Multivariate scale-free temporal dynamics: From spectral (Fourier) to fractal (wavelet) analysis

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spectral (Fourier) to fractal (wavelet) analysis

Patrice Abry, Herwig Wendt, Stéphane Jaffard, Gustavo Didier

To cite this version:

Patrice Abry, Herwig Wendt, Stéphane Jaffard, Gustavo Didier. Multivariate scale-free temporal

dynamics: From spectral (Fourier) to fractal (wavelet) analysis. Comptes Rendus Physique, Centre

Mersenne, 2019, 20 (5), pp.489-501. �10.1016/j.crhy.2019.08.005�. �hal-02346492�

Contents lists available atScienceDirect

Comptes

Rendus

Physique

www.sciencedirect.com

Fourier and the science of today / Fourier et la science d’aujourd’hui

Multivariate

scale-free

temporal

dynamics:

From

spectral

(Fourier)

to

fractal

(wavelet)

analysis

Dynamique temporelle multivariée invariante d’échelle : de l’analyse

spectrale (Fourier) à l’analyse fractale (ondelette)

Patrice Abry

a

,

∗

,

Herwig Wendt

b

,

Stéphane Jaffard

c

,

Gustavo Didier

d a_{UniversitédeLyon,ENSdeLyon,CNRS,Laboratoiredephysique,Lyon,France}

b_{IRIT,CNRS(UMR5505),UniversitédeToulouse,France}

c_{UniversitéParis-Est,LAMA(UMR8050),UPEM,UPEC,CNRS,Créteil,France} d_{DepartmentofMathematics,TulaneUniversity,NewOrleans,LA,USA}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Availableonlinexxxx

Keywords:

Fouriertransform,wavelettransform Multivariatesignals

Scale-freedynamics Self-similarity Multifractality

Mots-clés :

TransforméedeFourier,transforméeen ondelettes

Signauxmultivariés

Dynamiqueinvarianted’échelle Auto-similarité

Multifractalité

The Fourier transform (or spectral analysis) has become a universal tool for data analysis in many different real-world applications, notably for the characterization of temporal/spatial dynamics in data. The wavelet transform (or multiscale analysis) can beregardedas tailoringspectralestimation toclassesofsignalsorfunctions definedby scale-free dynamics. The present contribution first formally reviews these connections inthe contextofmultivariatestationaryprocesses, andseconddetailsthe abilityofthe wavelet transform to extend multivariate scale-freetemporal dynamics analysis beyond second-orderstatistics(Fourierspectrumandautocovariancefunction)tomultivariate self-similarityand multivariatemultifractality. Illustrations and qualitativediscussions ofthe relevanceofscale-freedynamicsformacroscopicbrainactivitydescriptionusingMEGdata areproposed.

©2019Académiedessciences.PublishedbyElsevierMassonSAS.Thisisanopenaccess articleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

r

é

s

u

m

é

LatransforméedeFourier(ouanalysespectrale)estaujourd’huidevenueunoutiluniversel pour l’analyse de données issues de nombreuses applications réelles de natures très différentes,particulièrementpertinentpourlacaractérisationdeladynamiquetemporelle ouspatiale.Latransforméeenondelettes(ouanalysemultéchelle)peutêtrevuecomme uneanalysespectraleadaptée àdesclassesde signauxoufonctionsdont ladynamique estinvarianted’échelle.Laprésentecontributionproposed’aborddefaireunétatdel’art des relations formelles entre ces deux analyses dans le cadre des processus aléatoires stationaires multivariés, puis de montrer la capacité de la transformée en ondelettes à étendrel’analysedel’invarianced’échellemultivariéeau-delàdesstatistiquesdesecond ordre (fonction de covariance et spectre de Fourier), à l’auto-similarité multivariée et à la multifractalité multivariée. Quelques illustrations et éléments de discussion sur la

*

Correspondingauthor.

E-mailaddresses:[email protected](P. Abry),[email protected](H. Wendt),[email protected](S. Jaffard),[email protected](G. Didier).

https://doi.org/10.1016/j.crhy.2019.08.005

1631-0705/©2019Académiedessciences.PublishedbyElsevierMassonSAS.ThisisanopenaccessarticleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

pertinencede cesconceptsetoutilspour l’analysedel’activité cérébralemacroscopique sontproposés.

©2019Académiedessciences.PublishedbyElsevierMassonSAS.Thisisanopenaccess articleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

1.1. Context:fromfrequencytoscalerepresentations

Fourier(spectral)transformandmultivariatetemporaldynamics. The Fouriertransformhas becomea universal tool in almost all fieldsofthe sciences. Notably,ithas beenusedto analyzedatafromnumerous real-worldapplications where the informationofinterest isconveyedby themultivariate temporalorspatial dynamicsof collectionsofsignals (images, fields,flowsofimages,video. . . ). Such dynamicsare oftenwell accounted forby theso-calledFourierspectrum, definedas theFouriertransformofauto- andcross-covariancefunctions,whichisthustiedtosecond-order(ortwopoint)statisticsof data[1].Besidescountlesssuccessesinreal-worldapplications,verydifferentinnature,rangingfromnaturalsciencesand engineeringtosocialsciences,thepopularityoftheFouriertransformreliesonasolidmathematicalfoundation[2–4] and fastandefficientcomputerimplementations(algorithms)[5,6].

Wavelet(multiscale)transformandmultivariatescale-freedynamics. TheFouriertransformandspectralanalysisare doc-umentedtobeofparticularrelevanceandinterestwhentemporaldynamicsarewellcharacterizedbyoscillatorybehavior, corresponding toenergyconcentration withinnarrowfrequencybands(forexample,macroscopicbrainactivitywith well-definedfrequencybandsknowntobeassociatedwithbrainactivity,e.g.,thealpha-bandandattention. . . [7]).Forsuchcases, dataarewelldescribedbymathematicalmodels(Markovprocesses,ARMAmodels. . . )whosedefinitionsaredeeplytiedto one (or a few)well-defined characteristic scale(s) of time, orequivalently, of frequencyranges. However, the “scale-free paradigm”isobservedtobeasuperiormodelingframeworkforagreatnumberofcomplexorlarge-dimensionaldatasets stemmingfromawiderangeofmodernfieldsofinvestigation.Thisincludesareassuchashumansocio-economicactivities (Internet traffic [8–11], finance [12–14], geography [15], artinvestigation [16,17]. . . ) and naturalphenomena (heartrate rhythms [18–22],neuroscience[23–26] or hydrodynamicturbulence[27–29] andgeophysics[30,31]. . . ). Theoriginal intu-itionintoscale-freebehaviormaylikelybetracedbacktotheseminalworksofB.Mandelbrot[32–34]:temporaldynamics aregovernedbyalargecontinuumoftimescales,andcrucialinformationisnolongerencodedintheidentificationofone orafewtimescales,butratherintherelationsamongsttimescales[8].Suchsituationshavebeenabundantlycharacterized, notably, by meansofconceptualmodelssuch as1

/

f -processes,self-similarityand(multi)fractality.Neuroscienceprovides a richcontextofparticularinterestwherescale-freedynamicshasbeenappliedtothemodelingofinfraslow spontaneous macroscopicbrainactivity[24,35].Ithasalsoprompted theuseofthewavelettransforminsteadoftheFouriertransform, that is, of scale-dependent rather than frequency-dependent analysis [36,37]. This was shown to efficiently reformulate spectralanalysis,boththeoreticallyandpractically[8,10,38–41].

BeyondFourierandsecond-orderstatistics. Multivariatetemporaldynamicscharacterizationisnaturally groundedin sec-ondorderstatisticalanalysis(Fourierspectraandauto- andcross- covariances).Bycontrast,itswaveletcounterpartpermits extensionsthataretheoreticallywell-groundedandefficientinpracticetohigherorderstatistics,andalsotosomeformsof non-stationarity[8].Thisisaccomplishedbymeansoftheconceptsofmultivariateself-similarity[42–44] andmultivariate multifractality[45,46].

1.2. Goals,contributions,andoutlines:waveletandhigher-ordermultivariatetemporaldynamics

The overallgoal ofthepresentcontributionisto showthe extenttowhich wavelettransforms encompassandenrich Fourier transformsfor theanalysis ofmultivariate scale-free dynamics. Fourier-basedandwavelet-based spectral estima-tionformultivariatestationarystochasticprocessesarethusreviewedandcomparedinSection2.Therichnessofwavelet eigenspectrum-based analysis for multivariate self-similarity is detailed inSection 3. The need for extending wavelet to waveletleaderanalysisinmultifractalanalysis,togetherwiththeinterestinmultivariate multifractalanalysis,isdescribed inSection4.

A Matlab toolbox implementingthewavelet-basedanalysisofscale-free dynamicsispubliclyandfreelyavailable with documentationathttp://www.ens-lyon.fr/PHYSIQUE/Equipe3/Multifractal/.

2. Fouriervs.waveletmultivariatespectralestimation

2.1. Fourier-basedspectralestimation 2.1.1. Stationarystochasticprocesses

Let

(

Xm

(

t

))

m=1,...,M,t∈RbeaM-variate,real-valued,finitevariancestationarystochasticprocess,withwell-defined

auto-and cross-covariance functions, Cm,n

(

τ

)

= E



Xm

(

t

)

Xn∗

(

t

+

τ

)



where

E

denotestheensembleaverage,∗ denotescomplexconjugation,and

˜

f

(

ν

)

= (

F

f

)(

ν

)

=



f

(

t

)

exp

(

−

2i

π

ν

t

)

dt isthe Fouriertransformof f .

2.1.2. MultivariateFourier-basedspectralestimation

Classicalnonparametricspectralestimation,oftenreferredtoasperiodogram orWelch spectralestimation[1],isgrounded inthe useofthe short-timeFourier(or Gabor)transform(STFT)[47].The STFTcoefficients gX

(,

k

)

are definedby

com-paring, by means of inner products, the signal to analyze, X

(

t

)

, against a collection oftranslated and frequency-shifted templates,

φ

,k

(

t

)

= φ(

t

−

kT0

)

exp

(

−

2i



ν

0t

)

ofareferencepattern

φ (

t

)

: gX

(,

k

)

= 

X

,

φ

,k



.The timeandfrequency

res-olutions T0 and

ν

0 are positivequantities thatcan be arbitrarilychosen, providedthat they satisfy T0

ν

0

≤

1

/(

4

π)

. Under

mild conditionsonthefinite-energy function

φ (

t

)

,the STFTcanbe inverted,andgX

(,

k

)

canbe interpreted asthe joint

timeandfrequencycontentofX aroundtimet

=

kT0andfrequency

ν

= 

ν

0 [47].

STFT-basedmultivariate spectral analysisamounts toestimatingthe frequencyspectra



m,n

(

ν

)

by time averages (thus

assumingergodicityofX ,inadditiontostationarity)ofSTFTcoefficientssquared-moduli:

ˆ

m,n

(

ν

= 

ν

0

)

=



k

gXm

(,

k

)

g∗Xn

(,

k

)

(1)

Straightforwardcalculationsyield

E ˆ



m,n

(

ν

= 

ν

0

)

=





m,n

(

f

− 

ν

0

)

| ˜φ(

f

)

|

2d f (2)

thus showingthat

ˆ

m,n

(

ν

)

estimates



m,n

(

ν

)

by averaging



m,n

(

f

)

over frequencies f within a spectral band controlled

by

˜φ

. The time and frequencyresolutions of the functions

φ

,k depend neither on



nor on k, and are only controlled

throughthechoiceofthefunction

φ

.STFTthusachievesafixedabsolute-frequencyresolution multivariatespectralanalysis. Ofparticularinterestinmultivariateanalysisisthepairwisecoherencefunction

Cohm,n

(

f

)

=



m,n

(

f

)





m,m

(

f

)

n,n

(

f

)

(3) whichconsistsofafrequency-dependentcorrelation coefficient.Byquantifyingwhich frequenciesare actuallyinvolvedin cross-temporaldynamics,itpermitsbetteranalysisoftheoveralltemporaldynamicsofa system.Forrealsignals, Cm,n

(

f

)

isalsorealandrangeswithin

−

1

≤

Cohm,n

(

f

)

≤

1.

2.2. Wavelet-basedspectralestimation 2.2.1. Multivariatewavelettransform

AsanalternativetotheSTFT,spectralestimationcanbereformulated usingthediscretewavelettransform(DWT). The coefficientsoftheDWT,dX

(

j

,

k

)

,aredefinedbycomparing,bymeansofinnerproducts,thesignaltoanalyze, X

(

t

)

,against

acollectionoftranslatedanddilatedtemplates,

ψ

j,k

(

t

)

=

1

/

a0j

ψ((

t

−

kT0a0j

)/

a

j

0

)

,ofafinite-energyreferencepattern

ψ(

t

)

:

dX

(

j

,

k

)

= 

X

,

ψ

j,k



.When

ψ

isa zero-mean function,



ψ(

t

)

dt

≡

0,andunderother mildconditions,the timeandscale resolutionsT0anda0canbechosensuchthattheDWTcanbeinvertedandthedX

(

j

,

k

)

canbeinterpretedasthejointtime

andscalecontentof X aroundtimet

=

ka₀jT0andscalea

=

a0j.Additionally,forappropriate

ψ

,theset

{

a

j/2

0

ψ

j,k

(

t

)

}

j,k∈R2 is

anorthonormalbasisofL2

(R)

,thusleadingtosimpleinversionalgorithms.ThegenericcaseofthedyadicDWT,usedhere, correspondstoselectinga0

=

2 [36,37].

Inadditiontobeingband-pass,themotherwavelet

ψ

isoftendesignedtohaveNψ vanishingmoments,withNψ defined

asthesmallestintegersuchthat

∀

k

=

0

. . . ,

Nψ

−

1

,



tk

ψ (

t

)

dt

≡

0 and



tNψ

_{ψ (}

_t

₎

_dt

=

_{0 (4)}

Nψ controlsthedecayof

˜ψ

around

ν

=

0 as:

| ˜ψ(

ν

)

|

Cψ

|

ν

|

Nψ,andthusthebandwidthof

˜ψ

[36,37].

Theband-passnatureof

ψ

permitstheassociationwithcharacteristiccentralfrequency

ν

ψ andbandwidth



ν

ψ:

ν

ψ

=



f

| ˜ψ(

f

)

|

2d f



| ˜ψ(

f

)

|

2_{d f} and



ν

ψ

=



|

f

−

ν

ψ

|

2

| ˜ψ(

f

)

|

2d f



| ˜ψ(

f

)

|

2_{d f} (5)

Thus,by remappingthescalea tothefrequency

ν

via

ν

=

ν

ψ

/

a, theDWTcoefficientsdX

(

j

,

k

)

canalsobe interpretedas

2.2.2. Multivariatewavelet-basedspectralestimation

DWT-basedmultivariatespectralanalysisamountstoestimating



m,n

(

ν

)

bytimeaveragesofDWTcoefficients

squared-moduli:

ˆ

(W) m,n

(

ν

=

ν

ψ

/

a0j

)

=



k dXm

(

j

,

k

)

d∗Xn

(

j

,

k

)

(6)

Straightforwardcalculationsyield:

E ˆ



(mW,n)

(

ν

=

ν

ψ

/

a0j

)

=





m,n

(

f

)

| ˜ψ(

f

/

a0j

)

|

2d f (7)

The central frequency,the time, andthefrequency resolutionof

ψ

j,k are relatedto thoseof

ψ

as

ν

j,k

=

ν

ψ

/

a0j,



ν

j,k

=



ν

ψ

/

a0j and



Tj,k

= 

Tψ

×

a0j. This showsthat

ˆ

(W)

m,n

(

ν

)

estimates



m,n

(

ν

ψ

/

a0j

)

by averaging



m,n over frequencies f

around

ν

ψ

/

a0j withina spectralbandcontrolled by



ν

ψ

/

a0j.DWTthusachievesafixedrelative-frequency resolution

multi-variatespectralanalysis.

Inanalogytothecoherencefunction,thewaveletcoherencecanbedefinedas[48]:

Coh(mW,n)

(

ν

=

ν

ψ

/

a

)

=



(mW,n)

(

ν

ψ

/

a

)



m(W,m)

(

ν

ψ

/

a

)

n(W,n)

(

ν

ψ

/

a

)

(8)

Forrealsignals,itrangeswithin

−

1

≤

Cohm(W,n)

≤

1 andquantifies,asascale-dependentcorrelationcoefficient,whichscales

are actually involvedin cross-temporaldynamics,andthus permitsto better analyzethe overalltemporal dynamicsofa scale-freesystem.

2.3. Fouriervs.wavelet:oscillatoryvs.scale-freedynamics

Theoretically,STFTandDWTleadtovalidrepresentationsofX thatdonotloseinformation.Thisissobecausetheycan be inverted,i.e. X canbeexactlyrecoveredfromtherepresentationcoefficientsgX

(,

k

)

ordX

(

j

,

k

)

.Thus,bothversionsof

spectralestimationprovidealternativeandconsistentestimatorsof



m,n

(

ν

)

:

ˆ

m,n

(

ν

=

l

ν

0

)

and

ˆ

(mW,n)

(

ν

=

ν

ψ

/

a0j

)

.

The equivalence between both estimation methods is illustrated empirically in Fig. 1. Based on several examples of bivariatetimeseries,thefigureshowsthattheplotoflog2

ˆ

m,n

(

ν

)

asafunctionoflog2

ν

= 

ν

0 superimposeswellonthe

plotoflog2

ˆ

m(W,n)

(

ν

)

asafunctionoflog2

ν

=

ν

ψ

/

a,when

ν

ψ

/

a

= 

ν

0.

Thus, both spectral estimation methods can be of interest, depending on the temporal dynamics of the data. When temporal dynamicsare bettercharacterized by oscillatory behaviors,STFT-basedspectral estimation providesaccurate es-timation ofenergies inthe corresponding frequencyband, while DWT-based spectral estimation is better suited forthe analysisofscale-freedynamics.ThisisillustratedinFig.1(toprow),showingsuperimposedspectralestimatesforsynthetic signals consistingof(additive) mixturesof oscillatory andscale-freedynamics.Spectral-based estimatesbetter locatethe frequencyoftheoscillatory modes,andbetterquantifythelowcoherence levelatcorrespondingfrequencies.Bycontrast, wavelet-basedestimatesbetterrevealpower-lawbehaviordowntolowerfrequencies.Thispermitsbetterestimationofthe scalingexponentsand,hence,moreaccurateanalysisofthescale-freedynamics.

2.4. Waveletsandscale-freedynamics

Historically, scale-free dynamics has beenmodeled by stationaryprocesses withFourier spectrasatisfying an asymp-totic power law(hence, scale-free)behavior in thelow-frequency limit:

∀

m

,

n

,



m,n

(

ν

)

γ

m,n

|

ν

|

βm,n for

ν

→

0.Forsuch

processes,asimplechangeofvariableinEq. (7) above, permittedbythedilationoperatorunderlyingthedefinitionofthe DWT,showsthattheDWT-basedspectralanalysisaccuratelyreproducesasymptoticpower-lawbehaviorinthecoarsescale limit,i.e.when j

→ +∞

:

E ˆ



(mW,n)

(

ν

=

ν

ψ

/

a0j

)

=

γ

(W) m,na 2 jβm,n 0 with

γ

(W) m,n

=

γ

m,n



|

f

|

βm,n

| ˜ψ(

_f

₎

|

2_{d f}



(9) The frequency-shift operator underlyingthe STFTdoesnot permit such achange ofvariable,which leads tosignificantly biased estimatesof power-lawbehavior. Lessaccurate estimation ofthe scaling exponents

β

m,n follows,in turnyielding

pooreranalysisoftemporal (scale-free)dynamics[8,10,38–41]. Thisisillustratedin Fig.1(secondrow) withmultivariate fractional Gaussian noise,used asa cornerstonemodel forscale-free dynamics,anddefined asthe incrementprocess of multivariatefractionalBrownianmotion(seeSection3foradefinition).Wavelet-basedspectralestimatesperfectlyrevealthe power-lawbehaviorsdowntolowfrequenciesandaconstantlevelofthewaveletcoherencefunctionacrossallfrequencies, thuspermittingarelevantcharacterizationofmultivariatescale-freedynamics.

Fig. 1. Fourierversuswaveletspectralestimation. SuperimposedFourier(blacklines)andwaveletspectra(reddottedlines)forsynthetic signalswith additivemixtureofscale-freeandoscillatorydynamics(toprow),withscale-freedynamics(secondrow),withscale-freedynamicsandsmoothslowly decaying(exponential)additivetrends(thirdrow),andforMEGdata(bottomrow).

2.5. Wavelet-basedspectralestimationandrobustnesstotrends

Besidesbeingbetter-suitedtotheanalysisofscale-freetemporaldynamics,DWT-basedspectralestimationbenefitsfrom furtherpracticalrobustness.Thisisnotablythecasewhensmoothtrendsaresuperimposedonthedataunderscrutiny.This isillustratedqualitativelyinFig.1(thirdrow)whereunrelateddeterministicsmooth(e.g.,algebraicorexponential)trends are added to each component ofa bivariate fractional Gaussian noise. Fourier-basedspectral analysis is clearly strongly biased at low frequencies by the smooth trends, and so is the coherence function. By contrast, wavelet-based spectral analysisperfectlyrevealsthemultivariatescale-freetemporaldynamicsacrossallfrequencies,aswellastheconstantlevel ofthecoherenceacrossscales,inperfectmatchwiththemultivariatefractionalGaussiannoisemodelsusedhere[41,49].

2.6. Macroscopicinfraslowbrainactivity,scale-freedynamicsandwavelet-basedspectralestimation

Fig.1(fourthrow)furthercomparesFourier-basedandwavelet-basedspectralestimationsforapairofMEGtimeseries recorded on a subject at rest.1 _{These spectral estimates show that brain macroscopic activity consists of a mixture of}

bothoscillatorybehaviorsinwell-establishedfrequencybands,each associatedwithspecificbrainfunctions,andinfraslow scale-free dynamics.The alpha-band, 8

≤

f

≤

12 Hz, corresponding to attention, displays significant powerinthis range, thatcanbewellanalyzedandquantifiedusingFourier-basedspectralestimation.Inaddition,theinfraslow( f

≤

3 Hz)brain dynamicsischaracterizedbytheabsenceofcharacteristicoscillations,hencebyscale-freedynamics.Initiallythoughttobe experimentalnoise orhead-movementinduced,thisinfraslowactivityhasnowbeenrecognized tobeassociatedwiththe rangeoffrequencieswheremostbrainenergyisconsumedandisnowviewedasthesignatureofspontaneousbrainactivity. Notably,it hasbeenconsistently shownthat thebrainatrest showsstronginfraslow scale-free dynamicsthat structures functionalconnectivity(e.g.,withtherestingstatenetwork)[23,24].Itisnowcommonlyconsideredthatthemodifications inducedby taskengagementinbrainactivitycan bequantifiedby departurefromrestingstateactivity,inparticularwith aregion-dependent decreaseofthescaling exponentsquantifyingthe scale-freedynamics(hence,ofthe overalltemporal correlations)[24,26,35].

3. Waveleteigenanalysisofmultivariateself-similarity

3.1. Multivariateself-similarity

Following the seminal intuitions ofB. Mandelbrot [51], scale-free dynamicshas oftenbeenmodeled as self-similarity [52]. The celebrated fractional Brownian motion (fBm), BH

(

t

)

, is definedas the only Gaussian self-similar process with

stationary increments. Together withits increment process, calledfractional Gaussian noise (fGn), it has been massively usedinthemodelingofscale-freetemporaldynamicsforunivariatedata.Forunivariatestochasticprocesses,self-similarity isdefinedasscaleinvarianceofallfinite-dimensionaldistributionsunderdilation,i.e.,foranydilationfactora

>

0,

{

BH

(

t

)

}

t∈Rfdd

= {

aHBH

(

t

/

a

)

}

t∈R (10)

where H istheself-similarityparameter,orscalingexponent andwherefdd

=

denotesequalityforallfinitedimensional distri-butions.TheassumptionsofstationaryincrementsandfinitevarianceconfineH insidetheinterval

(

0

,

1

]

.

Thoughseemingly intuitive, a canonicalmodel formultivariateself-similarity calledOperatorfractional Brownian mo-tion (OfBm) was only recently defined [42,53–55]. OfBm arises as a weak limit of multivariate time series displaying matrix-induced memoryproperties[56,57]. Italsoprovides anaturalframework forthemodeling ofInternet traffic[44], dendrochronology[58] andfractionalcointegration[59].Hereinafter,useismadeofalessgeneralyetmorepedagogicaland constructive definition,whichalso constitutesa specialyet broadsubclassofOfBm, referred tohereas multivariatefBm (MfBm).

First,let BH,

(

t

)

beacollectionof M fBmBHm

(

t

)

,each withapotentially differentself-similarityexponent Hm.These

M-components arepointwisecorrelatedaccordingtoa M

×

M symmetricpositive definite(covariance)matrix



.Second, let W denoteaM

×

M invertiblematrix.Then,MfBm, BH,,W

(

t

)

isdefinedbyalinearmixingofBH,

(

t

)

accordingtoW :

BH,,W

(

t

)

=

W BH,

(

t

)

(11)

Hence,MfBmisparametrizedbythevectorofscalingexponentsH

= (

H1

. . . ,

Hm

)

,themixingmatrixW ,andthecovariance

matrix



.

Multivariate self-similaritytranslatesintothefinite-dimensionalequalityofthejointormultivariate finite-dimensional distributions:

∀

a

>

0

,

{

BH,,W

(

t

)

}

t∈Rfdd

= {

aHBH,,W

(

t

/

a

)

}

t∈R (12)

withaH

:=



+∞_k=0logk

(

a

)

Hk

_/

_k

_!

_andwhereH

₌

_{W diagH W}−1 _isnowanM

_×

_{M matrixofscalingexponents.}

3.2. Waveletanalysisofmultivariateself-similarity 3.2.1. Multivariatewaveleteigenanalysis

Extendingspectral analysisto amultivariate settingandapplyingwaveletanalysistoMfBm leadto ascale-dependent collectionofM

×

M (wavelet)matricesS

(

j

)

withentries

Sm,n

(

j

)

=

1

/

nj



k

d(BH,,W)m

(

j

,

k

)

d(BH,,W)n

(

j

,

k

))

(13)

While theclassical analysisof multivariateself-similaritywouldconsist ofanalyzingeach entry Sm,n

(

j

)

independently

asa functionof scalesa

=

2j andpossibly estimatingthecorresponding scalingexponent,an original wavelet eigenanal-ysis approach was recentlyproposed [43,44]. The approach reverses the perspective on multivariate multiscale analysis: it firstconsiders thefull matrix S at agivenscale a

=

2j _{bycomputingthe eigenvalues}



1

(

j

)

. . . ,



M

(

j

)

,and, second, it

takesadvantageofthebehaviorofeach



m

(

j

)

asafunctionofscales2j bypossiblyestimatingthecorresponding scaling

exponents.

Notably,forMfBm,ithasbeenshownthat,intheasymptoticlimitofcoarsescales,theeigenfunctions



m

(

j

)

reproduce

multivariateself-similarityas[43,44]:



m

(

j

)

λ

m22 j Hm

,

2j

→ +∞

(14)

with

λ

mdependingon



,W and

ψ

.

3.2.2. Non-mixedmultivariateself-similarity

Todevelopmoreintuitionintothepotentialofthewaveleteigenanalysis,letusfirststudythesimplercasewherethere isnomixing,i.e.themixingmatrixreducestotheidentitymatrixW

≡

IM.Multivariateself-similaritythensimplifiestoM

Fig. 2. Multivariateself-similarity. Non-mixedOfBm(toprow),andmixedOfBm(bottomrow).Wavelet(cross)-spectra(centerleft),waveletcoherence function(centerleft),waveleteigenanalysiscomparedtowaveletspectra(right).

{

BH1

(

t

), . . . ,

BHM

(

t

)

}

t∈R

fdd

= {

aH1BH1

(

t

/

a

), . . . ,

aHMBHM

(

t

/

a

)

}

t∈R (15)

Hence,theentrywisecovariancefunctionsofMfBmreduceto

E

BHm

(

t

)

BHn

(

s

)

= 

m,n

(

|

t

|

Hm+Hn

+ |

_s

|

Hm+Hn

− |

_t

−

_s

|

Hm+Hn

₎

₍₁₆₎

BycombiningthedefinitionofDWTandthecovarianceofnon-mixedMfBm,onecanshowthat

∀

m

,

n

,

∀

j

>

0

,

E

Sm,n

(

j

)

=

wH,,ψ2j(Hm+Hn) (17)

withparameters wH,,ψ depending jointlyon H ,



and

ψ

.Calculationsandproofscloselyfollowthewaveletanalysisof

univariateself-similarity[43,44].

Thesepower-lawbehaviorsareillustratedwithbivariateMfBminFig.2(toprow,centerleft).Theycallforthefollowing comments.

First,becauseitisconstructedfromadilationoperator,waveletanalysisexactlyreproducesself-similarity(whileFourier analysis would not, cf.[40] for a more complete and richerdiscussion). Hence, this leads to efficient estimationof the scalingexponentsHm

+

Hn(cf.,e.g.,[8]).Intuitively,thiscanbeinterpretedasthefactthatwaveletanalysisextendsspectral

analysistomultivariatenonstationaryprocesses,yetwithstationaryincrements.Thiscanbegeneralizedtoprocesseswith stationaryhigher-orderincrements(increments ofincrements

. . .

), providedthat thenumberofvanishingmomentsofthe mother-wavelet Nψ isincreasedaccordingly.

Second,fornon-mixedMfBm,multivariate(oreigen)waveletanalysisisredundant.Thereare M

×

M potentiallyusable entriesinthematrix S,whereasmultivariate self-similarityanalysisreducestoestimatingonlyM scaling exponents.This isillustratedbythefactsthati

)

thewaveletcoherencefunctionisconstantacrossthescalesa

=

2j _{actuallymeasuringthe}

overallcorrelation coefficients(Fig. 2toprow,centerleft);andthat ii

)

theeigenfunctions



1

(

j

)

and



2

(

j

)

reproducethe

power-lawbehaviorofthewaveletautospectra S11

(

j

)

andS22

(

j

)

,withthesamescalingexponents2H1and2H2 (Fig.2top

row,left).Thiscanbeusedtorobustifytheestimationofscalingexponents[60],totesttheabsenceofmixing,ortoassess departuresfromnon-mixedmultivariate self-similarity(alsoreferredto asfractalconnectivity)[49]. Inparticular,thishas beenusedtoenrichthequantificationandassessmentoffunctionalconnectivityinneuroscience[35,61].

3.2.3. Mixedmultivariateself-similarity

Tofurthergainintuitionintothepotentialofwaveleteigenanalysis,letusnowconsidermixedMfBm,i.e.withamixing matrixW thatisunknownandnotnecessarilydiagonal.Inthatcase,multivariateself-similaritybecomesfarmoreintricate toanalyze.Indeed,thecovarianceofmixedMfBmBH,,W



BH,,W

=

W



BH,W (18)

mixestogetheradditively powerlaws,withdifferentscalingexponentsinvolvingpair-combinationsfromtheentirevector

H [43,44].ComputingthecollectionofM

×

M matricesS

(

j

)

frommixedMfBmresultsinthesameintricatesituation,

∀

m, n,

∀

j

>

0,

E

Sm,n

(

j

)

, consistsofadditive mixtures of M

(

M

+

1

)/

2 powerlaws,combiningall possible pairs ofexponents

(bottom row,center left) forbivariate MfBm, wherethe functionslog2S11

(

j

)

, log2S12

(

j

)

and log2S22

(

j

)

as functionsof

thelog-scaleslog₂a tendtosuperimpose.Classicalwaveletanalysisleads totheconclusionthatasingleexponent(i.e.the largest)drivesthetemporaldynamicsofdata.Theestimationofscalingexponentsbysolvingthenon-convexoptimization problem of fitting additive mixtures of power laws, though doable in principle, turns out to be not feasible in practice beyondthebivariatecase M

=

2 [62].However, thewaveletcoherencefunction,beingnon-constant acrossscales,already providesanindicationthatdatamaynotbedrivenbyasinglescalingexponent(Fig.2,bottomrow,centerright).

The wavelet-eigenanalysis approach actually permits the disentangling of mixed multivariate self-similarity. Indeed, the function log2



2

(

j

)

showsthe same (dominant) scalingbehavior as theone observed in log2S11

(

j

)

, log2S12

(

j

)

,and

log₂S22

(

j

)

. However, the evolution oflog2



1

(

j

)

asa function ofscales departs fromthe dominantbehavior. It actually

revealsthe non-dominantscalingexponent,which ishiddenbyforce ofmixing, butisstill presentinthejointstructure of data. Multivariate wavelet analysis thus permits the accurate disentangled analysisof multivariate self-similarity and efficient estimationofthe M scalingexponents [43,44] by operatinga changeofperspective: Theindependentunivariate analysisofthe M-componentsofmultivariateMfBmfirstinspectsthetemporaldynamicsofeachcomponentacrossscales andthencomparescomponentsonesagainsttheothers,resultinginpossiblystronglyinaccurateanalysis.Bycontrast, mul-tivariate wavelet eigenanalysis first investigates all components jointly at a given scale 2j (by computingan eigenvalue decomposition),andthenusesthebehavioroftheeigenvaluesacrossscalesasananalysistool.

Estimatingthemixingmatrixisgenerallydoableonlyinthecontextofindependentsources(when



isdiagonal).This isadifferenttopic,anditisnotaddressedhere;interestedreadersarereferredto[58].

4. Multivariatemultifractality

4.1. Beyondsecond-orderanalysis:multifractalanalysis

It can happen that data have exactly the same marginal distributions (one-point statistics)and the same covariance functionsorFourierspectrum(2-pointstatistics),butaredifferent.Distinguishingbetweensuchdatarequiresanalysistools designed to go beyondcovariance analysis. In the context of scale-free temporal dynamics characterization, multifractal analysisprovidessuchatool.

Inessence, multifractalanalysisaimstocharacterizethe fluctuationsalongtime oflocalregularityina signal X

(

t

)

,cf., e.g.,[63–65].Localregularitycanbequantifiedbypointwiseexponents,themostcommononebeingtheHölderexponent,

h

(

t

)

≥

0,defined asfollows: X belongsto C α

(

t

)

,

α

≥

0, ifthere exist apolynomial Pt with deg

(

Pt

)

<

α

and a constant

C

>

0 suchthat:

|

X

(

t

+

a

)

−

Pt

(

t

+

a

)

|

≤

C

|

a

|

α when

|

a

|

→

0.The Hölderexponentconsistsofthelargestsuch

α

: h

(

t

)



sup

{

α

:

X

∈

C α

(

t

)

}

≥

0.Essentially,thelargerh

(

t

)

,thesmoother X aroundt,andconversely,thecloserh

(

t

)

to 0,themore irregular X att.Other exponents,suchas p-exponents,generalize theuseofHölderexponents[66,67].Hereafter, weuse genericallyh

(

t

)

todenoteeitherHölderexponentorp-exponents.

Some processes display smooth regularity exponents. This is the case for MfBm: h

(

t

)

is constant for each compo-nent. However, in general, h

(

t

)

is so irregularthat one cannot base theanalysis onthe time evolutions ofthe functions

h1

(

t

)

. . . ,

hM

(

t

)

obtained independently fromeach component of multivariate data. Instead, multifractalanalysis aims to

provideaglobal,geometric,andmultivariatedescriptionofthetemporaldynamicsof X viatheso-calledmultivariate mul-tifractalspectrum D

(

h1

. . . ,

hM

)

,definedasthecollectionofHausdorffdimensionsdimH ofthesetsofpointst

∈ R

where

(

h1

(

t

)

. . . ,

hM

(

t

))

takesthesamevaluesh

≡ (

h1

. . . ,

hM

)

[46]:

D

(

h

)



dimH

t

: (

h1

(

t

) . . . ,

hM

(

t

))

=

h



(19) The multifractalspectrum

D(

h

)

canthusbeconsidered asanefficientsummaryofthemultivariatetemporaldynamics ofdataX .

4.2. Multifractalformalism

Standardmultifractalmodelsleadtohighlyirregularexponentshm

(

t

)

thatcannotbeestimatedinpractice[63–65] and

the numericalestimationprocedure for

D(

h

)

fromdata,referred toasthemultifractalformalism,requires theuseofnew

multiscalequantities,beyondwaveletcoefficients,whichmatchthepointwiseexponentchosentoquantifyregularity. Itisnowwelldocumented[65–67] thatmeasuringHölderexponents(orp-exponents)callsfortheuseofwaveletleaders

(orp-leaders).Thesearedefinedaslocall∞ orlp-normsofwaveletcoefficients:



X

(

j

,

k

)



sup 2j_k_∈3λ j,k



dX

(

j

,

k

)



or



(Xp)

(

j

,

k

)



⎛

⎜

⎝



2j_k_∈3λj,k



dX

(

j

,

k

)



p 2(j−j)

⎞

⎟

⎠

1/p (20)

where

λ

j,k

= [

k2j

,

(

k

+

1

)

2j

)

isadyadicintervalofsize2jand3

λ

j,k

 λ

j,k−1

∪ λ

j,k

∪ λ

j,k+1istheunionof

λ

j,kwithitstwo

Followingspectralestimationandextendingtohigherstatisticalordersandtop-leadersthewaveletspectrum S

(

j

)

,one canformacollectionofmultiscalefunctionsLq

(

j

)

parametrizedwithq

= (

q1

,

. . . ,

qM

)

,definedas

Lq

(

j

)

=

1 nj nj



k=1



(_X1p)

(

j

,

k

)

q1

× . . . × 

(p) XM

(

j

,

k

)

qM ₍₂₁₎

Fornumerousclassesofprocesseswithscale-freedynamics,itisexperimentallyobservedthat,inthelimitoffinescales, Lq

(

j

)

∼

Kq1...,qM2

jζ (q)

_,

₂j

_→

_{0 (22)}

Thescalingexponents

ζ (

q

)

canthusbeestimatedbylinearregression[65,67].

ThemultivariateLegendretransformcanbederivedfrom

ζ (

q

)

throughamultivariateLegendretransform

L

(

h

)

=

inf

q

(

1

+ 

q

,

h

 − ζ(

h

))

(23)

4.3. Limitations

In theunivariate setting,M

=

1, theLegendre spectrum always provides an upper-boundestimate of the multifractal spectrum,

L(

h

)

≥

D(

h

)

,wheretheinequalityturnsintoanequalityforlargeclassesofprocesses[64].

In the multivariate setting, it was recently shownthat the multivariate Legendre spectrum does not always yield an upper-bound estimate ofthe multivariatemultifractal spectrum,see[46,68] fora detailedanalysisof thisintricate issue. However, itisexpectedthat theinequality(and eventheequality)holdsforlargeclassesofprocesses,andthatit canbe usefulforreal-worlddatamodeling.Sometheoreticalguidelinesareprovidedin[68];additionally,genericresultsofvalidity areprovedin[69].

4.4. Multifractalformalisminpractice

Even though the Legendre spectrum does not necessarily estimate the multifractal spectrum, the scaling exponents

ζ (

q

)

conveyinformation of anystatistical order relatedto temporal dynamicsand are thus of interest in characterizing scale-free dynamics.However, becauseestimatinga multivariate functionis difficult,we proposea polynomial expansion thatgeneralizestothemultivariatesettingthestrategyproposed in[70,71] fortheunivariatecase. Foreaseofexposition, thediscussionhereisrestrictedtoabivariatesetting,M

=

2.Thescalingexponentscanthusbeapproximatedas

ζ (

q1

,

q2

)

=

c10q1

+

c01q2

+

c20q12

+

c02q22

+

c11q1q2

+ . . .

Undermildconditions,itcanbeshownthatthecoefficientscn1n2 (withn1

+

n2

=

n)entering theexpansioncanbe

re-latedtothemultivariatecumulantsofordern,Cn1n2

(

j

)

,ofthemultivariatevariables

{

log



(Xp1)

(

j

,

k

)

. . . ,

log



(p)

XM

(

j

,

k

)

}

.Indeed,

forcertainclassesofmultivariatemultifractalprocesses[68],itisobservedthat

Cn1n2

(

j

)

=

c

0

n1n2

+

cn1n2log 2

j ₍₂₄₎

The first-ordercumulants (n

=

1, C10

(

j

)

and C01

(

j

)

) conveyinformationmostly driven by thecovariancefunction ofthe

process X and,hence,arecloselyrelatedtothefunctionslog₂S10

(

j

)

andlog2S01

(

j

)

[72,73].

Thehigher-ordercumulants(n

≥

2,C20

(

j

)

,C02

(

j

)

,C11

(

j

)

. . . )conveyinformationontemporaldynamicsbeyond

second-orderstatistics,whichisnotalreadyencodedinthecovariancefunctions.

ThismaterializesthroughanapproximationofthebivariateLegendrespectrumas:

L

(

h1

,

h2

)

≈

1

+

c02b 2

h1

−

c10 b



2

+

c20b 2

h2

−

c01 b



2

−

c11b

h1

−

c10 b



h2

−

c01 b



(25) whereb



c20c02

−

c112

≥

0, thusshowing that theposition ofthemaximum ofthebivariate spectrum isgivenby hm

=

(

c10

,

c01

)

and, further,thatthecoefficientsc20,c02,andc11 characterizethemultifractalpropertiesof X ,notably withc11

encodingcross-multifractality.

Furthermore,by takinginspirationfrom thewaveletcoherence function (see Eq. (3)), we proposeto define a wavelet leadermultifractalcoherencefunctionas:

Coh(mf)

(

j

)

=

√

C11

(

j

)

C20

(

j

)

×

C02

(

j

)

(26) On a scale-by-scale basis, this quantifies cross-dependencies amongst the components of the data that are not already accountedforbythewaveletcoherencefunction.

Fig.3illustratesthetheoryandpracticeofmultifractalanalysisbasedonseveraldifferentsyntheticprocessesandMEG data.Use is madeofthe bivariatemultifractalrandom walk (bi-MRW),a cornerstonemultifractal process,designedhere

Fig. 3. Empiricalbivariatemultifractalanalysis. Fromtoptobottom:correlatedbivariateMRW,anticorrelatedbivariateMRW,uncorrelatedbivariateMRW. Fromlefttoright:incrementsofthetimeseriesandunivariateandbivariatemultifractalspectra;univariateanalysisforcomponent1withlog2L0,q(j),

C10(j)andC20(j)asfunctionsoflog2a univariateanalysisforcomponent2withlog2Lq,0(j),C01(j)andC02(j)asfunctionsoflog2a;bivariateanalysis

forcomponents1and2withlog2Lq1,q2(j),Coh

(mf)

(j)andC11(j)asfunctionsoflog2a.

as an extension of the univariate MRW construction in [74] by combining bivariate OfBm synthesis [75] with bivariate multifractalconstruction[72,73].

Fig. 3(toppair ofrows)showsthe incrementsofa correlatedbi-MRW andthe(wavelet p-leader-based) estimatesof univariate andbivariatemultifractalspectra (right),the univariatemultifractalanalysisofcomponent1 andcomponent2 (center rightand left),and thebivariate multifractal analysisof components 1 and2 withthe leader-based multifractal coherence function (left).The multifractalityofeach componentaswellascross-multifractalityare assessed bymeans of the linearbehaviorofthe functionslog2Lq,0

(

j

)

,log2L0,q

(

j

)

andlog2Lq1,q2

(

j

)

withrespecttothelog-scales log2a

=

j,or

equivalently,bythelinearbehavioroffunctionsC10

(

j

)

,C01

(

j

)

,C20

(

j

)

,C02

(

j

)

,andC11

(

j

)

.TheestimatedLegendrespectrum

L(

h1

,

h2

)

departsfromthesimpleform

L

1

(

h1

)

+

L

2

(

h2

)

−

1 (whichisexpectedforindependentprocesses);thisconstitutes

a strong indication of the presence of statistical dependencies not already quantified by the coherence functions. Such dependenciesarefurtherquantifiedbythewaveletleadermultifractalcoherencefunctionCoh(mf)

(

j

)

,whichshowsapositive constantbehavioracrossscales.Thisindicatestemporalcoincidencesofthelargestorsmallestregularityexponentswithin thetwocomponents.

Fig. 4. MEGDatabivariatemultifractalanalysis. MacroscopicbrainactivityMEGdata.Fromlefttoright:incrementsofthetimeseriesandunivariate andbivariatemultifractalspectra;univariateanalysisforcomponent1withlog2L0,q(j),C10(j)andC20(j)asfunctionsoflog2a;univariateanalysisfor

component2withlog2Lq,0(j),C01(j)andC02(j)asfunctionsoflog2a;bivariateanalysisforcomponents1and2withlog2Lq1,q2(j),Coh

(mf)

(j)andC11(j)

asfunctionsoflog2a.

Fig.3(secondpairofrows)displaysthesameplotsasaboveforanotherbi-MRW,withidenticalcorrelationasthefirst bi-MRW, butdifferentjoint statistics.Therefore,Fourier analysisandclassical correlation analysiswouldnot seeany dif-ference betweenthesetwo bi-MRW, whilebivariate multifractalanalysisclearly doeswithdifferentbivariate multifractal spectra(despiteidenticalunivariatemultifractalspectra).Also,thewaveletleadermultifractalcoherencefunctionCoh(mf)

(

j

)

showsaconstant, yetnegative, valueacross scales,indicating temporalcoincidencesbetweenthelargestregularity expo-nentsofoneofthecomponentsandthesmallestoftheothercomponents.

Finally,Fig.3(thirdpairofrows)usesanuncorrelatedbi-MRW. Inother words,Fourieranalysisorclassicalcorrelation wouldnot detect any correlation amongst the two components.By contrast,bivariate multifractalanalysisclearly shows statisticaldependenciesbeyondthesecondorder.Indeed,thebivariatemultifractalspectrumisthesameasthatofthefirst bi-MRW usedinthetop pairofrows, andso isthewaveletleader multifractalcoherence functionCoh(mf)

(

j

)

. Thisagain indicatestemporalcoincidencesofthelargestorsmallestregularityexponentswithinthetwocomponents.

4.5. Macroscopicbrainactivity:multifractalanalysis

Univariatemultifractalanalysishasbeenusedinthecharacterizationofbraintemporaldynamics.Notably,in[26],itwas appliedtoMEGdatatostudyinfraslowbrainactivity,i.e.brainactivitybelow1 Hz(oracrosslongtimeepochs,from1 sec-ondtoseveraltensofseconds).Itshowedthatbrainactivityatrestwascharacterizedbysignificantself-similarity(largeH

oftheorderof0

.

9),withasignificantoccipitalgradient,andlowornomultifractality.ThisessentiallymeansthattheMEG timeseriesrepresentingbrainactivityatrestarecharacterizedbyasignificantglobalcorrelation,whichislargerinfrontal regions thaninoccipitalones. Italsomeansthat thisoverall patternisobservednot tofluctuatelocallyovertime,which indicatesthat brainactivityatrestshowsaconstantovertime,simple,andstructuredtemporaldynamics.Bycontrast,La Roccaetal. [26] alsoshowthattaskengagementyieldsasignificantandoveralldecreaseofself-similarity,yetincreasingthe fronto-occipitalgradient:thedecreaseinself-similarityismoreeffectiveintheoccipitalregions(sensorialbrainactivity)of thebrainthanintheoccipitalones(integrated/processingbrainactivities).Thisglobaldecreaseinself-similaritycorrelates withanincreaseofmultifractalitythatremains,however,localandconfinedtoregionsofthebrainthatareinvolvedinthe task.Multifractalityindicatesburstyactivitywithsignificantfluctuationsovertimeofthestructuresinbrainactivity:itcan alsobe interpreted asfluctuationsinthewaytime flows inthedifferent partofthebrain, compared toan overall brain clock[26].

Multivariatebrainactivityanalysisremains tobeconductedsystematically overthewholebrainandanalyzed. Prelimi-narybivariatemultifractalanalysis,reportedinFig.4,performedonthesametwoMEGbrainactivitysignalsusedinFig.1

(bottomrow),suggestsmultifractality ineach componentandrevealsanon-trivial bivariatemultifractalspectrum poten-tiallyindicatingthemodulationofhigher-orderstatisticaldependenciesinbrainfromresttotask.Theseeffectsareunder systematicanalysis, andmay permit to enrichthe quantificationof functionalconnectivity. While usually basedon fMRI measurements andoncorrelation coefficients(hence staticproperties) betweenthecorresponding time series,functional connectivitycouldalsobeinvestigatedbyexploitingtherichertemporaldynamicsavailableinMEGdynamics,byusingthe behaviorsofwaveletandmultifractalcoherencefunctionswithrespecttotimescales.

5. Conclusionsandperspectives

Spectral estimation via Fourier transform (relying on a frequency translation operator) constitutes the classical cor-nerstone tool to assess cross-temporaldynamics inmultivariate time series.When such dynamics are scale-free, i.e.not

governed by anyparticularscales of timeplaying aspecific role, butratherby mechanismsthat bind alarge continuum of time scales together,spectral estimationcan beefficiently androbustlyconducted bymeans ofthe wavelettransform (multiscale,andrelyingonadilationoperator).

Beyond1

/

f orpower-lawdecreasingmultivariatefrequencyspectra,multivariatescale-freedynamicscanbebetter mod-eledbymultivariateself-similarity.Multivariatewaveleteigenanalysisisbasedonascale-by-scalewaveletdecompositionof estimatedwaveletcoefficientcovariancematrices.Itprovidesanoriginal,theoreticallysound,andpracticallyrobusttoolfor assessingscale-freedynamicsinmultivariatetemporaldynamics.Beyondthecurrenttheoreticalanalysisofestimation per-formance,severalissuesremainunderinvestigationsuchastestingthenumberofdifferentscalingexponentsthatactually existamongstmultivariatecomponents[76],oraddressinglarge-dimensionalframeworks,whenthenumberofcomponents maybeoftheorderof,orlargerthan,thenumberoftimesamples[77].

Furthermore,beyondthemodeling ofcovariance, thecharacterizationofscale-free dynamicsmayinvolvehigher-order statistics. Therefore, multifractal analysis can be regarded asa further extension to multivariate Fourier analysis in the context ofscale-freedynamics.Itrequirestheuseofmultiscalerepresentations constructedfromnonlinearandnon-local transformsofwaveletcoefficients.Itwasrecentlyshownthattheextensionfromunivariate tomultivariateisnot straight-forwardastheconditionsunderwhichthemultivariatemultifractalformalismyieldsthemultivariatemultifractalspectrum remain tobeworkedout (cf.[68] foran advanceddiscussion).However,preliminaryworktobe completedhasillustrated thatthemultifractalspectrumconveysinformationrelatedtotheco-occurrencesofsingularitiesamongstcomponents,and hence can, forsome cases, be related to statistical dependencies amongst components that are not already encoded in covariancefunctions[72,73].

6. Acknowledgments

Work supported by ANR-16-CE33-0020 MultiFracs, France. G.D. was partially supported by the prime award No. W911NF-14-1-0475 from the Biomathematics subdivision of the Army Research Office. G.D.’slong-term visits to France weresupportedbythe“ENSdeLyon”,theCNRS,andtheCarolLavinBernickfacultygrant.

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