Features modeling with an $\alpha$-stable distribution: Application to pattern recognition based on continuous belief functions

0.86 MB·English

by  Anthony Fiche

Features modeling with an α-stable distribution: application to pattern
recognition based on continuous belief functions
AnthonyFichea,∗,Jean-ChristopheCexusa,ArnaudMartinb,AliKhenchafa
aLabSticc,UMR6285,ENSTABretagne,2rueFranc¸oisVerny,29806BrestCedex9,France
bUMR6074IRISA,IUTLannion/Universite´deRennes1,rueE´douardBranlyBP30219,22302LannionCedex,France
Abstract
The aim of this paper is to show the interest in fitting features with an α-stable distribution to classify imperfect
5
data. Thesupervisedpatternrecognitionisthusbasedonthetheoryofcontinuousbelieffunctions,whichisawayto
1
considerimprecisionanduncertaintyofdata. Thedistributionsoffeaturesaresupposedtobeunimodalandestimated
0
2 byasingleGaussianandα-stablemodel.Experimentalresultsarefirstobtainedfromsyntheticdatabycombiningtwo
featuresofonedimensionandbyconsideringavectoroftwofeatures.Massfunctionsarecalculatedfromplausibility
n
a functionsbyusingthegeneralizedBayestheorem. Thesamestudyisappliedtotheautomaticclassificationofthree
J typesofseafloor(rock,siltandsand)withfeaturesacquiredbyamono-beamecho-sounder. Weevaluatethequality

2 oftheα-stablemodelandtheGaussianmodelbyanalyzingqualitativeresults,usingaKolmogorov-Smirnovtest(K-S
2 test),andquantitativeresultswithclassificationrates. Theperformancesofthebeliefclassifierarecomparedwitha

Bayesianapproach.
]
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A Keywords: Gaussianandα-stablemodel,unimodalfeatures,continuousbelieffunctions,supervisedpattern
recognition,Kolmogorov-Smirnovtest,Bayesianapproach.
.
s
c
[

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v 1. Introduction
2
1 The choice of a model has an important role in the problem of estimation. For example, the Gaussian model is
6 a very efficient model which fits data in many applications as it is very simple to use and saves computation time.
5 However,asisthecaseforalldistributionmodels,Gaussianlawshavesomeweaknessesandresultscanend-upbeing
0
skewed. Indeed,astheGaussianprobabilitydensityfunction(pdf)issymmetrical,itisnotvalidwhenthepdfisnot
1. symmetrical. It is therefore difficult to choose the right model which fits data for each application. The Gaussian
0 distribution belongs to a family of distributions called stable distributions. This family of distributions allows the
5
representationofheavytailsandskewness. Adistributionissaidtohaveaheavytailifthetaildecaysslowerthanthe
1
tailoftheGaussiandistribution. Therefore,thepropertyofskewnessmeansthatitisimpossibletofindamodewhere
:
v probabilitydensityfunctionissymmetrical. ThemainpropertyofstablelawsintroducedbyLe´vy[1]isthatthesum
Xi oftwoindependentstablerandomvariablesgivesastablerandomvariable. α-stabledistributionsareusedindifferent
fieldsofresearchsuchasradar[2,3],imageprocessing[4]orfinance[5,6],...
r
a The aim of this study is to show the interest in fitting data with an α-stable distribution. In [7], the author pro-
poses to characterize the sea floor from a vector of features modeled by Gaussian mixture models (GMMs) using
an Autonomous Underwater Vehicle (AUV). However, the problem with the Bayesian approach is the difficulty in
consideringtheuncertaintyofdata. Wethereforefavoredtheuseofanapproachbasedonthetheoryofbelieffunc-
tions[8,9]. In[10],theauthorscomparedaBayesianandbeliefclassifierwheredatafromsensorsaremodeledusing
GMMsestimatedviaanExpectation-Maximization(EM)algorithm[11]. Thisworkhasbeenextendedtodatamod-
eledbyα-stablemixturemodels[12]. However,itisdifficulttochoosebetweenthesetwomodelsbecausetheresults
∗Correspondingauthor.
Emailaddresses:[email protected](AnthonyFiche),[email protected]
(Jean-ChristopheCexus),[email protected](ArnaudMartin),[email protected](AliKhenchaf)
PreprintsubmittedtoInformationFusion January23,2015
areroughlythesame. Thispaperraisestwoproblems. Firstly,itisnecessarytoworkonadatasetwherefeaturesare
modeledbyα-stablepdfs. Thesecondproblemistoknowhowtoapplythetheoryofbelieffunctionswhendatafrom
sensorsaremodeledbyα-stabledistributions. Thispointhasbeendealtwithin[13].
The recent characterization of uncertain environments has taken an important place in several fields of applica-
tion such as the SOund Navigation And Ranging (SONAR). SONAR has therefore been used for the detection of
underwatermines[14]. In[15],theauthordevelopedtechniquestoperformautomaticclassificationofsedimentson
theseabedfromsonarimages. In[10],theauthorsclassifiedsonarimagesbyextractingfeaturesandmodelingthem
withGMMs. In[16],theauthorscharacterizedtheseafloorusingdatafromamono-beamecho-sounder. Adataset
representsanechosignalamplitudeaccordingtotime.Theycomparethetimeenvelopeofechosignalamplitudewith
asetoftheoreticalreferencecurves. Inthispaper,webuildaclassifierbasedonthetheoryofbelieffunctionswhere
avectoroffeaturesextractedfromamono-beamecho-sounderismodeledbyaGaussianandanα-stabledistribution.
The feature pdfs have only one mode. We finally show that it would be interesting to fit features with an α-stable
model.
Thispaperisorganizedasfollows: wefirstintroducethedefinitionofstability,themethodtoconstructα-stable
probability density functions and the methods of estimation in Section 2. The definitions in the multivariate case
arealsopresented. WeintroducethenotionofbelieffunctionsindiscreteandcontinuouscasesinSection3. Belief
functions are then calculated in the particular case of α-stable distributions and a belief classifier is constructed in
Section 4. We finally classify data by modeling data with Gaussian and α-stable distributions before the generated
dataareclassifiedbycomparingthetheoryofbelieffunctionswithGaussianandα-stabledistributionsinSection5.
WecomparetheresultsobtainedwiththetheoryofbelieffunctionsusingaBayesianapproach.
2. Theα-stabledistributions
Gaussintroducedtheprobabilitydensityfunctioncalled“Gaussiandistribution”(inhonorofGauss)inhisstudy
of astronomy. Laplace and Poisson developed the theory of characteristic function by calculating the analytical ex-
pressionofFouriertransformofaprobabilitydensityfunction.LaplacefoundthattheFouriertransformofaGaussian
lawisalsoaGaussianlaw. CauchytriedtocalculatetheFouriertransformofa“generalizedGaussian”functionwith
theexpression f (x)= 1(cid:82)+∞exp(−ctn)cos(tx)dt,buthedidnotsolvetheproblem.Whentheintegernisarealα,we
n π 0
definethefamilyofα-stabledistributions. However,Cauchydidnotknowatthattimeifhehaddefinedaprobability
density function. With the results of Polya and Bernstein, Janicki and Weron [17] demonstrated that the family of
α-stablelawsareprobabilitydensityfunctions. ThemathematicianPaulLe´vystudiedthecentrallimittheoremand
showed,withtheconstraintofinfinitevariance,thatthelimitlawisastablelaw[1]. Motivatedbythisproperty,Le´vy
calculatedtheFouriertransformforallα-stabledistributions. Thesestablelawswhichsatisfythegeneralizedcentral
limittheoremareinterestingastheyallowthemodelingofdataasimpulsivenoisewhentheGaussianmodelisnot
valid. Anotherpropertyoftheselawsistheabilitytomodelheavytailsandskewness. Inthefollowing,wedefinethe
notionsofstability,characteristicfunction,thewaytobuildapdffromcharacteristicfunction,thedifferentestimators
ofα-stabledistributionsandtheextensionofα-stabledistributiontothemultivariatecase.
2.1. Definitionofstability
Stability is when the sum of two independent random variables which follow α laws, gives an α law. Math-
ematically, this definition means the following: a random variable X is stable, denoted X ∼ S (β,γ,δ), if for all
α
(a,b)∈R+ × R+,therearec∈R+andd∈Rsuchthat:
aX +bX =cX+d, (1)
1 2
with X and X twoindependentα-stablerandomvariableswhichfollowthesamedistributionas X. IfEquation(1)
1 2
definesthenotionofstability,itdoesnotgiveanyindicationastohowtoparameterizeanα-stabledistribution. We
thereforeprefertousethedefinitiongivenbycharacteristicfunctiontorefertoanα-stabledistribution.
2
2.2. Characteristicfunction
Severalequivalentdefinitionshavebeensuggestedintheliteraturetoparameterizeanα-stabledistributionfrom
itscharacteristicfunction[18,19]. Zolotarev[19]proposedthefollowing:
 πα
φSα(β,γ,δ)(t)= eexxpp((jjttδδ−−||γγtt||α[1[1++jβjβ2tsaing(n(2t))losigg|nt|(]t))(|t|1−α−1)]) iiffαα(cid:44)=11,, (2)
π
whereeachfeaturehasspecificvalues:
• α∈]0,2]isthecharacteristicexponent.
• β∈[−1,1]istheskewnessparameter.
• γ∈R+∗representsthescaleparameter.
• δ∈Risthelocationparameter.
The advantage of this parameterization compared to [18] is that the values of the characterization and probability
density functions are continuous for all parameters. In fact, the parameterization defined by [18] is discontinuous
whenα=1andβ=0.
2.3. Theprobabilitydensityfunction
The representation of an α-stable pdf, denoted f , is obtained by calculating the Fourier transform of its
Sα(β,γ,δ)
characteristicfunction:
(cid:90) ∞
f (x)= φ (t)exp(−jtx)dt. (3)
Sα(β,γ,δ) Sα(β,γ,δ)
−∞
However, thisdefinitionisproblematicfortworeasons: whiletheintegrandfunctioniscomplex, itsboundsare
infinite. Nolan[20]thereforeproposedawaytorepresentnormalizedα-stabledistributions(i.e. γ = 1andδ = 0).
The main idea of Nolan [20] is to use variable modifications so that the integral has finite bounds. Each parameter
hasaninfluenceontheshapeofthe f . Thecurvehasalargepeakwhenαisnear0andaGaussianshapewhen
Sα(β,γ,δ)
α = 2 (Figure 1(a)). The shape of the distribution skews to the left if β = 1, to the right if β = −1 (Figure 1(b))
whilethedistributionissymmetricalwhenβ=0. Finally,thescaleparameterenlargesorcompressestheshapeofthe
distribution(Figure1(c))andthelocationparameterleadstothetranslationofthemodeofthe f (Figure1(d)).
Sα(β,γ,δ)
2.4. Anoverviewofα-stableestimators
Theestimatorsofα-stabledistributionsaredecomposedintothreefamilies:
• thesamplequantilemethods[21,22]
• thesamplecharacteristicfunctionmethods[23,24,25,26]
• theMaximumLikelihoodEstimation[27,28,29,30].
FamaandRoll[21]developedamethodbasedonquantiles. However,thealgorithmproposedbyFamaandRoll
suffersfromasmallasymptoticbiasinαandγandrestrictionsonα ∈]0.6,2]andβ = 0. McCulloch[22]extended
thequantilemethodtotheasymmetriccase(i.e. β=0). TheMcCullochestimatorisvalidforα∈]0.6,2].
Press [23, 24] proposed a method based on transformations of characteristic function. In [31], the author com-
paredtheperformancesofseveralestimators. Forexample,thePressmethodisefficientforspecificvalues. Koutrou-
velis[25]extendedthePressmethodandproposedaregressionmethodtoestimatetheparametersofα-stabledistri-
butions. Heproposedasecondversionofhisalgorithmwhichisdistinctthatitisiterative[26]. In[32],theauthors
provedthatthemethodproposedbyKoutrouvelisisbetterthanboththequantilemethodandthePressmethodbecause
itgivesconsistentandasymptoticallyunbiasedestimates.
TheMaximumLikelihoodEstimation(MLE)wasfirststudiedinthesymmetriccase[27,28]. Dumouchel[29]
developed an approximate maximum likelihood method. The MLE was also developed in the asymmetric case.
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Nolan[30]extendedtheMLEingeneralcase. TheproblemwiththeMLEisthecalculationoftheα-stableproba-
bility density function because there is no closed-form expression. Moreover, the computational algorithm is time-
consuming.
Consequently,weestimateaunivariateα-stabledistributionusingtheKoutrouvelismethod[26].
2.5. Multivariatestabledistributions
Itispossibletoextendtheα-stabledistributionstothemultivariatecase. ArandomvectorX ∈ Risstableiffor
alla,b∈R+,therearec∈R+andD∈Rd suchthat:
aX +bX =cX+D, (4)
1 2
whereX andX aretwoindependentandidenticallydistributedrandomvectorswhichfollowthesamedistribution
1 2
asX.
Thecharacteristicfunctionofanmultivariateα-stabledistribution,denotedX∼S (σ,δ),hastheform:
α,d
 (cid:32) (cid:90) (cid:33)
φSα,d(σ,δ)(t)= eexxpp(cid:32)−−(cid:90)Sd||<<tt,,ss>>||α(1(1+−jπ2jssggnn((<<tt,,ss>>))talnn((<π2αt,)sσ>(d))sσ)(+dsj)<+δj,<t>δ,t>(cid:33) iiffαα=(cid:44)11,, (5)
Sd
with
• Sd ={x∈Rd|||x||=1}thed-dimensionalunitsphere.
• σ(.)afiniteBorelmeasureonSd.
• δ,t∈Rd.
TheexpressionforthecharacteristicfunctioninvolvesanintegrationovertheunitsphereSd. Themeasureσiscalled
thespectralmeasureandδiscalledthelocationparameter. Theproblemwithamultivariateα-stabledistributionis
that characteristic function forms a non-parametric set. To avoid this problem, it is possible to consider a discrete
spectralmeasure[33]whichhastheform:
(cid:88)K
σ(.)= γδ (.), (6)
i si
i=1
withγ correspondingtoweightandδ theDiracmeasureins. Foranα-stabledistributioninR2withKmasspoints,
thequaintitys ={cos(θ),sin(θ)}withsiθ = 2π(i−1). i
i i i i K
Therearetwomethodstoestimateanα-stablerandomvector:
• thePROJectionmethod[34](PROJ)
• theEmpiricalCharacteristicFunctionmethod[35](ECF)
Thesetwoalgorithmshavethesameperformancesintermsofestimationandcomputationtime.
At this point, it is important to underline that the features extracted from mono-beam echo-sounders have the
propertiesofheavy-tailsandskewness. Consequently,wedecidedtoestimatethedatawithanα-stablemodel. These
data are however imprecise and uncertain: the imprecision and uncertainty of data can be linked to poor quality
estimationofthesedata. Totaketheseconstraintsintoconsideration,weusedanuncertaintheorycalledthetheoryof
belieffunctions. Consequently,thegoalofthenextsectionistopresentthetheoryofbelieffunctions.
3. Thebelieffunctions
The final objective of this paper is to classify synthetic and real data using the theory of belief functions. Data
obtainedfromsensorsaregenerallyimpreciseanduncertainasnoisescandisrupttheiracquisition. Itispossibleto
considertheseconstraintsbyusingthetheoryofbelieffunctions. Wewillfirstdevelopthetheoryofbelieffunctions
withinadiscreteframeworkbeforecharacterizingitinrealnumbers.
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3.1. Discretebelieffunctions
Thissectionoutlinesbasictoolsinrelationtothetheoryofbelieffunctions.
3.1.1. Definitions
Discrete belief functions were introduced by Dempster [8], and formalized by Shafer [9] where he considers a
discretesetofnexclusiveeventsC calledtheframeofdiscernment:
i
Θ={C ,...,C }. (7)
1 n
Θcanbeinterpretedasalltheassumptionstoaproblem. Belieffunctionsaredefinedas2Θ onto[0,1]. Theobjective
ofdiscretebelieffunctionsistoattributeaweightofbelieftoeachelementA∈2Θ. Belieffunctionsmustfollowthe
normalization:
(cid:88)
mΘ(A)=1, (8)
A⊆Θ
wheremΘiscalledthebasicbeliefassignment(bba).
AfocalelementisasubsetofAwheremΘ(A) > 0andseveralfunctionsinone-to-onecorrespondencearebuiltfrom
bba:
(cid:88)
belΘ(A) = mΘ(B), (9)
B⊆A,B(cid:44)∅
(cid:88)
plΘ(A) = mΘ(B), (10)
A∩B(cid:44)∅
(cid:88)
qΘ(A) = mΘ(B). (11)
B⊂Θ,B⊇A
ThecredibilityfunctionofAcalledbelΘ(A)isalltheelements B ⊆ AwhichbelievepartiallyinA. Thisfunction
canbeinterpretedasaminimumofbeliefinA.Onthecontrary,aplausibilityfunctionplΘ(A)illustratesthemaximum
beliefinAwhilecommonalityfunctionqΘ(A)representsthesumofbbaallocatedtothesupersetofA. Thisfunction
isveryusefulasshownbelow.
3.1.2. Combinationrule
We consider M different experts who give mass mΘ(i = 1,...,M) on each element A ⊆ Θ. Alternatively, it is
i
possibletocombinethemusingcombinationrules. Thereareseveralcombinationrules[36]intheliteraturewhich
differentlyaddressconflictsbetweensources. Themostcommonruleistheconjunctivecombination[37]wherethe
resultantmassofAisobtainedby:
(cid:88) (cid:89)M
mΘ(A)= mΘ(B) ,∀A∈2Θ. (12)
i i
B1∩...Bn=A(cid:44)∅ i=1
Themassoftheemptysetisgivenby:
(cid:88) (cid:89)M
mΘ(∅)= mΘ(B) ,∀A∈2Θ. (13)
i i
B1∩...Bn=∅ i=1
Thisruleallowsustostayintheopenworld. However,thisisnotpracticalascalculationsaredifficult. Itispossible
tocalculateresultantmasswithcommunalityfunctionswhereeachmassmΘ(i=1,...,M)mustbeconvertedintoits
i
communalityfunctiontocalculatetheresultantcommunalityfunctionasfollows:
(cid:89)M
qΘ(A)= qΘ(A). (14)
i
i=1
ThefinalmassisobtainedbycarryingouttheinverseoperationofEquation(11) [9].
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3.1.3. Pignisticprobability
To make a decision on Θ, several operators exist such that maximum credibility or maximum plausibility with
pignistic probability being the most commonly used operator [38]. This name comes from pignus, a bet, in Latin.
Thisoperatorapproachesthepair(bel,pl)byuniformlysharingamassoffocalelementsoneachsingletonC. This
i
operatorisdefinedby:
(cid:88) mΘ(A)
betP(C)= , (15)
i |A|(1−mΘ(∅))
A⊂Θ,Ci∈A
where|A|representsthecardinalityofA.
WechoosethedecisionC byevaluating maxbetP(C ).
i k
1≤k≤n
3.2. Continuousbelieffunctions
The basic description of continuous belief functions was accomplished by Shafer [9], then by Nguyen [39] and
Strat[40]. Recently,Smets[41]extendedthedefinitionofbelieffunctionstothesetofrealsR=R∪{−∞,+∞}and
massesareonlyattributedtointervalsofR.
3.2.1. Definitions
LetusconsiderI = {[x,y],(x,y],[x,y),(x,y);x,y ∈ R}asasetofclosed,half-openedandopenedintervalsofR.
FocalelementsareclosedintervalsofR. ThequantitymI(x,y)arebasicbeliefdensitieslinkedtoaspecificpdf. If
x>y,thenmI(x,y)=0. Withthesedefinitions,itispossibletodefinethesamefunctionsasinthediscretecase. The
interval[a,b]beingasetofRwitha≤b,thepreviousfunctionscanbedefinedasfollows:
(cid:90) x=b(cid:90) y=b
belR([a,b]) = mI(x,y)dydx, (16)
x=a y=x
(cid:90) x=b (cid:90) y=+∞
plR([a,b]) = mI(x,y)dydx, (17)
x=−∞ y=max(a,x)
(cid:90) x=a (cid:90) y=+∞
qR([a,b]) = mI(x,y)dydx. (18)
x=−∞ y=b
3.2.2. Pignisticprobability
Thedefinitionofpignisticprobabilityfora<bis:
(cid:90) x=+∞(cid:90) y=+∞ |[a,b]∩[x,y]|
Betf([a,b])= mI(x,y)dxdy. (19)
x=−∞ y=x |[x,y]|
Itispossibletocalculatepignisticprobabilitiestohavebasicbeliefdensities. However, manybasicbeliefdensities
existforthesamepignisticprobability. Toresolvethisissue,wecanusetheconsonantbasicbeliefdensity. Abasic
beliefdensityissaidtobe“consonant”whenfocalelementsarenested. FocalelementsI canbelabeledasanindex
u
u such that I ⊆ I(cid:48) with u(cid:48) > u. This definition is used to apply the least commitment principle, which consists in
u u
choosingtheleastinformativebelieffunctionwhenabelieffunctionisnottotallydefinedandisonlyknowntobelong
toafamilyoffunctions. Theleastcommitmentprinciplereliesonanorderrelationbetweenbelieffunctionsinorder
todetermineifabelieffunctionismoreorlesscommittedthananother. Forexample,itispossibletodefineonorder
basedonthecommonalityfunction:
(∀A⊆Θ,qΘ(A)≤qΘ(A))⇔(mΘ ⊆ mΘ). (20)
1 2 1 q 2
ThemassfunctionmΘislesscommittedthanmΘaccordingtothecommonalityfunction.
2 1
Thefunction Betf canbeinducedbyasetofisopignisticbelieffunctionsBiso(Betf). Manypapers[41,42,43]
deal with the particular case of continuous belief functions with nested focal elements. For example, Smets [41]
6
proved that the least committed basic belief assignment mR for the commonality ordering attributed to an interval
I =[x,y]withy> xrelatedtoabell-shaped1pignisticprobabilityfunctionisdeterminedby2:
mR([x,y])=θ(y)δ (x−ζ(y)), (21)
d
withx=ζ(y)satisfyingBetf(ζ(y))= Betf(y)andθ(y):
dBetf(y)
θ(y)=(ζ(y)−y) . (22)
dy
The resultant basic belief assignment mR is consonant and belongs to the set Biso(Betf). However, it is difficult to
buildbelieffunctionsintheparticularcaseofmultimodalpdfsbecausetheframeofdiscernmenthasconnectedsets.
In[44,45],theauthorsproposeawaytobuildbelieffunctionswithconnectedsetsbyusingacredalmeasureandan
indexfunction.
3.3. Credalmeasureandindexfunction
In[44],theauthorsproposeawaytocalculatebelieffunctionsfromanyprobabilitydensityfunction. Theyusean
indexfunction f andaspecificindexspaceItoscanthesetoffocalelementsF:
fI : I → F, (23)
y (cid:55)−→ fI(y). (24)
(cid:82)
The authors introduce a positive measure µΩ such that dµΩ(y) ≤ 1 describes unconnected sets. The pair (fI,µΩ)
I
definesabelieffunction. ForallA∈Ω,theydefinesubsetswhichbelongtotheBorelset:
F = {y∈ I|fI(y)⊆ A}, (25)
⊆A
F = {y∈ I|fI(y)∩A(cid:44)∅}, (26)
∩A
F = {y∈ I|fI(y)⊇ A}. (27)
⊇A
Fromthesedefinitions,theycomputebelieffunctions:
(cid:90)
belΩ(A) = dµΩ(y), (28)
(cid:90)F⊆A
plΩ(A) = dµΩ(y), (29)
(cid:90)F∩A
qΩ(A) = dµΩ(y). (30)
F⊇A
Theycontinuetheirstudybyconsideringconsonantbelieffunctions. ThesetoffocalelementsF mustbeordered
fromtheoperator⊆. Theydefineanindexfunction f fromR+toF suchthat:
y≥ x=⇒ f(y)⊆ f(x). (31)
TheygenerateconsonantsetsbyusingacontinuousfunctiongfromRd toI =[0,α [. Theα-cutsaretheset:
max
fI ={x∈Rd|g(x)≥α}. (32)
cs
Theyfinallydefinetheindexfunction:
fI : I =[0,α ] → {fI(α)|α∈ I}, (33)
cs max cs
α (cid:55)−→ fI(α). (34)
cs
1i.e.thepdfisunimodalwithamodeµ,continuousandstrictlymonotonousincreasing(decreasing)atleft(right)ofthemode.
2δdreferstotheDirac’smeasure.
7
TheinformationavailableistheconditionalpignisticdensityBetf. However,manybasicbeliefdensitiesexistforthe
samepignisticprobability Betf. Theleastcommitmentprincipleallowstheleastinformativebasicbeliefdensityto
bechosen,wherethefocalelementsaretheα-cutsofBetf suchthat3:
dµΩ(y)(α)=λ(fI(α))dλ(α). (35)
cs
4. Continuousbelieffunctionsandα-stabledistribution
In this section, we model data distributions as a single α-stable distribution. We must however introduce the
notionofplausibilityfunctionforanα-stabledistribution. Wefirstdescribehowtocalculatetheplausibilityfunction
knowingthepdfinRbeforeweextendtoinRd. Wefinallyexplainhowweconstructourbeliefclassifier.
4.1. LinkbetweenpignisticprobabilityfunctionandplausibilityfunctioninR
The information available is the conditional pignistic density Betf[C] with C ∈ θ. The function Betf[C] is
i i i
supposedtobebell-shapedforallα-stabledistributions(provedbyYamazato[46]).
TheplausibilityfunctionfromamassmRisobtainedbyanintegralofEquation(22)between[x,+∞[:
(cid:90) +∞ dBetf(t)
plR[C](I)= (ζ(t)−t) dt. (36)
i dt
x
ByassumingthatBetf issymmetrical,anintegrationbypartscansimplifyEquation(36):
(cid:90) +∞
plR[C](I)=2(x−µ)Betf(x)+2 Betf(t)dt. (37)
i
x
Nowletusconsideraparticularcasewheresymmetrical Betf isanα-stabledistribution(theparameterβ = 0). We
alreadyknowthatBetf(x)= f (x). Wecancalculate(cid:82)+∞Betf(t)dtbyusingtheChasles’theorem:
Sα(β,γ,δ) x
(cid:90) +∞ (cid:90) x (cid:90) +∞
f (t)dt= f (t)dt+ f (t)dt. (38)
Sα(β,γ,δ) Sα(β,γ,δ) Sα(β,γ,δ)
−∞ −∞ x
By definition, a pdf has the quantity (cid:82)+∞ f (t)dt = 1 and (cid:82) x f (t)dt represents the definition of the
−∞ Sα(β,γ,δ) −∞ Sα(β,γ,δ)
α-stablecumulativedensityfunctionF .
Sα(β,γ,δ)
Consequently,Equation(37)canbesimplified:
plR[C](I)=2(x−µ)f (x)+2(1−F (x)). (39)
i Sα(β,γ,δ) Sα(β,γ,δ)
Theplausibilityfunctionrelatedtoaninterval I = [x,y]canalsobeseenastheareadefinedundertheα -cutsuch
cut
thatα = Betf(x). Thenotation pl[C](x)isequivalentto pl[C](I).
cut i i
Now us let consider an asymmetric α-stable probability density function. We must proceed numerically to cal-
culate plausibility function at point x > µ, with µ the mode of the probability density function (Figure 2). The
1
plausibility function related to an interval I = [x ,y ] is defined by the area defined under the α -cut such that
1 1 1 cut
α = Betf(x ):
cut 1 plR[C](I )=(cid:90) x1 Betf(t)dt+(y −x )Betf(x )+(cid:90) +∞Betf(t)dt. (40)
i 1 1 1 1
−∞ y1
By definition, a pdf has the quantity (cid:82)+∞ f (t)dt = 1−F (y ) and (cid:82) x1 f (t)dt = F (x ). In
general, we know only one point y . Wy1e estSiαm(βa,γt,eδ)numerically xSα(sβu,γc,δh)th1at f −∞(ySα)(β,=γ,δ)f (xS)α.(βF,γi,δn)all1y, the
1 1 Sα(β,γ,δ) 1 Sα(β,γ,δ) 1
plausibilityfunctionrelatedtotheintervalI is:
1
plR[C](I )=1+F (x )−F(y )+(y −x )f(x ). (41)
i 1 Sα(β,γ,δ) 1 1 1 1 1
3λreferstotheLebesgue’smeasure.
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Inpractice,webasetheclassificationonseveralfeaturesdefinedfordifferentclassesΘ = {C ,..,C }. Forexample,
1 n
theHaralickparameters“contrast”and“homogeneity”havedifferentvaluesfortheclassesrock, siltandsand. The
featuresaremodeledbyaprobabilitydensityfunctionbecausethefeatureshavecontinuousvalues. Wecancalculate
a plausibility function related to its probability density functions by using the least commitment principle. Several
plausibilityfunctionsassociatedtothesamefeaturecanbecombinedbyusingthegeneralizedBayestheorem[47,48]
tocalculatemassfunctionsallocatedtoAofanintervalI:
(cid:89) (cid:89)
mR[x](A)= pl (x) (1−pl (x)). (42)
j j
Cj∈A Cj∈Ac
Forseveralfeatures,itispossibletocombinethemassfunctionswithcombinationrules.
Tovalidateourapproach,weclassifyplanesusingkinematicdataasin[42,43]andcomparethedecisionwiththe
approachofCaronetal.[43]. WeassociateaGaussianprobabilitydensityfunctionforeachspeed’starget(Figure3):
• CommercialdefinedbytheprobabilitydensityfunctionofspeedS (0,8,722.5).
2
• BomberdefinedbytheprobabilitydensityfunctionofspeedS (0,7,690).
2
• FighterdefinedbytheprobabilitydensityfunctionofspeedS (0,10,730).
2
WecanobservethatthedecisionisthesamewiththebothapproachesintheparticularcaseofGaussianprobability
densityfunctions.
4.2. LinkbetweenpignisticprobabilityfunctionandplausibilityfunctiontoRd
It is possible to extend plausibility function in Rd. In [43], the authors calculate plausibility function in the
GaussianpdfsituationofmodeδandmatrixofcovarianceΣ. Massfunctionisbuiltinsuchawaythatisoprobability
surfaces S with 1 ≤ i ≤ n are focal elements. In R2, isoprobability points of a multivariate Gaussian pdf of mean
i
δ and covariance matrix Σ are ellipses. In dimension d, the authors defined focal elements as the nested sets HV
α
enclosedbytheisoprobabilityhyperconicsHC ={x∈Rd|(x−δ)tΣ−1(x−δ)}. Theyobtainbbdbyapplyingtheleast
α
commitmentprinciple:
mRd(HV )= αd+22−1 exp(−1α)withα≥0. (43)
α 2d+22Γ(d+22) 2
Equation(43)definesaχ2 distributionwithd+2degreesoffreedom. Theplausibilityfunctionatpointxbelonging
to a surface S corresponds to the volume delimited by S (we give an example in Figure 4 for an α-stable pdf).
i i
Consequently,theplausibilityfunctionatpointxisdefinedby:
plRd(x∈Rd)=(cid:90) α=+∞ αd+22−1 exp(−1α)dα. (44)
α=(x−δ)tΣ−1(x−δ) 2d+22Γ(d+22) 2
Equation(44)canbesimplifiedby:
plRd(x∈Rd)=1−Fd+2((x−δ)(Σ)−1(x−δ)). (45)
The function Fd+2 is a cumulative density function of the χ2 distribution with d + 2 degrees of liberty (d is the
dimensionofvectorx).
TheauthorsalsocalculateplausibilityfunctionsintheparticularcaseofGMMswiththeirpdfsdenotedby p :
GMM
(cid:88)k=n
p (x)= w N(x,δ ,Σ ), (46)
GMM k k k
k=1
where N(x,δ ,Σ ) is the normal distribution of the k components with mean δ and matrix of covariance Σ and
k k k k
w is the weight of each mixture. They assign belief to nested sets belonging to a component. Consequently, the
k
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plausibilityfunctionoftheGMMscanbeseenasaweightedsumofplausibilityfunctionsdefinedbyeachcomponent
ofthemixture:
plRd(x∈Rd)=1−(cid:88)k=nwkFd+2((x−δk)(Σk)−1(x−δk)). (47)
k=1
Wewanttoextendthecalculationofplausibilityfunctionforanyα-stablepdf. However,thereisnoclosed-form
expressionforα-stablepdf. WeusetheapproachdevelopedbyDore´ etal.[44]tobuildbelieffunctions.
4.3. Beliefclassifier
Let us consider a data set with N samples from d sensors. For example, each feature of a vector x ∈ Rd can
beseenasapieceofinformationfromasensor. Theclassificationisdividedintotwosteps. N × p(with p ∈]0,1[)
samplesarefirstpickedoutrandomlyfromthedatasetforthelearningbase,notedX,suchthat:
X=xx...111 ......... xx1NN...××pp
d d
AllcolumnsofXbelongtoaclassC with1≤i≤n. Probabilitydensityfunctions(Gaussianorα-stablemodels)
i
are estimated from samples belonging to classes. The rest of the samples is used for the test base (N × (1 − p)
vectors). Weuseavalidationtesttodetermineifsamplesbelongtoanestimatedmodel. Ifthetestisnotvalid, we
stoptheclassification,otherwisewecontinuetheclassification(Figure5). Theclassificationstepdiagraminoneand
ddimensionsisshowninFigure6.
Plausibilityfunctionsknowingclassesforxarecalculatedeitherfromtheirprobabilitydensityfunctionsbyusing
Equation (41) for an unsymmetric α-stable probability density function or by using Equation (45) for a Gaussian
probabilitydensityfunction. Weobtaindmassfunctions(oneforeachfeature)atpoint xwiththegeneralizedBayes
theorem[47,48](Equation(42)). Thesedmassfunctionsarecombinedbyacombinationruletoobtainasinglemass
function(section3.1.2). Finally,themassfunctionistransformedintopignisticprobability(Equation(19))tomake
thedecision.
Itisalsopossibletoworkwithavectorxofddimensions.Foranα-stableprobabilitydensityfunction,plausibility
functions are calculated for each feature by using the approach of Dore´ et al [44] (section 3.3). For a Gaussian
probabilitydensityfunction,weuseEquation(45)tocalculateplausibiltyfunctions. Wecalculateonemassfunction
at point x by using the generalized Bayes theorem (Equation (42)). There is no combination step because we are
calculatingasinglemassfunction. WeuseEquation(19)totransformthemassfunctionintopignisticprobabilities.
Thedecisionischosenbyusingthemaximumnumberofpignisticprobabilities.
5. Applicationtopatternrecognition
The aim is to build a belief classifier and to perform a classification of synthetic and real data by estimating
features using a Gaussian and α-stable pdf. In [12], synthetic data are classified by modeling features using Gaus-
sian and α-stable mixture models. By observing confidence intervals, the hypothesis of α-stable mixture models is
significantlybetterthanthehypothesisofGaussianmixturemodels. However,whenthenumberofGaussiandistribu-
tionsincreases,classificationaccuraciesaresignificantlythesame. Imagesfromaside-scansonarareautomatically
classifiedbyextractingHaralickfeatures[49,50]. However,classificationaccuraciesareroughlythesame.
In this section, we limit algorithms with a vector of features in dimension d ≤ 2. Indeed, the generalization of
α-stablepdfindimensiond > 2increasesCPUtimebecausethereisnoclosed-formexpression. Consequently, we
distinguishtwocasesduringtheclassificationstep:
• Theonedimensioncase: eachdimensionisconsideredasafeature.
• Thetwocase: weconsideredavectoroftwofeatures.
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