Rough-fuzzy rule interpolation

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Rough-fuzzy rule interpolation

Chen, Chengyuan; MacParthalain, Neil; Li, Ying; Price, Christopher; Quek, Chai; Shen, Qiang

Published in:

Information Sciences

DOI:

10.1016/j.ins.2016.02.036

Publication date:

2016

Citation for published version (APA):

Chen, C., MacParthalain, N., Li, Y., Price, C., Quek, C., & Shen, Q. (2016). Rough-fuzzy rule interpolation.

Information Sciences, 351, 1-17. https://doi.org/10.1016/j.ins.2016.02.036

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ContentslistsavailableatScienceDirect

Information

Sciences

journalhomepage:www.elsevier.com/locate/ins

Rough-fuzzy

rule

interpolation

Chengyuan

Chen

a

,

Neil

Mac

Parthaláin

b

,

Ying

Li

c

,

Chris

Price

b

,

Chai

Quek

d

,

Qiang

Shen

b,∗

a School of Electrical and Information Engineering, Chongqing University of Science and Technology, Chongqing 401331, China b Department of Computer Science, Institute of Mathematics, Physics and Computer Science, Aberystwyth University, Aberystwyth SY23

3DB, UK

c School of Computer Science, Northwestern Polytechnical University, Xian 710129, China d School of Computer Engineering, Nanyang Technological University, 637457, Singapore

a

r

t

i

c

l

e

i

n

f

o

Article history: Received 1 July 2015 Revised 7 January 2016 Accepted 18 February 2016 Available online 27 February 2016

Keywords:

Fuzzy rule interpolation Rough-fuzzy sets

Transformation-based interpolation

a

b

s

t

r

a

c

t

Fuzzyruleinterpolationformsanimportantapproachforperforminginferencewith sys-temscomprisingsparserulebases.Even whenagivenobservation hasnooverlapwith theantecedentvaluesofanyexistingrules,fuzzyruleinterpolationmaystillderivea use-fulconclusion.Unfortunately,verylittleoftheexistingworkonfuzzyruleinterpolation canconjunctivelyhandlemorethanoneformofuncertaintyintherulesorobservations. Inparticular, thedifficulty indefiningtherequiredprecise-valuedmembershipfunctions forthefuzzysetsthatareusedinconventionalfuzzyruleinterpolationtechniques signif-icantlyrestrictstheirapplication.Inthispaper,anovelrough-fuzzyapproachisproposed inanattempttoaddresssuchdifficulties.Theproposedapproachallowsthe representa-tion,handlingandutilisationofdifferentlevelsofuncertaintyinknowledge.Thisallows transformation-basedfuzzyruleinterpolationtechniquestomodelandharnessadditional uncertaininformationinordertoimplementaneffectivefuzzyinterpolativereasoning sys-tem.Finalconclusionsarederivedbyperformingrough-fuzzyinterpolationoverthis rep-resentation.Theeffectivenessofthe approachisillustratedbyapracticalapplicationto thepredictionofdiarrhoealdiseaseratesinremotevillages.Itisfurtherevaluatedagainst arangeofotherbenchmarkcasestudies.Theexperimentalresultsconfirmtheefficacyof theproposedwork.

© 2016TheAuthors.PublishedbyElsevierInc. ThisisanopenaccessarticleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

Thecompositional rule ofinference [1]offers an effective mechanismforperforming fuzzy inference withdense rule bases.Givensucharulebaseandalsoanobservationthatisatleastpartiallycoveredbythatrulebase,aconclusioncanbe inferredfromcertainrulesthatintersectwiththeobservation.However,forthecaseswhereafuzzyrulebasecontains‘gaps’ (termed:sparserulebase[2]),ifagivenobservationhasnooverlapwiththeantecedent valuesofanyrule,conventional

Corresponding author.

E-mail addresses: ccy@cqust.edu.cn (C. Chen), ncm@aber.ac.uk (N.M. Parthaláin), lybyp@nwpu.edu.cn (Y. Li), cjp@aber.ac.uk (C. Price), ashcquek@ntu.edu.sg (C. Quek), qqs@aber.ac.uk (Q. Shen).

http://dx.doi.org/10.1016/j.ins.2016.02.036

0020-0255/© 2016 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license ( http://creativecommons.org/licenses/by-nc-nd/4.0/ ).

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Fig. 1. Different membership functions for a common underlying concept perceived by different people.

fuzzyinference methodscannot derive a conclusion.Fortunately,using fuzzyrule interpolation (FRI) [3,4], certain useful conclusionsmaystillbeobtained.

TheapplicationoftraditionalFRImethodsmayleadtoabnormalfuzzyconclusions,however.Oneparticularissueisthat the convexity ofthe derived fuzzyvalues isnot guaranteed[5,6],but convexity isoften a crucial requirementfor fuzzy inference inordertoattainimprovedinterpretabilityoftheresults.Anumberofsignificantextensionstothe originalFRI methodshavebeenproposedinliteratureinanattempttoaddressthisissue,including[7–13].Inparticular,thescaleand move transformation-basedFRI approach(abbreviatedto T-FRIhereafter) [14,15]andits generalisation[16,17] canhandle interpolationandextrapolationwhichinvolvemultiplefuzzyrules,whereeachruleconsistsofmultipleantecedents. Such workalsoguaranteestheuniqueness,aswellasthenormalityandconvexityoftheinterpolatedconclusion.Thisapproach hasrecentlybeenfurtherenhancedwithan adaptivemechanismsuch thatappropriate chainingoffuzzyinterpolative in-ferencescan beperformed[18].Furtherdevelopmenthasalsobeenreportedthat allows thefuzzyruleinterpolation and extrapolationtobeperformedinabackwardmanner,i.e.,fromruleconsequencetoantecedentvariables[19].

The aforementionedFRI techniquesprovide a basicmeans fordealingwithandinterpreting uncertaintyinrule-based reasoningby exploitingthe expressive poweroffuzzysets [20].However, thereis littlework inthe areaof FRIthat can handleuncertaintyinfuzzinessitself.Thisisbecausetheseapproachesareimplementedusingconventionalfuzzysettheory asa basis fortheunderlyingrepresentations [21].Whilstmembership functionsplayan important rolein defining fuzzy sets,it issometimesextremely difficult,ifnotimpossible, topreciselydefine suchmembership functions.Moregenerally, there maybe different types ofuncertainty infuzzy rule-basedsystems that need to be captured ormodelled [22]: (1) Thelinguisticvariablesthatareusedintheantecedentsandconsequencesofthegivenrulesmaybeindiscernible.(2)The interpretation of thevalues ofthe underlying linguisticvariables may be vague, e.g., the sameword can meandifferent thingstodifferentpeople.(3)Anelementcanbelongtoafuzzysetwithagivendegree,butthatdegreeofbelongingmay itselfbeuncertain.(4)Theobtainedrulesmaybeinconsistentwhenindividualviewsareprovidedfromagroupofexperts. (5)Observationsattainablebyinexactknowledgemaybenoisyandthereforerandomlydistributed.

Mostofthesetypesofuncertaintycanbedifficulttodealwithifcrispmembershipfunctionsofthefuzzysetshavetobe determined.Forinstance,thereare certainextremeweatherconditionsthatwouldbeconsideredto becoldbyallpeople, butotherlessextremeconditionsmaystillbeconsideredtobecoldonlybycertainindividuals.Themembershipfunctions fordifferentpeoplemaythereforebedifferent,dependingontheirperception,preference,experience,etc.Thisisshownin Fig.1.Thatis,bothsimilaritiesanddifferencesmayexistindefiningagivenconcept.Therefore,therepresentationofa con-ceptshouldsatisfytherequirementsofnotonlytheimprecisedescriptionbutalsobothcommonandindividualperceptions. Inthiscase,usingonlythemembershipvaluesofaconventional(type-1)fuzzysetmaynotbeadequatetocapture,reflect, andmodelthegivenconcept.Whenfacedwithsuchhigherorderuncertainty,whichisessentiallytheuncertaintyof eval-uationaboutuncertainty,ageneralapproachwouldbetosimplyignorethishigher-levelinformation.However,anobvious drawbackofthisisthatvaluableinformationmaybelostaboutboththeconceptthatisbeingmodelledandtheimpactthat uncertaininformationmayhaveuponthatconcept.This,inturn,mayleadtounacceptableinference conclusions. Alterna-tiverepresentationsarethereforerequiredinordertoachieveabetterunderstandingoftheconceptandmanipulatethese differentlevelsofuncertaintysuchthattheycanbehandledappropriately.Thus,itisdesirabletodevelopanewmodelto representthemembershipfunctionsoffuzzysets,inordertoprovideabettermeansofaddressinguncertaintyinFRI.

Theconceptofroughsets[23]wasoriginallyproposedasamathematicaltooltodealwithincompleteorimperfectdata andknowledgeininformationsystems.Aroughsetisitselfanapproximationofavagueconceptbyapairofprecisesets, calledlowerandupperapproximations[24,25].Thelowerapproximationcontainsallofthoseobjectswhichdefinitelybelong tothesetthatdenotesthegivenconcept,andtheupperapproximationcontainsallofthoseobjectswhichpossiblybelongto thatset.Assuch,roughsetsofferadistinctandcomplementaryapproachtofuzzysetsinsupportingapproximatereasoning. Inspiredbythisobservation,itispotentiallyusefultointegrateroughandfuzzytechniquesinordertoimprovetheability to handleuncertainty. This paperproposessuch an approachto rough-fuzzy set-based ruleinterpolation. A specification ofrough-fuzzy sets isintroduced to describe aparticular type ofhigherorder uncertainty,which ischaracterisedby the lower andupperapproximationmembership functions.This approachfacilitatesthe representationofuncertainfuzzyset membershipfunctionswithrough-fuzzyapproximations, therebyimprovingtheflexibilityofruleinterpolationinhandling differentlevels ofuncertainty that maybe presentinsparse rulebases andobservations.The work reflects theintuition thatthemoreusefulinformationavailabletotheinterpolationprocess,thebettertheinterpolatedresults.

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Therestofthepaperisstructuredasfollows.Section2introducesthebasicconceptsthatareusedforthedevelopment ofrough-fuzzyruleinterpolation.Section 3describesthemainsteps ofthe proposedrough-fuzzyrule interpolation algo-rithm.Section4presentsarealisticproblemapplicationthatshowsthepotentialforuseoftheproposed approachforthe predictionofdiarrhoealdiseasesinremotevillages.Section5providesfurtherexampleswhichillustratetheapplicabilityof theapproachandfurtherdemonstratetheefficacyofthework.ThepaperisconcludedinSection6,includingsuggestions forfurtherdevelopment.

2. Rough-fuzzysetsandtheirrepresentativevalues

2.1. Rough-fuzzysets

Thestartingpointfortheproposedapproachistheabilitytorepresentcomplicateduncertainknowledgeinaneffortto performFRI.Whenexactmembershipvaluesarenolongersuitablefordescribingtheunderlyinguncertainty,itisdesirable toutilise an alternativehigher-order representation.Arough-fuzzy representation offers suchcharacteristics, withthe first orderrepresentationembeddedwithin it.Thisdiffersfromtheconcept ofconventionalrough sets,whichischaracterised bythe lowerandupperapproximationswhose elementsare offull memberships [23].Ifhowever,the rough-fuzzy infor-mation or datadegenerates to the first orderrepresentation, the computational mechanismthat dealswith rough-fuzzy interpolationshouldalsonaturallydegeneratetothecorrespondingfirstordercalculus.

LetI=

(

U,A

)

beaninformationsystem, whereUisanon-empty setoffiniteobjects(namely,theunderlyinguniverse ofdiscourse)andAisanon-empty finitesetofattributessuch thata:U→Va forevery a∈AwithVabeingthedomain thatattributeatakesvaluesfrom.WithanyP⊆ AthereisacrispequivalencerelationIND(P)[23]:

IND

(

P

)

=

{

(

x,y

)

∈U2

|

aP, a

(

x

)

=a

(

y

)

}

(1)

If(x,y) IND(P), then xand y are indiscernible by attributes fromP.The equivalence class with respect to such an indiscernibilityrelationdefinedonPisdenotedby[x]P,x∈U.

LetX⊆ U,X beapproximated usingonly theinformationcontainedwithin P by constructingthe P-lower andP-upper

approximationsofX[23]:

PX=

{

x

|

[x]P⊆ X

}

PX=

{

x

|

[x]PX=∅

}

(2) Thetuple

PX,PX

iscalledaroughset.

Definition2.1. WithanyP⊆ A,analternative equivalencerelationIND(P) tothetraditionalone ofEq.(1)canbedefined

by

IND

(

P

)

=

{

(

x,y

)

∈U2

|

F

gP, Fg

(

x

)

C, Fg

(

y

)

C

}

(3)

whereFg,g

{

1,...,G

}

,arefuzzysetsthatjointlydefineaparticularconceptCinX,X⊆ U.

Eq.(3)expressestheequivalencerelationbetweenthemembershipsofxandytodifferentfuzzysetsofgivenconcept. Usingthisequivalencerelation,thelowerandupperapproximationsforasingleCinXcanberedefinedasfollows.

Definition2.2. LetIND(P)bean equivalencerelationonUandFg,g

{

1,...,G},befuzzysetsinC(CX),thelowerand

upperapproximationsareapairoffuzzysetswithmembershipfunctionsdefinedbythefollowing,respectively:

μ

PC

(

x∈[x]P

)

=inf

{

μ

Fg

(

x

)

,g

{

1,...,G

}|

x∈[x]P

}

μ

PC

(

x∈[x]P

)

=sup

{

μ

Fg

(

x

)

,g

{

1,...,G

}|

x∈[x]P

}

(4) Thetuple

PX,PX

iscalledarough-fuzzy(RF) set(whichdiffersfromthealternativeuseofthistermintheliterature [26]duetotheparalleldevelopmentoftheserelatedbutdifferentconcepts).

Remark2.1. Asdiscussedpreviously,similarbutdifferentfuzzysets,whichareconsideredtobelongtoanequivalenceclass,

maybeobtainedindescribingagivenconcept.Therefore,thelowerandupperapproximationsofanRFsetaredefinedon thebasisofthosefuzzysetswhichareknowntosharesuchanequivalencerelation.

Definition2.3. LetUbetheuniverse,anRFsetA˜onUishereindenotedbythelowerapproximation(LA)A˜Landtheupper

approximation(UA)A˜U suchthat ˜ A=

x,[

μ

LA˜

(

x

)

,

μ

U ˜ A

(

x

)

]

=

A˜ L,A˜U

,

xU (5) where0

μ

L ˜ A

(

x

)

μ

U ˜

A

(

x

)

≤ 1,andthelowerandupperapproximationsaretwoconventionalfuzzysets,namely,twofirst orderfuzzysets.

Remark2.2. The closerthe shapesof A˜L and A˜U are,the lessuncertainthe informationcontained within A˜is. When A˜L

coincideswithA˜U,theRFsetdegeneratestoaconventionalfuzzyset,i.e.,

μ

L

˜

A

(

x

)

=

μ

U

˜

(5)

Fig. 2. An RF set corresponding to the situation depicted by Fig. 1 .

Reconsider thesituationshowninSection1,wheredifferentpeoplemayinterpretthesameconceptdifferently.As re-flected in Fig. 1, it isdifficult to describe this situation usingconventional fuzzysets. However, RF sets can be adopted to representthisuncertain concept byexploiting thetwo approximations. In particular,the LA indicates theintersection amongst theregions that are agreedby individuals,whilethe UAindicates the unionofthe regionsthat are givenby at leastoneperson,asshowninFig.2.RFsetsthereforeutiliseLAsandUAstoexpressthehigherorderuncertaintyinvolved indescribingapieceofknowledgeordata.

2.2. Basicnotions

Animportantconcepttointroduceisthe‘lessthan’relationbetweentwofuzzysets[3].AnordinarysetA1 issaidtobe lessthananotherordinaryfuzzysetA2,denotedbyA1 ≺ A2,if

α

∈(0,1],thefollowingconditionshold:

inf

{

A1α

}

<inf

{

A2α

}

, sup

{

A1α

}

<sup

{

A2α

}

(6)

whereA1αandA2αarethe

α

-cutsetsofA1andA2,respectively,inf{Aiα}istheinfimumofAiα,andsup{Aiα}isthesupremum ofAiα,i=1,2.

Definition2.4. AnRFsetA˜1issaidtobelessthananotherRFsetA˜2,denotedasA˜1 ≺ ˜˜ A2,ifandonlyif

˜

A1L≺ ˜AL2, A˜U1 ≺ ˜AU2 (7)

Fromthis,thenotionofneighbouringrulesinvolvingRFsetscanbedefined.

Definition2.5. TwoRFrules

R1:Ifx1 isA˜11,x2 isA˜12,...,xM isA˜1M, thenyisB˜1 R2:Ifx1 isA˜21, x2isA˜22,..., xMisA˜2M, thenyisB˜2

are said to be neighbouring rules if and only if: (1) A˜1j ≺ ˜˜ A2j or A˜2j ≺ ˜˜ A1j, j

{

1,...,M

}

(where M is the number of antecedentvariablesinbothrules);and(2)thereisnoindividualrule“Ifx1 isA˜1,x2 isA˜2,...,xMisA˜M,thenyisB˜” such thatA˜1j≺ ˜˜ Aj≺ ˜˜ A2jifA˜1j≺ ˜˜ A2j,orA˜2j≺ ˜˜ Aj≺ ˜˜ A1jifA˜2j≺ ˜˜ A1j, j

{

1,...,M

}

.

RF ruleinterpolationcanthen beachievedby extendingtheconventionalFRI. Inthiscase,theinput andoutput ofan interpolativeprocessareRFsetsratherthanconventionalfuzzysets.

Definition 2.6. Given an RF rule baseandan RF observation, rough-fuzzyrule interpolationis a process throughwhich a

conclusionfromthegivenobservationisobtainedbyidentifyingtherulesintherulebasewhichflanktheobservationand interpolatingfromthoserules.

Notethat intheabovedefinition,tworules (e.g.,theR1 andR2 givenpreviously)aresaid toflank agivenobservation

[3],say,O=

(

A˜∗1,A˜∗2,...,A˜∗M

)

,ifA˜1j≺ ˜˜ Aj ≺ ˜˜ A2j,orA˜2j≺ ˜˜ Aj ≺ ˜˜ A1j, j

{

1,...,M

}

.

2.3. Representativevalues

In orderto supportthe interpolation ofrules involving RF sets followingthe transformation-basedapproach [14],the ruleswhichhaveminimaldistancesfromagivenobservationneedtobeselectedfirst.Here,adistancebetweenagiven ob-servationandaruleintherulebaseismeasuredonthebasisofrepresentativevalue(Rep).TheconceptofRepisintroduced below.

TheRepvaluecapturesimportantinformationsuchastheoveralllocationofanRFsetwithinthedefinitiondomain,and iscomputedandthenutilisedastheguidetoperformsubsequentinferenceduringtheinterpolationprocess.Forsimplicity, inthiswork,itisassumedthatonlypolygonalRFsetsareconsidered;thatis,boththelowerandtheupperapproximation are each represented by a polygonal-shaped first order fuzzy set. Note that in the existing T-FRI [15], the Rep(A) of an

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Fig. 3. LA ˜ AL and UA ˜ AU of a polygonal RF set ˜ A .

ordinaryfuzzy setA isdefinedby the weightedaverageofthe xcoordinatevalues ofall oddpointsai,i

{

0,...,k− 1

}

,

suchthat: Rep

(

A

)

= k−1  i=0 wiai (8)

withA=

(

a0,. . .,ak−1

)

beingapolygonalfuzzysetofkoddpoints,andwidenotingtheweightassignedtothepointai.

Definition2.7. SupposethatapolygonalRFsetA˜isgiven,asshowninFig.3,whoselowerandupperapproximationsare:

˜ A=

(

a˜L 0,...,a˜Ll−1; ˜HAL˜1,...,H˜ L ˜ Al−2

)

,

(

a˜ U 0,...,a˜Uu−1; ˜HUA˜1,...,H˜ U ˜

Au−2

)

. The lower andupperReps Rep

(

A˜

L

)

andRep

(

A˜U

)

ofA˜ are definedby

Rep

(

A˜L

)

x= l−1  i=0 wL ia˜Li Rep

(

A˜L

)

y= l−2  i=1 wL iH˜AL˜i

Rep

(

A˜U

)

x= u−1  j=0 wUja˜Uj Rep

(

A˜U

)

y= u−2  j=1 wU jH˜UA˜j (9) wherewV

v (V∈{L,U},

v

{

i,j

}

)istheweightassignedtopointa˜Vv anditscorrespondingmembershipvalueH˜VA˜v,andxand

ydenoteacertainvariabledimensionandthecorrespondingmembershipdistribution,respectively.

Ingeneral,differentdefinitionscanbe adoptedforderiving differentRepvalues.Forinstance,thesimplestcaseisthat allpointstake the sameweight value,i.e., wL

i=1/l and wUj =1/u.The centreofcorecan alsobe usedasan alternative. In thiscase, the Repmay be solely determined by those pointswith a membership value of1: Rep

(

A˜U

)

x= 12

(

a˜U(l/2)−1 +a˜U l−(l/2)

)

andRep

(

A˜ U

)

y= 12

(

H˜AU˜ (l/2)−1+ ˜ HU ˜

Al(l/2)

)

.ThelowerRepsareomittedhere,whichcanbecalculatedinasimilar wayinvolvingthose pointsofthe maximummembership value. Other alternative definitionscan be found in[15].For a givenapplication,oneoftheseweightingschemesneedstobetakenforimplementation.

Remark2.3. In theexisting T-FRI,the Rep(A)y ofa givenconventional fuzzysetA isa constant, onlythe xcoordinateis

thereforeconsideredastheRep(A).However,thisisnolongerthecaseinthisworkduetotheintroductionofhigherorder uncertainty,bothxandydimensionsmustbeconsidered.ThecalculationofRep

(

A˜

)

yfollowsthatusedtocalculateRep

(

A˜

)

x inanefforttomaintainconsistency.

Todistinguish amongst differentRF set shapes, the shape diversity factor fis also introduced here. The present work definesthisconceptbyfollowingtheconventionaldefinitionofstatisticalstandarddeviation(althoughthismaybedefined differentlyifdesiredforaparticularimplementation).

Definition2.8. Theloweranduppershapediversityfactors fL

˜ A and f U ˜ A aredefinedby

fL ˜ A=



l−1 i=0(a˜Li−Rep(A˜L)x)2 l fU ˜ A =



u−1 j=0(a˜Uj−Rep(A˜U)x)2 u (10)

Remark2.4. A smallshapediversity factorimpliesthatthe oddpointsofA˜L (A˜U) tendtobe closeto thoseofthe lower

(upper)Rep.Thatis,thesmallertheshapediversityfactor,thesmallertheareaofthelower(upper)approximation. Toextendthemethodologyofconventional T-FRIto FRIinvolvingRF sets,a singleoverallRepofa givenRFset is re-quired.Forthis, theweightfactorwofthelower (upper)approximation isdefinedfirst,whichreflects therelative contri-butionofthelower(upper)shapediversityindepictingtheunderlyingRF set.Theintroductionoftheselower andupper

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shapediversityfactors helpsminimisethepossibilitythat theuseofRFsets ofdifferentshapesleadstothesameoverall Repvalues.

Definition2.9. ThelowerandupperweightfactorswL

˜

A andw U

˜

A aredefinedastheweightsoftheshapediversityfactors,in termsoftheareasofthelowerandupperapproximations,suchthat:

wV ˜ A = fV ˜ A fL ˜ A+f U ˜ A , V=L,U (11) Remark2.5. Ingeneral, fL ˜ A+f U ˜ A =0.Ifhowever, f L ˜ A+f U ˜ A =0,i.e., f L ˜ A=0and f U ˜

A =0,theRFsetdegenerates toasingleton value,wL ˜ A=w U ˜ A =1/2.

Definition2.10. TheoverallRepofagivenRFsetA˜isdefinedby

Rep

(

A˜

)

=  V∈{L,U}

wV ˜ A  e∈{x,y} Rep

(

A˜V

)

e

(12) wherewV ˜

A istheweightassignedtoRep

(

A˜

V

)

ofA˜V,V{L,U}.

3. Rough-fuzzyruleinterpolation

3.1. SelectionofclosestNrules

AswithconventionalFRIapproaches,ingeneral,multipleruleswithmultipleantecedentsneedtobeconsideredinorder toobtainaninterpolatedconclusion.Forthis,thefirststepthatneedstobeconsideredistochoosetheclosestN(N≥ 2) rulesfromtherulebasewithrespecttothegivenobservation.Adistancemeasureisthusutilisedtomeasuretheproximity oftherulesbyexploitingsuchRepvaluesthatcapturespecificinformationembeddedinRFsets.

Withoutanylossofgenerality,supposethattherearenRFrulesinanRF rulebase.AruleRi,anobservationOandthe conclusionCarerepresentedbythefollowing,respectively:

Ri:Ifx1 isA˜i1,...,xj isA˜i j,...,xMisA˜iM,thenyisB˜i

O:x1 isA˜∗1,...,xjisA˜∗j,...,xM isA˜∗M

C:yisB˜∗

whereA˜i j denotestheRF setthatactsasthe valueofthejthantecedent ofRi,A˜∗j isthe observationofthevariablexj,B˜∗ is thedesiredinterpolated conclusion, andB˜i denotesthe consequent RF set ofRi with j

{

1,...,M

}

, withMbeing the numberofantecedentvariables.

Definition3.1. ThedistancedijbetweenapairofRFsetsA˜i j andA˜∗j isdefinedasfollows:

di j=d

(

A˜i j,A˜∗j

)

=d

(

Rep

(

A˜i j

)

,Rep

(

A˜∗j

))

(13) whered(.,.)ishereincomputedusingtheEuclideandistancemetric(thoughanyotherdistancemetricmaybeusedasan alternative).

Definition 3.2. The distance di betweenthe rule Ri andtheobservation O isdeemed to be theaverage of thedistances

betweentheRFsetsofeachruleantecedentandthecorrespondingvariableinO:

di=

M j=1 di j2, di j= di j maxj− minj (14)

where maxj andminj are the maximumandminimum value inthe domainof thevariable xj, j

{

1,...,M

}

. Theuse of normalised distance measure di j is to ensure that the resulting distances are compatible witheach other over different domains.

Giventheabovedefinition,thedistancesbetweenagivenobservationandallrulesintherulebasecanbecalculated.The

NruleswhichhaveminimaldistancesarechosenastheclosestNruleswithrespecttothegivenobservation.Thechoiceof alargerNwillhelpconsiderawiderrangeofneighbouringrulesinperforminginterpolation,therebymorelikelytoresult inglobalresultsbutrequiringsignificantlymorecomputation.Onthecontrary,thechoiceofasmallerNwilltendtotake into accountonly neighbouring rulesand henceinvolvelesscomputationtime. Since FRI isin generalusedto derive an approximateresultinthefirstplace,inpracticalapplication,Ncanbechosentobe2.Thisisthecaseforconventionalrule interpolationalso.However,inthefollowingtheoreticaldevelopmenttomaintaingenerality, thenumberofclosestrulesis settoN(N≥ 2)unlessotherwisestated.

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3.2.Constructionofintermediaterule

As witha number ofconventional FRI approaches, the approach in thiswork is developedfollowing the principle of analogicalreasoning[27].First,anartificiallycreatedintermediateruleisinterpolatedsuchthattheantecedentofthe inter-mediateruleisas‘close’tothegivenobservationaspossible.Then,aconclusionisworkedoutfromthegivenobservation byfiringthisgeneratedintermediaterulethroughacertainanalogicalreasoningmechanism.

Definition3.3. SupposethatNclosestrulesarechosenwithrespecttoagivenobservation.Theserulesarerepresentedas

Ri,i

{

1,...,N

}

,each havingMantecedent variablesA˜i j, j

{

1,...,M

}

,andare usedtoderive theintermediate rule.Let

wA˜i j denote the weight to which the jth antecedent of the ith closest rule contributes to the emerging intermediate rule,

whichisdefinedasthereciprocalofthecorrespondingdistancemeasure:

wA˜i j= 1 di j = 1 d

(

A˜i j,A˜∗j

)

(15) whereA˜∗jdenotestheobservedRFsetofantecedentvariablej.Thenormalisedweightw˜

Ai j isthendefinedby wA˜ i j= wA˜i j N i=1wA˜i j (16)

Remark3.1. Thisdefinitionreflectstheintuitionthatthelargerthedistanceis,thelessrelevantthecorrespondingattribute

istotheobservation.Ingeneral, dij =0.Ifhowever,di j=0,then Rep

(

A˜i j

)

=Rep

(

A˜∗j

)

.Inthiscase, theobservationis con-sideredtobeidenticaltothecorrespondingantecedentoftheruleRi,intermsoftheirRepvalues.Thus,wA˜i j issetto1for

theidenticalcaseswiththerestsetto0. TheantecedentA˜IFT

j oftheintermediateruleisconstructedfromtheantecedentsoftheidentifiedclosestrules.Aprocess

shiftisthenutilisedtomodifyA˜IFT

j sothattheantecedentoftheintermediaterulewillhavethesameRepasA˜∗j: ˜ Aj=A˜IFT j +

δ

A˜j

(

maxj− minj

)

, ˜ AIFT j = N  i=1 wA˜ i j ˜ Ai j (17)

where

δ

A˜j isaconstantdefinedby

δ

A˜j=

Rep

(

A˜∗j

)

− Rep

(

A˜IFT j

)

maxj− minj

(18) TheconsequenceoftheintermediateruleB˜iscalculatedbyanalogytothecomputationoftheantecedent,suchthat:

˜ B=B˜IFT+

δ

˜ B

(

max− min

)

, B˜IFT= N  i=1 wB˜ i ˜ Bi (19)

whereB˜IFT istheconsequenceoftheintermediatefuzzyrule,maxandminarethemaximumandminimumvalueswithin thedomainoftheconsequentvariable,w˜

Bi and

δ

B˜arethemeansofwAi j and

δ

A˜j,i

{

1,...,N

}

, j

{

1,...,M

}

,respectively, whicharedefinedby

wB˜ i= 1 M M  j=1 wA˜ i j,

δ

B˜= 1 M M  j=1

δ

A˜j (20)

3.3.Interpolationthroughsimilarity-constrainedtransformations

Theaforementionedartificiallyconstructedintermediateruleisderivedfromthechosenclosestruleswithrespecttoan observation.Itcan be usedto performinference without furtherreferenceto its originals.Suppose that a certain degree ofsimilaritybetweentheantecedent partofthisruleandtheobservationis established,itis intuitive torequirethat its consequentpartandtheeventualconclusionshouldattainthesamesimilaritydegree.Thatis,foranintermediaterule:“Ifx1

isA˜1,...,xjisA˜j,...,xMisA˜M,thenyisB˜”,andagivenobservationO=

(

A˜∗1,...,A˜∗j,...,A˜∗M

)

,theshapedistinguishability betweenB˜andtheinterpolatedconsequenceB˜∗ isanalogoustothecombinationoftheshapedistinguishabilitiesbetween

˜

AjandA˜∗j, j=1,2,...,M.Inordertoensurethis,thefollowingthreetransformationsaredesigned.

Notethatallthreetransformationsareseparatelyimplementedoneachdimensionandseparately calculatedoneachof thelowerandupperbounds.However,theunderlyingcomputationalmechanismsareidentical.Forpresentationalsimplicity, thedescriptionofthesetransformationsisgivenwithoutthesubscriptjandthesuperscriptLorU.

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3.3.1. Scaletransformation

Consider the lower (upper) approximation of A˜ and that of A˜∗, respectively represented as A˜=

(

a˜0,. . .,a˜k−1; ˜H˜ A1,. . ., ˜ H˜ Ak−2

)

and A˜ ∗=

(

a˜∗ 0,. . .,a˜∗k−1; ˜HA∗˜1,. . ., ˜ H˜

Ak−2

)

. The following parameters, termed the scale rates

sp(p=0,. . .,



(

k/2

)



− 1)rescalethepthsupportofA˜inordertoapproximatethecorrespondingsupportofA˜∗:

sp= ˜ ak−p−1− ˜ap ˜ ak−p−1− ˜ap (21) From these scale rates, thefollowing scale ratios Sq (q=1,...,



(

k/2

)



− 1) modify the rescaled qth support of A˜ to furtherapproximatethecorrespondingsupportofA˜∗suchthattheresultingRFsetA˜isofthesamescaleasthatofA˜∗:

Sq=

˜ ak−q−1− ˜aq ˜ ak−q− ˜aq−1− ˜ a k−q−1− ˜aq ˜ a k−q− ˜aq−1 1−a˜k−q−1− ˜aq ˜ a k−q− ˜aq−1 ifsq≥ sq−1 ˜ ak−q−1− ˜aq ˜ ak−q− ˜aq−1− ˜ ak−q−1− ˜aq ˜ a k−q− ˜aq−1 ˜ a k−q−1− ˜aq ˜ a k−q− ˜aq−1 ifsq−1>sq (22)

Fromthis, byimposingtherequiredsimilarities,thecorrespondingscale ratesspthat willhelprescalethepthsupport ofB˜intotheemergingB˜∗canbeobtainedsuchthat

sp=

sp ifp=0 sp−1(sp−sp−1) b˜ k−p−˜bp−1 ˜ b k−p−1−˜bp−1  sp−1 a˜ k−p− ˜ap−1 ˜ a k−p−1− ˜ap−1  +sp−1 ifsp≥ sp−1, p>0 sp−1sp sp−1 ifsp−1>sp, p>0 (23)

Theaboveshowsonlythesituationwhereoneantecedentvariableisconsidered(foreitheranLAoranUA).Ingeneral, foreach antecedent variablej andeachapproximation V, V{L,U},such a scaletransformation isrepeatedly appliedto transformA˜V

j totheintermediatetermsA˜

V

j withsVjp andSVjq.B˜V is thengenerated fromB˜

V usingthe aggregatedsV

˜ Bp and SV ˜ Bq,wheres V ˜ Bp= 1 M M j=1sjpV andS V ˜ Bq= 1 M M j=1SjqV. 3.3.2. Movetransformation

ThemoveratiosMr(r=0,...,



(

k/2

)



− 2)shiftthelocationsofthesupportsofA˜(r−1)tothatofA˜∗(whereA˜(r−1)isthe termobtainedafterthe

(

r− 1

)

thsub-movewiththeinitialisationA˜−1=A˜):

Mr=

˜ ar− ˜a( r−1) r min  ˜ a(rr−1)+···+a˜(r−1) (k/2)−1 (k/2)−r − ˜a( r−1) r ,a˜( r−1) k−r − ˜a( r−1) k−r−1  ifa˜∗r≥ ˜a(rr−1) ˜ ar− ˜a( r−1) r min  ˜ a(r−1) k−r−1− ˜ a(r−1) k−(k/2)+···+a˜(r−1) k−r−1 (k/2)−r ,a˜( r−1) r − ˜a(r−1r−1)  ifa˜(rr−1)>a˜∗r (24)

wherea˜(rr−1) isthenewpositionofa˜r after the

(

r− 1

)

thsub-move.Initially,when r=0,a˜0(−1)=a˜0, a˜(

r−1)

k−r − ˜a(

r−1)

k−r−1 and

˜

ar(r−1)− ˜a(r−1r−1)arenotconsideredwithinthedenominatorsmin{.,.}.

Ingeneral, foreach antecedentvariablejandeachapproximation V,V∈{L,U},thismovetransformationisrepeatedly applied toobtain A˜(jr)V=

{

a˜(j0r)V,...,a˜(jr()kV −1)

}

fromA˜(jr−1)V usingM V jr.B˜( r)V =

{

b˜(r)V 0 ,...,b˜( r)V

k−1

}

isthen obtainedfromB˜(r−1)V

usingtheaggregatedMV

˜ Br,whereM V ˜ Br= 1 M M j=1MVjr,resultinginA˜ ((k/2)−2)V j =A˜∗Vj andB˜((k/2)−2)V =B˜∗V. 3.3.3. Heighttransformation

Theheightratesho(o=1,...,k− 2)areutilisedtoadjusttheheightsH˜A˜L

oof ˜ ALtotheheightsH˜∗L ˜ Ao of ˜ A∗L: ho= ˜ H∗L˜ Ao ˜ HL ˜ Ao (25) where0<H˜∗L ˜ Ao≤ ˜H ∗U ˜ Ao =1and0< ˜ HL ˜ Ao≤ ˜H U ˜

Ao=1.Thisconstraintappliestotheinterpolatedconclusionaswell.Thatis,ifthe

heightofB˜∗LisgreaterthantheheightofB˜∗U aftertheheighttransformation,thenH˜∗L˜

Bo=

˜

H∗U˜

Bo.

Ingeneral,foreach antecedentvariablejandeachapproximationV,V∈{L,U},thisheight transformationisrepeatedly appliedtotransformtheheightsofA˜L

j tothoseofA˜∗Lj withhjo.Theheightoftheinterpolatedconclusionisthenobtained usingtheaggregatedhB˜o,wherehB˜o=

1

M M

(10)

Fig. 4. Causal diagram of a simplified application problem.

Remark3.2. ScaletransformationscalesA˜jupordowntoA˜j,retainingtheratiosbetweenleftandrightslopes,buthaving

differentsupportwidths.Thecloserthescaleratiosto0,themoresimilarA˜jandA˜j.Move transformationshiftsA˜j toA˜∗j

whichhasthesamesupportwidth,buthavingdifferentlocations.Thecloserthemoveratiosto0,themoresimilarA˜j and ˜

Aj.HeighttransformationadjuststheheightofA˜jtothatofA˜∗jwhiletheothercharacteristicsremainthesame.Thecloser theheightratesto1,themoresimilarA˜jandA˜∗j.

Scale,moveandheighttransformationsguaranteethat thetransferred setshavethesametypeofshapesasthatofthe original.Thatis,thesethree transformationsallow thesimilarity degreebetweenB˜ andB˜∗ tobe determined fromthose betweenA˜jandA˜∗j.

4. Applicationcasestudy

Inthis section, a practical problemconcerning the prediction ofdiarrhoeal disease in remote villages is employed in ordertodemonstratethepotentialoftheproposedwork.Itshowshowtheimplementedtechniquescanhelptorepresent theunderlyinghigherorderuncertaininformationandinterpolateafinalconclusion.

4.1. Problemoverview

Environmentalchangeinfluencesdiseaseburden[28,29].Intensivestudieshavebeenmadeinanefforttoidentifylogical relationshipswhichliebehindsuchinfluences,predictingtheconsequencesofaparticularenvironmentalchange.Thisisof significantimportance inthe assessmentofthepotential impactof suchchanges upon theenvironmentandsociety, e.g., priortoproposinganylarge-scaleinfrastructureprojects.

Oneparticularapplicationinthisarea hasrecentlybeeninvestigatedin[17,18],whichis basedonthestudyof[30].It addressestheissueofmeasuringhowtheconstructionofanewroadorrailwaynetworkinapreviouslyroadlessareamay affecttheepidemiologyofinfectiousdiseasesinnortherncoastalEcuador.Apredictivemodelhasbeenbuiltwheremanyof thefactorsarenotlinearlyrelated,butinteractwitheachotherinagridnetwork.Addressingthisapplicationproblem,an illustrativeexampleispresentedheretodemonstratetheapplicationoftheRF setsandRF ruleinterpolation.Theoriginal problemof[18]issimplifiedsuchthatallofthefactorsunderconsiderationarelinearlyconnected.Theresultingsimplified causalmodelisshowninFig.4.

Thiscausaldiagramshowsthatthediarrhoealdiseaserateofaremotevillageisdirectlyaffectedbytwofactors.First,low socialconnectednesstendstoleadtofailureincreatingadequatewaterandsanitationinfrastructurebecausetheresidents areunlikelytoknowoneanotherwellandsharesocialnorms[31,32],therebyusuallyresultinginahighdiarrhoealdisease rate.Second,morefrequentcontactbetweentheresidentswithin avillageandthoseoutsidetendstoincrease therateof introductionofpathogens,therebyalsoraisingthediarrhoealdiseaserate.

Allfactorsconsideredinthisexamplearerepresentedassystemvariablesandeach relationbetweentwodirectly con-nectedfactorsisrepresentedasaruleassociatingtherelevantvariables.Insummary,therearefivevariablesintheproblem: contactoutsideofthevillage,reintroductionofpathogenicstrains,social connectedness,hygieneandsanitation infrastruc-ture,andinfectious diseaserate,denoted asx1,. . .,x5,respectively. Aselectionofthe originalrules containedintherule

basearegiveninTable1,wheretworulesaresufficientforinterpolation.Inthismodel,fuzzinessisnaturallyobtainedfrom thepresenceofthelinguistictermsthatdescribethereal-valueddomainvariables.Notethatdifferentvariablesaredefined upon differentdomains. Tosimplify knowledge representation,variabledomains are mapped onto thereal lineand nor-malised.Theillustrativefuzzysets thatrepresentthenormalisedlinguistictermsexpressedbyacertainexpertareshown

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Table 1 Example rules.

Contact Reintroduction Connectedness Infrastructure Reintroduction Infrastructure Rate

( x 1 ) ( x 2 ) ( x 3 ) ( x 4 ) ( x 2 ) ( x 4 ) ( x 5 )

R1 L L L LH VL MH L

R2 H H MH H M L MH

Fig. 5. Definition of the linguistic terms for domain variables from a certain expert.

inFig.5,wherethetriangleswithdashedlinesindicatethetworulesgeneratedfromthisparticularexpert’sopinion.Itis importanttonotethattheoriginalrulebasecontainssubstantialgaps,whichmakesinterpolationessential.

Inordertoevaluatethefinaldiseaserate,agroupofexpertsareselectedtoexpresstheirindividualviewoneachfactor. However,thisprocedurereliesheavilyupon theopinionsofexperts,who musthaveacomprehensiveanddetailed under-standingof theunderlying problem. Such opinionsare oftenbiasedandsubjectiveand/or inconsistentbetweendifferent individuals. Suppose that the opinions fromsix experts, denoted as T1,...,T6, in the group are shown in Fig. 6, where

subsetsofrules(onesubset percausalimplication):AB,CDandBDEareestablishedbytheexpertswitheach supportedbytwoofthem.NotethatA=x1,B=x2,C=x3,D=x4andE=x5.Thisrevealstheunderlyinguncertaintyabout

theopinionsfromdifferentexpertsandreinforcestheneedforRF representation.In particular,opinionfromexpertT1 is

theonedisplayedinFig.5.Forbrevity,othersareomittedhere.

4.2. Experimentationanddiscussion 4.2.1. Motivationrevisited

Givendifferentexpertrulesandobservations,one waytoresolvethe problemmight beto useaconventional FRI ap-proach,sayT-FRItoimplementtherequiredinterpolationseparately.Supposethat twopairsofexpertrulesarecontained inasub-rulebase:A1→B1 andA2→B2,whereA11→B11andA21 →B21areprovidedbyexpertT1,whileA12→B12and A22 →B22areprovidedbyexpertT2.PresentedwithtwoobservationsA∗1andA∗2,theinterpolatedresultbytheuseofT-FRI

isasetwhichcontains4elements.Thecomputationwithrespecttotheremainderofthesubsetsofrulesfollowsthesame procedure,resultinginaconsequencesetof32interpolatedresults,aslistedinTable2.

Notethat thecardinality ofthesetofinterpolated consequentresultsincreasesrapidly alongwiththeincrease ofthe cardinalityofrulesubsetsandthenumberofobservations.Thisresultsinhighcomputationalcomplexity.Asoutlined previ-ously,thefirststepofinterpolationrequiresthecomputationoftheclosestrulesfromagivenrulebase.Adistancemeasure needs tobeemployed inordertoestimatetheproximity betweeneach ruleantecedentandtheobservation.Thisimplies a time complexity of O(xyz), where x is the number of observations to be interpolated, y is the number of antecedent variables,andzisthenumberoffuzzyrulesinvolvedintherelevantrulesubset.

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0.5

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

1

µ A

T

1

T

2

T

2

T

1

0.5

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

3

µ C

T

4

T

3

T

3

T

4

0.5

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

2

µ B

T

5

T

6

T

2

T

1

T

6

T

5

T

1

T

2

0.5

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

4

µ D

T

6

T

5

T

3

T

4

T

5

T

6

T

4

T

3

0.5

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

5

µ E

T

5

T

6

T

6

T

5

Fig. 6. Interpolated results from conventional FRI.

Table 2

Interpolated results with T-FRI.

Values Values Values Values

E∗ 1 (0.771,0.845,0.953) E2∗ (0.805,0.914,0.999) E∗3 (0.767,0.830,0.945) E4∗ (0.799,0.890,0.989) E∗ 5 (0.775,0.867,0.952) E6∗ (0.809,0.936,0.998) E∗7 (0.772,0.851,0.945) E8∗ (0.802,0.913,0.988) E9∗ (0.786,0.852,0.975) E10∗ (0.821,0.918,0.991) E∗11 (0.781,0.838,0.966) E12∗ (0.814,0.894,0.981) E∗ 13 (0.792,0.871,0.980) E14∗ (0.826,0.941,0.995) E∗15 (0.788,0.856,0.972) E16∗ (0.819,0.917,0.985) E∗ 17 (0.678,0.721,0.798) E18∗ (0.710,0.775,0.840) E∗19 (0.675,0.713,0.791) E20∗ (0.698,0.750,0.826) E∗ 21 (0.682,0.733,0.799) E22∗ (0.712,0.790,0.840) E∗23 (0.679,0.725,0.792) E24∗ (0.701,0.765,0.825) E∗ 25 (0.685,0.724,0.810) E26∗ (0.717,0.777,0.854) E∗27 (0.681,0.717,0.804) E28∗ (0.705,0.754,0.838) E∗ 29 (0.708,0.753,0.833) E30∗ (0.742,0.812,0.879) E∗31 (0.704,0.745,0.826) E32∗ (0.730,0.786,0.863)

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Table 3

Relevant RF sets for sub-rule base A → B .

A1 = (0 . 14 , 0 . 17 , 0 . 17 , 0 . 2 ;0 . 75 , 0 . 75) , (0 . 1 , 0 . 16 , 0 . 18 , 0 . 26 ;1 , 1) Attribute values A2 = (0 . 67 , 0 . 71 , 0 . 71 , 0 . 75 ;0 . 62 , 0 . 62) , (0 . 61 , 0 . 68 , 0 . 73 , 0 . 8 ;1 , 1) B1 = (0 . 22 , 0 . 24 , 0 . 24 , 0 . 27 ;0 . 56 , 0 . 56) , (0 . 16 , 0 . 21 , 0 . 25 , 0 . 34 ;1 , 1) B2 = (0 . 8 , 0 . 87 , 0 . 87 , 0 . 94 ;0 . 74 , 0 . 74) , (0 . 78 , 0 . 85 , 0 . 9 , 0 . 98 ;1 , 1) Observation A= (0 . 5 , 0 . 54 , 0 . 54 , 0 . 58 ;0 . 62 , 0 . 62) , (0 . 43 , 0 . 51 , 0 . 56 , 0 . 66 ;1 , 1) Table 4

Relevant RF sets for sub-rule base B ∧ D → E .

B3 = (0 . 05 , 0 . 08 , 0 . 08 , 0 . 11 ;0 . 67 , 0 . 67) , (0 . 02 , 0 . 07 , 0 . 10 , 0 . 17 ;1 , 1) Rule 1 D3 = (0 . 10 , 0 . 12 , 0 . 12 , 0 . 18 ;0 . 8 , 0 . 8) , (0 . 05 , 0 . 11 , 0 . 13 , 0 . 22 ;1 , 1) E1 = (0 . 13 , 0 . 16 , 0 . 16 , 0 . 18 ;0 . 71 , 0 . 71) , (0 . 1 , 0 . 15 , 0 . 17 , 0 . 25 ;1 , 1) B4 = (0 . 38 , 0 . 42 , 0 . 42 , 0 . 45 ;0 . 7 , 0 . 7) , (0 . 32 , 0 . 4 , 0 . 43 , 0 . 5 ;1 , 1) Rule 2 D4 = (0 . 38 , 0 . 51 , 0 . 51 , 0 . 56 ;0 . 86 , 0 . 86) , (0 . 37 , 0 . 50 , 0 . 53 , 0 . 59 ;1 , 1) E2 = (0 . 58 , 0 . 60 , 0 . 60 , 0 . 62 ;0 . 4 , 0 . 4) , (0 . 53 , 0 . 57 , 0 . 63 , 0 . 73 ;1 , 1) Observation B= (0 . 612 , 0 . 682 , 0 . 682 , 0 . 733 ;0 . 64 , 0 . 64) , (0 . 565 , 0 . 639 , 0 . 697 , 0 . 811 ;1 , 1) D= (0 . 582 , 0 . 679 , 0 . 679 , 0 . 708 ;0 . 39 , 0 . 39) , (0 . 531 , 0 . 564 , 0 . 722 , 0 . 808 ;1 , 1)

Suppose thatthere arem1 rulesinthe formofAB,m2 rules inCD,m3 rulesin BDE, n1 observationsfor

theantecedent ofthefirst rule,andn2 observationsforthesecond. Fromthis, thetime complexitiesfortherule subsets

depictingthe relationsAB,CD andBDE are O(m1n1),O(m2n2) andO(m1n1m2n2m3), respectively.Apartfrom

thecomputationalcomplexity,thisleadsto difficultyindeterminingorinterpretingthefinalresult. Forexample,consider twointerpolatedresultsE8=

(

0.802,0.913,0.988

)

andE25=

(

0.685,0.724,0.810

)

.Usingthemethodin[33],thesimilarity betweenthesetwo fuzzysets is0.002.In thiscase,it isdifficult tomake achoice giventhealmost completely different conclusions. Clearly, it is important to be able to obtain a single consensus opinion which summarises the information containedindisparateordivergingopinions.Fortunately,theproposed RFapproachcanbeappliedwithoutsufferingfrom theabovedifficulty.AllsuchuncertainrelationscanbecapturedusingRFsetsandtheconclusioncanbederivedbyRFrule interpolation.

4.2.2. ApplicationofRF-basedFRI

AnRFrulebaseandobservationcanbebuiltontopofthosesingle-expertrulebasesandobservationsusingEq.(4),with examplesshowninFig.7.Thatis,allfuzzyvaluesforasingleunderlyingvariableareaggregatedintoanRF set,wherethe uncertaintyisdescribedbythelowerandupperapproximations.Forthisexample,forsimplicity,eachvariableisassociated withtwofuzzyvalues.SucharulebaseandobservationincludesdifferentlevelsofuncertaintyandrepresentsthemasRF sets.Also,theyareconsideredintheprocessofinterpolationinordertoobtainconsensusinferenceconclusions.

TheresultantRFsetsforABarelistedinTable3.Theinterpolationprocessisdescribedasfollows:

1. ThelowerandupperReps,shapediversityfactorsandweightfactorsarecalculatedaccordingtoEqs.(9),(10)and(11), respectively.

2. The overall Reps, Rep

(

A1

)

=0.486, Rep

(

A

)

=0.842, Rep

(

A2

)

=1.002, are calculated from Eq. (12). A=

(

0.505,0.542,0.542,0.579; 0.66,0.66

)

, (0.452, 0.519, 0.559, 0.632; 1, 1)

and B=

(

0.620,0.674,0.674,0.732; 0.68, 0.68),(0.588,0.651,0.698,0.781;1,1)

arethencomputed.

3. The scale rates sL

B0=1.084,sUB0=1.273, sUB1=1.229,the move ratios MLB0=−0.179, MUB0=0.093 andthe height rate

hB1=0.939intheintegratedtransformation fromA andA∗ arecalculatedwithregard toEqs. (21),(24)and(25),

re-spectively.

4. The scale rates, move ratios and height rate are used to transform B to the interpolated conclusion B∗=

(

0.612,0.682,0.682,0.733; 0.64,0.64

)

,(0.565,0.639,0.697,0.811;1,1)

,assummarisedinFig.7.

Since theinterpolationforcalculatingD∗ fromCD issimilarto theabove process,it isomittedheretoavoid rep-etition.HavinggeneratedB∗andD∗,BDEisthen utilisedtoderive thefinalresult. TheRF setsinvolvedarelisted in Table4.

Note that both given rules inthe form ofBDE lie on one side ofthe observation,the problem ofinterpolation thereforebecomesthatofextrapolation.However,theextensionofRFruleinterpolationtoextrapolationisstraightforward. 1. Forthefirstantecedent,thedistancesbetweenB3,B4 andthepreviouslyinterpolatedB∗arecalculatedbyEq.(13).The

weights arecalculatedandnormalised usingEqs.(15)and(16),respectively, resultinginthenewweights of0.30and 0.70.The normalisedweights togetherwiththeparameter

δ

B=0.60,whichiscomputedby Eq.(18),are thenusedto generatetherequiredintermediatefuzzysetB=

(

0.634,0.671,0.671,0.701; 0.69,0.69),(0.584,0.654, 0.684,0.754;1, 1)

,accordingtoEq.(17).Dcanthenbecalculatedinthesameway.

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0.5

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

1

µ A

A

1

A

*

A

2

0.5

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

3

µ C

C

1

C

*

C

2

0.5

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

2

µ B

B

3

B

1

B

4

B

*

B

2

0.5

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

4

µ D

D

3

D

1

D

4

D

*

D

2

0.5

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

5

µ E

E

1

E

2

E

*

Fig. 7. Interpolated results from RF rule interpolation.

2.Tocomputetheconsequent,theaveragedweightsof0.26and0.74,and

δ

E=0.49arecalculatedusingEq.(20).Fromthis, the intermediateoutput E=

(

0.772,0.794,0.794,0.814; 0.48,0.48

)

,

(

0.727,0.770,0.819,0.914; 1, 1)

isobtainedusing Eq.(19).

3.The average of two support scale rates (1.81 and 0.79) of the LAs is computed according to Eqs. (21), (22) and (23), resulting in sL

E0=1.30 and forming the scale rate of the aggregated LA. The scale rates of the ag-gregated UA sU

E0=1.38 and sUE1=3.31 can then be generated following the same procedure. Similarly, the aggre-gated move ratios ML

E0=0.81 and M

U

E0=1.41 are calculated from two move ratios (0.24 and 1.37) of the LAs and two move ratios (0.60 and 2.22) of the UAs using Eq. (24). These, together with the aggregated height rate, namely, the average hE1=0.46 from Eq. (25), are employed to transform E, achieving the final result E∗=

(

0.773,0.779,0.779,0.829; 0.22,0.22

)

,

(

0.690,0.702,0.855,0.961; 1,1

)

. The interpolated result is again illustrated in Fig.7.

(15)

Table 5

Involved RF sets for Singleton-valued Interpolation. ˜ A1 = (3 , 3 , 3 ;1) , (3 , 3 , 3 ;1) Attribute values A˜ 2 = (12 , 13 , 13 . 5 ;0 . 6) , (11 , 13 , 14 ;1) ˜ B1 = (4 , 4 , 4 ;1) , (4 , 4 , 4 ;1) ˜ B2 = (10 . 5 , 11 . 5 , 12 ;0 . 5) , (10 , 11 . 5 , 13 ;1) Observation A˜ ∗= (6 , 7 , 8 ;0 . 6) , (5 , 7 , 9 ;1)

1

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 x

µ

A

~

1

A

~

2

A

~

*

1

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 y

µ

B

~

1

B

~

2

B

~

*

Fig. 8. A single antecedent case with singleton-valued conditions.

NotethatthisfigurereflectsthedistributionofthoseresultsshowninFig.6.Inparticular,theshapeoftheresultantRF setissimilartotheshapedistributionofthose32interpolatedsets,whereasthecomputationalcomplexityoftheformeris muchlowerthanthatofthelatter.Thiscanbeseenbycomparingthecalculatedtimecomplexitiesoftheformer,whichare

O(m1), O(m2)andO(m3), respectively. Itisobvious thatthe reductionin computationcomplexityissignificant,especially

whenthenumberofobservationsbecomeslargeforagivenapplication.Inaddition,sinceamajorityofthe32resultsare closerto the second rule(see Table 4and Fig.6), theresultant RF setis alsocloserto the consequent ofthisrule. This demonstratesthatthepresentapproachcorrespondstotheintuitionforitsdevelopment.

5. Furtherevaluation

5.1. Specificcases

TheRFruleinterpolationandextrapolationprocesseshavebeenillustratedintheprevioussection.However,the partic-ularapplicationdoesnotofferascenariototestspecificcaseswhererulesinvolvesingleton‘fuzzy’valuesorwhereRFsets degeneratetoconventionalfuzzysets.Thesecasesareempiricallycheckedbelow.

5.1.1. Singleton-valuedinterpolation

Thiscaseconsiders onesingleantecedent variableinvolvingsingleton-valuedconditions.TherelevantRFsets arelisted inTable5.

TheinterpolatedconclusionB˜∗=

(

5.98,7.04,7.98; 0.57

)

,

(

5.27,6.66,9.27; 1)

iscalculatedfollowingtheprocedures de-scribedinSections2and3,asshowninFig.8.Itfollowsthatifcertaincomponentsinvolvedinthegivenrulesare singleton-valued,theinterpolatedconclusionremainsanRFsetgivenanRFobservation.Thishasaclearintuitiveappeal.

5.1.2. Degeneratedinterpolation

TheconceptofRFsetsextendsthatofconventionalfuzzysets,whiletheRFruleinterpolationextendsfromtheexisting T-FRI. When there is no higher order uncertainty involved,i.e., the LA coincides withthe LU, an RF set degenerates to a conventional fuzzy set. If all of the sets under consideration in the implementation of interpolation/extrapolation are conventionalfuzzysets,thentheresultsobtainedbytheproposedapproachareidenticaltothoseofT-FRI.Theexamplein Fig.9illustratesthis.

Consider a casewhereall ofthe RF sets concerned degenerateto conventional fuzzysets, i.e.,A˜∗L=A˜∗U,A˜L k=A˜

U k and ˜

BL

(16)

1

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 x

µ

A

~

1

A

~

*

A

~

2

1

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 y

µ

B

~

1

B

~

*

B

~

2

Fig. 9. Interpolation when RF sets degenerate to conventional fuzzy sets.

Table 6

Involved RF sets for Degenerated Interpolation. ˜ A1 = (0 , 4 , 5 , 6 ;1 , 1) , (0 , 4 , 5 , 6 ;1 , 1) Attribute values A˜ 2 = (11 , 12 , 13 , 14 ;1 , 1) , (11 , 12 , 13 , 14 ;1 , 1) ˜ B1 = (0 , 2 , 3 , 4 ;1 , 1) , (0 , 2 , 3 , 4 ;1 , 1) ˜ B2 = (10 , 11 , 12 , 13 ;1 , 1) , (10 , 11 , 12 , 13 ;1 , 1) Observation A˜ ∗= (6 , 6 , 9 , 10 ;1 , 1) , (6 , 6 , 9 , 10 ;1 , 1) Table 7

RMSE% for the benchmark datasets.

Yacht Servo

Partitions Expert 1 Expert 2 Expert 3 RF Expert 1 Expert 2 Expert 3 RF 2 30.90( ∗) 30.46( ∗) 30.02( ∗) 25.80 26.20( ∗) 25.60( ∗) 25.00( ∗) 23.83 3 19.50( ∗) 19.93( ∗) 19.64( ∗) 19.10 15.13(-) 15.45( ∗) 15.74( ∗) 15.29 4 18.72( ∗) 19.82( ∗) 21.20( ∗) 17.31 12.59( ∗) 12.78( ∗) 13.22( ∗) 12.19 5 16.53(-) 18.21( ∗) 18.52( ∗) 16.59 11.60( ∗) 11.78( ∗) 12.29( ∗) 11.36 6 14.72(v) 16.10(-) 17.87( ∗) 16.10 11.50( ∗) 11.50( ∗) 12.07( ∗) 11.24 7 15.23(v) 16.35( ∗) 16.01( ∗) 15.52 11.29( ∗) 11.30( ∗) 11.59( ∗) 11.01 Concrete Housing 2 18.10( ∗) 18.06(-) 18.03(-) 18.05 17.26(-) 17.22(-) 17.13(v) 17.21 3 16.82(-) 16.94( ∗) 17.01( ∗) 16.85 16.65( ∗) 16.10( ∗) 15.54(-) 15.75 4 15.09( ∗) 14.84(-) 14.62(v) 14.85 15.15( ∗) 14.70(-) 14.44(v) 14.73 5 13.54(v) 14.65(-) 14.62(-) 14.63 14.09( ∗) 13.89(-) 13.61(v) 13.86 6 13.47( ∗) 13.34(-) 14.66( ∗) 13.36 13.51(-) 13.57(-) 13.66(-) 13.56 7 13.14(v) 13.49( ∗) 14.42( ∗) 13.28 12.98(v) 13.19(-) 13.42( ∗) 13.16

Usingtheproposedapproach,theinterpolatedconclusionB˜∗=

(

5.23,5.23,7.61,8.32;1,1),(5.23,5.23,7.61,8.32;1,1)

isobtained,asshowninFig.9.Thedetailsofthecalculationareomittedheretoavoidrepetition.Itfollowsthatifallgiven setsareconventionalfuzzysets,theinterpolatedresultisindeedthesameasthatachievedusingtheclassicalT-FRI.

5.2.Additionalapplicationcasestudies

Furthertotheexamplespreviously shown,theproposed approachhasalsobeenevaluatedagainst furtherapplications fordecisionsupport, byadapting UCI-MLR benchmark datasets[34],including YachtHydrodynamics, Servo,Concrete Com-pressiveStrength,andHousing.

Table 7 shows the results of the averaged root-mean-square error (RMSE) values computed over 10 times 10-fold cross-validation [35,36],in relation to K (K=2,...,7) partitions (i.e.,the underlyingdomain ofeach variable is divided into6fuzzyvalues)andN (N=6) closestrules that areusedto interpolatetheconclusions.The resultsare comparedto thoseachievedby individualT-FRIinterpolations thateachimplementtheinterpolationwiththeopinionprovidedbyone

Figure

Updating...

References