小波(Wavelet)乐园

小波分析的基础知识, 小波分析的软件实现, 小波分析应用的现状与前景
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A Practical Guide to Wavelet 4
(2004-12-17 05:15:16)

MonteCarloMethodImportantNote:Whenapplyingthesignificanceandconfidencetestsfrom"APracticalGuidetoWaveletAnalysis",youdonotneedtouseaMonteCarlosimulation.Analyticalformulaearegiveninthepaperforthestatisticaldistributionofwaveletpower.Theseformulaearecorrect,assumingthattheunderlyingdistributionoftheoriginaltimeseriesisGaussian.ThefollowingdescriptionofMonteCarloisgivenforthosewhowantmorebackgroundo...[阅读全文]
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A Practical Guide to Wavelet 3
(2004-12-17 05:13:28)

ScalesIntheprevioussectionwesawhowonecanuseatypicalwavelet(theMorlet)todecomposeatimeseriesintotime-frequencyphasespace.Wenowturntotheactualcomputationofthewavelettransform.Thefirstthingtoconsideristheshapeofthewavelet.FordecomposingtheNINO3SSTdata,wechosetheMorletwaveletbecause:itiscommonlyused,it'ssimple,itlookslikeawave.Thereareaninfinitenumberofothermotherwaveletsthatcouldbechosen(seeFarge1992...[阅读全文]
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A Practical Guide to Wavelet 2
(2004-12-17 05:12:49)

WaveletsIntheprevioussectionwesawhowonemeasureofthestationarityofatimeserieswastocalculatetherunningvarianceusingafixed-widthwindow.Althoughwepointedoutthedisadvantageofusingafixed-widthwindow,onecouldrepeattheanalysiswithavarietyofwindowwidths.Bysmoothlyvaryingthewindowwidthonecouldthenbuildapictureofthechangesinvarianceversusbothtimeandwindowwidth.Theobviousproblemwiththistechniqueisthesimple"bo...[阅读全文]
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A Practical Guide to Wavelet
(2004-12-17 05:11:53)

IntroductionManytimeseriesingeophysicsexhibitnon-stationarityintheirstatistics.Whiletheseriesmaycontaindominantperiodicsignals,thesesignalscanvaryinbothamplitudeandfrequencyoverlongperiodsoftime.AnexamplewouldbeseasurfacetemperaturesintheequatorialPacificOcean(Figure1).Thedominant"mode"ofvariabilityistheElNiño-SouthernOscillation(ENSO),shownbythehigh-frequency"spikes"onatimescaleof2-7years....[阅读全文]
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The Wavelet Tutorial 4
(2004-12-17 05:10:42)

MULTIRESOLUTIONANALYSIS:THEDISCRETEWAVELETTRANSFORMWhyistheDiscreteWaveletTransformNeeded?Althoughthediscretizedcontinuouswavelettransformenablesthecomputationofthecontinuouswavelettransformbycomputers,itisnotatruediscretetransform.Asamatteroffact,thewaveletseriesissimplyasampledversionoftheCWT,andtheinformationitprovidesishighlyredundantasfarasthereconstructionofthesignalisconcerned.Thisredundanc...[阅读全文]
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The Wavelet Tutorial 3
(2004-12-17 05:09:25)

MULTIRESOLUTIONANALYSIS
&
THECONTINUOUSWAVELETTRANSFORMby
RobiPolikarMULTIRESOLUTIONANALYSISAlthoughthetimeandfrequencyresolutionproblemsareresultsofaphysicalphenomenon(theHeisenberguncertaintyprinciple)andexistregardlessofthetransformused,itispossibletoanalyzeanysignalbyusinganalternativeapproachcalledthemultiresolutionanalysis(MRA).MRA,asimpliedbyitsname,analyzesthesignalatdiffer...[阅读全文]
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Wavelet Tutorial 2
(2004-12-17 05:07:50)

FUNDAMENTALS:
THEFOURIERTRANSFORM
AND
THESHORTTERMFOURIERTRANSFORMFUNDAMENTALSLet'shaveashortreviewofthefirstpart.
WebasicallyneedWaveletTransform(WT)toanalyzenon-stationarysignals,i.e.,whosefrequencyresponsevariesintime.IhavewrittenthatFourierTransform(FT)isnotsuitablefornon-stationarysignals,andIhaveshownexamplesofittomakeitmoreclear.Foraquickrecall,letmegivethefollowingexample.S...[阅读全文]
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The Wavelet Tutorial 1
(2004-12-17 05:05:38)

FUNDAMENTALCONCEPTS
&
ANOVERVIEWOFTHEWAVELETTHEORY
SecondEdition
Welcometothisintroductorytutorialonwavelettransforms.Thewavelettransformisarelativelynewconcept(about10yearsold),butyettherearequiteafewarticlesandbookswrittenonthem.However,mostofthesebooksandarticlesarewrittenbymathpeople,fortheothermathpeople;stillmostofthemathpeopledon'tknowwhattheothermathpeoplearetalkingabou...[阅读全文]
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Wavelet Introduction (7)
(2004-12-17 04:57:48)

WaveletsintheFutureWiththefoundationsofwavelettheorysecurelyinplace,thefieldhasgrownrapidlyoverthelastdecade.Adistributionlistonwaveletsthatbeganwith40namesin1990isnowanonlinenewsletterwithmorethan17,000subscribers.Moreover,ithascontinuedtoevolvethroughahealthymixoftheoryandpractice.Engineersarealwaystryingnewapplications,andformathematicians,therearestillimportanttheoreticalquestionstobeanswered....[阅读全文]
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Wavelet Introduction (6)
(2004-12-17 04:57:13)

HowDoWaveletsWork?Sofar,the“killerapp”forwaveletshasbeendigitalimagecompression.TheyarecentraltothenewJPEG-2000digitalimagestandardandtheWSQ(waveletscalarquantization)methodthattheFBIusestocompressitsfingerprintdatabase.Inthiscontext,waveletscanbethoughtofasthebuildingblocksofimages.Animageofaforestcanbemadefromthebroadestwavelets:abigswathofgreenfortheforest,asplashofblueforthesky.Moredetailed,...[阅读全文]
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