LMC and SDL Complexity Measures: A Tool to Explore Time Seriesdownloads.hindawi.com/journals/complexity/2019/2095063.pdf · ResearchArticle LMC and SDL Complexity Measures: A Tool

Research ArticleLMC and SDL Complexity Measures A Tool toExplore Time Series

Joseacute Roberto C Piqueira 1 and Seacutergio Henrique Vannucchi Leme de Mattos 2

1Escola Politecnica da Universidade de Sao Paulo Avenida Prof LucianoGualberto travessa 3 n 158 05508-900 Sao Paulo SP Brazil2Universidade Federal de Sao Carlos Rod Washington Luıs km 235 SP-310 13565-905 Sao Carlos SP Brazil

Correspondence should be addressed to Jose Roberto C Piqueira piqueiralacuspbr

Received 21 September 2018 Revised 24 November 2018 Accepted 10 December 2018 Published 2 January 2019

Guest Editor Jose Garcia-Rodriguez

Copyright copy 2019 Jose Roberto C Piqueira and Sergio Henrique Vannucchi Leme de Mattos This is an open access articledistributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction inany medium provided the original work is properly cited

This work is a generalization of the Lopez-Ruiz Mancini and Calbet (LMC) and Shiner Davison and Landsberg (SDL) complexitymeasures considering that the state of a systemor process is representedby a continuous temporal series of a dynamical variable Asthe two complexity measures are based on the calculation of informational entropy an equivalent information source is defined byusing partitions of the dynamical variable rangeDuring the time intervals the information associatedwith themeasured dynamicalvariable is the seed to calculate instantaneous LMC and SDLmeasures To show how themethodology works generating indicatorstwo examples one concerning meteorological data and the other concerning economic data are presented and discussed

1 Introduction

The word complexity in the common sense meaning repre-sents systems that are difficult to describe design or under-stand However since Kolmogorov presented the concept ofcomputational complexity [1] new ideas have been associatedwith this wordmainly in life sciences [2] relating complexityand information [3]

As a consequence complexity started to be associatedto with systems and with the emergence of unexpectedbehaviors due to nonlinearities [4 5] and concerning systemtheory [6] a new meaning was carved postulating thatcomplexity is half way of the equilibrium and disequilibrium[7]

Developing this idea in a seminal paper [8] Lopez-Ruiz Mancini and Calbet proposed the LMC (Lopez-RuizMancini and Calbet) complexity measure for a randomdistribution by using informational entropy [9] to evaluateequilibrium and the quadratic deviation from the uniformdistribution to evaluate disequilibrium

However there has been some criticism about the LMCmeasure considering that it is inaccurate for some classes ofsystems obeying Markovian chains and cannot be considered

to represent an extensive variable Feldman and Crutchfield[10] proposed a correction for the disequilibrium termreplacing it by the relative entropywith respect to the uniformdistribution

Shiner Davison and Landsberg proposed another mod-ification of the LMC measure replacing the disequilibriumterm by the complement of the equilibrium term Thismeasure is called SDL (Shiner Davison and Landsberg) [11]and presents conclusions similar to that obtained by usingLMC for the majority of usual statistical distributions [2]

The main restriction to LMC and SDL complexity mea-sures is due to Crutchfield Feldman and Shalizi as theyargue that an equilibrium system can be structurally complex[12] but this problem could be solved by weighting order anddisorder according to the specific problem to be analyzed

Since the early 2000s the idea of adapting LMC and SDLto dynamical systems was successfully applied to differenttypes of time evolution problems bird songs [13] neural plas-ticity [14] interactions between species in ecological systems[2] physiognomies of landscapes [15] economic series [16]spread depression [17] and quantum information [18]

With these ideas in mind this article presents a system-atization of the methodology used in the referred papers

HindawiComplexityVolume 2019 Article ID 2095063 8 pageshttpsdoiorg10115520192095063

2 Complexity

based on LMC and SDL measures to be applied to temporalseries by defining and calculating the dynamic complexitymeasures

The procedure applied to a temporal series representingsome organizational or functional aspect of a system pro-vides insights regarding the evolution of its complexity

As the LMC and SDL dynamical measures are based oninformational entropy [16] the first task described in thenext section is to define an alphabet source associating aprobability distribution with the possible system states

Following the definition of the probability distribution anew section defines how dynamical LMC and SDL measurescan be calculated at each time based on the individualinformation associated with the system state at this timegenerating temporal series for LMC and SDL measures

To illustrate the calculation procedure two examples arepresented one related to a meteorological time series andthe other to an economic time series In both cases sectiona practical discussion about how to divide the range of thevalues assumed by the system state is presented

The examples were chosen to show that the methodologycan be applied to different types of phenomena precipitation(first example) with strong periodic component and eco-nomic time series (second example) that seems to be random

Thework is closedwith a conclusion section emphasizingthat the same procedure can be applied to any kind oftemporal real numbers series even with different temporalscale to calculate complexity measures

2 Defining Source and ProbabilityDistribution for a Temporal Series

Considering Shannonrsquos model [9] for an information sourcea time series 119909(119899) is considered to be a function of thenonnegative integers into a real interval ie 119909(119899) 119885+ 997888rarr(119886 119887) associating with each time 1199050 + 119899119879 a real numberbelonging to (119886 119887) with 1199050 gt 0 being the initial instant and119879 gt 0 an arbitrary period depending on the data availability

The set 119909(1199050) 119909(1199050 + 119879) 119909(1199050 + 119899119879) is assumed to be asequence of independent random variables and the stochasticprocess 119909(119899) as a whole is stationary [19]

The first step is to divide the interval (119886 119887) into Nsubintervals For the sake of simplicity N is chosen equal to2119896 119896 isin 119885+

At this point it could be asked how to choose N as thereis a compromise between precision (high values of N) andspeed of calculation (low values of N) This question will notbe addressed theoretically however in the example sectionpractical hints about this choice are presented

Consequently the source alphabet is defined by theintervals 119860 119894 119894 = 1 119873 with⋃119873119894=1 119860 119894 = (119886 119887) and119860 119894cap119860119895 =120601 forall119894 = 119895

Then a time interval defined by a given nmust be chosenand for the time sequence 1199050 1199050+119879 1199050+119899119879 the values of thevariable 119909(119899) must be read and associated with the intervals119860 119894 containing their respective value

Therefore for the whole set 1199050 1199050 + 119879 1199050 + 119899119879 eachinterval119860 119894 belonging to the source alphabet is associatedwith

119909(119899) a certain number of times 119899119894 which defines a relativefrequency 119901119894 = 119899119894(119899 + 1)

As 119901119894 ge 0 andsum119873119894=1 119901119894 = 1 it can be taken as a probabilityassociated with each interval 119860 119894

Following the definition for each subinterval 119860 119894 sub (119886 119887)its individual contribution to the whole information entropyis given by 119878119894 = minus119901119894 log2 119901119894 and the maximum value of theinformational entropy for the whole source 119878119898119886119909 = log2119873 =119896 can be calculated [9]

3 Dynamical LMC and SDL

As the source alphabet and individual information weredefined the instantaneous values of 119909(119899) are associated withtheir respective 119878119894 allowing the calculation of the instanta-neous value of the equilibrium (disorder) term

Δ (119899) = 119878119894119896 (1)

Combining (1) with the different definitions of the dise-quilibrium (order) terms dynamical LMC and SDLmeasuresare defined

31 LMC Dynamical Measure As indicated by Lopez-RuizMancini and Calbet [8] the dynamic disequilibrium (order)term can be calculated as the quadratic deviation of thesource alphabet probability distribution from the uniformdistribution and consequently the individual contribution ofeach interval 119860 119894 is

119863(119899) = (119901119894 minus 1119873)2

(2)

Extending the definition of LMC measure dynamicalLMC calculated in 1199050 + 119899119879 is given by

119862119871119872119862 (119899) = Δ (119899)119863 (119899) (3)

32 SDL Dynamical Measure As proposed by Shiner Davi-son and Landsberg [11] the dynamic disequilibrium (order)term can be calculated as the complement of the dynamicequilibrium term

119863 (119899) = (1 minus Δ (119899)) (4)

Extending the definition of SDLmeasure dynamical SDLcalculated in 1199050 + 119899119879 is given by

119862119878119863119871 (119899) = Δ (119899)119863 (119899) (5)

4 Applying the Method toMeteorological Data

A monthly meteorological temporal series is studied in thissection showing that the described method can be appliedindependently of the natural time scale and periodicity of thephenomenon

The meteorological data series relative to rain precip-itation in Dourados-MS-Brazil [20] is analyzed only in a

Complexity 3

20 40 60 80 100 1200time (months)(from 012004 to 122012)

0

50

100

150

200

250

300

350

Prec

ipita

tion

inde

x (m

m)

Figure 1 Precipitation index (Dourados-MS-Brasil)

methodological point of view without any meteorologicalconjecture about the results

The monthly precipitation index temporal series fromJanuary 2004 to September of 2012 shown in Figure 1 rep-resents the value of 119909(119899) [20] whose complexity is analyzed

Consequently the interval (119886 119887) related to the excursionof 119909 is (0 338) It is divided into 8(119896 = 3) 16(119896 = 4) 32(119896 =5) and 64(119896 = 6) subintervals to build the sources and theirrespective probability distributions

Based on these probability distributions 119862119871119872119862(119899) and119862119878119863119871(119899) are calculated and plotted giving an idea about howthemeasure choice and the interval division affect the results

41 Equivalence between LMC and SDL Dividing the rangeof 119909(119899) into 8 parts the results of the calculation of 119862119871119872119862(119899)and 119862119878119863119871(119899) measures are shown in Figures 2(a) and 2(b)respectively

As Figures 2(a) and 2(b) show in spite of the numericaldifferences the time evolutions of 119862119871119872119862(119899) and 119862119878119863119871(119899) arerepresented by similar curves in the eight-part division case

Observing the figures it is possible to infer that the LMCmeasure captures the periodic character of the precipitationalong the years in a better way However the SDL measureassumes its maximum value (25)

If the range of 119909(119899) is divided into 16 parts Figures 3(a)and 3(b) show the results for 119862119871119872119862(119899) and 119862119878119863119871(119899)

It can be observed that in this case (sixteen-divisioncase)119862119871119872119862(119899) and 119862119878119863119871(119899) differ by a scale factor with LMCmeasures presenting better accuracy to express the periodiccharacter of the rain seasons SDL measure presents highvalue of peaks but the maximum value (25) is not reached

Comparing Figures 2(a) and 3(a) 119862119871119872119862(119899) for differentrange partitions the global aspects of the curves are the sameand by increasing the number of divisions the dynamicalrange of the measures decreases and some rapid oscillatoryvariations similar to noise appear

Comparing Figures 2(b) and 3(b) 119862119878119863119871(119899) for differentrange partitions the whole aspects of the curves are the same

and the noisy aspect due to the increasing number of intervaldivisions is similar to the presented by 119862119871119872119862(119899)

42 Range Interval Partition As it was observed the dynam-ical range of the both measures decreases as the numberof divisions increases as a consequence of the fact that thenumber of elements of the source increases provoking moreuniform distribution of the possible measures

To better understand this phenomenon the measuresare recalculated by increasing the number of intervals of119909(119899) and the result for a thirty-two partition is shown inFigure 4(a) for 119862119871119872119862(119899) and in Figure 4(b) for 119862119878119863119871(119899)

By analyzing the results from Figures 2(a) 3(a) and4(a) it could be observed that by increasing the numberof intervals the dynamical range of 119862119871119872119862(119899) decreases butapparently for this long series the temporal evolution of119862119871119872119862(119899)maintains its qualitative behavior mixing noise withaccuracy

By analyzing the results from Figures 2(b) 3(b) and4(b) it could be observed that by increasing the numberof intervals the dynamical range of 119862119878119863119871(119899) decreases andits maximum value (25) is not reached Apparently for thislong series the temporal evolution of 119862119878119863119871(119899) maintains itsqualitative behavior mixing noise with accuracy

5 Applying the Method to Economic Data

In this section the economic series relative to the conversionof currencies studied in [16] is taken as an example showingthe applicability of the methodology for random phenomena

The temporal series related to the daily dollar to Brazil-ian real (USDBR) conversion rate from January 1999 toSeptember of 2015 shown in Figure 5 [16] is analyzed onlyin a methodological point of view without any economicconjecture about the results

This conversion rate represents the value of 119909(119899) whosecomplexity is analyzed

Consequently the interval (119886 119887) related to the excursionof 119909 is (1207 4178) It is divided into 8(119896 = 3) 16(119896 = 4)32(119896 = 5) and 64(119896 = 6) subintervals to build the sourcesand the respective probability distributions



As Figures 6(a) and 6(b) show in spite of the numericaldifferences the time evolution of 119862119871119872119862(119899) and 119862119878119863119871(119899) arequalitatively the same and represented by very similar curvesin the eight-part division case


It can be observed that in this case (sixteen-divisioncase) 119862119871119872119862(119899) and 119862119878119863119871(119899) differ only by a scale factor withthe same qualitative time evolution

4 Complexity

times10-3

0

1

2

3

4

5

6

7LM

C


(a) LMC


01

015

02

025

SDL

(b) SDL

Figure 2 Temporal evolution of complexity (8-part division)

times10-3

0

05

1

15

2

25

3

35

4

45

LMC


(a) LMC


01

015

02

025SD

L

(b) SDL


times10-3

0

05

1

15

2

25

3

LMC


(a) LMC

006

008

01

012

014

016

018

02

022

024

SDL


(b) SDL


Complexity 5

1

15

2

25

3

35

4

USD

BR

500 1000 1500 2000 2500 3000 3500 4000 45000time (days) (start0199minusend0815)

Figure 5 USA Dollar-Brazilian real exchanging rate

x 10minus4

0

1

2

LMC

500 1000 1500 2000 2500 3000 3500 4000 45000time (days) (01041999 to 09222016)

(a) LMC

x 10minus3

500 1000 1500 2000 2500 3000 3500 4000 45000time (days)(01041999 to 09222016)

0

1

2

3

4

5

6

7

8

9

SDL

(b) SDL


x 10minus5

500 1000 1500 2000 2500 3000 3500 4000 45000time (days)(01041999 t0 09222016)

0

05

1

15

2

25

3

35

4

45

LMC

(a) LMC

x 10minus3

0

05

1

15

2

25

3

35

4

45

SDL

500 1000 1500 2000 2500 3000 3500 4000 45000time (days)(01041999 to 09222016)

(b) SDL


6 Complexity

x 10minus6

0

1

2

3

4

5

6

7

8LM

C

500 1000 1500 2000 2500 3000 3500 4000 45000time (days)(01041999 to 09222016)

(a) 32-part partition

x 10minus6

500 1000 1500 2000 2500 3000 3500 4000 45000time (days)(0104199 to 09222016)

0

02

04

06

08

1

12

LMC

(b) 64-part partition

Figure 8 Temporal evolution of LMC complexity

Comparing Figures 6(a) and 7(a) 119862119871119872119862(119899) for differentrange partitions the whole qualitative aspects of the curvesare the same and by increasing the number of divisions thedynamical range of the measures changes implying somerapid oscillatory variations similar to noise

Comparing Figures 6(b) and 7(b) 119862119878119863119871(119899) for differentrange partitions the whole qualitative aspects of the curvesare the same and the noisy aspect due to the increasingnumber of interval divisions is maintained

Consequently from now on only LMC measure will beanalyzed since SDL presents the same qualitative dynamicalbehavior and partition sensitivity

52 Range Interval Partition By increasing the number ofintervals of 119909(119899) and recalculating 119862119871119872119862(119899) the result for athirty-two partition is shown in Figure 8(a) and for a sixty-four partition in Figure 8(b)

By analyzing the results from Figures 6(a) 7(a) 8(a)and 8(b) it could be observed that by increasing the num-ber of intervals the maximum value of 119862119871119872119862(119899) decreasesimproving the precision but apparently for this long seriesthe temporal evolution of 119862119871119872119862(119899) maintains its qualitativebehavior mixing noise with accuracy

Attempting to be more precise about how the rangeinterval partition 119862119871119872119862(119899) is calculated for the severalpartitions but considering a shorter time period for the dataThe interval between July and December of 2002 is chosenbecause as explained in [16] it is critical concerning theconversion rates in Brazil

Figures 9(a) 9(b) 9(c) and 9(d) show the LMC dynam-ical measure calculated for the initial set of data with therange interval divided into 8 16 32 and 64 parts respectively

It can be observed from these results that for shorterintervals the general qualitative characteristics of the timeevolution appear independently on the partition Howeveras the number of subintervals increases the instantaneous

numerical values change but the precision increases allowingmore accurate analysis

6 Conclusions

A methodology for calculating LMC and SDL dynamicalcomplexity was developed starting with the construction of asource and a probability distribution for any temporal seriesThe contribution is just concerning to extend ideas mainlyapplied to static situations to temporal evolution of variablesrepresenting some kind of organization phenomenon

LMC and SDL measures were observed to be equivalentin some temporal analyses but when there is a strongoscillatory component the LMC measure seems to be moreaccurate to express the temporal evolution of the complexityas the meteorological data analysis shows

For more randomly distributed data the two measures(LMC and SDL) present the same accuracy as the economicdata analysis shows

A point that is always an object of discussion is the rangeinterval partition The choice of the number of subintervals isa matter of experience

Long time intervals are not so sensitive to the increaseof the number of divisions in spite of meteorological databeing more sensitive than economic However for short timeintervals increasing the number of divisions produces a lessprecise analysis introducing noise

The examples presented were just to illustrate themethodological approach without any compromise withmeteorologic or economic conclusions that can be inferredby a specialist by using the developed tool

Data Availability

All the data used in this paper are available in the Internet byfollowing the links given in the references

Complexity 7

x 10minus4

0

1

2

LMC

50 100 150 2000time (days)(07022002 to 12022002)


x 10minus5

50 100 150 2000time (days)(07022002 to 12022002)

05

1

15

2

25

3

35

4

LMC


x 10minus6

20 40 60 80 100 120 140 160 1800time (days)(03022002 to 12032002)

05

1

15

2

25

3

35

4

45

5

LMC

(c) 32-part partition

x 10minus7

20 40 60 80 100 1200time (days)(02072002 to 02122002)

05

1

15

2

25

3

35

4

45

5

LMC

(d) 64-part partition

Figure 9 LMC temporal evolution for shorter time intervals

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

All the data used in this paper are available in the Internetby following the links given in the references This work wassupported by CNPq Brazil

References

[1] A N Kolmogorov ldquoThree approaches to the definition of theconcept ldquoquantity of informationrdquordquo Akademiya Nauk SSSRInstitut Problem Peredachi Informatsii Akademii Nauk SSSRProblemy Peredachi Informatsii vol 1 no vyp 1 pp 3ndash11 1965

[2] M Anand and L Orloci ldquoComplexity in plant communitiesThe notion and quantificationrdquo Journal of Theoretical Biologyvol 179 no 2 pp 179ndash186 1996

[3] HHaken Information and Self-Organization Springer Series inSynergetics Springer-Verlag Berlin Second edition 2000

[4] E MorinOn Complexity Hampton Press New York NY USA2008

[5] GNicolis and I Prigogine Self-Organization inNonequilibriumSystems John Wiley amp Sons USA 1977

[6] L Von Bertalanffy General SystemTheory Foundations Devel-opment Applications George Braziller Inc New York NYUSA 1968

[7] K Kaneko and I Tsuda Complex Systems chaos and beyondSpringer Verlag Berlin Germany 2001

[8] R Lopez-Ruiz H L Mancini and X Calbet ldquoA statisticalmeasure of complexityrdquo Physics Letters A vol 209 no 5-6 pp321ndash326 1995

[9] C E Shannon and W Weaver The Mathematical Theory ofCommunication llini Books Edition Chicago USA 1993

[10] D P Feldman and J P Crutchfield ldquoMeasures of statisticalcomplexity Whyrdquo Physics Letters A vol 238 no 4-5 pp 244ndash252 1998

8 Complexity

[11] J S Shiner M Davison and P T Landsberg ldquoSimple measurefor complexityrdquo Physical Review E Statistical Nonlinear andSoft Matter Physics vol 59 no 2 pp 1459ndash1464 1999

[12] J P Crutchfield D P Feldman and C R Shalizi ldquoCommentI on ldquoSimple measure for complexityrdquordquo Physical Review EStatistical Physics Plasmas Fluids and Related InterdisciplinaryTopics vol 62 no 2 pp 2996-2997 2000

[13] M L Da Silva J R C Piqueira and J M E Vielliard ldquoUsingShannon entropy on measuring the individual variability inthe Rafous-bellied thrush Turdus rufiventris vocal communi-cationrdquo Journal of Theoretical Biology vol 207 no 1 pp 57ndash642000

[14] M Pinho M Mazza J R C Piqueira and A C RoqueldquoShannons entropy applied to the analysis of tonotopic reor-ganization in a computational model of classic conditioningrdquoNeurocomputing pp 923ndash928 2002

[15] S H V L De Mattos L E Vicente A P Filho and J R CPiqueira ldquoContributions of the complexity paradigm to theunderstanding of Cerradorsquos organization and dynamicsrdquo Anaisda Academia Brasileira de Ciencias vol 88 no 4 pp 2417ndash24272016

[16] L P D Mortoza and J R C Piqueira ldquoMeasuring complexityin Brazilian economic crisesrdquo PLOSONE vol 12 no 3 ArticleID e0173280 2017

[17] J R C Piqueira V M F De Lima and C M Batistela ldquoCom-plexity measures and self-similarity on spreading depressionwavesrdquo Physica A Statistical Mechanics and Its Applications vol401 pp 271ndash277 2014

[18] Y C Campbell-Borges and J R C Piqueira ldquoComplexityMeasure A Quantum Information Approachrdquo vol 10 p 192012

[19] A Papoulis and S U Pillai Probability Random Variables andStochastic Processes Mc Graw Hill USA 4th edition 2002

[20] ldquoEMBRAPA (Empresa Brasileira de Produ cao Agropecuaria)rdquohttpswwwembrapabragropecuaria-oestebibliotecaacervo2018

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2 Complexity

based on LMC and SDL measures to be applied to temporalseries by defining and calculating the dynamic complexitymeasures

The procedure applied to a temporal series representingsome organizational or functional aspect of a system pro-vides insights regarding the evolution of its complexity

As the LMC and SDL dynamical measures are based oninformational entropy [16] the first task described in thenext section is to define an alphabet source associating aprobability distribution with the possible system states

Following the definition of the probability distribution anew section defines how dynamical LMC and SDL measurescan be calculated at each time based on the individualinformation associated with the system state at this timegenerating temporal series for LMC and SDL measures

To illustrate the calculation procedure two examples arepresented one related to a meteorological time series andthe other to an economic time series In both cases sectiona practical discussion about how to divide the range of thevalues assumed by the system state is presented

The examples were chosen to show that the methodologycan be applied to different types of phenomena precipitation(first example) with strong periodic component and eco-nomic time series (second example) that seems to be random

Thework is closedwith a conclusion section emphasizingthat the same procedure can be applied to any kind oftemporal real numbers series even with different temporalscale to calculate complexity measures

2 Defining Source and ProbabilityDistribution for a Temporal Series

Considering Shannonrsquos model [9] for an information sourcea time series 119909(119899) is considered to be a function of thenonnegative integers into a real interval ie 119909(119899) 119885+ 997888rarr(119886 119887) associating with each time 1199050 + 119899119879 a real numberbelonging to (119886 119887) with 1199050 gt 0 being the initial instant and119879 gt 0 an arbitrary period depending on the data availability

The set 119909(1199050) 119909(1199050 + 119879) 119909(1199050 + 119899119879) is assumed to be asequence of independent random variables and the stochasticprocess 119909(119899) as a whole is stationary [19]

The first step is to divide the interval (119886 119887) into Nsubintervals For the sake of simplicity N is chosen equal to2119896 119896 isin 119885+

At this point it could be asked how to choose N as thereis a compromise between precision (high values of N) andspeed of calculation (low values of N) This question will notbe addressed theoretically however in the example sectionpractical hints about this choice are presented

Consequently the source alphabet is defined by theintervals 119860 119894 119894 = 1 119873 with⋃119873119894=1 119860 119894 = (119886 119887) and119860 119894cap119860119895 =120601 forall119894 = 119895

Then a time interval defined by a given nmust be chosenand for the time sequence 1199050 1199050+119879 1199050+119899119879 the values of thevariable 119909(119899) must be read and associated with the intervals119860 119894 containing their respective value

Therefore for the whole set 1199050 1199050 + 119879 1199050 + 119899119879 eachinterval119860 119894 belonging to the source alphabet is associatedwith

119909(119899) a certain number of times 119899119894 which defines a relativefrequency 119901119894 = 119899119894(119899 + 1)

As 119901119894 ge 0 andsum119873119894=1 119901119894 = 1 it can be taken as a probabilityassociated with each interval 119860 119894

Following the definition for each subinterval 119860 119894 sub (119886 119887)its individual contribution to the whole information entropyis given by 119878119894 = minus119901119894 log2 119901119894 and the maximum value of theinformational entropy for the whole source 119878119898119886119909 = log2119873 =119896 can be calculated [9]

3 Dynamical LMC and SDL

As the source alphabet and individual information weredefined the instantaneous values of 119909(119899) are associated withtheir respective 119878119894 allowing the calculation of the instanta-neous value of the equilibrium (disorder) term

Δ (119899) = 119878119894119896 (1)

Combining (1) with the different definitions of the dise-quilibrium (order) terms dynamical LMC and SDLmeasuresare defined

31 LMC Dynamical Measure As indicated by Lopez-RuizMancini and Calbet [8] the dynamic disequilibrium (order)term can be calculated as the quadratic deviation of thesource alphabet probability distribution from the uniformdistribution and consequently the individual contribution ofeach interval 119860 119894 is

119863(119899) = (119901119894 minus 1119873)2

(2)

Extending the definition of LMC measure dynamicalLMC calculated in 1199050 + 119899119879 is given by

119862119871119872119862 (119899) = Δ (119899)119863 (119899) (3)

32 SDL Dynamical Measure As proposed by Shiner Davi-son and Landsberg [11] the dynamic disequilibrium (order)term can be calculated as the complement of the dynamicequilibrium term

119863 (119899) = (1 minus Δ (119899)) (4)

Extending the definition of SDLmeasure dynamical SDLcalculated in 1199050 + 119899119879 is given by

119862119878119863119871 (119899) = Δ (119899)119863 (119899) (5)

4 Applying the Method toMeteorological Data

A monthly meteorological temporal series is studied in thissection showing that the described method can be appliedindependently of the natural time scale and periodicity of thephenomenon

The meteorological data series relative to rain precip-itation in Dourados-MS-Brazil [20] is analyzed only in a

Complexity 3


0

50

100

150

200

250

300

350

Prec

ipita

tion

inde

x (m

m)




























4 Complexity

times10-3

0

1

2

3

4

5

6

7LM

C


(a) LMC


01

015

02

025

SDL

(b) SDL


times10-3

0

05

1

15

2

25

3

35

4

45

LMC


(a) LMC


01

015

02

025SD

L

(b) SDL


times10-3

0

05

1

15

2

25

3

LMC


(a) LMC

006

008

01

012

014

016

018

02

022

024

SDL


(b) SDL


Complexity 5

1

15

2

25

3

35

4

USD

BR



x 10minus4

0

1

2

LMC

500 1000 1500 2000 2500 3000 3500 4000 45000time (days) (01041999 to 09222016)

(a) LMC

x 10minus3

500 1000 1500 2000 2500 3000 3500 4000 45000time (days)(01041999 to 09222016)

0

1

2

3

4

5

6

7

8

9

SDL

(b) SDL


x 10minus5

500 1000 1500 2000 2500 3000 3500 4000 45000time (days)(01041999 t0 09222016)

0

05

1

15

2

25

3

35

4

45

LMC

(a) LMC

x 10minus3

0

05

1

15

2

25

3

35

4

45

SDL

500 1000 1500 2000 2500 3000 3500 4000 45000time (days)(01041999 to 09222016)

(b) SDL


6 Complexity

x 10minus6

0

1

2

3

4

5

6

7

8LM

C

500 1000 1500 2000 2500 3000 3500 4000 45000time (days)(01041999 to 09222016)


x 10minus6

500 1000 1500 2000 2500 3000 3500 4000 45000time (days)(0104199 to 09222016)

0

02

04

06

08

1

12

LMC












6 Conclusions







Data Availability


Complexity 7

x 10minus4

0

1

2

LMC

50 100 150 2000time (days)(07022002 to 12022002)


x 10minus5

50 100 150 2000time (days)(07022002 to 12022002)

05

1

15

2

25

3

35

4

LMC


x 10minus6

20 40 60 80 100 120 140 160 1800time (days)(03022002 to 12032002)

05

1

15

2

25

3

35

4

45

5

LMC


x 10minus7

20 40 60 80 100 1200time (days)(02072002 to 02122002)

05

1

15

2

25

3

35

4

45

5

LMC





Acknowledgments


References











8 Complexity


















Journal of












Journal of







Volume 2018






Volume 2018








Complexity 3


0

50

100

150

200

250

300

350

Prec

ipita

tion

inde

x (m

m)




























4 Complexity

times10-3

0

1

2

3

4

5

6

7LM

C


(a) LMC


01

015

02

025

SDL

(b) SDL


times10-3

0

05

1

15

2

25

3

35

4

45

LMC


(a) LMC


01

015

02

025SD

L

(b) SDL


times10-3

0

05

1

15

2

25

3

LMC


(a) LMC

006

008

01

012

014

016

018

02

022

024

SDL


(b) SDL


Complexity 5

1

15

2

25

3

35

4

USD

BR



x 10minus4

0

1

2

LMC

500 1000 1500 2000 2500 3000 3500 4000 45000time (days) (01041999 to 09222016)

(a) LMC

x 10minus3

500 1000 1500 2000 2500 3000 3500 4000 45000time (days)(01041999 to 09222016)

0

1

2

3

4

5

6

7

8

9

SDL

(b) SDL


x 10minus5

500 1000 1500 2000 2500 3000 3500 4000 45000time (days)(01041999 t0 09222016)

0

05

1

15

2

25

3

35

4

45

LMC

(a) LMC

x 10minus3

0

05

1

15

2

25

3

35

4

45

SDL

500 1000 1500 2000 2500 3000 3500 4000 45000time (days)(01041999 to 09222016)

(b) SDL


6 Complexity

x 10minus6

0

1

2

3

4

5

6

7

8LM

C

500 1000 1500 2000 2500 3000 3500 4000 45000time (days)(01041999 to 09222016)


x 10minus6

500 1000 1500 2000 2500 3000 3500 4000 45000time (days)(0104199 to 09222016)

0

02

04

06

08

1

12

LMC












6 Conclusions







Data Availability


Complexity 7

x 10minus4

0

1

2

LMC

50 100 150 2000time (days)(07022002 to 12022002)


x 10minus5

50 100 150 2000time (days)(07022002 to 12022002)

05

1

15

2

25

3

35

4

LMC


x 10minus6

20 40 60 80 100 120 140 160 1800time (days)(03022002 to 12032002)

05

1

15

2

25

3

35

4

45

5

LMC


x 10minus7

20 40 60 80 100 1200time (days)(02072002 to 02122002)

05

1

15

2

25

3

35

4

45

5

LMC





Acknowledgments


References











8 Complexity


















Journal of












Journal of







Volume 2018






Volume 2018








4 Complexity

times10-3

0

1

2

3

4

5

6

7LM

C


(a) LMC


01

015

02

025

SDL

(b) SDL


times10-3

0

05

1

15

2

25

3

35

4

45

LMC


(a) LMC


01

015

02

025SD

L

(b) SDL


times10-3

0

05

1

15

2

25

3

LMC


(a) LMC

006

008

01

012

014

016

018

02

022

024

SDL


(b) SDL


Complexity 5

1

15

2

25

3

35

4

USD

BR



x 10minus4

0

1

2

LMC

500 1000 1500 2000 2500 3000 3500 4000 45000time (days) (01041999 to 09222016)

(a) LMC

x 10minus3

500 1000 1500 2000 2500 3000 3500 4000 45000time (days)(01041999 to 09222016)

0

1

2

3

4

5

6

7

8

9

SDL

(b) SDL


x 10minus5

500 1000 1500 2000 2500 3000 3500 4000 45000time (days)(01041999 t0 09222016)

0

05

1

15

2

25

3

35

4

45

LMC

(a) LMC

x 10minus3

0

05

1

15

2

25

3

35

4

45

SDL

500 1000 1500 2000 2500 3000 3500 4000 45000time (days)(01041999 to 09222016)

(b) SDL


6 Complexity

x 10minus6

0

1

2

3

4

5

6

7

8LM

C

500 1000 1500 2000 2500 3000 3500 4000 45000time (days)(01041999 to 09222016)


x 10minus6

500 1000 1500 2000 2500 3000 3500 4000 45000time (days)(0104199 to 09222016)

0

02

04

06

08

1

12

LMC












6 Conclusions







Data Availability


Complexity 7

x 10minus4

0

1

2

LMC

50 100 150 2000time (days)(07022002 to 12022002)


x 10minus5

50 100 150 2000time (days)(07022002 to 12022002)

05

1

15

2

25

3

35

4

LMC


x 10minus6

20 40 60 80 100 120 140 160 1800time (days)(03022002 to 12032002)

05

1

15

2

25

3

35

4

45

5

LMC


x 10minus7

20 40 60 80 100 1200time (days)(02072002 to 02122002)

05

1

15

2

25

3

35

4

45

5

LMC





Acknowledgments


References











8 Complexity


















Journal of












Journal of







Volume 2018






Volume 2018








Complexity 5

1

15

2

25

3

35

4

USD

BR



x 10minus4

0

1

2

LMC

500 1000 1500 2000 2500 3000 3500 4000 45000time (days) (01041999 to 09222016)

(a) LMC

x 10minus3

500 1000 1500 2000 2500 3000 3500 4000 45000time (days)(01041999 to 09222016)

0

1

2

3

4

5

6

7

8

9

SDL

(b) SDL


x 10minus5

500 1000 1500 2000 2500 3000 3500 4000 45000time (days)(01041999 t0 09222016)

0

05

1

15

2

25

3

35

4

45

LMC

(a) LMC

x 10minus3

0

05

1

15

2

25

3

35

4

45

SDL

500 1000 1500 2000 2500 3000 3500 4000 45000time (days)(01041999 to 09222016)

(b) SDL


6 Complexity

x 10minus6

0

1

2

3

4

5

6

7

8LM

C

500 1000 1500 2000 2500 3000 3500 4000 45000time (days)(01041999 to 09222016)


x 10minus6

500 1000 1500 2000 2500 3000 3500 4000 45000time (days)(0104199 to 09222016)

0

02

04

06

08

1

12

LMC












6 Conclusions







Data Availability


Complexity 7

x 10minus4

0

1

2

LMC

50 100 150 2000time (days)(07022002 to 12022002)


x 10minus5

50 100 150 2000time (days)(07022002 to 12022002)

05

1

15

2

25

3

35

4

LMC


x 10minus6

20 40 60 80 100 120 140 160 1800time (days)(03022002 to 12032002)

05

1

15

2

25

3

35

4

45

5

LMC


x 10minus7

20 40 60 80 100 1200time (days)(02072002 to 02122002)

05

1

15

2

25

3

35

4

45

5

LMC





Acknowledgments


References











8 Complexity


















Journal of












Journal of







Volume 2018






Volume 2018








6 Complexity

x 10minus6

0

1

2

3

4

5

6

7

8LM

C

500 1000 1500 2000 2500 3000 3500 4000 45000time (days)(01041999 to 09222016)


x 10minus6

500 1000 1500 2000 2500 3000 3500 4000 45000time (days)(0104199 to 09222016)

0

02

04

06

08

1

12

LMC












6 Conclusions







Data Availability


Complexity 7

x 10minus4

0

1

2

LMC

50 100 150 2000time (days)(07022002 to 12022002)


x 10minus5

50 100 150 2000time (days)(07022002 to 12022002)

05

1

15

2

25

3

35

4

LMC


x 10minus6

20 40 60 80 100 120 140 160 1800time (days)(03022002 to 12032002)

05

1

15

2

25

3

35

4

45

5

LMC


x 10minus7

20 40 60 80 100 1200time (days)(02072002 to 02122002)

05

1

15

2

25

3

35

4

45

5

LMC





Acknowledgments


References











8 Complexity


















Journal of












Journal of







Volume 2018






Volume 2018








Complexity 7

x 10minus4

0

1

2

LMC

50 100 150 2000time (days)(07022002 to 12022002)


x 10minus5

50 100 150 2000time (days)(07022002 to 12022002)

05

1

15

2

25

3

35

4

LMC


x 10minus6

20 40 60 80 100 120 140 160 1800time (days)(03022002 to 12032002)

05

1

15

2

25

3

35

4

45

5

LMC


x 10minus7

20 40 60 80 100 1200time (days)(02072002 to 02122002)

05

1

15

2

25

3

35

4

45

5

LMC





Acknowledgments


References











8 Complexity


















Journal of












Journal of







Volume 2018






Volume 2018








8 Complexity


















Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








Documents

LMC and SDL Complexity Measures: A Tool to Explore Time Seriesdownloads.hindawi.com/journals/complexity/2019/2095063.pdf · ResearchArticle LMC and SDL Complexity Measures: A Tool