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Estimation of Periodicity in Non-Uniformly Sampled Astronomical Data - an Approach Using Spatio-Temporal Kernel Based Co...

Permanent Link: http://ufdc.ufl.edu/UFE0043072/00001

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Title: Estimation of Periodicity in Non-Uniformly Sampled Astronomical Data - an Approach Using Spatio-Temporal Kernel Based Correntropy
Physical Description: 1 online resource (52 p.)
Language: english
Creator: MISHRA,BIBHU PRASAD
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

Subjects

Subjects / Keywords: ANALYSIS -- ASTRONOMICAL -- CORRENTROPY -- ESTIMATION -- FOLDING -- KERNEL -- PERIODICITY -- PERIODOGRAM -- SERIES -- SPATIO -- TEMPORAL -- TIME
Electrical and Computer Engineering -- Dissertations, Academic -- UF
Genre: Electrical and Computer Engineering thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Period estimation in non-uniformly sampled time series data is frequently a goal in astronomical data analysis. There are various problems faced during estimation of period. Firstly, data is sampled non-uniformly which makes it difficult to use simple techniques such as Fourier transform for performing spectral analysis. Secondly, there are large gaps in data which makes it difficult to interpolate the signal for re-sampling. Thirdly, in data sets with smaller time periods, the non-uniformity in sampling and noise in data pose even greater problems because of the lesser number of samples per period. Lastly, one of the biggest problem is that the time period of these light curves can not be easily identified by periodogram techniques because of the inherent modulations in the light curve within a single instance of time period which must be accounted for while estimating the period. Generally periodogram techniques give a peak at fundamental frequency which may not be the frequency corresponding to the true period but rather correspond to sub-multiple of the true period. In the present work we first discuss few of the existing methods such as Fourier transform, Lomb periodogram, Dirichlet transform for period estimation, then shifting our focus to kernel based methods. A new spatio-temporal kernel based cost function has been proposed which works directly on the non-uniformly sampled data. Furthermore, a spatio-temporal kernel for correntropy on transformed space has been proposed to estimate the time period of the data with enhanced accuracy. Finally, comparison of proposed methods has been done to the existing techniques to highlight the improvement provided by the kernel based methods.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by BIBHU PRASAD MISHRA.
Thesis: Thesis (M.S.)--University of Florida, 2011.
Local: Adviser: Principe, Jose C.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2011-10-31

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Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2011
System ID: UFE0043072:00001

Permanent Link: http://ufdc.ufl.edu/UFE0043072/00001

Material Information

Title: Estimation of Periodicity in Non-Uniformly Sampled Astronomical Data - an Approach Using Spatio-Temporal Kernel Based Correntropy
Physical Description: 1 online resource (52 p.)
Language: english
Creator: MISHRA,BIBHU PRASAD
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

Subjects

Subjects / Keywords: ANALYSIS -- ASTRONOMICAL -- CORRENTROPY -- ESTIMATION -- FOLDING -- KERNEL -- PERIODICITY -- PERIODOGRAM -- SERIES -- SPATIO -- TEMPORAL -- TIME
Electrical and Computer Engineering -- Dissertations, Academic -- UF
Genre: Electrical and Computer Engineering thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Period estimation in non-uniformly sampled time series data is frequently a goal in astronomical data analysis. There are various problems faced during estimation of period. Firstly, data is sampled non-uniformly which makes it difficult to use simple techniques such as Fourier transform for performing spectral analysis. Secondly, there are large gaps in data which makes it difficult to interpolate the signal for re-sampling. Thirdly, in data sets with smaller time periods, the non-uniformity in sampling and noise in data pose even greater problems because of the lesser number of samples per period. Lastly, one of the biggest problem is that the time period of these light curves can not be easily identified by periodogram techniques because of the inherent modulations in the light curve within a single instance of time period which must be accounted for while estimating the period. Generally periodogram techniques give a peak at fundamental frequency which may not be the frequency corresponding to the true period but rather correspond to sub-multiple of the true period. In the present work we first discuss few of the existing methods such as Fourier transform, Lomb periodogram, Dirichlet transform for period estimation, then shifting our focus to kernel based methods. A new spatio-temporal kernel based cost function has been proposed which works directly on the non-uniformly sampled data. Furthermore, a spatio-temporal kernel for correntropy on transformed space has been proposed to estimate the time period of the data with enhanced accuracy. Finally, comparison of proposed methods has been done to the existing techniques to highlight the improvement provided by the kernel based methods.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by BIBHU PRASAD MISHRA.
Thesis: Thesis (M.S.)--University of Florida, 2011.
Local: Adviser: Principe, Jose C.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2011-10-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2011
System ID: UFE0043072:00001


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First,mysinceregratitudegoestomyadvisorDr.JoseC.Prncipeforhiswonderfulguidanceandremarkablepatiencethroughoutmyresearch,mycommitteemembersDr.JohnHarrisandDr.JohnM.Sheafortheirguidanceandhelpthroughoutmygraduatestudies.IwouldliketothankourcollaboratorsDr.PavlosProtopapasandDr.PabloA.Estevezfortheirvaluableinsight.IwouldalsoliketothankAlexandAbhishekfortheirhelpduringtheinitialpartoftheproject,Rakeshforhissuggestions,AustinforhisdiscussionsonvarioustopicsandallmembersofCNELfortheirknowledgeonvarietyoftopics.LastbutnottheleastIwouldliketothankmyparents,mysisterandfriendsfortheirconstantsupportandencouragementthroughoutmylife. 4

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page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 6 LISTOFFIGURES ..................................... 7 ABSTRACT ......................................... 8 CHAPTER 1ANINTRODUCTIONTOPERIODICITYESTIMATIONINASTRONOMICALDATAANALYSIS ................................... 10 1.1OverviewoftheAstronomicalData ...................... 10 1.2IntroductiontoEstimationTechniques .................... 11 2PERIODICITYESTIMATIONTECHNIQUES:AREVIEW ............ 14 2.1Theory ...................................... 14 2.1.1SplineInterpolation ........................... 14 2.1.2LombPeriodogram ........................... 17 2.1.3DirichletTransform ........................... 18 2.2Results ..................................... 18 3PERIODICITYESTIMATIONUSINGKERNELBASEDMETHODS ....... 21 3.1Correntropy ................................... 21 3.2Spatio-temporalKernelbasedProposedMethod .............. 23 3.3KernelSize ................................... 27 3.4Results ..................................... 28 4PERIODICITYESTIMATIONUSINGSPATIO-TEMPORALKERNELBASEDCORRENTROPYONFOLDEDTIMESERIESDATA ............... 33 4.1PeriodEstimation ................................ 35 4.2KernelSize ................................... 36 4.3Results ..................................... 38 5CONCLUSION .................................... 42 APPENDIX:VARIABLESTEPSIZE ........................... 48 REFERENCES ....................................... 50 BIOGRAPHICALSKETCH ................................ 52 5

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Table page 2-1Comparativeperformanceusinginterpolationbasedtechniques,LombperiodogramandDirichlettransformalongwithresultspublishedbyHarvardUniversity,TimeSeriesCenter. ................................. 19 3-1Comparativeperformanceusingproposed2Dkernelbasedtechniqueandcorrentropyoninterpolatedlightcurvedueto[ 4 ]alongwithresultspublishedbyHarvardUniversity,TimeSeriesCenter.Correctlyidentiedvaluesaremarkedinbold. ........................................ 30 4-1ComparativeperformanceusingproposedcorrentropybasedtechniquealongwithresultspublishedbyHarvardUniversity,TimeSeriesCenter.Correctlyidentiedvaluesaremarkedinbold. ........................ 40 5-1Performanceevaluationoftheexistingtechniquesandtheroposedtechniques.TheresultspublishedbyHarvardUniversity,TimeSeriesCenterhasbeenusedasthegoldenstandardfortheevaluation. .................. 45 6

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Figure page 2-1Thesampledvaluesatnon-uniformlyspacedtimeintervalsareinblueandinterpolatedanduniformlyre-sampledvaluesinredfordifferentvaluesofp. .. 15 2-2Frameselectionfromthelightcurve1.3804.164.Notethatthey-axisrepresentsthebrightnessmagnitudeofthestarsystem.Howeverthebrightertheobjectappears,thelowerthevalueofitsmagnitudeasitiscustomaryinastronomytoplotthemagnitudescalereversed. ........................ 16 3-1ContourandsurfaceplotofCIM(X,Y)withY=0in2DspacewithaGaussiankernelandakernelsizeequalto1. ........................ 23 3-2Figureillustratingthereasonwhysimplecorrentropycannotbedirectlyusedincaseofnon-uniformlysampleddata. ...................... 25 3-3Plotof2Dkernelbasedmeasurewithvaryingstandarddeviationvaluesfortimekernel.Inthiscaselightcurve1.3810.19hasantimeperiodof88.9406daysandlightcurve1.3449.27hasatimeperiodof4.0349days. ....... 28 3-4Plotof2Dkernelbasedmeasurewithvaryingstandarddeviationvaluesformagnitudekernel.Inthiscaselightcurve1.3810.19isthedatasetusedandhasatruetimeperiodequalto88.9406days. .................. 29 4-1Reconstructionofsingleperiodofthesignalbybreakingtheoriginalsignalintoframesoflengthequaltothetruetimeperiodofthesignal. ......... 34 4-2Foldingperformedonanon-uniformlysampledsignalwithtrueperiodequalto1unit.Foldinghasbeenperformedwithtrailperiodequalto1unitand1.3units. ......................................... 35 4-3Plotofcorrentropyoftransformedspacewithvaryingstandarddeviationvaluesfortimekernel. .................................... 38 4-4Plotofcorrentropyoftransformedspacewithvaryingstandarddeviationvaluesformagnitudekernel. ................................ 39 5-1Magnitudeplotoflightchannel1.3810.19.Notethatthey-axisrepresentsthemagnitudeofthestar.Magnitudemeasuresthebrightnessofacelestialobject,howeverthebrightertheobjectappears,thelowerthevalueofitsmagnitude.Itiscustomaryinastronomytoplotthemagnitudescalereversed. ....... 43 5-2Plotofspatio-temporalkernelbasedcorrentropyforlightcurve1.3448.153. .. 44 5-3Plotofspatio-temporalkernelbasedcorrentropyforlightcurve1.3810.19. .. 44 7

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11 ]. 1 ]isoperatedwiththepurposeofsearchingforthemissingdarkmatterinthegalactichalo,likebrowndwarfsorplanets.InMACHOthelightamplicationiscausedbybendingofspacearoundaheavyobjectduetothephenomenonknownasmicrolensing.CurrentexitingtechniquesmostlyuseLomb-Scargle(LS)periodogram[ 7 ],[ 9 ]whichisanextensionofclassicalperiodogramtechniquesbutitworkswithnon-uniformlysampleddata.TheestimatedperiodgivenbytheLSperiodogramisusedtofoldthetimeseriesmodulotheestimatedvaluefortheperiodsothattheperiodicnatureofdataisclearlyseen.Then,theestimatedperiodistrimmedsuchthatthescatterofthefoldedplotisreduced.Oncethisisachieveditispossibletoperformcalculationstoobtainamorepreciseestimateoftheperiod.Thisnalstepknownasanalysisofvariance(AoV)in 10

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15 ].ThesevaluespublishedbyTimeSeriesCenteratHarvardUniversityareusedasthegoldenstandardforcomparisonofthevariousalgorithmsasthenalvalueshavebeenmanuallyinspectedbytheTimeSeriesCenterteam.Thisprocessiscomputationallyintensiveandwithdatabeingcollectedfrombillionsofastronomicalobjectsweneedatechniquewhichismoreefcientandaccurateatthesametime.Thisinherentdifcultyoftheproblemrequirescomputationallyintelligenttechniques[ 2 ],[ 3 ],[ 12 ],[ 16 ],[ 18 ]tosolvetheproblem.Thepresentworkproposestwoalgorithm,oneusinga2DGaussiankernelonallpairsofsamplepointsandtheotherusinginformationtheoreticapproachbasedoncorrentropy[ 6 ],[ 10 ],[ 13 ],[ 17 ]withanewspatio-temporalkernel.Wewillbecomparingtheresultsofthecurrentworkwiththealgorithmproposedin[ 4 ]andalsowiththeexistingmethodsinvolvinginterpolation,Lombperiodogrametc. 11

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7 ]andDirichlettransform[ 8 ].AlthoughframingandinterpolatingenableuseofsimplestandardtechniquessuchascorrelationorFouriertransformthismethodnolongerusestheoriginaldatapointsdirectlyastheoriginalinformationiscombinedwithinterpolationnoise.Interpolationnoisealongwithinherentnoiseinthecollecteddatafurthercompromisestheprecisionofperiodestimationofthelightcurve.Hencefromanengineeringpointofviewitisbettertousesamplesdirectlyforperiodicityestimationpurposesratherthanusinginterpolateddata.ThemostpopulartechniqueswhichworkonthedatasamplesdirectlywithoutinvolvinguseofinterpolationareLombperiodogramandDirichlettransform.Aswewillseelaterevenperiodogrambasedtechniqueshavedrawbacksowingtothenatureofthelightcurves.Inthecurrentworkthedatasetusedhasbeenobtainedfromeclipsingbinarystarsystems.Inthesesystemstherearetwoeclipsespercyclewhichgivesrisetothemodulationeffectandhenceperiodogrambasedmethodstendtogivepeakatfrequenciescorrespondingtothesub-multiplevaluesofthetruetimeperiod.Theseproblemsareaddressedbytheproposedkernelbasedmethods.Therestofthechaptersareorganizedasfollows:Chapter 2 dealswithmethodswhichinvolvesinterpolationandre-samplingofdataandalsothetechniquessuchasLombperiodogramandDirichlettransformwhichworkdirectlyonthenon-uniformlysampleddata.Chapter 3 introducestheconceptofCorrentropyanddealswiththe 12

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4 ]andanewproposedtechniquebasedonspatio-temporalGaussiankernel.Chapter 4 dealswiththenalproposedalgorithmwhichusesspatio-temporalkernelbasedcorrentropyonatransformedspace.Chapter 5 concludestheworkanddiscussesthepotentialproblemswhichcanbeaddressedinthefuturetofurtherimprovetheperiodestimationtechniques. 13

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5 ]andthenre-samplethedataatuniformlyspacedintervals.Therearevariouskindsofinterpolationmethodssuchaslinearinterpolation,polynomialinterpolationandsplineinterpolationofwhichsplineinterpolationisthemostcommonlyused.Forexperimentalpurposeswehaveusedacubicsplineinterpolationtointerpolatethesignalfromthedatagiven.TheexpressioninEquation 2 givesthecostfunctionwhichistobeminimizedforinterpolationofthe 14

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B(p=0.5)Figure2-1. Thesampledvaluesatnon-uniformlyspacedtimeintervalsareinblueandinterpolatedanduniformlyre-sampledvaluesinredfordifferentvaluesofp. data. 2-1 .Itiscomparedtotheplotforp=0.5. 15

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Frameselectionfromthelightcurve1.3804.164.Notethatthey-axisrepresentsthebrightnessmagnitudeofthestarsystem.Howeverthebrightertheobjectappears,thelowerthevalueofitsmagnitudeasitiscustomaryinastronomytoplotthemagnitudescalereversed. TheinterpolateddatathusobtainedhasbeenusedforFouriertransformandAuto-correlationfunction(ACF)toestimatetheperiod.Inboththecasesrstframeswerechosenfromthelightcurvesuchthattherewereatleast100pointsintheframe 16

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2-2 showstheframeselectionfromthelightcurve1.3804.164.Theninterpolationwasperformedasdescribedaboveandthenre-samplingwasdoneattherateof20samplesperday.ThenincaseofFouriertransformthere-sampleddataisHammingwindowed.ThenN-pointFFTisperformedsuchthatNisthelowestpowerof2suchthatitisgreaterthanorequaltototalnumberofpointsintheresampleddataset.ThepeakintheFFTplotisusedtoestimatethetimeperiodofthelightcurvebysimplyinvertingthefrequencyvalueatthepeak.IncaseofACFasthenamesuggestsautocorrelationisperformedontheinterpolateddata.Thenthelargestpeakotherthanthepeakatzerolagisidentiedandthelagvalueatthatpeakistheestimatedvalueofthetimeperiod. 7 ]doesnotneedthesamplestobeevenlyspacedandhencecouldbeuseddirectlyondataforourcase.ItalsoallowsexaminingfrequencieshigherthanthemeanNyquistfrequencyi.e.theNyquistfrequencyobtainedbyevenlyspacingthesamenumberofdatapointsatthemeansamplingrate.Thesolereasonforusingperiodogramanalysisisthatitprovidesareasonablygoodapproximationtothespectrumobtainedbyttingsinewavesbyleastsquarestothedataandplottingthereductioninthesumofresidualsagainstfrequency.ThisleastsquaresspectrumprovidesthebestmeasureofthepowercontributedbythedifferentfrequenciestothevarianceofdataandcanberegardedasnaturalextensionofFouriermethodstonon-uniformdata.ItreducestoFourierpowerspectruminthelimitofequalspacing.TheLombperiodogramforzeromeantimeseriesx(tn)isdenedasfollows; 22fC(!)+S(!)g(2) 17

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2!arctanPNn=1sin2!tn 8 ]preservesinformationaboutsamplinginstantsbecauseitdoesnotsimplyconsiderx(tn)assequenceofsamplesbutasafunctionoftimeinstantstn.Dirichlettransformisdenedasfollows; 2.1.1 .Thelasttwotechniquesi.e.Lombperiodogram 18

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Comparativeperformanceusinginterpolationbasedtechniques,LombperiodogramandDirichlettransformalongwithresultspublishedbyHarvardUniversity,TimeSeriesCenter. LightcurveHarvardFourierAuto-LombDirichletbluechannel-TSCtransformcorrelationperiodogramtransform 1.3810.1988.940644.281189.0544.521744.52171.4411.61245.114322.5986473.622.505522.50551.4168.43443.930122.140587.522.021522.02151.3809.105828.907314.499128.8514.422514.42251.4652.56527.571813.768127.6513.74513.7451.4288.97517.61318.808635.258.78978.78971.4539.77816.250215.753831.28.12708.12701.4173.140914.153414.1241113.87.08657.08651.3449.94814.00647.062170.057.01377.01371.4174.1048.49298.533385.054.2494.2491.4538.815.534381.9250.02.76762.76761.3564.1634.715534.1333198.21.1791.1791.3804.1644.187538.071862.9752.09412.09411.3449.274.0349102.410.352.01772.01771.3448.1533.276517.066716.13.27683.27681.4539.372.995568.266740.21.49821.49821.3442.1721.0205922.755629.93330.51030.51031.3325.930.9517619.504820.150.95170.95171.3444.8800.9028619.061529.02504.7084.7081.3447.7830.83615159.939076.960.71830.7183 andDirichlettransformworkdirectlyonthenon-uniformlysampleddataasmentionedearlier.DatasethasbeenobtainedbyMACHOsurveyandtheunitoftimeinthedatasetisequaltoaday.Forapplyingthetechniquesbasedoninterpolationmentionedabove,onadatasetrstaframeofdataisselectedhavingatleast100samplepointswithoutgapsgreaterthan10days.Onethingwenoticeisthatitispossibletoobtainmorethanoneframeofdatafromeachlightcurvedataset.Inthosecaseswesimplyaverageoutoverthevariousvaluesobtainedfromaparticulartechnique.Thenformethodsinvolvinguseofinterpolation,acubicsplineisusedtoapproximatethelightcurveandthentheinterpolatedcurveisre-sampleduniformlywithFs=20samplesperday.ForFourieranalysiswesimplyusetheuniformlysampleddata 19

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2048.Inboththecasesagainthepeakcorrespondstotheperiodicityofthelightcurve.TheTable 2-1 givestheperiodvaluesestimatedforvariouslightcurvesbyusingthefourmethodsdescribedabove.Thevalueswhicharecorrectestimateofthetrueperiodarementionedusingboldfont.IntheTable 2-1 weobservethatinmanycasesthevalueoftheestimatedperiodishalfofthetrueperiod.Thereasonbehindthisisthattheeclipsingbinarystarsystemshavetwoeclipsespercycleandhenceitgivesrisetomodulationeffectwhichcanbeseeninthesignal.TheeffectissimilartoasshowninFigure 5-1 .Inthiscasethesignalhasancyclelengthofaround88daysbutduetothemodulationweseeatroughinaboutevery44days.Alsoasweareusingperiodogramtechniqueswhichaimatttingsinewavesintothesignalandhencetendtogiveapeakatfrequencycorrespondingtotwicethecycleratewhichisexpected.EvenACFfailsduetothismodulationeffect.Hencekeepinginmindthedrawbacksofperiodogrambasedmethodsandchallengesposedbythenatureofdatawemovetowardstheuseofkernelbasedmethods. 20

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4 ]utilizinginterpolationandcorrentropyisusedforsimulationpurposesandcomparedtothenewlyproposedspatio-temporalkernelbasedtechnique. 13 ].Itcanbedenedbyinnerproductofvectors,whichcanbecomputedbyusingapositivedenitekernelfunction,,satisfyingMercersconditionsisdenedinEquation 3 3 .Thisfreeparameterischosenfromthedata-setitself.FordeningcorrentropyaGaussiankernelhasbeenused.Givenarandomprocessfxt,tTgwheretdenotestimeandTdenotestheindexsetofinterest,thenthecorrentropyfunctionisdenedasshownbelow; 21

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3 .ThevariableinEquation 3 determinestheemphasisgiventohigherordermomentsascomparedtosecondordermoment.Thuscorrentropyisafunctionoftwoargumentssimilartocorrelationbutwiththeadditionofhigherordermomentsintroducedbythekernelfunction.Asincreasesthehigherordermomentsdecaycausingthesecondordermomenttodominateandhencecorrentropyapproachescorrelation.Duetointroductionofhigherordermomentscorrentropyhasbeenfoundtoproducesharperandnarrowerpeakscorrespondingtosimilarityestimationcomparedtosimplecorrelationfunction.Anotherimportantpropertywhichcorrentropyinducesintheinputspaceisawelldenedmetricknownasthecorrentropyinducedmetric(CIM)[ 14 ].Itisdenedas; 22

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BSurfaceplotFigure3-1. ContourandsurfaceplotofCIM(X,Y)withY=0in2DspacewithaGaussiankernelandakernelsizeequalto1. actsasL2orL0normisdirectlyrelatedtothekernelsize.ThisisillustratedinFigure 3-1 .ThisuniquepropertyinCIMisveryusefulinrejectingtheoutliers.Inthisaspectitisdifferentfromsimplecorrelationwhichprovidesaglobalmeasure.Anotherimportantconceptassociatedwiththatofcorrentropyiscorrentropyspectraldensity(CSD).Itisdenedas; Fs(3)whereisthemeanvalueofcorrentropy.ThisisequivalenttoFouriertransformofthecenteredcorrentropyfunction. 23

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4 ].Butinthismethodasweshallseetheoriginaldatapointsareneverusedandlotsofsampledpointsaredroppedwhilendingframesofdatawithouthavinglargegapsinthem.Henceifwecouldusenon-uniformlysampleddatadirectlyinameasuresimilartocorrentropyitwouldbemoreuseful.Beforeproceedingtosolvetheproblemweanalyzetheproblemfacedwhileimplementingsimplecorrentropyonnon-uniformlysampleddata.InFigure 3-2 weseethatincaseofregularlysampleddataforeverylagvalue,samplepoint'A'inthereferencesignalhasacorrespondingvalueinthetimeshiftedsignalwhichisnottrueforthenon-uniformlysampleddata.HenceinuniformlysampledcasewehavepairsofpointswhicharepassedintothekernelandthekerneloutputissummedoverallpairstogivethecorrentropyvalueasinEquation 3 .Thereforeinthepresentscenarioofnon-uniformdatainsteadofpairingeachsamplepointwithanothersamplevalueataxedlagwepaireachsamplepointwitheveryothersamplepointbutassignacertainweighttoeachofthosepairingasshowninFigure 3-2 .Henceforaparticularlagvalueiftwosamplesweresampledattimeinstantswhicharedifferentfromthenthatpairisassignedaweightast(t,s+).Wecanclearlyseethatifthesamplesareexactlyspacedatintervalthentheweightassignedismaximum.HenceEquation 3 dealswiththenewkerneloutputforthesamplevalueinreferencesignalsampledattime't'.Equation 3 showsthesummationoverallsamplesinthedatasetandnallyEquation 3 dealswiththenormalizationtogivetheexpectedvalueofthekerneloutputoverallsamplescomparedtosimplecorrentropy.InEquation 3 3 and 3 thelefthandsideistheexpressionrelatedtosimplecorrentropyandtherighthandsideshowstheexpressionfornewtwodimensionalkernelbasedtechnique.AlsoinEquations 3 3 and 3 24

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Figureillustratingthereasonwhysimplecorrentropycannotbedirectlyusedincaseofnon-uniformlysampleddata. 3 denesthenalexpressionforusingkernelstonon-uniformdatainasimilarfashiontocorrentropy.WeuseGaussiankernelfortwhichisthetimekernel.Inordertoimplementtheideaofusingkernelsdirectlytonon-uniformlysampleddataweproposeanewspatio-temporalkernel.Itisdenedon2Dvectorsandtheinnerproductofvectorscanbecomputedusingapositivedenitekernelfunction,,denedinEquation 3 .IntheEquation 3 wehaveusedasingledimensionalvaluebutinourcasewearedealingwith2dimensionalvectors.Moreconcretelywedeneatwodimensionalvectorhwhichhastimevalueinonedimensionandmagnitudevalueinthe 25

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3 denedontime(t)andmagnitude(x)componentofthedatasetrespectively.Thiskernelisstillpositivedenite,beingeffectivelyatwodimensionalGaussiankernelwithdiagonalcovariancematrixwithrstdiagonalcomponent1dealingwithtimecomponenttkandseconddiagonalcomponent2dealingwithmagnitudeofdataxkatthattimeinstant.Thusthenewcostfunctionisdenedasfollows; 1. LetH=fhk=[tk,xk]T,1
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3.2 .WecanobserveinFigure 3-3 thatthepeakbecomesmoreprominentandplotbecomesmoresmoothbyincreasing1.Wealsotakeintoconsiderationthefactthatforlightcurveswhichhaveasmalltimeperiod,usingalargervalueof1wouldattenthepeak.Thishappensduetothereasonthatforsmallertimeperiodtherateofchangeofmagnitudevalueoveraxedperiodtimeislargercomparedtoadatasethavingalargertimeperiod.Hencefordatasetswithsmallertimeperiodsmanypairswhichshouldhavebeengivenlessweightduetothedifferenceinthevaluesoftheirsamplinginstants,aregivenmoreweight.Thisdistortstheplotandsuppressesthepeakatthetrueperiod.Soatrade-offisconsideredbetweenthesetwoopposingfactorsandwehave1=0.4.Forthecaseofmagnitudekernelthevalueof2isconsideredw.r.t.amplitudedynamicrange.Choosingaverylargekernelsizemeansanytwomagnitudevaluesfromthecorrespondingvectorspassedthroughthekernelwillgivesimilaroutputasthekerneltapersveryslowly.Choosingaverysmallkernelsizewouldgiveanoutputof1onlywhenwehaveequalmagnitudevaluesandgiveoutputclosetozeroforanyotherpairofamplitudevalues.ThisisclearlyreectedinFigure 3-4 whereintheplotof2=(Dynamicrangeofmagnitude)vsTrialperiodweseealargerkernelsizegivesaatplothavingavalueclosetooneirrespectiveoftheassumedperiodvalueandwhereasmallkernelsizegivesaplothavingvalueclosetozero.Thereforetoobtainasharperpeakatthetrueperiodwechoose2=(Dynamicrangeofmagnitude)=0.1astheoptimumvalue.Forsimplicityandtohavetheplotvaluesrestrictedbetween0and1wedropthenormalizingfactorforunitintegralintheGaussiankernel. 27

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Plotof2Dkernelbasedmeasurewithvaryingstandarddeviationvaluesfortimekernel.Inthiscaselightcurve1.3810.19hasantimeperiodof88.9406daysandlightcurve1.3449.27hasatimeperiodof4.0349days. 4 ].Thistechniqueproposedby[ 4 ]basicallyinvolvesinterpolationofdata,followedbycorrentropyandthencalculating 28

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Plotof2Dkernelbasedmeasurewithvaryingstandarddeviationvaluesformagnitudekernel.Inthiscaselightcurve1.3810.19isthedatasetusedandhasatruetimeperiodequalto88.9406days. theCSD.FromtheCSDthepeaksareidentiedandareusedtoestimatetheperiodofthelightcurve.TheresultsarecomparedinTable 3-1 .IntheTable 3-1 weseethatbothcorrentropyoninterpolateddataandtheproposedmethodusing2DkernelsperformbetterthanthemethodsdescribedinChapter 2 .Firstweobservethatfortheinterpolationbasedmethodsincaseofauto-correlationandFFT 29

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Comparativeperformanceusingproposed2Dkernelbasedtechniqueandcorrentropyoninterpolatedlightcurvedueto[ 4 ]alongwithresultspublishedbyHarvardUniversity,TimeSeriesCenter.Correctlyidentiedvaluesaremarkedinbold. LightcurveHarvardProposedCorrentropyonbluechannel-TSCtechniqueinterpolateddata 1.3810.1988.940688.8344.46851.4411.61245.114322.8922.48891.4168.43443.930143.9321.87461.3809.105828.907314.3114.34891.4652.56527.571827.9113.71021.4288.97517.613117.978.68451.4539.77816.250265.0316.20761.4173.140914.153414.067.04861.3449.94814.006414.036.99941.4174.1048.492916.994.24121.4538.815.534311.015.48451.3564.1634.715518.894.61791.3804.1644.187520.974.18781.3449.274.03494.0234.02391.3448.1533.276522.883.22111.4539.372.99553.02.95701.3442.1721.0205922.9822.75561.3325.930.9517620.02215.041.3444.8800.9028619.18243.88571.3447.7830.8361517.975275.5491 wegetonlythreehitseachbutwegetsevencorrectidenticationsfortheCSDbasedmethod.AlthoughinterpolationweredoneonsameframesofdataweseethatCSDperformsbetter.Interpolationintroducessignicantamountoferrorwhenthegapsintheframearecomparabletothatofthetruevalueofthetimeperiod.ThiscanbeseenfromthefactthatinTable 2-1 forthecaseofauto-correlationandFFTalmostallthedatasetscorrectlyidentiedhavealargertimeperiod.Especiallyauto-correlationhasallthecorrectidenticationsforlightcurveswhichhaveatimeperiodgreaterthan25 30

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5-1 .Againintheproposedtechniquewhichusesa2Dkernelwegetbetterperformancecomparedtotheexistingtechniquesasitisabletoget8correctidentications.Thisproposedmethodisalsoabletoidentifycorrectlyfordatasetshavingatimeperiodaslowas3days.Theuseofagaussiankernelhelpsincorrectlyidentifyingtheusefulsamplepairswhilecalculatingthenalmeasure.Incaseof6lightcurveswendthattheproposedmethodgivesavalueoftheperiodwhichisamultipleofthetrueperiod.Thereasonbehindthisisthatthemethodisunabletondenoughsamplepairswhichhavetimedifferenceclosertothetruetimeperiod.Thusthe2Dkernelbasedmeasuredoesnotproducepeakatthetruepeiodbutratherproducespeaksatmultiplesofthetrueperiodforwhichitisabletondsufcientnumberofsamplepairs.In2casesweseethat2Dkernelbasedmethodalsogivesavalueofperiodwhichishalfofthetrueperiod.Thiscanbeattributedtothemodulationeffectasexplainedearlier.Althoughthesekernelbasedmethodsperformmuchbetterthantheexistingmethodswestillseethatthesetwotechniquesfailtoproduceanyresultfordatasetshavingatimeperiodlessthanorcloseto1day.ThiscanbeseenintheresultsTable 3-1 forthelastfourlightcurves.Thereasonbehindthisisthataveragesamplingrateinthedatasetsarealwaysgreaterthan1.Thusweneedtodevelopaalgorithmwhichwouldbeabletodetectthecorrectvalueoftimeperiodevenwhentheaveragesamplingrateismorethanthatofthetruetimeperiodofdata.Wehavenon-uniformlysampled 31

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4-1 wherethesignalwithatimeperiodof10unitsandaveragesamplingtimeof1unitisusedtoreconstructasingleperiod.IfwefoldthedatausingavalueofTwhichisnotamultipleofthetrueperiodthentheactualsignalwouldnotbeobtained.ItiseasytoseethattheperiodTwillyieldthesmoothestrepresentationintheprincipalargumentdomainwhereasavaluewhichisnotanintegralmultipleofTwillyieldanoisyrepresentationasillustratedinFigure 4-2 .Thereforeoneneedstondamethodologytocomparethesimilarityofthesamplesbothintimeandinamplitude,whichwillbeimplementedwithatwodimensionalkernel.Wecanseehowwecancreateasingleperiodofthesignalbyknowingthetrueperiod.Unfortunatelythismethodisgreedy,andmanydifferenttrialperiodvalueneedstobeevaluatedtoobtaintheperiodforwhichthesimilarityisthehighest.Moreconcretely,wedeneatwodimensionvectorhwhichhastimevalueinonedimensionandmagnitudevalueintheother.Itisexpressedasha=[ta,xa]Tand 33

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Reconstructionofsingleperiodofthesignalbybreakingtheoriginalsignalintoframesoflengthequaltothetruetimeperiodofthesignal. 3 denedontime(t)andmagnitude(x)componentofthedatasetrespectively.Thiskernelisstillpositivedenite,beingeffectivelyaGaussiankernelwithdiagonalcovariancematrix 34

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BFigure4-2. Foldingperformedonanon-uniformlysampledsignalwithtrueperiodequalto1unit.Foldinghasbeenperformedwithtrailperiodequalto1unitand1.3units. withrstdiagonalcomponent1dealingwithtimecomponenttkandseconddiagonalcomponent2dealingwithmagnitudeofdataxkatthattimeinstant.Usingthenewlydenedkernelthecorrentropyequationisdenedasfollows; 4.1 dealswiththeproposedtechniqueforestimationoftimeperiodofthesignal. 1. LetH=fhk=[tk,xk]T,1
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ForthetrialperiodT=p,thetransformationponHissuchthatp(H)=YwhereY=fk=[k,xk]T,1
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4-4 whereintheplotof2=(Dynamicrangeofmagnitude)vsTrialperiodweseealargerkernelsizegivesaatplothavingavalueclosetooneirrespectiveoftheassumedperiodvalueandwhereasmallkernelsizegivesaplothavingvalueclosetozero.Thereforetoobtainasharperpeakatthetrueperiodwechoose2=(Dynamicrangeofmagnitude)=0.1astheoptimumvalue.Forsimplicityandtohavetheplotvaluesrestrictedbetween0and1wedropthenormalizingfactorforunitintegralintheGaussiankernel. 37

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Plotofcorrentropyoftransformedspacewithvaryingstandarddeviationvaluesfortimekernel. 38

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Plotofcorrentropyoftransformedspacewithvaryingstandarddeviationvaluesformagnitudekernel. on2Dkernelandhenceisabletoexploittheadvantageprovidedbythekernelbasedmethodsinrejectingtheoutliers.Alsothiscorrentropyisperformedonfoldedtimeseriesdatawhichmakesitrobusttotheeffectsofaveragesamplingrateascomparedtointerpolationbasedmethodsorthekernelbasedmethodsdescribedearlier.InTable 4-1 weseethattheproposedalgorithmgives15correctidenticationsandfortherest5casesitgivesavalueequaltohalfofthetrueperiod. 39

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ComparativeperformanceusingproposedcorrentropybasedtechniquealongwithresultspublishedbyHarvardUniversity,TimeSeriesCenter.Correctlyidentiedvaluesaremarkedinbold. LightcurveHarvard2Dcorrentropybluechannel-TSCbasedtechnique 1.3810.1988.940644.50171.4411.61245.114345.14411.4168.43443.930143.93131.3809.105828.907314.45461.4652.56527.571827.57481.4288.97517.613117.61161.4539.77816.250216.25081.4173.140914.153414.15091.3449.94814.006414.00591.4174.1048.49298.49281.4538.815.53435.53441.3564.1634.71554.71561.3804.1644.18754.18761.3449.274.03494.03471.3448.1533.27653.27641.4539.372.99551.49771.3442.1721.020590.51031.3325.930.951760.951761.3444.8800.902860.902861.3447.7830.836150.41807 Therobustnesstoaveragesamplingratecanbeseenfromthefactthatforthelastfourlightcurveswithtimeperiodscloseto1daythemethodgives2hitsandfortwocasesitgiveshalfthevalueoftruetimeperiod.The5caseswherewegetestimatedvaluesashalfofthetruetimeperiod,canbeattributedtomodulationeffectasdescribedinearlierchapters.Howeveraninterestingthingtonoteisthatinallthesevecasesthepeaksatthetruetimeperiodwerelargerthanthepeaksathalfthevalue.Thusifinsomewaywecannetunethethresholdthenperhapswewouldbeabletoget100%accuracy. 40

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2 3 and 4 inmanycasesweseethattheidentiedpeakisatavaluewhichishalfofthetrueperiod.Thereasonforgettingapeakatsub-multipleoftrueperiodisduetotheshapeofthesignalwhichcanbeseeninFigure 5-1 .Weseethemodulationeffectinsideaperiodwhichisresponsibleforthepeakatavaluewhichishalfofthetrueperiod.ThismodulationeffecttendstoaffecttheperiodogramtechniquessuchasLombperiodogram,Dirichlettransformthemost,asthesemethodstrytotsinewavesintothedata.Thesemethodstendtogivearesultintermsofthefundamentalfrequencyratherthanlookingattheactualnumberofperiodiccyclesperunittime.Outofthe20datasetsin15instancesLombperiodogramandDirichlettransformidentifythetimeperiodashalfofthetruetimeperiod.ThiscanbeseeninTable 2-1 .ThisisalsoseeninthescenariowhenFFTisusedoninterpolateddatawherein7instancesthedetectedperiodishalfofthetrueperiod.AswearedealingwitheclipsingbinarystarsystemsandthemagnitudewaveformisasshowninFigure 5-1 fundamentalfrequencyidentiedbythespectralmethodsalmostalwaysturnsouttobetwicetheperiodcyclerate.InChapter 2 interpolationbasedmethodshavetheaddeddisadvantagethatfordatasetswithsmallertimeperiodtheaveragenumberofsamplesavailableperperiodforinterpolationislessandhencethequalityofinterpolationisaffectedtoagreatextent.Thiseffectcanbeseenbythefactthatasthetimeperiodofthedatasetusedfortestingdecreasestheinterpolationbasedmethodstendtoproducemoreerroneousresults.ThiscanbeseenfromthefactthatincaseofFFTandAutocorrelationmethodsfortherst10datasetswegetthreecorrectidenticationbutforthenal10datasetswhichhavesmallertimeperiodswegetnocorrectidentication.InChapter 3 thenewproposedmethodgives8correctestimationswhereasthemethodproposedin[ 4 ]gives7hits.Againonethingtoobservehereisthatforthe4

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Magnitudeplotoflightchannel1.3810.19.Notethatthey-axisrepresentsthemagnitudeofthestar.Magnitudemeasuresthebrightnessofacelestialobject,howeverthebrightertheobjectappears,thelowerthevalueofitsmagnitude.Itiscustomaryinastronomytoplotthemagnitudescalereversed. datasetswhichhavetimeperiodclosetoonedayneitherofthemethodactuallygivesacorrectestimation.Theproblemarisesasthedatasetsonanaveragehaveonesampleperperiod.Thismakesitdifculttoeitherinterpolatethesamplesorestimateperiod.InChapter 4 theproposedmethodusesfoldingtoreconstructasingleperiod.Thuseventhoughdatasetshavefewersamplesperperiodonanaverageitdoesnotaffectthemethod.Hencethisproposedtechniqueevenestimateswithveryhighdegreeofaccuracytheperiodofdatasetshavingtimeperiodclosetoorlessthanaday.Infactitisabletogive15accurateestimatesandtheremaining5givevalueswhicharehalfofthetrueperiod.Ifwecomparethismethodwiththemethodsdescribedearlier,the 43

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Plotofspatio-temporalkernelbasedcorrentropyforlightcurve1.3448.153. Figure5-3. Plotofspatio-temporalkernelbasedcorrentropyforlightcurve1.3810.19. 44

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Performanceevaluationoftheexistingtechniquesandtheroposedtechniques.TheresultspublishedbyHarvardUniversity,TimeSeriesCenterhasbeenusedasthegoldenstandardfortheevaluation. IndexMethodusedCorrectAverageabsoluteidenticationrelativeerrorfor(20lightcorrectlyidentiedcurvesused)timeperiod 1FFToninterpolateddata30.012462Auto-correlationon30.00202Interpolateddata3Lombperiodogram20.000084Dirichlettransform20.000085CSDoninterpolateddata70.0058162Dkernelbasedmeasure80.00927(Proposed)72Dkernelbasedcorrentropy150.00009onfoldedtimeseriesdata(Proposed) valueoftheestimategivenbythe2Dkernelbasedcorrentropytechniquearemoreaccurate.Anotherthingtonotehereisthatpeaksobtainedinthiscaseareverysharpespeciallyfordatasetswithasmallertimeperiod.ThiscanbeeasilyseeninFigure 5-2 whichhasasharperpeakascomparedtothatofFigure 5-3 .Thuswecanseethatthespatio-temporalkernelbasedcorrentropymethodissuperiortotheexistingmethodsnotonlyintermsofnumberofhitswhereitestimatestheperiodcorrectlybutalsointhedegreeofaccuracyofthosehits.ThiscanbeseenfromtheTable 5-1 45

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4.1 whiledescribingtheprocedure.Althoughwehaveusedaadaptivethresholdforvariousdatasetstoidentifysignicantpeaksthefractionofdynamicrangeisusedisxed.Thisactuallydependsontheamountofmodulation.Thisalsoholdsfortherstproposedmethodbasedonspatio-temporalkernel.Weneedtolookintoawaytoidentifythedegreeofmodulationorinotherwordstheshapeofthecurve.Anotherdrawbackisthatwehavetoscanoveraxedrangeofvaluestoidentifythetrueperiod.Thusdevelopmentofanefcientalgorithmtodetecttherangeofvaluesortheorderofthe 46

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ThischapterdealswiththestepsizeusedwhilescanningoverarangeofvaluestoimplementtheperiodestimationalgorithmdescribedinChapter 4 .Thevariablestepsizeisnecessitatedbythefactthatusingasmallerstepsizeincreasesthecomputationalcomplexitybutusingalargerstepsizeespeciallymightcausethealgorithmtomissthepeakiftrueperiodofthelightcurveissmall.Ifperiodicityofthelightcurveislargethenwecanaffordtousealargerstepsizewhilescanningtherangeofvalueswithoutmissingthepeakcorrespondingtothetimeperiod.ThiseffectcanbeseenintheFigure 5-2 and 5-3 .Forlightcurve1.3448.153whichhasatimeperiodequalto3.2764daysweseepeaksaremuchsharperandforlightcurve1.3810.19whichhasatimeperiodequalto88.94063daysthepeaksarewider.Thismeanswecanaffordtouselargerstepsizeforlightcurve1.3810.19andyetbeabletoidentifythepeakswhereaswecannotusealargerstepsizeforlightcurve1.3448.153withouttheriskofmissingoutthepeak.Wepresentbelowaproofonhowchoosingalargerstepsizecancausethealgorithmtofail.Let,Numberofdaysoverwhichlightcurvedataiscollected=TTrueperiod(unknown)=pTrialperiod=q(variableforourexperiment)Now(say)stepsizebeingusedinneighborhoodofp=rThisvalueristheresolutionbeingusedtoscantherangeofvaluesinneighborhoodofpErrorisintroducedasthetrueperiodmaynotbepresentinthesetoftrialperiodvalues.Thiserrorisminimizedwhenatrialperiodclosesttothetruevalueoftheperiodisusedwhilescanningoveraxedrange.Lettheerror=Iteasytoseethatjj
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A and A weget2(r=2)T q2=0.01Solvingweget; A Henceinthealgorithmasmallerstepsizeisusedwhilescanningoversmallertrialperiodvaluesandalargerstepsizeforlargetrialperiodvalues. 49

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[1] C.Alcock,R.A.Allsman,D.R.Alves,T.S.Axelrod,A.C.Becker,D.P.Bennett,K.H.Cook,N.Dalal,A.J.Drake,K.C.Freeman,M.Geha,K.Griest,M.J.Lehner,S.L.Marshall,D.Minniti,C.A.Nelson,B.A.Peterson,P.Popowski,M.R.Pratt,P.J.Quinn,C.W.Stubbs,W.Sutherland,A.B.Tomaney,T.VandeheiandD.Welch,TheMACHOProject:MicrolensingResultsfrom5.7YearsofLMCObservations,AstrophysicalJournal,vol.542,pp.281-307,2000. [2] B.E.Boser,I.M.GuyonandV.N.Vapnik,ATrainingAlgorithmforOptimalMarginClassiers,Proceedingsofthe5thAnnualACMWorkshoponCOLT,pp.144-152,Pittsburgh,USA,1992. [3] J.Debosscher,L.M.Sarro,C.Aerts,J.Cuypers,B.Vandenbussche,R.GarridoandE.Solano,AutomatedSupervisedClassicationofVariableStars.I.Methodology,AstronomyandAstrophysics,vol.475,pp.1159-1183,December,2007. [4] PabloA.Estevez,SeniorMember,IEEE,PabloHuijse,PabloZegers,SeniorMember,IEEE,JoseC.Principe,FellowMember,IEEE,andPavlosProtopapas,PeriodDetectioninLightCurvesfromAstronomicalObjectsUsingCorrentropy,IJCNN,July18-23,2010. [5] PrabhuBabuandPetreStoica,Spectralanalysisofnon-uniformlysampleddata-areview,DigitalSignalProcessing,2009,Elsevier. [6] Jian-WuXu,PuskalP.Pokharel,AntonioR.C.PaivaandJoseC.Prncipe,Non-LinearComponentAnalysisbasedonCorrentropy,IJCNN,July16-21,2006. [7] N.R.Lomb,LeastSquareFrequencyAnalysisofUnequallySpacedData,Astro-physicsandSpaceScience,vol.39,Feb.1976,p.447-462. [8] AndrzejWojtkiewiczandMichaiTustytiski,ApplicationoftheDirichletTransforminAnalysisofNon-UniformlySampledSignals,ProceedingoftheinternationalconferenceonAcoustic,SpeechandSignalProcessing.p.V.25-V.28,1992. [9] J.D.Scargle,StudiesinAstronomicalTimeSeriesAnalysis.II.StatisticalAspectsofSpectralAnalysisofUnevenlySpacedData,TheAstrophysicalJournals,vol.263,pp.835-853,December,1982. [10] A.GunduzandJ.C.Prncipe,CorrentropyasaNovelMeasureforNonlinearityTests,SignalProcessing,vol.89,pp.147-23,2009. [11] M.Petit,VariableStars(NewYork:Wiley),1987. [12] P.Protopapas,J.M.Giammarco,L.Faccioli,M.F.Struble,R.DaveandC.Alcock,FindingOutlierLightCurvesinCataloguesofPeriodicVariableStars,MonthlyNoticesoftheRoyalAstronomicalSociety,vol.369,pp.677-696,June,2006. 50

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I.Santamarya,P.P.PokharelandJ.C.Prncipe,GeneralizedCorrelationFunction:Denition,Properties,andApplicationtoBlindEqualization,IEEETransactionsonSignalProcessing,vol.54,no.6,pp.2187-2197,June,2006. [14] W.Liu,P.P.Pokharel,andJ.C.Prncipe,Correntropy:Propertiesandapplicationsinnon-gaussiansignalprocessing,IEEETransactionsonSignalProcessing,vol.55,no.11,pp.52865298,2007. [15] A.Schwarzenberg-Czerny,Ontheadvantageofusinganalysisofvarianceforperiodsearch,MonthlyNoticesoftheRoyalAstronomicalSociety(MNRAS),vol.241,pp.153-165,1989. [16] G.Wachman,R.Khardon,P.ProtopapasandC.Alcock,KernelsforPeriodicTimeSeriesArisinginAstronomy,ProceedingsoftheEuropeanConferenceonMachineLearning,LectureNotesinComputerScience,Vol.5782,pp.489-505,2009. [17] J.-W.XuandJ.C.Prncipe,APitchDetectorBasedonaGeneralizedCorrelationFunction,IEEETransactionsonAudio,Speech,andLanguageProcessing,vol.16,no.8,pp.1420-1432,November,2008. [18] T.-F.Wu,C.-J.LinandR.C.Weng,ProbabilityEstimatesforMulti-ClassClassicationbyPairwiseCoupling,JournalofMachineLearningResearch,vol.5,pp.975-1005,2004. 51

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BibhuPrasadMishrawasborninIndiain1987.HedidhisschoolinginRourkelabeforejoiningengineeringschool.HestartedhisengineeringonAugust,2005atIITKharagpur.In2009hegraduatedwithHonorsandreceivedhisBachelorofTechnology(B.Tech)inElectronicsandECE.UpongraduationhejoinedUniversityofFloridatopursueMasterofSciencedegreeinElectricalandComputerEngineering.HehasbeenworkingwithDr.PrncipeinComputationalNeuroEngineeringLaboratory(CNEL)sinceSpring2010.HereceivedhisMasterofSciencedegreeinDepartmentofElectricalandComputerEngineeringinUniversityofFloridain2011. 52