrBoltzmann nor Herr Planck has given a definition ofWrdquo Essayreviewrdquo Astrophysics and Space Science vol 290 no 3-4 pp241ndash245 2004

[53] P N Rathie and S da Silva ldquoShannon Levy and Tsallis a noterdquoApplied Mathematical Sciences vol 2 no 28 pp 1359ndash13632008

[54] R M Gray Entropy and Information Theory Springer NewYork NY USA 2009

[55] J A Tenreiro Machado A C Costa and M F M Lima ldquoDy-namical analysis of compositionsrdquoNonlinear Dynamics vol 65no 4 pp 339ndash412 2011

[56] J A T Machado A C Costa and M D Quelhas ldquoEntropyanalysis of the DNA code dynamics in human chromosomesrdquoComputers amp Mathematics with Applications vol 62 no 3 pp1612ndash1617 2011

[57] J A TenreiroMachado A C Costa andM D Quelhas ldquoShan-non Renyie and Tsallis entropy analysis of DNA using phaseplanerdquo Nonlinear Analysis Real World Applications vol 12 no6 pp 3135ndash3144 2011

Mathematical Problems in Engineering 7

1980 1985 1990 1995 2000 2005 201005

05506

06507

07508

08509

0951

Year

Nor

mal

ized

entro

py

119878(119879)05

119878(119877)05

119878(119880)05

119878(119879)2

119878(119877)2

119878(119880)2

119878

Figure 7 Total Shannon Tsallis Renyi and Ubriaco normalizedentropies as a function of the year of the events 119884

119903 119902 = 05 and

119902 = 2

Tsallis entropy reduces to Renyi entropy when 119902 rarr 1 Tsallisentropy was applied to diffusion equations [42] and Fokker-Planck systems [43] Renyi entropy has an inverse power lawequilibrium distribution [44] and satisfies the zeroth law ofthermodynamics [45] The two parameters Sharma-Mittalentropy [46] is accepted as a generalization of Tsallis Renyiand Boltzmann-Gibbs entropies for limiting cases of theparameters [47]

Recently more general entropy measures have been pro-posed in the literature where the additivity axiom has beenrelaxed For instance Ubriaco [48] proposed the followingformula for the fractional entropy

119878(119880)

119902=

119882

sum

119894=1

(minus ln119901119894)119902

119901119894

(4)

that has the same properties as the Shannon entropy exceptadditivity

Applications of entropy in distinct complex systems canbe found in [49ndash57]

In Figure 7 the total Shannon normalized entropy (119878) isdepicted as well as Tsallisrsquo (119878(119879)

119902) Renyirsquos (119878(119877)

119902) and Ubriacorsquos

(119878(119880)

119902) for 119902 = 05 and 119902 = 2 as a function of the year of

the events 119884119903 Figures 9 10 and 11 show 119878

(119879)

119902 119878(119877)119902

and 119878(119880)119902

normalized entropies as a function of the year 119884119903 and

entropy parameter 01 le 119902 le 10In Figure 7 we observe two types of behavior namely

short- and long-term phenomena In what concerns shorttime behavior we verify peaks during 1983ndash1985 1997-19982004ndash2007 and minima at 1980 1995 and 2010 In whatconcerns long-time relationships a smooth decreasing isobserved for 119878(119879)

05 119878(119877)05

and 119878(119880)2

Removing the maxima andminima we get a time series for years 1981 1982 1986ndash1992 1994ndash1996 1999 2000 2002 2003 2008 and 2009(Figure 8) Largersmaller entropies correspond to chartscloserafar uniform distributions therefore seemingly wehave lessmore organized terrorist events in global terms

1980 1985 1990 1995 2000 2005 2010075

08

085

09

095

1

Year

Nor

mal

ized

entro

py

119878(119879)05

119878(119877)05

119878(119880)2

Lin Reg 119878(119879)05Lin Reg 119878(119877)05Lin Reg 119878(119880)2

Figure 8 Total Tsallis and Renyi normalized entropies (119902 = 05)

and Ubriacorsquos (119902 = 2) as a function of the events in years 1981 19821986ndash1992 1994ndash1996 1999 2000 2002 2003 2008 2009

1980 1985 1990 1995 2000 2005 2010

01234567

8910040506070809

1

Year

119902

119878(119879)119902

Figure 9 Total Tsallis normalized entropy 119878(119879)119902 as a function of the

year of the events 119884119903 and parameter 119902

These conclusions remain invariant for Figures 9 to 11 wherewe vary both the entropy definition and the parameter tuningTherefore we conclude that such results are robust againstsuch type of variations

As discussed in the previous section the statistical dis-tributions of real data can be approximated by either singleor double PLs In the latter case we study not only the totalentropy but also the entropy associated to each part of thedistributions Therefore we compute the entropy associatedto DPL1 and DPL2 that approximate the first and secondpart of the distributions respectively When adopting thisprocedure we are restricted to the Shannon and Ubriacoentropies as Tsallisrsquo and Renyirsquos do not admit the associativitydescribed above

In the sequel we present several results of the analysistaking into account the grouping criteria 119884

119903 119882119894 119877119895 119879119896 and

119860119897 Figure 12 depicts the Shannon entropy versus parameter

for all statistical distributions The black squares (denotedldquoSPLrdquo) correspond to the plot of versus the total entropyof the respective distributions 119878 The black circles (denotedldquoDPL1rdquo) are the plot of

1versus the entropy associated to the

first parts of the distributions 1198781 The white circles (denoted

ldquoDPL2rdquo) represent the plot of 2versus the entropy associated

8 Mathematical Problems in Engineering

1980 1985 1990 1995 2000 2005 2010

012345678910040608

1

Year

119902

119878(119877)119902

Figure 10 Total Renyi normalized entropy 119878(119877)119902 as a function of the

year of the events 119884119903 and parameter 119902

1980 1985 1990 1995 2000 2005 2010

012345678910

0402

0608

1

Year

119902

119878(119880)119902

Figure 11 Total Ubriaco normalized entropy 119878(119880)119902

as a function ofthe year of the events 119884

119903 and parameter 119902

to the second parts of the distributions 1198782 As can be seen for

the distributions that behave as single PLs higher entropiescorrespond to the lower values of the parameter and the twoparameters are linearly related A similar pattern is observedfor the parameters corresponding to DPL1 For DPL2 theparameter

2increases with entropy but the almost linear

relation between both parameters remainsFigure 13 shows identical results for the Ubriaco entropy

Theplot corresponds to 119902 = 05 nevertheless identical resultsare obtained for other values

For all DPLs related to terrorist events 119899 (119899 = 1 60)we find that the parameters (119862

119894 119894) 119894 = 1 2 obey the follow-

ing relation

1119899

10038161003816100381610038161003816log1198621119899|119901= 2119899

10038161003816100381610038161003816log1198622119899|119901+ 120598119899 (5)

where 119901 = minus1612 The mean value of 120598119899is 120598 = 0002 and the

corresponding standard deviation is 120590120598= 0036 Moreover

for the analyzed data we find 119894119899| log119862

119894119899|119901= 119896119894119899 Parameters

119896119894119899are approximately constant with mean value 119896 = 0277

and standard deviation 120590119896= 006

It is worth noticing that (5) is similar to Poissonrsquos law ofan adiabatic reversible process involving ideal gases given by

1198751119881120574

1= 1198752119881120574

2 (6)

where variables 119875119894and 119881

119894represent pressure and specific

volume respectively Equation (6) implies that 119875119881120574 = constParameter 120574 is called Poissonrsquos coefficient taking values 120574 =

53 ≃ 167 and 120574 = 75 = 140 for monoatomic and diatomicgases respectively Additionally it should be noticed that theabsolute value of the exponent 119901 is very similar to the oneobserved for an ideal gas undergoing a reversible adiabaticprocess

0

05

1

15

2

25

3

0 05 1 15 2 25Shannon entropy

SPLDPL1

DPL2

Figure 12 Parameters versus Shannon entropies for the dis-tributions corresponding to the criteria 119884

119903 119882119894 119877119895 119879119896 and 119860

119897

ldquoSPLrdquo corresponds to (119878 ) ldquoDPL1rdquo corresponds to (1198781 1) ldquoDPL2rdquo

corresponds to (1198782 2)

0

05

1

15

2

25

3

0 02 04 06 08 1 12 14 16

SPL DPL1DPL2

Ubriaco entropy (119902 = 05)

Figure 13 Parameters versus Ubriaco entropies for the distri-butions corresponding to the criteria 119884

119903 119882119894 119877119895 119879119896 and 119860

119897 ldquoSPLrdquo

corresponds to (119878(119880)119902 ) ldquoDPL1rdquo corresponds to (119878(119880)

1199021 1) ldquoDPL2rdquo

corresponds to (119878(119880)1199022 2) 119902 = 05

5 Conclusions

PLs have been widely reported in the modeling of distinctphenomena and have been associated with long memorybehavior self-similarity fractal structures and fractionalcalculus

In this paper we reviewed interesting and importantresults on PLs distributions and their applications to the

Mathematical Problems in Engineering 9

modeling of the number of victims in catastrophic eventsWe found double PL behavior in real data of catastrophicoccurrences in particular terrorismWe have plotted the twoPLs parameters (119862

119894 119894) 119894 = 1 2 corresponding to certain

events and observed an interesting pattern in the chartwhere the lines that connect each pair of points defining thedouble PLs are almost aligned to each other We have alsocomputed the entropy of the data sets This complementaryanalysis of the numerical data revealed extra relationships butthe fact is that these phenomena have a dense and rich volumeof characteristics and further research efforts are needed to adeeper understanding

References

[1] J Tenreiro Machado F B Duarte and G M Duarte ldquoPowerlaw analysis of financial index dynamicsrdquo Discrete Dynamics inNature and Society vol 2012 Article ID 120518 12 pages 2012

[2] J Tenreiro Machado ldquoAnd I say to myself lsquoWhat a fractionalworlrsquordquo Fractional Calculus and Applied Analysis vol 14 no 4pp 635ndash654 2011

[3] E Scalas R Gorenflo and FMainardi ldquoFractional calculus andcontinuous-time financerdquo Physica A vol 284 no 1ndash4 pp 376ndash384 2000

[4] F Mainardi M Raberto R Gorenflo and E Scalas ldquoFractionalcalculus and continuous-time finance II the waiting-time dis-tributionrdquo Physica A vol 287 no 3-4 pp 468ndash481 2000

[5] B J West ldquoFractal physiology and the fractional calculus aperspectiverdquo Frontiers in Physiology vol 1 p 12 2010

[6] B Gutenberg and R F Richter ldquoFrequency of earthquakes incaliforniardquo Bulletin of the Seismological Society of America vol34 pp 185ndash188 1944

[7] L F Richardson ldquoVariation of the Frequency of Fatal QuarrelswithMagnituderdquo Journal of the American Statistical Associationvol 43 pp 523ndash546 1948

[8] J M Carlson and J S Langer ldquoMechanical model of an earth-quake faultrdquo Physical Review A vol 40 no 11 pp 6470ndash64841989

[9] D C Roberts and D L Turcotte ldquoFractality and self-organizedcriticality of warsrdquo Fractals vol 6 no 4 pp 351ndash357 1998

[10] D R Davis and D E Weinstein ldquoBones bombs and breakpoints the geography of economic activityrdquo The AmericanEconomic Review vol 92 no 5 pp 1269ndash1289 2002

[11] L E Cederman ldquoModeling the size of wars from billiard ballsto sandpilesrdquo American Political Science Review vol 97 no 1pp 135ndash150 2003

[12] J C Bohorquez S Gourley A R Dixon M Spagat and N FJohnson ldquoCommon ecology quantifies human insurgencyrdquoNature vol 462 no 7275 pp 911ndash914 2009

[13] V Pareto Cours drsquoEconomie Politique Droz Geneva Switzer-land 1896

[14] F Auerbach ldquoDas Gesetz der BelvolkerungskoncentrationrdquoPetermanns GeographischeMitteilungen vol 59 pp 74ndash76 1913

[15] J B Estoup Gammes Stenographiques Institut de France 1916[16] G Zipf Selective Studies and the Principle of Relative Frequency

in Language Harvard University Press Cambridge Mass USA1932

[17] G Zipf Human Behavior and the Priciple of Least EffortAddison-Wesley Cambridge Mass USA 1949

[18] W Li ldquoReferences on Zipf rsquos lawrdquo httpwwwsorinsolomonnetsimsorinccszipfreferences20on20zipf27s20lawhtm

[19] D Sornette Critical Phenomena in Natural Sciences chapter 14Springer Heidelberg Germany 2nd edition 2003

[20] C M A Pinto A M Lopes and J A Tenreiro Machado ldquoAreview of power laws in real life phenomenardquo Communicationsin Nonlinear Science and Numerical Simulations vol 17 no 9pp 3558ndash3578 2012

[21] J Tuldava ldquoThe frequency spectrum of text and vocabularyrdquoJournal of Quantitative Linguistics vol 3 no 1 pp 38ndash50 1996

[22] GMartınez-Mekler R AlvarezMartınezM Beltran del Rıo RMansilla P Miramontes and G Cocho ldquoUniversality of rank-ordering distributions in the arts and sciencesrdquo PLoS One vol4 no 3 Article ID e4791 2009

[23] R F I Cancho and R V Sole ldquoTwo regimes in the frequency ofwords and the origins of complex lexicons Zipf rsquos law revisitedrdquoJournal of Quantitative Linguistics vol 8 no 3 pp 165ndash1732001

[24] R Alvarez-Martinez G Martinez-Mekler and G CochoaldquoOrder-disorder transition in conflicting dynamics leading torank-frequency generalized beta distributionsrdquo Physica A vol390 no 1 pp 120ndash130 2011

[25] L F Richardson Statistics of Deadly Quarrels QuadrangleBooks Chicago Ill USA 1960

[26] F Guzzetti ldquoLandslide fatalities and the evaluation of landsliderisk in Italyrdquo Engineering Geology vol 58 no 2 pp 89ndash1072000

[27] S N Jonkman ldquoGlobal perspectives on loss of human lifecaused by floodsrdquo Natural Hazards vol 34 no 2 pp 151ndash1752005

[28] N F Johnson M Spagat J A Restrepo et al ldquoUniversal pat-terns underlying ongoingwars and terrorismrdquo httparxivorgabsphysics0605035

[29] D S Geller and J David Singer Nations at War A ScientificStudy of International Conflict Cambridge Studies in Interna-tional Relations Cambridge University Press Cambridge UK1998

[30] O Becerra N Johnson P Meier J Restrepo and M Spa-gat ldquoNatural disasters casualties and power laws a com-parative analysis with armed conflictrdquo in Proceedings of theAnnual Meeting of the American Political Science AssociationLoews Philadelphia and the Pennsylvania Convention Cen-ter Philadelphia Pa USA 2006 httpwwwallacademiccommetap151714 indexhtml

[31] M E J Newman ldquoPower laws Pareto distributions and Zipf rsquoslawrdquo Contemporary Physics vol 46 pp 323ndash351 2005

[32] A Clauset andM Young ldquoScale invariance in global terrorismrdquohttparxivorgabsphysics0502014

[33] K T Bogen and E D Jones ldquoRisks of mortality and morbidityfrom worldwide terrorism 1968ndash2004rdquo Risk Analysis vol 26no 1 pp 45ndash59 2006

[34] A Clauset M Young and K S Gleditsch ldquoOn the frequency ofsevere terrorist eventsrdquo Journal of Conflict Resolution vol 51 no1 pp 58ndash87 2007

[35] EM-DAT ldquoTheOFDACRED International Disaster DatabaserdquoUniversite catholique de Louvain Brussels Belgium httpwwwemdatnet

[36] National Consortium for the Study of Terrorism and Responsesto Terrorism (START) ldquoGlobal Terrorism Databaserdquo 2011httpwwwstartumdedugtd

[37] R ClausiusTheMechanicalTheory of Heat with Its Applicationsto the Steam Engine and to Physical Properties of Bodies Johnvan Voorst London UK 1865

[38] L Boltzmann Vorlesungen ber Gastheorie J A Barth LeipzigGermany 1896

10 Mathematical Problems in Engineering

[39] C E Shannon ldquoAmathematical theory of communicationrdquoTheBell System Technical Journal vol 27 pp 379ndash423 623ndash6561948

[40] E T Jaynes ldquoInformation theory and statistical mechanicsrdquo vol106 pp 620ndash630 1957

[41] A I Khinchin Mathematical Foundations of Information The-ory Dover New York NY USA 1957

[42] A R Plastino M Casas and A Plastino ldquoA nonextensive max-imum entropy approach to a family of nonlinear reaction-diffusion equationsrdquo Physica A vol 280 no 3 pp 289ndash3032000

[43] T D Frank and A Daffertshofer ldquo119867-theorem for nonlinearFokker-Planck equations related to generalized thermostatis-ticsrdquo Physica A vol 295 no 3-4 pp 455ndash474 2001

[44] E K Lenzi R SMendes and L R da Silva ldquoStatisticalmechan-ics based onRenyi entropyrdquo Physica A vol 280 no 3-4 pp 337ndash345 2000

[45] A S Parvan andT S Biro ldquoExtensive Renyi statistics fromnon-extensive entropyrdquo Physics Letters A vol 340 no 5-6 pp 375ndash387 2005

[46] B D Sharma and D P Mittal ldquoNew nonadditive measures ofentropy for discrete probability distributionsrdquo Journal of Math-ematical Sciences vol 10 pp 28ndash40 1975

[47] T D Frank and A Daffertshofer ldquoExact time-dependentsolutions of the Renyi Fokker-Planck equation and the Fokker-Planck equations related to the entropies proposed by Sharmaand Mittalrdquo Physica A vol 285 no 3 pp 351ndash366 2000

[48] M R Ubriaco ldquoEntropies based on fractional calculusrdquo PhysicsLetters A vol 373 no 30 pp 2516ndash2519 2009

[49] P T Landsberg and V Vedral ldquoDistributions and channelcapacities in generalized statistical mechanicsrdquo Physics LettersA vol 247 no 3 pp 211ndash217 1998

[50] A Plastino and A R Plastino ldquoTsallis Entropy and Jaynesrsquoinformation theory formalismrdquo Brazilian Journal of Physics vol29 no 1 pp 50ndash60 1999

[51] X Li C Essex M Davison K H Hoffmann and C SchulzkyldquoFractional diffusion irreversibility and entropyrdquo Journal ofNon-Equilibrium Thermodynamics vol 28 no 3 pp 279ndash2912003

[52] H J Haubold A M Mathai and R K Saxena ldquoBoltzmann-Gibbs entropy versus Tsallis entropy recent contributions toresolving the argument of Einstein concerning ldquoneither HerrBoltzmann nor Herr Planck has given a definition ofWrdquo Essayreviewrdquo Astrophysics and Space Science vol 290 no 3-4 pp241ndash245 2004

[53] P N Rathie and S da Silva ldquoShannon Levy and Tsallis a noterdquoApplied Mathematical Sciences vol 2 no 28 pp 1359ndash13632008

[54] R M Gray Entropy and Information Theory Springer NewYork NY USA 2009

[55] J A Tenreiro Machado A C Costa and M F M Lima ldquoDy-namical analysis of compositionsrdquoNonlinear Dynamics vol 65no 4 pp 339ndash412 2011

[56] J A T Machado A C Costa and M D Quelhas ldquoEntropyanalysis of the DNA code dynamics in human chromosomesrdquoComputers amp Mathematics with Applications vol 62 no 3 pp1612ndash1617 2011

[57] J A TenreiroMachado A C Costa andM D Quelhas ldquoShan-non Renyie and Tsallis entropy analysis of DNA using phaseplanerdquo Nonlinear Analysis Real World Applications vol 12 no6 pp 3135ndash3144 2011

8 Mathematical Problems in Engineering

1980 1985 1990 1995 2000 2005 2010

012345678910040608

1

Year

119902

119878(119877)119902

Figure 10 Total Renyi normalized entropy 119878(119877)119902 as a function of the

year of the events 119884119903 and parameter 119902

1980 1985 1990 1995 2000 2005 2010

012345678910

0402

0608

1

Year

119902

119878(119880)119902

Figure 11 Total Ubriaco normalized entropy 119878(119880)119902

as a function ofthe year of the events 119884

119903 and parameter 119902

to the second parts of the distributions 1198782 As can be seen for

the distributions that behave as single PLs higher entropiescorrespond to the lower values of the parameter and the twoparameters are linearly related A similar pattern is observedfor the parameters corresponding to DPL1 For DPL2 theparameter

2increases with entropy but the almost linear

relation between both parameters remainsFigure 13 shows identical results for the Ubriaco entropy

Theplot corresponds to 119902 = 05 nevertheless identical resultsare obtained for other values

For all DPLs related to terrorist events 119899 (119899 = 1 60)we find that the parameters (119862

119894 119894) 119894 = 1 2 obey the follow-

ing relation

1119899

10038161003816100381610038161003816log1198621119899|119901= 2119899

10038161003816100381610038161003816log1198622119899|119901+ 120598119899 (5)

where 119901 = minus1612 The mean value of 120598119899is 120598 = 0002 and the

corresponding standard deviation is 120590120598= 0036 Moreover

for the analyzed data we find 119894119899| log119862

119894119899|119901= 119896119894119899 Parameters

119896119894119899are approximately constant with mean value 119896 = 0277

and standard deviation 120590119896= 006

It is worth noticing that (5) is similar to Poissonrsquos law ofan adiabatic reversible process involving ideal gases given by

1198751119881120574

1= 1198752119881120574

2 (6)

where variables 119875119894and 119881

119894represent pressure and specific

volume respectively Equation (6) implies that 119875119881120574 = constParameter 120574 is called Poissonrsquos coefficient taking values 120574 =

53 ≃ 167 and 120574 = 75 = 140 for monoatomic and diatomicgases respectively Additionally it should be noticed that theabsolute value of the exponent 119901 is very similar to the oneobserved for an ideal gas undergoing a reversible adiabaticprocess

0

05

1

15

2

25

3

0 05 1 15 2 25Shannon entropy

SPLDPL1

DPL2

Figure 12 Parameters versus Shannon entropies for the dis-tributions corresponding to the criteria 119884

119903 119882119894 119877119895 119879119896 and 119860

119897

ldquoSPLrdquo corresponds to (119878 ) ldquoDPL1rdquo corresponds to (1198781 1) ldquoDPL2rdquo

corresponds to (1198782 2)

0

05

1

15

2

25

3

0 02 04 06 08 1 12 14 16

SPL DPL1DPL2

Ubriaco entropy (119902 = 05)

Figure 13 Parameters versus Ubriaco entropies for the distri-butions corresponding to the criteria 119884

119903 119882119894 119877119895 119879119896 and 119860

119897 ldquoSPLrdquo

corresponds to (119878(119880)119902 ) ldquoDPL1rdquo corresponds to (119878(119880)

1199021 1) ldquoDPL2rdquo

corresponds to (119878(119880)1199022 2) 119902 = 05

5 Conclusions

PLs have been widely reported in the modeling of distinctphenomena and have been associated with long memorybehavior self-similarity fractal structures and fractionalcalculus

In this paper we reviewed interesting and importantresults on PLs distributions and their applications to the

Mathematical Problems in Engineering 9

modeling of the number of victims in catastrophic eventsWe found double PL behavior in real data of catastrophicoccurrences in particular terrorismWe have plotted the twoPLs parameters (119862

119894 119894) 119894 = 1 2 corresponding to certain

events and observed an interesting pattern in the chartwhere the lines that connect each pair of points defining thedouble PLs are almost aligned to each other We have alsocomputed the entropy of the data sets This complementaryanalysis of the numerical data revealed extra relationships butthe fact is that these phenomena have a dense and rich volumeof characteristics and further research efforts are needed to adeeper understanding

References

[1] J Tenreiro Machado F B Duarte and G M Duarte ldquoPowerlaw analysis of financial index dynamicsrdquo Discrete Dynamics inNature and Society vol 2012 Article ID 120518 12 pages 2012

[2] J Tenreiro Machado ldquoAnd I say to myself lsquoWhat a fractionalworlrsquordquo Fractional Calculus and Applied Analysis vol 14 no 4pp 635ndash654 2011

[3] E Scalas R Gorenflo and FMainardi ldquoFractional calculus andcontinuous-time financerdquo Physica A vol 284 no 1ndash4 pp 376ndash384 2000

[4] F Mainardi M Raberto R Gorenflo and E Scalas ldquoFractionalcalculus and continuous-time finance II the waiting-time dis-tributionrdquo Physica A vol 287 no 3-4 pp 468ndash481 2000

[5] B J West ldquoFractal physiology and the fractional calculus aperspectiverdquo Frontiers in Physiology vol 1 p 12 2010

[6] B Gutenberg and R F Richter ldquoFrequency of earthquakes incaliforniardquo Bulletin of the Seismological Society of America vol34 pp 185ndash188 1944

[7] L F Richardson ldquoVariation of the Frequency of Fatal QuarrelswithMagnituderdquo Journal of the American Statistical Associationvol 43 pp 523ndash546 1948

[8] J M Carlson and J S Langer ldquoMechanical model of an earth-quake faultrdquo Physical Review A vol 40 no 11 pp 6470ndash64841989

[9] D C Roberts and D L Turcotte ldquoFractality and self-organizedcriticality of warsrdquo Fractals vol 6 no 4 pp 351ndash357 1998

[10] D R Davis and D E Weinstein ldquoBones bombs and breakpoints the geography of economic activityrdquo The AmericanEconomic Review vol 92 no 5 pp 1269ndash1289 2002

[11] L E Cederman ldquoModeling the size of wars from billiard ballsto sandpilesrdquo American Political Science Review vol 97 no 1pp 135ndash150 2003

[12] J C Bohorquez S Gourley A R Dixon M Spagat and N FJohnson ldquoCommon ecology quantifies human insurgencyrdquoNature vol 462 no 7275 pp 911ndash914 2009

[13] V Pareto Cours drsquoEconomie Politique Droz Geneva Switzer-land 1896

[14] F Auerbach ldquoDas Gesetz der BelvolkerungskoncentrationrdquoPetermanns GeographischeMitteilungen vol 59 pp 74ndash76 1913

[15] J B Estoup Gammes Stenographiques Institut de France 1916[16] G Zipf Selective Studies and the Principle of Relative Frequency

in Language Harvard University Press Cambridge Mass USA1932

[17] G Zipf Human Behavior and the Priciple of Least EffortAddison-Wesley Cambridge Mass USA 1949

[18] W Li ldquoReferences on Zipf rsquos lawrdquo httpwwwsorinsolomonnetsimsorinccszipfreferences20on20zipf27s20lawhtm

[19] D Sornette Critical Phenomena in Natural Sciences chapter 14Springer Heidelberg Germany 2nd edition 2003

[20] C M A Pinto A M Lopes and J A Tenreiro Machado ldquoAreview of power laws in real life phenomenardquo Communicationsin Nonlinear Science and Numerical Simulations vol 17 no 9pp 3558ndash3578 2012

[21] J Tuldava ldquoThe frequency spectrum of text and vocabularyrdquoJournal of Quantitative Linguistics vol 3 no 1 pp 38ndash50 1996

[22] GMartınez-Mekler R AlvarezMartınezM Beltran del Rıo RMansilla P Miramontes and G Cocho ldquoUniversality of rank-ordering distributions in the arts and sciencesrdquo PLoS One vol4 no 3 Article ID e4791 2009

[23] R F I Cancho and R V Sole ldquoTwo regimes in the frequency ofwords and the origins of complex lexicons Zipf rsquos law revisitedrdquoJournal of Quantitative Linguistics vol 8 no 3 pp 165ndash1732001

[24] R Alvarez-Martinez G Martinez-Mekler and G CochoaldquoOrder-disorder transition in conflicting dynamics leading torank-frequency generalized beta distributionsrdquo Physica A vol390 no 1 pp 120ndash130 2011

[25] L F Richardson Statistics of Deadly Quarrels QuadrangleBooks Chicago Ill USA 1960

[26] F Guzzetti ldquoLandslide fatalities and the evaluation of landsliderisk in Italyrdquo Engineering Geology vol 58 no 2 pp 89ndash1072000

[27] S N Jonkman ldquoGlobal perspectives on loss of human lifecaused by floodsrdquo Natural Hazards vol 34 no 2 pp 151ndash1752005

[28] N F Johnson M Spagat J A Restrepo et al ldquoUniversal pat-terns underlying ongoingwars and terrorismrdquo httparxivorgabsphysics0605035

[29] D S Geller and J David Singer Nations at War A ScientificStudy of International Conflict Cambridge Studies in Interna-tional Relations Cambridge University Press Cambridge UK1998

[30] O Becerra N Johnson P Meier J Restrepo and M Spa-gat ldquoNatural disasters casualties and power laws a com-parative analysis with armed conflictrdquo in Proceedings of theAnnual Meeting of the American Political Science AssociationLoews Philadelphia and the Pennsylvania Convention Cen-ter Philadelphia Pa USA 2006 httpwwwallacademiccommetap151714 indexhtml

[31] M E J Newman ldquoPower laws Pareto distributions and Zipf rsquoslawrdquo Contemporary Physics vol 46 pp 323ndash351 2005

[32] A Clauset andM Young ldquoScale invariance in global terrorismrdquohttparxivorgabsphysics0502014

[33] K T Bogen and E D Jones ldquoRisks of mortality and morbidityfrom worldwide terrorism 1968ndash2004rdquo Risk Analysis vol 26no 1 pp 45ndash59 2006

[34] A Clauset M Young and K S Gleditsch ldquoOn the frequency ofsevere terrorist eventsrdquo Journal of Conflict Resolution vol 51 no1 pp 58ndash87 2007

[35] EM-DAT ldquoTheOFDACRED International Disaster DatabaserdquoUniversite catholique de Louvain Brussels Belgium httpwwwemdatnet

[36] National Consortium for the Study of Terrorism and Responsesto Terrorism (START) ldquoGlobal Terrorism Databaserdquo 2011httpwwwstartumdedugtd

[37] R ClausiusTheMechanicalTheory of Heat with Its Applicationsto the Steam Engine and to Physical Properties of Bodies Johnvan Voorst London UK 1865

[38] L Boltzmann Vorlesungen ber Gastheorie J A Barth LeipzigGermany 1896

10 Mathematical Problems in Engineering

[39] C E Shannon ldquoAmathematical theory of communicationrdquoTheBell System Technical Journal vol 27 pp 379ndash423 623ndash6561948

[40] E T Jaynes ldquoInformation theory and statistical mechanicsrdquo vol106 pp 620ndash630 1957

[41] A I Khinchin Mathematical Foundations of Information The-ory Dover New York NY USA 1957

[42] A R Plastino M Casas and A Plastino ldquoA nonextensive max-imum entropy approach to a family of nonlinear reaction-diffusion equationsrdquo Physica A vol 280 no 3 pp 289ndash3032000

[43] T D Frank and A Daffertshofer ldquo119867-theorem for nonlinearFokker-Planck equations related to generalized thermostatis-ticsrdquo Physica A vol 295 no 3-4 pp 455ndash474 2001

[44] E K Lenzi R SMendes and L R da Silva ldquoStatisticalmechan-ics based onRenyi entropyrdquo Physica A vol 280 no 3-4 pp 337ndash345 2000

[45] A S Parvan andT S Biro ldquoExtensive Renyi statistics fromnon-extensive entropyrdquo Physics Letters A vol 340 no 5-6 pp 375ndash387 2005

[46] B D Sharma and D P Mittal ldquoNew nonadditive measures ofentropy for discrete probability distributionsrdquo Journal of Math-ematical Sciences vol 10 pp 28ndash40 1975

[47] T D Frank and A Daffertshofer ldquoExact time-dependentsolutions of the Renyi Fokker-Planck equation and the Fokker-Planck equations related to the entropies proposed by Sharmaand Mittalrdquo Physica A vol 285 no 3 pp 351ndash366 2000

[48] M R Ubriaco ldquoEntropies based on fractional calculusrdquo PhysicsLetters A vol 373 no 30 pp 2516ndash2519 2009

[49] P T Landsberg and V Vedral ldquoDistributions and channelcapacities in generalized statistical mechanicsrdquo Physics LettersA vol 247 no 3 pp 211ndash217 1998

[50] A Plastino and A R Plastino ldquoTsallis Entropy and Jaynesrsquoinformation theory formalismrdquo Brazilian Journal of Physics vol29 no 1 pp 50ndash60 1999

[51] X Li C Essex M Davison K H Hoffmann and C SchulzkyldquoFractional diffusion irreversibility and entropyrdquo Journal ofNon-Equilibrium Thermodynamics vol 28 no 3 pp 279ndash2912003

[52] H J Haubold A M Mathai and R K Saxena ldquoBoltzmann-Gibbs entropy versus Tsallis entropy recent contributions toresolving the argument of Einstein concerning ldquoneither HerrBoltzmann nor Herr Planck has given a definition ofWrdquo Essayreviewrdquo Astrophysics and Space Science vol 290 no 3-4 pp241ndash245 2004

[53] P N Rathie and S da Silva ldquoShannon Levy and Tsallis a noterdquoApplied Mathematical Sciences vol 2 no 28 pp 1359ndash13632008

[54] R M Gray Entropy and Information Theory Springer NewYork NY USA 2009

[55] J A Tenreiro Machado A C Costa and M F M Lima ldquoDy-namical analysis of compositionsrdquoNonlinear Dynamics vol 65no 4 pp 339ndash412 2011

[56] J A T Machado A C Costa and M D Quelhas ldquoEntropyanalysis of the DNA code dynamics in human chromosomesrdquoComputers amp Mathematics with Applications vol 62 no 3 pp1612ndash1617 2011

[57] J A TenreiroMachado A C Costa andM D Quelhas ldquoShan-non Renyie and Tsallis entropy analysis of DNA using phaseplanerdquo Nonlinear Analysis Real World Applications vol 12 no6 pp 3135ndash3144 2011

Mathematical Problems in Engineering 9

modeling of the number of victims in catastrophic eventsWe found double PL behavior in real data of catastrophicoccurrences in particular terrorismWe have plotted the twoPLs parameters (119862

119894 119894) 119894 = 1 2 corresponding to certain

events and observed an interesting pattern in the chartwhere the lines that connect each pair of points defining thedouble PLs are almost aligned to each other We have alsocomputed the entropy of the data sets This complementaryanalysis of the numerical data revealed extra relationships butthe fact is that these phenomena have a dense and rich volumeof characteristics and further research efforts are needed to adeeper understanding

References

[1] J Tenreiro Machado F B Duarte and G M Duarte ldquoPowerlaw analysis of financial index dynamicsrdquo Discrete Dynamics inNature and Society vol 2012 Article ID 120518 12 pages 2012

[2] J Tenreiro Machado ldquoAnd I say to myself lsquoWhat a fractionalworlrsquordquo Fractional Calculus and Applied Analysis vol 14 no 4pp 635ndash654 2011

[3] E Scalas R Gorenflo and FMainardi ldquoFractional calculus andcontinuous-time financerdquo Physica A vol 284 no 1ndash4 pp 376ndash384 2000

[4] F Mainardi M Raberto R Gorenflo and E Scalas ldquoFractionalcalculus and continuous-time finance II the waiting-time dis-tributionrdquo Physica A vol 287 no 3-4 pp 468ndash481 2000

[5] B J West ldquoFractal physiology and the fractional calculus aperspectiverdquo Frontiers in Physiology vol 1 p 12 2010

[6] B Gutenberg and R F Richter ldquoFrequency of earthquakes incaliforniardquo Bulletin of the Seismological Society of America vol34 pp 185ndash188 1944

[7] L F Richardson ldquoVariation of the Frequency of Fatal QuarrelswithMagnituderdquo Journal of the American Statistical Associationvol 43 pp 523ndash546 1948

[8] J M Carlson and J S Langer ldquoMechanical model of an earth-quake faultrdquo Physical Review A vol 40 no 11 pp 6470ndash64841989

[9] D C Roberts and D L Turcotte ldquoFractality and self-organizedcriticality of warsrdquo Fractals vol 6 no 4 pp 351ndash357 1998

[10] D R Davis and D E Weinstein ldquoBones bombs and breakpoints the geography of economic activityrdquo The AmericanEconomic Review vol 92 no 5 pp 1269ndash1289 2002

[11] L E Cederman ldquoModeling the size of wars from billiard ballsto sandpilesrdquo American Political Science Review vol 97 no 1pp 135ndash150 2003

[12] J C Bohorquez S Gourley A R Dixon M Spagat and N FJohnson ldquoCommon ecology quantifies human insurgencyrdquoNature vol 462 no 7275 pp 911ndash914 2009

[13] V Pareto Cours drsquoEconomie Politique Droz Geneva Switzer-land 1896

[14] F Auerbach ldquoDas Gesetz der BelvolkerungskoncentrationrdquoPetermanns GeographischeMitteilungen vol 59 pp 74ndash76 1913

[15] J B Estoup Gammes Stenographiques Institut de France 1916[16] G Zipf Selective Studies and the Principle of Relative Frequency

in Language Harvard University Press Cambridge Mass USA1932

[17] G Zipf Human Behavior and the Priciple of Least EffortAddison-Wesley Cambridge Mass USA 1949

[18] W Li ldquoReferences on Zipf rsquos lawrdquo httpwwwsorinsolomonnetsimsorinccszipfreferences20on20zipf27s20lawhtm

[19] D Sornette Critical Phenomena in Natural Sciences chapter 14Springer Heidelberg Germany 2nd edition 2003

[20] C M A Pinto A M Lopes and J A Tenreiro Machado ldquoAreview of power laws in real life phenomenardquo Communicationsin Nonlinear Science and Numerical Simulations vol 17 no 9pp 3558ndash3578 2012

[21] J Tuldava ldquoThe frequency spectrum of text and vocabularyrdquoJournal of Quantitative Linguistics vol 3 no 1 pp 38ndash50 1996

[22] GMartınez-Mekler R AlvarezMartınezM Beltran del Rıo RMansilla P Miramontes and G Cocho ldquoUniversality of rank-ordering distributions in the arts and sciencesrdquo PLoS One vol4 no 3 Article ID e4791 2009

[23] R F I Cancho and R V Sole ldquoTwo regimes in the frequency ofwords and the origins of complex lexicons Zipf rsquos law revisitedrdquoJournal of Quantitative Linguistics vol 8 no 3 pp 165ndash1732001

[24] R Alvarez-Martinez G Martinez-Mekler and G CochoaldquoOrder-disorder transition in conflicting dynamics leading torank-frequency generalized beta distributionsrdquo Physica A vol390 no 1 pp 120ndash130 2011

[25] L F Richardson Statistics of Deadly Quarrels QuadrangleBooks Chicago Ill USA 1960

[26] F Guzzetti ldquoLandslide fatalities and the evaluation of landsliderisk in Italyrdquo Engineering Geology vol 58 no 2 pp 89ndash1072000

[27] S N Jonkman ldquoGlobal perspectives on loss of human lifecaused by floodsrdquo Natural Hazards vol 34 no 2 pp 151ndash1752005

[28] N F Johnson M Spagat J A Restrepo et al ldquoUniversal pat-terns underlying ongoingwars and terrorismrdquo httparxivorgabsphysics0605035

[29] D S Geller and J David Singer Nations at War A ScientificStudy of International Conflict Cambridge Studies in Interna-tional Relations Cambridge University Press Cambridge UK1998

[30] O Becerra N Johnson P Meier J Restrepo and M Spa-gat ldquoNatural disasters casualties and power laws a com-parative analysis with armed conflictrdquo in Proceedings of theAnnual Meeting of the American Political Science AssociationLoews Philadelphia and the Pennsylvania Convention Cen-ter Philadelphia Pa USA 2006 httpwwwallacademiccommetap151714 indexhtml

[31] M E J Newman ldquoPower laws Pareto distributions and Zipf rsquoslawrdquo Contemporary Physics vol 46 pp 323ndash351 2005

[32] A Clauset andM Young ldquoScale invariance in global terrorismrdquohttparxivorgabsphysics0502014

[33] K T Bogen and E D Jones ldquoRisks of mortality and morbidityfrom worldwide terrorism 1968ndash2004rdquo Risk Analysis vol 26no 1 pp 45ndash59 2006

[34] A Clauset M Young and K S Gleditsch ldquoOn the frequency ofsevere terrorist eventsrdquo Journal of Conflict Resolution vol 51 no1 pp 58ndash87 2007

[35] EM-DAT ldquoTheOFDACRED International Disaster DatabaserdquoUniversite catholique de Louvain Brussels Belgium httpwwwemdatnet

[36] National Consortium for the Study of Terrorism and Responsesto Terrorism (START) ldquoGlobal Terrorism Databaserdquo 2011httpwwwstartumdedugtd

[37] R ClausiusTheMechanicalTheory of Heat with Its Applicationsto the Steam Engine and to Physical Properties of Bodies Johnvan Voorst London UK 1865

[38] L Boltzmann Vorlesungen ber Gastheorie J A Barth LeipzigGermany 1896

10 Mathematical Problems in Engineering

[39] C E Shannon ldquoAmathematical theory of communicationrdquoTheBell System Technical Journal vol 27 pp 379ndash423 623ndash6561948

[40] E T Jaynes ldquoInformation theory and statistical mechanicsrdquo vol106 pp 620ndash630 1957

[41] A I Khinchin Mathematical Foundations of Information The-ory Dover New York NY USA 1957

[42] A R Plastino M Casas and A Plastino ldquoA nonextensive max-imum entropy approach to a family of nonlinear reaction-diffusion equationsrdquo Physica A vol 280 no 3 pp 289ndash3032000

[43] T D Frank and A Daffertshofer ldquo119867-theorem for nonlinearFokker-Planck equations related to generalized thermostatis-ticsrdquo Physica A vol 295 no 3-4 pp 455ndash474 2001

[44] E K Lenzi R SMendes and L R da Silva ldquoStatisticalmechan-ics based onRenyi entropyrdquo Physica A vol 280 no 3-4 pp 337ndash345 2000

[45] A S Parvan andT S Biro ldquoExtensive Renyi statistics fromnon-extensive entropyrdquo Physics Letters A vol 340 no 5-6 pp 375ndash387 2005

[46] B D Sharma and D P Mittal ldquoNew nonadditive measures ofentropy for discrete probability distributionsrdquo Journal of Math-ematical Sciences vol 10 pp 28ndash40 1975

[47] T D Frank and A Daffertshofer ldquoExact time-dependentsolutions of the Renyi Fokker-Planck equation and the Fokker-Planck equations related to the entropies proposed by Sharmaand Mittalrdquo Physica A vol 285 no 3 pp 351ndash366 2000

[48] M R Ubriaco ldquoEntropies based on fractional calculusrdquo PhysicsLetters A vol 373 no 30 pp 2516ndash2519 2009

[49] P T Landsberg and V Vedral ldquoDistributions and channelcapacities in generalized statistical mechanicsrdquo Physics LettersA vol 247 no 3 pp 211ndash217 1998

[50] A Plastino and A R Plastino ldquoTsallis Entropy and Jaynesrsquoinformation theory formalismrdquo Brazilian Journal of Physics vol29 no 1 pp 50ndash60 1999

[51] X Li C Essex M Davison K H Hoffmann and C SchulzkyldquoFractional diffusion irreversibility and entropyrdquo Journal ofNon-Equilibrium Thermodynamics vol 28 no 3 pp 279ndash2912003

[52] H J Haubold A M Mathai and R K Saxena ldquoBoltzmann-Gibbs entropy versus Tsallis entropy recent contributions toresolving the argument of Einstein concerning ldquoneither HerrBoltzmann nor Herr Planck has given a definition ofWrdquo Essayreviewrdquo Astrophysics and Space Science vol 290 no 3-4 pp241ndash245 2004

[53] P N Rathie and S da Silva ldquoShannon Levy and Tsallis a noterdquoApplied Mathematical Sciences vol 2 no 28 pp 1359ndash13632008

[54] R M Gray Entropy and Information Theory Springer NewYork NY USA 2009

[55] J A Tenreiro Machado A C Costa and M F M Lima ldquoDy-namical analysis of compositionsrdquoNonlinear Dynamics vol 65no 4 pp 339ndash412 2011

[56] J A T Machado A C Costa and M D Quelhas ldquoEntropyanalysis of the DNA code dynamics in human chromosomesrdquoComputers amp Mathematics with Applications vol 62 no 3 pp1612ndash1617 2011

[57] J A TenreiroMachado A C Costa andM D Quelhas ldquoShan-non Renyie and Tsallis entropy analysis of DNA using phaseplanerdquo Nonlinear Analysis Real World Applications vol 12 no6 pp 3135ndash3144 2011

10 Mathematical Problems in Engineering

[39] C E Shannon ldquoAmathematical theory of communicationrdquoTheBell System Technical Journal vol 27 pp 379ndash423 623ndash6561948

[40] E T Jaynes ldquoInformation theory and statistical mechanicsrdquo vol106 pp 620ndash630 1957

[41] A I Khinchin Mathematical Foundations of Information The-ory Dover New York NY USA 1957

[42] A R Plastino M Casas and A Plastino ldquoA nonextensive max-imum entropy approach to a family of nonlinear reaction-diffusion equationsrdquo Physica A vol 280 no 3 pp 289ndash3032000

[43] T D Frank and A Daffertshofer ldquo119867-theorem for nonlinearFokker-Planck equations related to generalized thermostatis-ticsrdquo Physica A vol 295 no 3-4 pp 455ndash474 2001

[44] E K Lenzi R SMendes and L R da Silva ldquoStatisticalmechan-ics based onRenyi entropyrdquo Physica A vol 280 no 3-4 pp 337ndash345 2000

[45] A S Parvan andT S Biro ldquoExtensive Renyi statistics fromnon-extensive entropyrdquo Physics Letters A vol 340 no 5-6 pp 375ndash387 2005

[46] B D Sharma and D P Mittal ldquoNew nonadditive measures ofentropy for discrete probability distributionsrdquo Journal of Math-ematical Sciences vol 10 pp 28ndash40 1975

[47] T D Frank and A Daffertshofer ldquoExact time-dependentsolutions of the Renyi Fokker-Planck equation and the Fokker-Planck equations related to the entropies proposed by Sharmaand Mittalrdquo Physica A vol 285 no 3 pp 351ndash366 2000

[48] M R Ubriaco ldquoEntropies based on fractional calculusrdquo PhysicsLetters A vol 373 no 30 pp 2516ndash2519 2009

[49] P T Landsberg and V Vedral ldquoDistributions and channelcapacities in generalized statistical mechanicsrdquo Physics LettersA vol 247 no 3 pp 211ndash217 1998

[50] A Plastino and A R Plastino ldquoTsallis Entropy and Jaynesrsquoinformation theory formalismrdquo Brazilian Journal of Physics vol29 no 1 pp 50ndash60 1999

[51] X Li C Essex M Davison K H Hoffmann and C SchulzkyldquoFractional diffusion irreversibility and entropyrdquo Journal ofNon-Equilibrium Thermodynamics vol 28 no 3 pp 279ndash2912003

[52] H J Haubold A M Mathai and R K Saxena ldquoBoltzmann-Gibbs entropy versus Tsallis entropy recent contributions toresolving the argument of Einstein concerning ldquoneither HerrBoltzmann nor Herr Planck has given a definition ofWrdquo Essayreviewrdquo Astrophysics and Space Science vol 290 no 3-4 pp241ndash245 2004

[53] P N Rathie and S da Silva ldquoShannon Levy and Tsallis a noterdquoApplied Mathematical Sciences vol 2 no 28 pp 1359ndash13632008

[54] R M Gray Entropy and Information Theory Springer NewYork NY USA 2009

[55] J A Tenreiro Machado A C Costa and M F M Lima ldquoDy-namical analysis of compositionsrdquoNonlinear Dynamics vol 65no 4 pp 339ndash412 2011

[56] J A T Machado A C Costa and M D Quelhas ldquoEntropyanalysis of the DNA code dynamics in human chromosomesrdquoComputers amp Mathematics with Applications vol 62 no 3 pp1612ndash1617 2011

[57] J A TenreiroMachado A C Costa andM D Quelhas ldquoShan-non Renyie and Tsallis entropy analysis of DNA using phaseplanerdquo Nonlinear Analysis Real World Applications vol 12 no6 pp 3135ndash3144 2011