Oddness/evenness-based classifiers for Boolean or numerical data

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to the repository administrator: tech-oatao@listes-diff.inp-toulouse.fr

This is an author’s version published in: http://oatao.univ-toulouse.fr/24811

To cite this version: Bounhas, Myriam and Prade, Henri and

Richard, Gilles Oddness/evenness-based classifiers for Boolean or

numerical data. (2017) International Journal of Approximate

Reasoning, 82. 81-100. ISSN 0888-613X

Official URL:

https://doi.org/10.1016/j.ijar.2016.12.002

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Oddness/evenness-based

classiﬁers

for

Boolean

or

numerical data

Myriam

Bounhas

a

,

b

,

Henri

Prade

c

,

d

,

∗

,

Gilles

Richard

c a_LARODEC_Laboratory,_ISG_de_Tunis,₄₁_rue_de_la_Liberté,₂₀₀₀_Le_Bardo,_Tunisia

b_Emirates_College_of_Technology,_P.O._Box:_41009,_Abu_Dhabi,_United_Arab_Emirates c_IRIT,_UPS-CNRS,₁₁₈_route_de_Narbonne,₃₁₀₆₂_Toulouse_Cedex,_France d_QCIS,_University_of_Technology,_Sydney,_Australia

a

b

s

t

r

a

c

t

Keywords: Logicalproportion Classiﬁcation Booleandata Numericaldata

k-Nearestneighborsmethod

In this paper, we propose two viewpoints for estimating to what extent a new item, described intermsofbinary-valued features, fits with asetof existingitems. Theyare respectively basedonan oddnessindexand an evennessindex, whichinspite oftheir names, are not exactly the oppositeof each other. Both indicators, whichreferto one feature,arebuiltfromheterogeneouslogicalproportions,andinvolvefouritems,thenew itemandthreeothers.LogicalproportionsareBooleanfunctionsthatrelatefourvariables through comparisonsbetween pairs of them.Heterogeneous onesexpress thatthere is anintruderamongfourtruthvalues,whichisforbiddentoappearinaspecificposition. Global oddnessand evennessfunctions of anitemwith respectto aset are built from thecorrespondingindexesbytakingallfeaturesintoaccount,andthenbyconsideringall triplesofitemsintheset.Moreovertheoddnessfunctionnaturallyextendstonumerical featuresandtosubsetsofitemsofdifferentsizes(pairs,triples,etc.).Simpleclassification procedures canbebasedontheseglobalfunctions: anewitemisassignedto theclass thatminimizesoddnessormaximizesevenness.Experimentsonclassicalbenchmarkswith Boolean,ornumericaldata(foroddness)showthattheresultsarecompetitivewithother classificationmethods.

1. Introduction

Ithasbeenacknowledgedforalongtimethat proportionsplayan importantroleinourperceptionandunderstanding of reality. Indeed proportions are a matter of comparisons expressed by differencesor ratios that are equated to other differencesorratios.Twocenturiesago,Gergonne[10,11]wastheﬁrsttoexplicitlyrelatenumerical(geometric)proportions totheideasofinterpolationandregression.

Itisonlyinthelastdecadethatanalogical proportions,i.e., statementsoftheform A isto B asC isto D,whereeach capitalletter refers to asituation described by a vectorof featurevalues, havebeenformalized ﬁrstin termsof subsets of properties that hold true in a given situation [12,26], andthen in a logical manner [16]. Quite early, it was shown

*

Correspondingauthorat:IRIT,UPS-CNRS,118routedeNarbonne,31062ToulouseCedex,France.

E-mailaddresses:Myriam_Bounhas@yahoo.fr(M. Bounhas),prade@irit.fr(H. Prade),richard@irit.fr(G. Richard).

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that a formal view of analogical proportions maybe the basis of a new type of classifier that performs well on some difficult benchmarks [1,15].This was confirmed by other implementations directly based on a logicalview ofanalogical proportions[3].

Besides, itwas shownthat analogical proportionsbelong toa larger familyofso-calledlogicalproportions thatrelate a 4-tupleofBooleanvariables[17],wherethe8code-independentlogicalproportionsareofparticularinterestsincetheir truth status remain unchangedif a property is encoded positively or negatively.These 8 logical proportions divide into 4 homogeneous proportions,which includethe analogical proportionand3 relatedproportions,and4 heterogeneous

pro-portions [20].A heterogeneous proportionexpresses the idea that thereis an intruder amongthe 4truth values, which is forbiddentoappearina speciﬁcposition.Intuitivelyspeaking,an item properlyassignedtoa classshouldnot be (too much)anintruderinthisclass.Itsuggeststhatheterogeneousproportionsmaybealsoofinterestasabasisfordesigning anewtypeofclassiﬁer.Thisisthetopicofthepaper.

It isa commonsenseprincipletoconsiderthataclass cannotbereasonablyassignedtoanewitem ifthisitemwould appear to be at odds with respectto the known members of theclass. On the contrary, theitem should be even with respect to theseclass members forentering the class. The useof proportions leads tothe ideaof considering triples of elements ina class asa basis forestimating theevenness orthe oddness ofa new item withrespect tothe class. This departs from theusual view wherethe estimationof the(non) agreementof anewitem withrespect toa set ofitems amountstocomparetheitem,featurebyfeature,withadistributionofthefeaturevaluesinthewholeset.

Anoddnessindexandanevennessindex,whicharenottheexactoppositeofeachother,areproposed.Intheevenness view,triplesaretheonlysubsetswherewhenthenewitemconformswiththeminorityforagivenBooleanfeature,there isnolongeranymajority(withrespecttothisfeature)inthetripleaugmentedwiththenewitem.Thenonecanestimate towhatextentanewitemﬁtswiththemajorityofelementsinanytripleofmembersofaclassonasetoffeatures.

However, it isunclear howto extendtheevenness-based approachto numericaldata.A slightly differentview,based onthedirectestimationofoddness,whichcanstillberelatedtoheterogeneouslogicalproportions,canbealsoconsidered. Thisleadstoanoddnessmeasurethatcanbeextendedtonumericalfeaturesinastraightforwardmanner,andthatcanbe alsogeneralizedtosubsetsofanysizeandnotonlyontriples.Thusinthispaper,theevaluationoftheevennessorofthe oddness ofanitem withrespecttoa classreliesonalocal view,wherethe newitem, shouldappeareven/notappearat oddswithamaximumnumberof(small)subsetsofaconsideredclass.

The paperisorganizedasfollows.Thenext sectionprovides thenecessarybackgroundonBooleanlogicalproportions, introducingthetwotypesofproportions:thehomogeneousonesandtheheterogeneousones,byespeciallyemphasizinga code independencyproperty.Results areestablished that singleout theseproportionsintermsofparticular featuresthat are meaningfulwhenitcomesto classification.Then,basedonheterogeneousproportions,oddnessandevennessindexes are introducedinSection 3,andtheirexact relationshipestablished.The extensionofoddness andevenness indexestoa numericalfeatureisdiscussed.Section4describesheterogeneousproportions-basedclassifiers.Weshowhowtheproposed oddness andevennessindexescanprovidea basisforestimatingtheoddnessortheevenness ofanitem withrespectto a wholeclass. Therelatedwork Section5 providesa briefoverviewofanalogicalproportions-basedclassifiers, which are based on ahomogeneous logical proportion, butwork quite differentlyfromheterogeneous proportions-basedclassifiers. Section 6isdevotedtoa setofexperimentsonstandardbenchmarkscomingfromtheUCIrepository.Theyarecompared both with classical classifiersandwithanalogical proportions-basedclassifiers. Finally, we provide some hintsfor future worksandconcludingremarksinSection7.

Thispapergathersandsubstantiallyextendsapproachesandresultspartiallyreportedinthreeconferencepapers[5–7].

2. Heterogeneousproportionsvs.homogeneousproportions

Logical proportionsare thebasicingredientsofourapproach.TheseproportionsareBooleanformulasinvolving4 vari-ables.Theyhavebeendeeplyinvestigatedin[19].Inthefollowingsection,weﬁrstrecallhowtheyarebuilt,thenwefocus ontheproportionsthatwewilluseinthispaper.

Asfornotations,thepaperusesthestandardonesforBooleanconnectives,namely

∨,

∧,

≡,

→

fordisjunction, conjunc-tion,equivalenceandmaterialimplicationrespectively.

2.1. Backgroundonlogicalproportions

Alogicalproportionstatesarelationbetween4itemsthatisexpressedintermsofcomparisonsbetweenpairsofitems, eachitembeingrepresentedasasetofBooleanfeatures.Considering2Booleanvariablesa andb correspondingtothesame feature attachedto2items A and B,a

∧

b anda

∧

b indicatethat A and B behavesimilarlyw.r.t.thegivenfeature(they are called“similarity”indicators),a

∧

b anda

∧

b thefactthat A and B behave differently(they arecalled“dissimilarity” indicators).Whenwehave4items A

,

B

,

C

,

D,forcomparingtheirrespectivebehaviorinapairwisemanner,weareledto considerlogicalequivalencesbetweensimilarity,ordissimilarityindicators,suchasa

∧

b

≡

c

∧

d for instance.Thisenables ustodeﬁnealogicalproportion[19]:

Deﬁnition1.A logicalproportion T

(

a

,

b

,

c

,

d

)

isthe conjunctionoftwo equivalencesbetweenindicatorsfor

(

a

,

b

)

on one sideandindicatorsfor

(

c

,

d

)

ontheotherside.

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Forinstance,

((

a

∧

b

)

≡ (

c

∧

d

))

∧ ((

a

∧

b

)

≡ (

c

∧

d

))

is alogicalproportion. It hasbeenestablished thatthere are120 syntacticallyandsemanticallydistinctlogicalequivalences.Therearetwowaysfordistinguishingremarkablesubsetsamong the120proportions:eitherbyinvestigatingtheirstructure,orbyinvestigatingtheirsemantics(i.e.theirtruthtable).Inthis section, weshallsee thatboth investigationslead tothe sameconclusion:there isa classof8proportions whichstands outofthecrowd.Thisclasscanbesubdividedinto2sub-groupsof4proportions.

Indeed a propertythat appears tobe paramountin manyreasoning tasks is codeindependency: there should be no distinction when encoding information positively or negatively. In other words, encoding truth (resp. falsity) with 1 or with 0(resp.with0and 1)isjustamatterofconvention,andshouldnotimpacttheﬁnalresult.Whendealingwithlogical proportions,thispropertyiscalledcodeindependency andcanbeexpressedas

T

(

a

,

b

,

c

,

d

)

→

T

(

a

,

b

,

c

,

d

)

Fromastructuralviewpoint,rememberthataproportionisbuiltupwithapairofequivalencesbetweenindicatorschosen among16equivalences. So,toensure codeindependency,the onlywaytoproceed istoﬁrst choosean equivalencethen topairitwithitscounterpartwhereeveryliteralisnegated:forinstancea

∧

b

≡

c

∧

d shouldbepairedwitha

∧

b

≡

c

∧

d

inordertogetacodeindependentproportion.Thissimplereasoningshowsthatwehaveonly16/2

=

8 codeindependent proportionswhoselogicalexpressionsaregivenbelow.

A

: ((

a

∧

b

)

≡ (

c

∧

d

))

∧ ((

a

∧

b

)

≡ (

c

∧

d

))

R

: ((

a

∧

b

)

≡ (

c

∧

d

))

∧ ((

a

∧

b

)

≡ (

c

∧

d

))

P

: ((

a

∧

b

)

≡ (

c

∧

d

))

∧ ((

a

∧

b

)

≡ (

c

∧

d

))

I

: ((

a

∧

b

)

≡ (

c

∧

d

))

∧ ((

a

∧

b

)

≡ (

c

∧

d

))

H1

: ((

a

∧

b

)

≡ (

c

∧

d

))

∧ ((

a

∧

b

)

≡ (

c

∧

d

))

H2

: ((

a

∧

b

)

≡ (

c

∧

d

))

∧ ((

a

∧

b

)

≡ (

c

∧

d

))

H3

: ((

a

∧

b

)

≡ (

c

∧

d

))

∧ ((

a

∧

b

)

≡ (

c

∧

d

))

H4

: ((

a

∧

b

)

≡ (

c

∧

d

))

∧ ((

a

∧

b

)

≡ (

c

∧

d

))

Only4amongtheseproportionsmakeuseofsimilarityanddissimilarityindicatorswithoutmixingthesetypesofindicators insideone equivalence:forthisreason, these4proportions A

,

R

,

P

,

I are calledhomogeneousproportions.Forinstance,an informal readingof A would be:“a differs fromb asc differsfromd and viceversa.” Thisexpresses themeaning ofan analogical proportion, i.e., a statement ofthe form“a is to b asc isto d”. Wecan consider A as aBoolean counterpart tothe ideaofnumericalproportion, eithergeometric,i.e., a_b

=

_dc,orarithmetic a

−

b

=

c

−

d.It mayalsobeviewedasa qualitativeformofcomparisonofdifferences,reminiscenttotheconceptofderivative wherewestudytheratio f(a_a)−₋_bf(b) with f

(

a

)

=

c and f

(

b

)

=

d,whichisclosetonumericalproportions.

Theideaofaproportionsuggeststhatsome stabilitypropertiesholdw.r.t.permutations.Indeed,wecanpermute vari-ables and check, for instance,if a given proportion still holds when permuting the 2 ﬁrst variables. We denote pi j the permutationofvariableinpositioni withvariableinposition j.Forinstance,p14 permutesthevariablesinextreme posi-tions1 and4,while p23permutesvariablesinmeanpositions.And p12

(

a

)

=

b

,

p12

(

b

)

=

a

,

p12

(

c

)

=

c

,

p12

(

d

)

=

d.

Deﬁnition2.AproportionT isstablew.r.t.permutationpi j iff T

(

a

,

b

,

c

,

d

)

→

T

(

pi j

(

a

),

pi j

(

b

),

pi j

(

c

),

pi j

(

d

))

Itcan becheckedthat A isstablew.r.t.theextremes p14 orthemeans p23 permutations. P isstableforp12and p34 permutations,while R is stablefor p13 andp24.Moreover A

,

R

,

P

,

I are symmetrical(i.e. T

(

a

,

b

,

c

,

d

)

→

T

(

c

,

d

,

a

,

b

)).

This isobservableontheirtruthtables:SeethetoppartofTable 1,whereonlythe6patternsthatmakethelogicalproportions true appear). Besides, I is the only logical proportion that is stable w.r.t. any permutation of two ofits variables. This noticeableresultisprovedin[18].

Moreover, R andP arecloselyrelatedto A viapermutations.Namelywehave

A

(

a

,

b

,

c

,

d

)

≡

P

(

c

,

b

,

a

,

d

)

≡

R

(

b

,

a

,

c

,

d

)

Infact,whend isﬁxed,exchangingthevariablesa

,

b

,

c amountstomovefromonehomogeneousproportiontoanother, orto remainstable( A(a

,

b

,

c

,

d

)

=

A

(

a

,

c

,

b

,

d

);

P

(

c

,

b

,

a

,

d

)

=

P

(

b

,

c

,

a

,

d

);

R

(

b

,

a

,

c

,

d

)

=

R

(

c

,

a

,

b

,

d

)),

with I remainingan exception. Thus A

,

R

,

P collectively maintain a formof exchangeability property withrespect to a

,

b

,

c, while I ensures

it byitself. Theseexchangeabilitypropertiesare ofparticular interestwhen applyinghomogeneous logicalproportionsto classiﬁcation.

The 4 remaining code independent logical proportions H1

,

H2

,

H3

,

H4 are calledheterogeneousproportions: it is clear fromtheirlogicalexpressionthattheymixsimilarityanddissimilarityindicatorsinsideeachequivalence.Theirtruthtables

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Table 1

Homogeneous/heterogeneousproportionsvalidpatterns.

A R P I 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 0 0 1 1 0 0 1 1 1 0 0 1 1 0 0 1 1 1 0 0 1 1 0 0 0 1 1 0 0 1 1 0 0 1 0 1 0 1 1 0 0 1 0 1 0 1 0 1 1 0 1 0 1 0 0 1 1 0 1 0 1 0 1 0 H1 H2 H3 H4 1 1 1 0 1 1 1 0 1 1 1 0 1 1 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 1 0 1 1 0 1 1 1 0 1 1 0 1 1 1 0 1 1 0 0 1 0 0 0 1 0 0 1 0 0 0 1 0 0 1 0 1 1 0 1 1 1 0 1 1 1 0 1 1 1 0 1 0 0 1 0 0 0 1 0 0 0 1 0 0 0

areshowninthebottompartofTable 1,whereonlythe6patternsthatmakethemtrueappear.Theindexi inHirefersto apositioninsidetheformulaHi

(

a

,

b

,

c

,

d

).

Namely,ascanbechecked,ineachofthese6patternsthereisaminorityvalue (i.e.,thevaluehavingthesmallestnumberofoccurrencesinthepattern,thenthisvalueislikeanintruderamongtheother values), andi is theonlypositionwherethe minorityvalue neverappears amongthe64-tuplesofvaluesthat make Hi true.

ByexaminingthetruthtableoftheheterogeneousproportionsinTable 1,wegetthefollowingproperties:

A

(

a

,

b

,

c

,

d

)

∧

R

(

a

,

b

,

c

,

d

)

∧

P

(

a

,

b

,

c

,

d

)

∧

I

(

a

,

b

,

c

,

d

)

= ⊥

togetherwith

(

A

(

a

,

b

,

c

,

d

)

∧

R

(

a

,

b

,

c

,

d

)

∧

P

(

a

,

b

,

c

,

d

))

≡

Eq

(

a

,

b

,

c

,

d

)

where Eq

(

a

,

b

,

c

,

d

)

=

1 ifa

=

b

=

c

=

d and Eq

(

a

,

b

,

c

,

d

)

=

0 otherwise.Similarly,forheterogeneousproportions,wehave

H1

(

a

,

b

,

c

,

d

)

∧

H2

(

a

,

b

,

c

,

d

)

∧

H3

(

a

,

b

,

c

,

d

)

∧

H4

(

a

,

b

,

c

,

d

)

= ⊥

whichimplies:

(

H1

(

a

,

b

,

c

,

d

)

∧

H2

(

a

,

b

,

c

,

d

)

∧

H3

(

a

,

b

,

c

,

d

))

→ ¬

H4

(

a

,

b

,

c

,

d

)

(1) Obviously, we have similar properties by permuting the indexes of the Hi’s. The meaning of the conjunction

H1

(

a

,

b

,

c

,

d

)

∧

H2

(

a

,

b

,

c

,

d

)

∧

H3

(

a

,

b

,

c

,

d

)

will be discussed in the following section, devoted to heterogeneous propor-tions.

2.2. Speciﬁcityofheterogeneousproportions

In order to get a clear understanding of the heterogeneous proportions and to extract relevant properties, we now investigatetheirtruthtables.

2.2.1. Heterogeneityandexchangeability

StillwithinTable 1,anobvioussemanticsappears:Hiholdswhenthereareexactly3parameterswithidenticalBoolean values(

=

1 forexample)andtheparameterinpositioni isoneoftheseidenticalvalues.

Deﬁnition3.Given4 Boolean values a

,

b

,

c

,

d in thisorder such that 3 of them are identical and the remaining one is different,thepositioni

∈ [

1,4

]

ofthisremainingvalueiscalledtheintruderposition ortheintruderforbrevity.

Then, Hiholdsiffthereisanintruderamongthe4valuesa

,

b

,

c

,

d andtheintruderpositionisnoti.Thissuggeststhat

Hishouldbestablew.r.t.thepermutationswhichdonotaffectpositioni.Infactalittlebitmorecanbeestablished:

Property1. ApartfromI,Hiaretheonlylogicalproportionsstablew.r.tanypermutationwhichdoesnotaffectpositioni.

ThespecialcaseofI stablew.r.t.anypermutationhasbeenalreadyprovedin[19].Table 1allowstocheckthattheHi’s arestablew.r.t.thepermutationswhichdonotaffectpositioni.Showingthattheyaretheonlyonesamongthe120logical propertiesstablew.r.t.thesepermutationsrequiresatediouscheckingprocedurethatcannotbesummarizedhere.

Property 1isquitesatisfactoryandconﬁrmstheinformalsemanticsofHi.Thiswillbeusefulwhenusingheterogeneous proportionstoclassiﬁcation.Moreimportantly,thisgivesaclearsemanticstotheconjunctionH1

(

a

,

b

,

c

,

d

)

∧

H2

(

a

,

b

,

c

,

d

)

∧

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H3

(

a

,

b

,

c

,

d

)

above: H1

(

a

,

b

,

c

,

d

)

∧

H2

(

a

,

b

,

c

,

d

)

∧

H3

(

a

,

b

,

c

,

d

)

holdsiffamongthe4valuesa

,

b

,

c

,

d,thereisan intruder andthisintruderisd.Thiswillbethebasisofouroddnessmeasure.Inthenextsubsection,weestablishsomeresultsabout theparityofthenumberof1or0intruthtablesforheterogeneousproportions,whicharecontrastedwithhomogeneous proportions.Thisleadstoamodelofoddnessofagivenvalue,amongamultisetof4values.

2.2.2. Parityofthenumberof1or0intables

SincelogicalproportionsareBooleanformulasinvolving4variables,theirtruthtableshave16rows,whereonly6lead to1(see[19]foracompleteinvestigation).Onecouldaskifanytruthtablehaving6linesleadingto1and10linesleading to0corresponds toalogicalproportion.A simplenumberingargumentshowsthatthisisnotthecase.Ontopofthat,we canbuildclassesofpatternswhichcannotbevalidforanyproportion:

Property2. Thereisnologicalproportionthatistrueforthefourelementsofthesetofvaluations

{

0111,1011,1101,1110

}

.The sameholdsfor

{

1000,0100,0010,0001

}

.

Proof. Anequivalencebetweenindicatorsisoftheforml1

∧

l2

≡

l3

∧

l4.Ifthisequivalenceisvalidfor

{

0111,1011

}

,itmeans that itstruth value doesnot changewhenwe switch thetruth value ofthe2 ﬁrstliteralsfrom0 to1: thereare only2 indicatorsfora andb satisfyingthisrequirement:a

∧

b anda

∧

b.Ifthisequivalenceisstillvalidfor

{

1101,1110

}

,itstruth valuedoesnotchangewhenweswitchthetruthvalueofthe2lastliteralsfrom0 to1:thereareonly2indicatorsforc and d satisfyingthisrequirement:c

∧

d andc

∧

d.Thentheequivalencel1

∧

l2

≡

l3

∧

l4isjusta

∧

b

≡

c

∧

d,a

∧

b

≡

c

∧

d,a

∧

b

≡

c

∧

d ora

∧

b

≡

c

∧

d.Noneoftheseequivalencesistrueforthefourelementsofthesetofvaluations

{

0111,1011,1101,1110

}

. Thesamereasoningisstillapplicablefortheotherclass.

2

Applying a similar reasoning, we can build other set of valuations which cannot make simultaneously true a logical proportion.

Property3. Alogicalproportioncannotbemadetruebyasetof4valuationsincluding3valuationsofoneoftheclassesappearingin

Property 2andwherethe4thvaluationisjustthenegationcomponentwiseoftheremainingvaluationoftheclass.

Forinstance,thereisnologicalproportiontruefor

{

0111,1011,1101,0001

}

orfor

{

0111,0100,1101,1110

}

. Thisremarkhelpsestablishingthefollowingresult:

Property4. Heterogeneousproportionsaretheonlyproportionswhoseallvalidpatternshaveanoddnumberof1.

Proof. Fromthetruthtables, weobservethatonlyvalidpatternsforheterogeneousproportionshaveanoddnumberof1. Letusnowconsideraproportionwhose6validpatternscarry anodd numberof1. Asthereare exactly8patternswith anoddnumberof1,andthankstothepreviousproperty,thisproportionincludesnecessarily3patternsfromeach ofthe previousclasses.Ifthevalidpatternsinoneclassareobtainedfromthevalidpatternsfromtheotherclassjustbynegating all the variables,the proportionis codeindependent andthen, it isa heterogeneous proportion. Inthe opposite case, it means that we have atleast one pattern inthe ﬁrst class withno negated counterpart in the other class: for instance, 1110,1101,1011 arevalidbut0001 isnota validpattern, leavingonly1000,0100,0010 tocompletethetruth tableofa logicalproportion.ThenProperty 3tellsthatthereisnoproportionvalidfor1110,1101,1011,1000.

2

Asimilarpropertyholdsforhomogeneousproportions:

Property5. Homogeneousproportionsaretheonlyproportionswhoseallvalidpatternshaveanevennumberof1.

Wenowshowthespeciﬁcityofheterogeneous(andhomogeneous)proportionsfromareasoningpointofview.

2.2.3. Inferenceandunivocalproportions

Thereisawaytoinferunknownpropertiesofapartiallyknownobject D startingfromtheknowledge wehaveabout itsotherspeciﬁedproperties,andassumingthatalogicalproportionT holdscomponentwisewiththreeotherobjectsA, B, C ,alsorepresentedintermsofthesamen Booleanfeatures.Thiscanbedoneviaaninductionprinciplethatcanbestated asfollows(where J isasubsetof

[

1,n

]

,andxidenotesthetruthvalueoffeaturei forobjectX

∈ {

A

,

B

,

C

,

D

}

):

∀

i

∈ [

1

,

n

] \

J

,

T

(

ai

,

bi

,

ci

,

di

)

∀

i

∈

J

,

T

(

ai

,

bi

,

ci

,

di

)

Thiscan be seen asa continuityprinciple assuming that ifit isknown thata proportionholds forsome attributes, this proportionshouldstillholdfortheotherattributes.Itgeneralizestheinferenceprincipleusedwiththeanalogicalproportion

(7)

no guaranteethatthe conclusionholdswhen thepremisseshold. Nevertheless,speciallywhentheratio |_nJ| is closeto 1, which meansthat proportionsholdon alarge numberofattributes, itis naturalto considerthat such aproportionmay alsoholdonthesmallnumberofremainingattributes.

This principle requires the unicity of the solution of equation T

(

a

,

b

,

c

,

x

)

=

1 where x is unknown, when it exists. Namely,given3Booleanvaluesa

,

b

,

c,wewanttodetermineforwhatlogicalproportionT theequation T

(

a

,

b

,

c

,

x

)

=

1 is solvable,andinsuchacase,ifthesolutionisunique.

Deﬁnition4.If, whentheequation T

(

a

,

b

,

c

,

x

)

=

1 issolvable, thesolutionisunique, thentheproportionT is saidto be 4-univocal. Inasimilarmanner,onemaydeﬁneproportionsthatare1,2,or3-univocal.T isunivocal whenitisi-univocal

foreveryi

∈ [

1,4

]

.

Firstofall,itiseasytoseethattherearealwayscaseswheretheequationT

(

a

,

b

,

c

,

x

)

=

1 hasnosolution,whateverthe proportionT .Indeed,thetriplea

,

b

,

c maytake23

₌

_{8 values,}_while_any_proportion_{T is}_true_only_for₆_distinct_valuations, leavingatleast2caseswithnosolution.Forinstance,whenwedealwithH4,theequationsH4

(0,

0,0,x

)

andH4

(1,

1,1,x

)

havenosolution.

Wehavethefollowingresult:

Property6. Thehomogeneousandtheheterogeneousproportionsaretheonlyproportionswhichareunivocal.

Proof. From thetruthtables, weseethat the2typesofproportionssatisfytheproperty. Now,a proportionwhich isnot

i-univocalis necessarilyvalidboth forapatternwithan odd numberof1,andfora patternwithan evennumber of1. Then,Properties 4 and5excludehomogeneousandheterogeneousproportions.

2

Atthisstep,weseethatheterogeneousproportionsallowtosingleoutaparticularpositionamonganorderedlistof4 values.Thispositiontargetsthevaluewhichisdeﬁnitelynotanintruderamongthemultisetof4items.Forinstance,when

Hiisvalid,thevalueinpositioni isnotanintruder.Weshallseeinthefollowingsectionthatthispropertycanbeusedto checktheoddnessofagivenitemw.r.t.amultisetofelements.

3. Evaluatingthepresenceortheabsenceofanintruderinamultiset

Based onheterogeneousproportions,we deﬁnean oddnessindexandthenan evennessindex, beginningineach case with Booleanvalues, beforeinvestigating their extension to dealwith gradedtruth values.We also laybare the relation betweenevennessandoddness.

TheseoddnessandevennessindexespertaintoaBoolean,oranumericalvalue(denoted x ord inthefollowing)w.r.t. a multiset S ofsuch values.Forreasonsexplainedinthissection,it willappeardesirableto keepthismultisetsmall(i.e.,

|

S

|

=

3, ormaybe2intheoddnesscase).Moreover, inthissectionthe valuesx orin S maybe thoughtofasthevalues ofthesamefeaturefordifferentitems.InSection4,weshallbuildoddnessandevennessmeasuresfromtheseindexesby cumulatingthemoverfeatures,andbyconsideringcollectionsofmultisetsS withintheexamplesdescribingthesameclass

C

inatrainingset.

3.1. Oddnessindex

The ideaofoddness introducedbelowdirectlyreliesontheevaluationoftheextenttowhichanewitemtobeadded toasubsetreinforcesitsheterogeneityandsoappearsasanintruder init.

3.1.1. AnoddnessmeasureforBooleandata

Inthefollowing,weﬁrstdeﬁnean indextoevaluatethe oddness ofanewcomer w.r.t.amultisetofBooleanvaluesvia theheterogeneousproportions,wherethemultisetisatriple.Then,weextendthisindextomulti-valuedlogicinthenext subsection,beforegeneralizingtheextendedoddnessindextomultisetsofanysize.

Let usrememberthe meaning of Hi: Hi holds iffthere isan intruder amonga

,

b

,

c

,

d and the parameterin position

i isnot thisintruder.As showninTable 2,each proportion Hi providesa pieceof knowledgeonthe intruderandwhen combined withother pieces, wecan pickout whichoneis theintruderamonga

,

b

,

c and d.Forexample H1

(

a

,

b

,

c

,

d

)

=

H2

(

a

,

b

,

c

,

d

)

=

H3

(

a

,

b

,

c

,

d

)

=

1 meansthatthereisanintruderwhichisoutofthemultiset

{

a

,

b

,

c

}

. Thenwedeﬁnetheoddnessofd w.r.t.

{

a

,

b

,

c

}

bythefollowingformula:

O dd

(

{

a

,

b

,

c

},

d

)

=

de f H1

(

a

,

b

,

c

,

d

)

∧

H2

(

a

,

b

,

c

,

d

)

∧

H3

(

a

,

b

,

c

,

d

)

(2) Asanimmediateconsequenceofequation(1),wehave:

(8)

Table 2

H1,H2,H3andO dd truthvalues.

a b c d H1 H2 H3 Odd 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 1 0 1 1 0 0 0 0 1 1 0 0 0 0 0 1 0 0 1 0 1 0 0 1 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 1 0 1 1 0 1 0 0 0 0 1 1 0 1 0 0 1 0 0 0 0 1 0 1 0 0 0 0 0 1 0 1 1 1 0 1 0 1 1 0 0 0 0 0 0 1 1 0 1 1 1 0 0 1 1 1 0 1 1 1 1 1 1 1 1 0 0 0 0

DuetothepermutationpropertiesoftheHi’s,therighthandsideofthisdeﬁnitionisstablew.r.t.anypermutationofa

,

b

,

c, thenthemultisetnotationonthelefthandsideisjustiﬁed.Thetruthtableof O dd isgiveninTable 2.

Itisclearthat O dd holdsonlywhenthevalueofd isseenasodd amongtheothervalues:d istheintruder.Moreover

O dd does not hold inthe opposite situation where there is a majority amongvalues in a

,

b

,

c

,

d and d belongs to this majority(e.g. O dd

(

{

0,1,0

},

0)

=

0),orthereisnomajorityatall(e.g. O dd

(

{

0,1,1

},

0)

=

0).

AsimpleobservationofTable 2showsthattheoddnessindexcanberewrittenas

O dd

(

{

a

,

b

,

c

},

d

)

≡ ((

a

∨

b

∨

c

)

≡

d

)

∧ (

a

∧

b

∧

c

)

≡

d

))

(3) Givenamultiseta

,

b

,

c of3identicalBooleanvalues, O dd

(

{

a

,

b

,

c

},

d

)

canthenactasaﬂagindicatingifthe4thvalued

isdifferentfromthecommonvalueofa

,

b

,

c.Thenthevalued isatoddsw.r.t.theothervalues.

3.1.2. Extensiontonumericaldata

It is possible to extend the previous oddness measure in order to handlevariables with graded values (i.e. variables whosevaluesbelongto

[

0,1

]

,afterasuitablenormalizationofnumericaldata).Fornow,thisoddnessisjust0 or1 (i.e.the truthvalue of O dd

(

{

a

,

b

,

c

},

d

)),

butwe wouldliketoconsidertuplessuch as

(0.1,

0.2,0.1,0.8)andstillconsiderthatthe 4thvalueissomewhatoddw.r.t.the3otherones.

Adirecttranslationofformula(2),takingmin for

∧

,max for

∨

,and1

− |

· −

· |

for

≡

asinŁukasiewiczlogic[23],leads to:

O dd

(

{

a

,

b

,

c

},

d

)

=

de f min

(

|

max

(

a

,

b

,

c

)

−

d

|, |

min

(

a

,

b

,

c

)

−

d

|)

(4) Firstofall,itisaneasygametocheckthat O dd remainscodeindependentwithgradedvalues,i.e.changingvaluesinto theircomplementto1.

Letusexamine some examples toget aprecise understanding ofthe formulafornumericaldata andtocheck ifthis

oddness measureﬁtswiththeintuition.

•

We see that O dd

(

{

u

,

u

,

u

},

v

)

= |

u

−

v

|

.Indeed, ifu

=

v then obviouslythe 4thvalue isnot an intruder. The larger

|

u

−

v

|

,themorev isatoddsw.r.tthe3valuesequaltou.

•

Weseealsothat O dd

(

{

v

,

u

,

u

},

v

)

=

0 whichisconsistentwiththeexpectedsemanticsof O dd.

•

Generally,O dd

(

{

u

,

v

,

w

},

max(u

,

v

,

w

))

=

O dd

(

{

u

,

v

,

w

},

min(u

,

v

,

w

))

=

0,andinanycase, O dd

(

{

u

,

v

,

w

},

u

)

≤

0.5.

•

Letusnowconsider anumericalsituation with4differentnumericalvalues,forinstance:

(

{

0,0.1,0.2

},

0.9). Wefeel that d

=

0.9 appears asan intruder in the multiset

(

{

0,0.1,0.2

})

. This is consistent with the obtained truth value

O dd

(

{

0,0.1,0.2

},

0.9)

=

0.7.Moreover, O dd

(

{

0,0.1,0.1

},

0.9)

=

0.8 andO dd

(

{

0,0.1,0.3

},

0.9)

=

0.6,whichﬁtswiththe intuition.

•

Conversely,thepattern

(

{

0.7,1,1

},

0.9)doesnotstronglysuggest0.9 asanintrudervalue.IndeedO dd

(

{

0.7,1,1

},

0.9)

=

0.1. O dd

(

{

0.9,1,1

},

0.7)

=

0.2 isabithigher,asexpectedsincewehavemovedtowardsmoreuniformityamonga

,

b

,

c

andslightlyincreasedthedifferencesbetweend andtheelementsof

{

a

,

b

,

c

}

.Moreover,notethatO dd

(

{

0.7,1,1

},

0.9)

=

0.1, and O dd

(

{

0,0,1

},

0.9)

=

0.1, since the two cases illustrate two different ways of not being really an intruder. Indeed, although 0.9 is close to the majority value in

{

0.7,1,1

}

in the ﬁrst case, and far from the majority value in

{

0,0,1

}

in thesecond case, closeness to majorityvalue in

{

a

,

b

,

c

}

isnot atall what O dd estimates.Ratherit is expectedtoﬁndsimilarestimatesinthetwoabovecases,sincetheyarerespectivelycloseto O dd

(

{

1,1,1

},

1)

=

0 and to O dd

(

{

0,0,1

},

1)

=

0 asshowninTable 2.

•

Finally, O dd

(

{

a

,

b

,

c

},

d

)

does not behave as

|

d

−

average

(

{

a

,

b

,

c

})|

: the cases

{

a

,

b

,

c

}

= {

0.5,0.5,0.5

}

, d

=

0.5, and

(9)

deﬁni-tion, O dd

(

{

0.5,0.5,0.5

},

0.5)

=

0, while O dd

(

{

0,0.5,1

},

0.5)

=

0.5. Thus O dd

(

{

a

,

b

,

c

},

d

)

is a more accurate oddness measurethan

|

d

−

average

(

{

a

,

b

,

c

})|

when theset

{

a

,

b

,

c

}

containsheterogeneousvalues.

From the previous examples,we understandthat theproposed deﬁnitionﬁts withtheinitial intuitionandprovides high truth valueswhend appearstobe atodds w.r.t.themultiset

{

a

,

b

,

c

}

andlowtruth valuesintheopposite casewhered

isnot verydifferentfromtheother values.Ontopofthat,theexpression ofO dd givenhereisnolongertheconjunction ofthemultiple-valuedextensions ofH1

,

H2

,

H3asgivenin[20],whichwouldleadtoalesssatisfactorymeasureofoddness. Indeed, we are here interested in the oddness ofd w.r.t. a multiset

{

a

,

b

,

c

}

, andnot in picking out an intruder in the multiset

{

a

,

b

,

c

,

d

}

asin[20].

3.1.3. Oddnesswithrespecttomultisetsofvarioussize

Since weare interested incheckingifd seemsan intruderina givenmultiset,wemayconsidermultisets ofanysize whendeﬁning theoddnessindex.Indeed,theprevious oddnessindexisnotlimitedtomultisets

{

a

,

b

,

c

}

with3elements, andcan beeasily generalizedtoan indexofoddness O dd

(

S

,

x

)

ofan item x w.r.t.a multiset S ofvalues in

[

0,1

]

ofany size,asfollows:

O dd

(

S

,

x

)

=

de f min

(

|

max

(

S

)

−

x

|, |

min

(

S

)

−

x

|)

As canbe seen,weonlycompare x totheupperandlowervaluesin S,whichmaybe reallyconsidered asameaningful summary of S onlyifS isvery small(when wehavenoadditionalinformationaboutthedistributionofvaluesin S),i.e.

|

S

|

=

1,2,3 ormaybe 4.Clearly,thecomputationof O dd

(

S

,

x

)

reﬂectsthesmallestdistanceofx toanelementin S only

for

|

S

|

=

1,2.Indeed,thisisnotthe caseassoonas

|

S

|

≥

3 ascan beseen onthefollowing example: O dd

(

{

a

,

b

,

c

},

x

)

=

O dd

(

{

0,0.5,1

},

0.5)

=

0.5,while

|

b

−

x

|

= |

0.5

−

0.5

|

=

0.

As itmaybe the caseforrealdatasets,we mayhavemissingvalues.Obviously,whenthere isa missingvalue inthe multiset S ofsizen,thenasimpleoptionistoconsiderthemultiset S ofsizen

−

1 andtoconsider O dd

(

S

,

x

)

=

O dd

(

S

,

x

)

where S hasnolongeranymissingvalue.

3.2. Evennessindex

Adopting a dual viewpoint, we may want to know if adding a newelement to a given subset of items keeps it as homogeneousasitis,i.e.,thenewcomerdoesnotappearasanintruderinthissubset,andratheragreeswithitsmajority. Homogeneity can be considered as a kind of evenness of the newcomer w.r.t. the existing items of the subset. In this subsection,we advocatea wayofjudgingevenness onthe basisofthe majority,ifany,insidethe triples.Contrary tothe oddness deﬁnition, whereall Hi

,

i

=

1,2,3 arerequired to deﬁne the oddnessindex, only H4 is neededfordeﬁning an evennessindex,thusdenotedE ven4.

3.2.1. AnevennessmeasureforBooleandata

Since theidea istoagree withamajority, wenoticethat thesmallestmultisets S ofelements wheremajoritymakes senseareclearlytriples.LetconsiderthreeBooleanvaluesa,b,c inS.Then,inaBooleanworld,therearetwopossibilities, eithera

=

b

=

c,ortwoofthethreeareequal.Inbothcases,a strictmajoritytakesplace.Letm denotethemajorityvalue. Now consider the newcomer d,either d

=

m, and m remains the majorityvalue in

{

a

,

b

,

c

,

d

}

, ord

=

m, and thereisno longeranymajorityin

{

a

,

b

,

c

,

d

}

(twovaluesareequalto1andtwovaluesto0).Onlywiththeﬁrstcase,d conformstothe majority.

Notethat ifweconsiderlarger subsets S,evenwithonly4elements ratherthan3,itbecomes possiblethatthe new-comer increasesan existing minority,withoutchanging themajority.Indeed,themajorityvalue that maybe sharedby 3 elements inthe 4-elementsmultisetwillthen remainunchangedinthe5-elements multisetresultingfromthearrival of a ﬁfthelementwhateveritsvalue. A similarphenomenontakesplaceifwe startwithlarger subsetsS having5elements ormore.Sowearelosingadistinctivepropertyof3-elementssubsetswhichhaveadifferentmajoritybehaviordepending ifd conformsornottothemajorityinthe3-elementssubset. Thismeansthattriplesaretheonlysubsetssuchthataddingan itemthatconformstothetripleminoritydestroysthemajority.Thus,3-elementssubsetsareabletoclearlydiscriminate,among differentd thosethatconformtothemajorityofthetriple.

Theideaofmajorityjustdescribedhelpsustodeﬁneanewevenness measureviatheheterogeneousproportions.Letus recallthesemanticsof Hi: Hi holdsiffthereisanintruderamonga

,

b

,

c

,

d andtheparameterinpositioni isnot this in-truder.Asaconsequence,Hiimpliesthatthereisamajorityofvaluesamong

(

a

,

b

,

c

,

d

)

andthevalueinpositioni conforms tothemajorityofvaluesappearingamongthe3otherpositions(i.e.themultisetofvalues

{

a

,

b

,

c

,

d

}

ismoreorlesseven). But thereverse implicationdoesnot holdsince whenthe 4parameters haveidenticalvalue,

∀

i

∈ [

1,4

],

Hi

(

a

,

b

,

c

,

d

)

=

0. Then,tohaveaconciseBooleandeﬁnitionfor“thereisamajorityofvaluesamongtheparametersa

,

b

,

c

,

d andthe param-eter inposition i belongsto thismajorityof values”,weneed to considerthecasewhere all thevaluesare identicalby usingthefollowingformula:

(10)

Table 3

H4,Eq andE ven4truthvalues.

H4 Eq Even4 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0 0 1 0 1 0 1 0 0 1 1 0 0 0 0 1 0 0 1 0 1 0 1 0 1 0 0 0 0 1 1 0 0 0 0 0 1 1 1 1 0 1 1 0 0 0 1 0 1 1 0 0 1 0 0 0 1 0 1 0 0 0 0 1 0 1 1 1 0 1 1 1 0 0 0 0 0 1 1 0 1 1 0 1 1 1 1 0 0 0 0 1 1 1 1 0 1 1

where Eq

(

a

,

b

,

c

,

d

)

=

de f

(

a

≡

b

)

∧ (

b

≡

c

)

∧ (

c

≡

d

).

Thus,with E veni we take intoaccountthe specialcasewhereallthe valuesareequal.ThetruthtableofE ven4 isgiveninTable 3.Itisclearthat E ven4 holdsonlywhenthevalueofd belongs to amajorityofthe parameter’svalues.And E ven4 doesnot holdinan opposite situationwherethere isnomajority of valuesasitisthecaseforE ven4

(0011)

orE ven4

(0110).

The situations where E ven4

(

a

,

b

,

c

,

d

)

=

1 exactly coverthe two cases already mentioned whered is identicalto the majorityvalue inthe triple

{

a

,

b

,

c

}

(is notthe intruder),namelyeither a

=

b

=

c, ortwo ofthethree areequal tod.So thefactthat d joins

{

a

,

b

,

c

}

,when E ven

(

a

,

b

,

c

,

d

)

=

1,leavestheresultingsubset aseven asitwas,hencethename,and infactthemajorityisreinforcedbythearrival ofd.Notealsothat E ven4

(

a

,

b

,

c

,

d

)

isleftunchangedbyanypermutation of

{

a

,

b

,

c

}

.Thismeans thatthe orderinginsidetriples doesnotmatter. Besides, E ven4

(

a

,

b

,

c

,

d

)

=

E ven4

(

a

,

b

,

c

,

d

)

where

x

=

1 if x

=

0 andx

=

0 ifx

=

1,expressingthat E ven4

(

a

,

b

,

c

,

d

)

doesnotdependonthewaytheinformationisencoded. Fromnowon, E ven4willbedenotedE ven asthisistheonlyoptionweusewithi

=

4.

3.2.2. RelationbetweenoddnessandevennessintheBooleancase

The oddness andevenness Boolean functionshave been built by truth tablesinspection. However, these 2 functions exhibitnoticeablelinks.Despitethefactthattheirnamemightsuggestthatoddnessandevennesscapturedualconcepts,it isnotthecasethat E ven

(

a

,

b

,

c

,

d

)

≡ ¬

O dd

(

{

a

,

b

,

c

},

d

).

Infact,therelationsbetweenthe2measuresareasfollows(where

I denotestheinverseparalogydeﬁnedinsection2.1):

Property7.

E ven

(

a

,

b

,

c

,

d

)

≡ ¬

O dd

(

{

a

,

b

,

c

},

d

)

∧ ¬

I

(

a

,

b

,

c

,

d

)

O dd

(

{

a

,

b

,

c

},

d

)

≡ ¬

E ven

(

a

,

b

,

c

,

d

)

∧ ¬

I

(

a

,

b

,

c

,

d

)

I

(

a

,

b

,

c

,

d

)

≡ ¬

E ven

(

a

,

b

,

c

,

d

)

∧ ¬

O dd

(

{

a

,

b

,

c

},

d

)

Thiscanbe easily checkedonthetruth tables.Thisreﬂects thefact that O dd and E ven and I are mutuallyexclusive. Let usnote that E ven

(

a

,

b

,

c

,

d

)

→ ¬

O dd

(

{

a

,

b

,

c

},

d

),

that

¬

E ven

(

a

,

b

,

c

,

d

)

→ ¬

H4

(

a

,

b

,

c

,

d

)

andthat O dd

(

{

a

,

b

,

c

},

d

)

→

¬

H4

(

a

,

b

,

c

,

d

)

aswell.

Amorecompletediscussionoftherelationbetweenoddnessandevennesscanbefoundin[21].

3.2.3. Extensiontonumericaldata

Inordertodealwithgradedtruthvalues,weneedtoextendthepreviousdeﬁnitionofEvengiveninformula(4)tothe casewherethetruthvaluesbelongto

[

0,1

]

,aswehavedonefortheoddnessfunction.

Adirecttranslationofformula(4),takingmin for

∧

,max for

∨

,and1

− |

· −

· |

for

≡

asinŁukasiewicslogic,leadsto thefollowingexpressionforE ven4:

max

(

min

(

1

− |

min

(

a

,

b

)

−

min

(

1

−

c

,

d

)

|,

1

− |

min

(

1

−

a

,

1

−

b

)

−

min

(

c

,

1

−

d

)

|),

1

− |

max

(

a

,

b

,

c

,

d

)

−

min

(

a

,

b

,

c

,

d

)

|)

Letusexaminethebehaviorofthisdeﬁnition.Inordertogetaclearpicture,weconsiderthe2followingcurves:

•

f

(

x

)

=

E ven

(0,

x

,

x

,

x

):

we wouldexpect f toget theconstant value 1,since, whateverits value,the last element x,

(11)

Fig. 1. f and g functions with standard deﬁnition of E ven.

Fig. 2. f and g functions with the new deﬁnition of E ven.

•

g

(

x

)

=

E ven

(0,

x

,

x

,

0): weexpect afunction decreasingfrom1 to 0 when x goesfrom0 to 1.Indeed,thesmaller x,

thecloserto 1 E ven(0,x

,

x

,

0) shouldbe,whilethelarger x themore0 appears tobe equaltotheminorityvalue in themultiset

{

0,x

,

x

}

.

The corresponding curves are inFig. 1. Black square dots are forfunction f , empty circles forfunction g (mindthat an emptycirclemaypartiallyhideablacksquarewhen f and g coincide).Ascanbeseen, f isnotaconstantfunctionandg

isnotmonotonicallydecreasing.ThiscontrastswithO dd

(0,

x

,

x

,

x

)

=

0 and O dd

(0,

x

,

x

,

0)

=

0.

It appears that a direct translationof the Booleandeﬁnition (4) doesnot ﬁt withthe expectedmeaning ofevenness in the caseof graded truth values. But obviously, we could start from the property E ven

≡ ¬

O dd

∧ ¬

I to get another translationas:

E ven

(

a

,

b

,

c

,

d

)

=

min

(

1

−

O dd

(

a

,

b

,

c

,

d

),

1

−

I

(

a

,

b

,

c

,

d

))

ThisnewdeﬁnitionleadsviaaneasycomputationtoE ven

(0,

x

,

x

,

x

)

=

1

−

x ifx

≤

0.5 and1

−

min

(

x

,

2

−

2x)whenx

≥

0.5. Withthisnewdeﬁnitionof E ven, thecurvescorresponding to f and g aregiveninFig. 2(weuse blacksquaredots for function f andemptycirclesforfunction g again).

Observethatthebehavior ofg

(

x

)

=

E ven

(0,

x

,

x

,

0)issatisfactory sincewegetthedecreasingfunction1

−

x.However

f

(

x

)

=

E ven

(0,

x

,

x

,

x

)

maybefarfrom1(inparticular,E ven

(0,

2₃

,

2₃

,

2₃

)

=

1₃).Suchabehaviorisnotsatisfactoryatall.Itis still anopenquestiontoﬁndabetterdeﬁnitionfor E ven inthegradedcase,whichwouldcoincidewiththeBooleancase whena

,

b

,

c

,

d

∈ {

0,1

}

.Forthisreason,weshallnotexperimenttheevennessfunctionwithnumericaldata.

3.2.4. Dealingwithmissingvalues

Missinginformationisquitecommoninreallifedatasetsandawaytoextendthesemanticsofanalogicalproportionto dealwiththisissuehasbeendeeplyinvestigatedin[20]forinstance.Infact,suchanapproachcanbealsoappliedhere,as explainednow.

Still keepingalogical approachandconsidering that‘?’ denotes amissingvalue (i.e.an informationisunknown),the idea is toextend thetruth table of the E ven formula asfollows: E ven

(?,

0,0,0)

=

E ven

(0,

?,0,0)

=

E ven

(0,

0,?,0)

=

1,

E ven

(?,

1,1,1)

=

E ven

(1,

?,1,1)

=

E ven

(1,

1,?,1)

=

1, and E ven

(

x

,

y

,

z

,

t

)

=

0 for any other patternincluding atleasta missing value ‘?’. It isclearthat, withthe 6 ﬁrstpatterns, whateverthe candidatevalue of themissing feature,the 4th argument belongs to the majority and cannot be an intruder. In all the remaining cases, where we have no certainty regardingthestatusofd,weadoptacautiousbehaviorbyconsideringthat E ven doesnothold.

(12)

4. Heterogeneousproportions-basedclassiﬁers

Intheprevioussection,wehavedefinedtwonewindexesthatevaluatetheoddness orevenness ofanitemwithrespect toa multiset S of afixed size(especially S is a tripleincaseofevenness). Inthe contextofclassification, ouraimisto maintainhomogeneityorevennessinsideeachclass

C

andtoavoidoddnesswhenclassifyinganewitem.Forthispurpose, weﬁrstextendO dd and E ven indexestodealwithvectorsinsteadofsimpleBooleanornumericalvalues,andthen,build upaglobaloddness/evennessmeasureofanitemx w.r.t.aclass

C

.

Inthefollowing,weproposeafamilyofclassifiersbasedontheseglobalevenness/oddnessmeasuresandindexedbythe sizeofthesubsets(usedinthecomparisonprocessfortheoddness-basedclassifiers).ButwefirstrecalltheBayesianview oftheconformityofanitemw.r.t.aclass,whichcontrastswiththeviewsproposedintherestofthissection.

4.1. TheBayesianviewpoint

Inaclassical Bayesianview,wehave P rob

(C|

x

)

=

1_Z

·

P rob

(C)

·

n_i₌₁P rob

(

xi

|

C)

assumingthat then featuresare inde-pendent.Theevidence Z dependsonlyon

x, P rob

(

C)

reﬂectssomecharacteristicsof

C

suchasitssize,and

n_i₌₁P rob

(

xi

|

C)

evaluatestheconformityof

x

= (

x1

,

· · · ,

xn

)

with

C.

Undersomeconditionalindependenceassumptions,thisprobabilitycan berewrittenasaweightedproductofP rob

(

xi

|

C),

i.e.theconditionalprobabilitytogetvalue xiforfeaturei intheclass

C

. Usually, P rob

(

xi

|

C)

isestimatedasthefrequencyofelementshavingvalue xi forfeaturei inthewholeclass

C

.Thus,the expressionof P rob

(C|

x

)

involvestheproductof P rob

(C)

withakindofconjunctivecombinationexpressedastheproduct oftheproportionofelementsof

C

identicalto

x foreachfeaturei.A counterpartofthisevaluationexistsinthesettingof possibilitytheory[2].InthecaseofBooleanfeatures,let p

(

C,

i

)

betheproportionofthemajorityvalueforfeaturei in

C

. Then,anelementaryestimationof P rob

(

C|

x

)

is:

1 Z

·

P rob

(

C

)

·

i∈Mp

(

C

,

i

)

·

j∈M

(

1

−

p

(

C

,

j

))

whereM

⊆ {

1,

· · ·,

n

}

isthesubsetoffeatureswhere

x isconformtothemajorityin

C

,andM isthecomplementarysubset where

x is notconform tothemajority. Theideaofconformityin thisapproach isthusrelatedtothenotion ofmajority w.r.t. thewhole set

C

itself.Inthefollowing,weinvestigatetheideaofjudgingconformityw.r.t. acollectionofsmallersubsets

S ⊂ C

,andthentocumulatetheresultsofthecomparisonof

x withthedifferentsubsets

S

.

4.2. Classiﬁcationindexes

When it comes to real life application, it is not enough to represent individualswith a single Boolean or realvalue. Generally,individualsareencodedbyasetoffeatures.Basedonthepreviouslydeﬁnedoddnessandevennessmeasures,we havetodeﬁnenewmeasuressuitableforvectors.

4.2.1. Oddnessandevennessmeasuresforvectors

Whendealing withvectors

→

x

∈ [

0,1

]

n,Booleanvectorsarealsocoveredasaparticularcase.The O dd and E ven

mea-sures,deﬁnedrespectivelyby(1)and(4),areusedtoestimatetowhatextentavalue x canbe consideredasodd oreven

amonga multiset S of values.Thanks tothetwo latterformulas,assuming the independenceoffeatures,it isnaturalto computetheoddness orevenness ofavector

→

x asthesum oftheoddness orevenness foreachfeaturexi

∈ →

x ,asfollows:

O dd

(

S

, →

x

)

=

de f

in=1O dd

(

Si

,

xi

)

∈ [

0

,

n

]

E ven

(

S

, →

x

)

=

de f

ni=1E ven

(

Si

,

xi

)

∈ [

0

,

n

]

wherexi isthei-thcomponentof

→

x ,Siisthemultisetgatheringthei-thcomponentsofthevectorsin

S

.Notethatinthe caseofE ven,theset

S

hasexactly3elements,butwekeepthesetnotationforsakeofnotationuniformity.

Ifouraimistomeasure oddness,highvaluesof O dd

(

S,

→

x

)

(closeton)meansthat,formany features,

→

x appearsas an intruderandmayreducethehomogeneitywhengoing from

S

tothemultiset

S ∪ {→

x

}

.If O dd

(

S,

→

x

)

=

0,nofeature indicatesthat

→

x behavesasanintruderandthereisnoobstaclefor

→

x tojointhemultiset S.1

Ontheopposite, ifouraimisto computeevenness ofthesubset

S

when

→

x isbeingadded,the bigger E ven

(S,

→

x

),

thelargerthenumberoffeatures forwhich

→

x conformstothe majorityin

S

,thebetter

→

x conformstovectorsin

S

.If

E ven

(S,

→

x

)

=

n,theredoesnotexista featurewhere

→

x behavesasanintruder.Then, itisacceptablefor

→

x to jointhe set

S

.

1 _It_is_clear_that_when_dealing_with_{classiﬁcation}_task,_S_is_just_a_set_of_examples,_without_any_repetition,_but_obviously,_its_projections_{componentwise,} theSi’saremultisetsofBooleanorrealvalues.