NeutrosophicAssociationRuleMiningAlgorithmfor BigDataAnalysis
MohamedAbdel-Basset 1,* ID ,MaiMohamed 1,FlorentinSmarandache 2,* ID andVictorChang 3
1 DepartmentofOperationsResearch,FacultyofComputersandInformatics,ZagazigUniversity, Sharqiyah44519,Egypt;analyst_mohamed@yahoo.com
2 Math&ScienceDepartment,UniversityofNewMexico,Gallup,NM87301,USA
3 InternationalBusinessSchoolSuzhou,Xi’anJiaotong-LiverpoolUniversity,Wuzhong, Suzhou215123,China;ic.victor.chang@gmail.com
* Correspondence:analyst_mohamed@zu.edu.eg(M.A.-B.);smarand@unm.edu(F.S.)
Received:5March2018;Accepted:9April2018;Published:11April2018
Abstract: BigDataisalarge-sizedandcomplexdataset,whichcannotbemanagedusingtraditional dataprocessingtools.Miningprocessofbigdataistheabilitytoextractvaluableinformationfrom theselargedatasets.Associationruleminingisatypeofdataminingprocess,whichisindentedto determineinterestingassociationsbetweenitemsandtoestablishasetofassociationruleswhose supportisgreaterthanaspecificthreshold.Theclassicalassociationrulescanonlybeextractedfrom binarydatawhereanitemexistsinatransaction,butitfailstodealeffectivelywithquantitative attributes,throughdecreasingthequalityofgeneratedassociationrulesduetosharpboundary problems.Inordertoovercomethedrawbacksofclassicalassociationrulemining,weproposein thisresearchanewneutrosophicassociationrulealgorithm.Thealgorithmusesanewapproach forgeneratingassociationrulesbydealingwithmembership,indeterminacy,andnon-membership functionsofitems,conductingtoanefficientdecision-makingsystembyconsideringallvague associationrules.Toprovethevalidityofthemethod,wecomparethefuzzyminingandthe neutrosophicmining.Theresultsshowthattheproposedapproachincreasesthenumberofgenerated associationrules.
Keywords: neutrosophicassociationrule;datamining;neutrosophicsets;bigdata
1.Introduction
Theterm‘BigData’originatedfromthemassiveamountofdataproducedeveryday.Eachday, Googlereceivescca.1billionqueries,Facebookregistersmorethan800millionupdates,andYouTube countsupto4billionviews,andtheproduceddatagrowswith40%everyyear.Othersourcesof dataaremobiledevicesandbigcompanies.Theproduceddatamaybestructured,semi-structured, orunstructured.Mostofthebigdatatypesareunstructured;only20%ofdataconsistsinstructured data.Therearefourdimensionsofbigdata:
Consequently,Gartner[1]outlinesthatbigdata’slargevolumerequirescost-effective,innovative formsforprocessinginformation,toenhanceinsightsanddecision-makingprocesses.
Prominentdomainsamongapplicationsofbigdataare[2,3]:
Symmetry 2018, 10,106;doi:10.3390/sym10040106 www.mdpi.com/journal/symmetry
(1) Businessdomain.
(2) Technologydomain.
(3) Healthdomain.
(4) Smartcitiesdesigning.
Thesevariousapplicationshelppeopletoobtainbetterservices,experiences,orbehealthier, bydetectingillnesssymptomsmuchearlierthanbefore[2].Somesignificantchallengesofmanaging andanalyzingbigdataare[4,5]:
(1) AnalyticsArchitecture:Theoptimalarchitecturefordealingwithhistoricandreal-timedataat thesametimeisnotobviousyet.
(2) Statisticalsignificance:Fulfillstatisticalresults,whichshouldnotberandom.
(3) Distributedmining:Variousdataminingmethodsarenotfiddlingtoparalyze.
(4) Timeevolvingdata:Datashouldbeimprovedovertimeaccordingtothefieldofinterest.
(5) Compression:Todealwithbigdata,theamountofspacethatisneededtostoreishighlyrelevant.
(6) Visualization:Themainmissionofbigdataanalysisisthevisualizationofresults.
(7) Hiddenbigdata:Largeamountsofbeneficialdataarelostsincemoderndataisunstructureddata.
Duetotheincreasingvolumeofdataatamatchlessrateandofvariousforms,weneedtomanage andanalyzeuncertaintyofvarioustypesofdata.Bigdataanalyticsisasignificantfunctionofbigdata, whichdiscoversunobservedpatternsandrelationshipsamongvariousitemsandpeopleinterestona specificitemfromthehugedataset.Variousmethodsareappliedtoobtainvalid,unknown,anduseful modelsfromlargedata.Associationruleminingstandsamongbigdataanalyticsfunctionalities. Theconceptofassociationrule(AR)miningalreadyreturnstoH’ajeketal.[6].Eachassociation ruleindatabaseiscomposedfromtwodifferentsetsofitems,whicharecalledantecedentand consequent.Asimpleexampleofassociationruleminingis“iftheclientbuysafruit,he/sheis 80%likelytopurchasemilkalso”.Thepreviousassociationrulecanhelpinmakingamarketing strategyofagrocerystore.Then,wecansaythatassociationrule-miningfindsallofthefrequent itemsindatabasewiththeleastcomplexities.Fromalloftheavailablerules,inordertodeterminethe rulesofinterest,asetofconstraintsmustbedetermined.Theseconstraintsaresupport,confidence, lift,andconviction.Supportindicatesthenumberofoccurrencesofaniteminalltransactions, whiletheconfidenceconstraintindicatesthetruthoftheexistingruleintransactions.Thefactor “lift”explainsthedependencyrelationshipbetweentheantecedentandconsequent.Ontheother hand,theconvictionofaruleindicatesthefrequencyratioofanoccurringantecedentwithouta consequentoccurrence.Associationrulesminingcouldbelimitedtotheproblemoffindinglarge itemsets,wherealargeitemsetisacollectionofitemsexistinginadatabasetransactionsequaltoor greaterthanthesupportthreshold[7–20].In[8],theauthorprovidesasurveyoftheitemsetmethods fordiscoveringassociationrules.Theassociationrulesarepositiveandnegativerules.Thepositive associationrulestaketheform X → Y , X ⊆ I, Y ⊆ I and X ∩ Y = ϕ,where X, Y areantecedent andconsequentand I isasetofitemsindatabase.Eachpositiveassociationrulemayleadtothree negativeassociationrules, → Y , X → Y ,and X → Y .Generatingassociationrulesin[9]consists oftwoproblems.Thefirstproblemistofindfrequentitemsetswhosesupportsatisfiesapredefined minimumvalue.Then,theconcernistoderivealloftherulesexceedingaminimumconfidence, basedoneachfrequentitemset.Sincethesolutionofthesecondproblemisstraightforward,mostofthe proposedworkgoesinforsolvingthefirstproblem.Anapriorialgorithmhasbeenproposedin[19], whichwasthebasisformanyoftheforthcomingalgorithms.Atwo-passalgorithmispresented in[11].Itconsumesonlytwodatabasescanpasses,whileaprioriisamulti-passalgorithmand needsuptoc+1databasescans,wherecisthenumberofitems(attributes).Associationrulesmining isapplicableinnumerousdatabasecommunities.Ithaslargeapplicationsintheretailindustryto improvemarketbasketanalysis[7].Streaming-Rulesisanalgorithmdevelopedby[9]toreportan associationbetweenpairsofelementsinstreamsforpredictivecachinganddetectingthepreviously
undetectablehitinflationattacksinadvertisingnetworks.Runningminingalgorithmsonnumerical attributesmayresultinalargesetofcandidates.Eachcandidatehassmallsupportandmanyrules havebeengeneratedwithuselessinformation,e.g.,theageattribute,salaryattribute,andstudents’ grades.Manypartitioningalgorithmshavebeendevelopedtosolvethenumericalattributesproblem. Theproposedalgorithmsfacedtwoproblems.Thefirstproblemwasthepartitioningofattribute domainintomeaningfulpartitions.Thesecondproblemwasthelossofmanyusefulrulesdueto thesharpboundaryproblem.Consequently,somerulesmayfailtoachievetheminimumsupport thresholdbecauseoftheseparatingofitsdomainintotwopartitions.
Fuzzysetshavebeenintroducedtosolvethesetwoproblems.Usingfuzzysetsmakethe resultedassociationrulesmoremeaningful.Manyminingalgorithmshavebeenintroducedtosolve thequantitativeattributesproblemusingfuzzysetsproposedalgorithmsin[13–27]thatcanbe separatedintotwotypesrelatedtothekindofminimumsupportthreshold,fuzzyminingbasedon single-minimumsupportthreshold,andfuzzyminingbasedonmulti-minimumsupportthreshold[21]. Neutrosophictheorywasintroducedin[28]togeneralizefuzzytheory.In[29–32],theneutrosophic theoryhasbeenproposedtosolveseveralapplicationsandithasbeenusedtogenerateasolution basedonneutrosophicsets.Single-valuedneutrosophicsetwasintroducedin[33]totransferthe neutrosophictheoryfromthephilosophicfieldintothemathematicaltheory,andtobecomeapplicable inengineeringapplications.In[33],adifferentiationhasbeenproposedbetweenintuitionisticfuzzy setsandneutrosophicsetsbasedontheindependenceofmembershipfunctions(truth-membership function,falsity-membershipfunction,andindeterminacy-membershipfunction).Inneutrosophic sets,indeterminacyisexplicitlyindependent,andtruth-membershipfunctionandfalsity-membership functionareindependentaswell.Inthispaper,weintroduceanapproachthatisbasedonneutrosophic setsforminingassociationrules,insteadoffuzzysets.Also,acomparisonresultedassociation rulesinbothofthescenarioshasbeenpresented.In[34],anattempttoexpresshowneutrosophic setstheorycouldbeusedindatamininghasbeenproposed.TheydefineSVNSF(single-valued neutrosophicscorefunction)toaggregateattributevalues.In[35],analgorithmhasbeenintroduced tominingvagueassociationrules.Itemspropertieshavebeenaddedtoenhancethequalityofmining associationrules.Inaddition,almostsolditems(itemshasbeenselectedbythecustomer,butnot checkedout)wereaddedtoenhancethegeneratedassociationrules.AH-pairDatabaseconsisting ofatraditionaldatabaseandthehesitationinformationofitemswasgenerated.Thehesitation informationwascollected,dependingonlineshoppingstores,whichmakeiteasiertocollectthattype ofinformation,whichdoesnotexistintraditionalstores.Inthispaper,wearethefirsttoconvert numericalattributes(items)intoneutrosophicsets.Whilevagueassociationrulesaddnewitemsfrom thehesitatinginformation,ourframeworkaddsnewitemsbyconvertingthenumericalattributesinto linguisticterms.Therefore,thevagueassociationruleminingcanberunontheconverteddatabase, whichcontainsnewlinguisticterms.
ResearchContribution
Detectinghiddenandaffinitypatternsfromvarious,complex,andbigdatarepresentsasignificant roleinvariousdomainareas,suchasmarketing,business,medicalanalysis,etc.Thesepatternsare beneficialforstrategicdecision-making.Associationrulesminingplaysanimportantroleaswell indetectingtherelationshipsbetweenpatternsfordeterminingfrequentitemsets,sinceclassical associationrulescannotusealltypesofdatafortheminingprocess.Binarydatacanonlybeusedto formclassicalrules,whereitemseitherexistindatabaseornot.However,whenclassicalassociation rulesdealwithquantitativedatabase,nodiscoveredruleswillappear,andthisisthereasonfor innovatingquantitativeassociationrules.Thequantitativemethodalsoleadstothesharpboundary problem,wheretheitemisbeloworabovetheestimationvalues.Thefuzzyassociationrulesare introducedtoovercometheclassicalassociationrulesdrawbacks.Theiteminfuzzyassociationrules hasamembershipfunctionandafuzzyset.Thefuzzyassociationrulescandealwithvaguerules, butnotinthebestmanner,sinceitcannotconsidertheindeterminacyofrules.Inordertoovercome
drawbacksofpreviousassociationrules,anewneutrosophicassociationrulealgorithmhasbeen introducedinthisresearch.Ourproposedalgorithmdealseffectivelyandefficientlywithvague rulesbyconsideringnotonlythemembershipfunctionofitems,butalsotheindeterminacyandthe falsityfunctions.Therefore,theproposedalgorithmdiscoversallofthepossibleassociationrulesand minimizesthelosingprocessesofrules,whichleadstobuildingefficientandreliabledecision-making system.Bycomparingourproposedalgorithmwithfuzzyapproaches,wenotethatthenumber ofassociationrulesisincreased,andnegativerulesarealsodiscovered.Theseparationofnegative associationrulesfrompositiveonesisnotasimpleprocess,andithelpsinvariousfields.Asan example,inthemedicaldomain,bothpositiveandnegativeassociationruleshelpnotonlyinthe diagnosisofdisease,butalsoindetectingpreventionmanners.
Therestofthisresearchisorganizedasfollows.Thebasicconceptsanddefinitionsofassociation rulesminingarepresentedinSection 2.Aquickoverviewoffuzzyassociationrulesisdescribed inSection 3.TheneutrosophicassociationrulesandtheproposedmodelarepresentedinSection 4 AcasestudyofTelecomEgyptCompanyispresentedinSection 5.Theexperimentalresultsand comparisonsbetweenfuzzyandproposedassociationrulesarediscussedinSection 6.Theconclusions aredrawninSection 7
2.AssociationRulesMining
Inthissection,weformulatethe|D|transactionsfromtheminingassociationrulesforadatabase D. Weusedthefollowingnotations:
(i) I = {i1, i2,... im } representsallthepossibledatasets,calleditems.
(ii) Transactionset T isthesetofdomaindataresultingfromtransactionalprocessingsuchas T ⊆ I (iii) Foragivenitemset X ⊆ I andagiventransaction T,wesaythat T contains X ifandonlyif X ⊆ T. (iv) σX :thesupportfrequencyof X,whichisdefinedasthenumberoftransactionsoutof D that contain X. (v) s:thesupportthreshold. X isconsideredalargeitemset,if σX ≥|D|× s.Further,anassociationruleisanimplicationof theform X ⇒ Y ,where X ⊆ I, Y ⊆ I and X ∩ Y = ϕ Anassociationrule X ⇒ Y isaddressedin D withconfidence c ifatleast c transactionsoutof D containboth X and Y.Therule X ⇒ Y isconsideredasalargeitemsethavingaminimumsupport s if: σX∪Y ≥|D|× s
Foraspecificconfidenceandspecificsupportthresholds,theproblemofminingassociation rulesistofindoutalloftheassociationruleshavingconfidenceandsupportthatislargerthanthe correspondingthresholds.Thisproblemcansimplybeexpressedasfindingallofthelargeitemsets, wherealargeitemset L is: L = {X|X ⊆ I ∧ σX ≥|D|× s}
3.FuzzyAssociationRules
Miningofassociationrulesisconsideredasthemaintaskindatamining.Anassociationrule expressesaninterestingrelationshipbetweendifferentattributes.Fuzzyassociationrulescandealwith bothquantitativeandcategoricaldataandaredescribedinlinguisticterms,whichareunderstandable terms[26].
Let T = {t1,..., tn } beadatabasetransactions.Eachtransactionconsistsofanumberofattributes (items).Let I = {i1,..., im } beasetofcategoricalorquantitativeattributes.Foreachattribute ik, (k = 1,..., m),weconsider {n1,..., nk } associatedfuzzysets.Typically,adomainexpertdetermines themembershipfunctionforeachattribute.
Thetuple < X, A > iscalledthefuzzyitemset,where X ⊆ I (setofattributes)and A isasetof fuzzysetsthatisassociatedwithattributesfrom X
Followingisanexampleoffuzzyassociationrule:
IFsalaryishighandageisoldTHENinsuranceishigh
Beforetheminingprocessstarts,weneedtodealwithnumericalattributesandpreparethemfor theminingprocess.Themainideaistodeterminethelinguistictermsforthenumericalattributeand definetherangeforeverylinguisticterm.Forexample,thetemperatureattributeisdeterminedbythe linguisticterms{verycold,cold,cool,warm,hot}.Figure 1 illustratesthemembershipfunctionofthe temperatureattribute.
Figure1. Linguistictermsofthetemperatureattribute.
Themembershipfunctionhasbeencalculatedforthefollowingdatabasetransactionsillustrated inTable 1
Table1. MembershipfunctionforDatabaseTransactions.
TransactionTemp.MembershipDegree T1181cool
T2130.6cool,0.4cold
T3120.4cool,0.6cold T4330.6warm,0.4hot
T5210.2warm,0.8cool T6251warm
Weaddthelinguisticterms{verycold,cold,cool,warm,hot}tothecandidatesetandcalculate thesupportforthoseitemsets.Afterdeterminingthelinguistictermsforeachnumericalattribute, thefuzzycandidatesethavebeengenerated.
Table 2 containsthesupportforeachitemsetindividualone-itemsets.Thecountforevery linguistictermhasbeencalculatedbysummingitsmembershipdegreeoverthetransactions.Table 3 showsthesupportfortwo-itemsets.Thecountforthefuzzysetsisthesummationofdegreesthat resultedfromthemembershipfunctionofthatitemset.Thecountfortwo-itemsethasbeencalculated bysummingtheminimummembershipdegreeofthe2items.Forexample,{cold,cool}hascount0.8, whichresultedfromtransactionsT2andT3.FortransactionT2,membershipdegreeofcoolis0.6and membershipdegreeforcoldis0.4,sothecountforset{cold,cool}inT2is0.4.Also,T3hasthesame countfor{cold,cool}.So,thecountofset{cold,cool}overalltransactionsis0.8.
Table2. 1-itemsetsupport.
1-itemsetCountSupport Verycold00 Cold10.17 Cool2.80.47 Warm1.60.27 Hot0.60.1
Table3. 2-itemsetsupport.
2-itemsetCountSupport {Cold,cool}0.80.13 {Warm,hot}0.40.07 {warm,cool}0.20.03
Insubsequentdiscussions,wedenoteanitemsetthatcontains k itemsas k-itemset.Thesetofall k-itemsetsin L isreferredas Lk
4.NeutrosophicAssociationRules
Inthissection,weoverviewsomebasicconceptsoftheNSsandSVNSsovertheuniversalset X, andtheproposedmodelofdiscoveringneutrosophicassociationrules.
4.1.NeutrosophicSetDefinitionsandOperations
Definition1([33]). Let X beaspaceofpointsandx∈X.Aneutrosophicset(NS) A in X isdefinitebya truth-membershipfunction TA (x),anindeterminacy-membershipfunction IA (x) andafalsity-membership function FA (x) TA (x), IA (x) and FA (x) arerealstandardorrealnonstandardsubsetsof] 0,1+[.Thatis TA(x):X → ] 0,1+[, IA (x):X → ] 0,1+[and FA (x):X → ] 0,1+[.Thereisnorestrictiononthesumof TA (x),IA (x) andFA (x),so0 ≤ supTA (x) +supIA (x) +supFA (x) ≤ 3+
Neutrosophicisbuiltonaphilosophicalconcept,whichmakesitdifficulttoprocessduring engineeringapplicationsortouseittorealapplications.Toovercomethat,Wangetal.[31],definedthe SVNS,whichisaparticularcaseofNS.
Definition2. Let X beauniverseofdiscourse.Asinglevaluedneutrosophicset(SVNS) A over X isanobject takingtheform A ={ x,TA(x), IA (x), FA (x) :x∈X},where TA (x):X → [0,1], IA (x):X → [0,1]and FA (x): X → [0,1]with0 ≤ TA (x) + IA (x) + FA (x) ≤ 3forallx∈X. Theintervals TA(x), IA(x) and FA(x) represent thetruth-membershipdegree,theindeterminacy-membershipdegreeandthefalsitymembershipdegreeof x to A, respectively.Forconvenience,aSVNnumberisrepresentedby A =(a,b,c),wherea,b,c∈[0,1]anda+b+c ≤ 3.
Definition3(Intersection)([31]). FortwoSVNSs A = TA(x), IA(x), FA(x) and B = TB(x), IB(x), FB(x) , theintersectionoftheseSVNSsisagainanSVNSswhichisdefinedas C = A ∩ B whosetruth,indeterminacyand falsitymembershipfunctionsaredefinedas TC(x)= min(TA(x), TB(x)), IA(x)= min(IA(x), IB(x)) and FC(x)= max(FA(x), FB(x))
Definition4(Union)([31]). FortwoSVNSs A = TA (x), IA (x), FA (x) and B = TB (x), IB (x), FB (x) , theunionoftheseSVNSsisagainanSVNSswhichisdefinedas C = A ∪ B whosetruth,indeterminacyand falsitymembershipfunctionsaredefinedas TC (x)= max(TA (x), TB (x)), IA (x)= max(IA (x), IB (x)) and FC (x)= min(FA (x), FB (x)).
Definition5(Containment)([31]). Asinglevaluedneutrosophicset A containedintheotherSVNS B, denotedby A ⊆ B ifandonlyif TA (x) ≤ TB (x), IA (x) ≤ IB (x) and FA (x) ≥ FB (x) forall x in X.
Next,weproposeamethodforgeneratingtheassociationruleundertheSVNSenvironment.
4.2.ProposedModelforAssociationRule
Inthispaper,weintroduceamodeltogenerateassociationrulesofform: X → Y where X ∩ Y = ϕ and X, Y areneutrosophicsets.
Ouraimistofindthefrequentitemsetsandtheircorrespondingsupport.Generatingan associationrulefromitsfrequentitemsets,whicharedependentontheconfidencethreshold,arealso discussedhere.Thishasbeendonebyaddingtheneutrosophicsetinto I,where I isallofthe possibledatasets,whicharereferredasitems.So I = N ∪ M where N isneutrosophicsetand M is classicalsetofitems.Thegeneralformofanassociationruleisanimplicationoftheform X → Y , where X ⊆ I, Y ⊆ I, X ∩ Y = ϕ
Therefore,anassociationrule X → Y isaddressedinDatabase D withconfidence‘c’ifatleast c transactionsoutof D containsboth X and Y.Ontheotherhand,therule X → Y isconsideredalarge itemsethavingaminimumsupport s if σX∪Y ≥|D|× s.Furthermore,theprocessofconvertingthe quantitativevaluesintotheneutrosophicsetsisproposed,asshowninFigure 2
Figure2. Theproposedmodel.
Theproposedmodelfortheconstructionoftheneutrosophicnumbersissummarizedinthe followingsteps:
Step1 Setlinguistictermsofthevariable,whichwillbeusedforquantitativeattribute.
Step2 Definethetruth,indeterminacy,andthefalsitymembershipfunctionsforeachconstructed linguisticterm.
Step3 Foreachtransaction t in T,computethetruth-membership,indeterminacy-membershipand falsity-membershipdegrees.
Step4 ExtendeachlinguistictermlinsetoflinguistictermsLintoTL,IL,andFL todenotetruthmembership,indeterminacy-membership,andfalsity-membershipfunctions,respectively.
Step5 Foreach k-itemsetwhere k = {1,2,..., n},and n numberofiterations.
• calculatecountofeachlinguistictermbysummingdegreesofmembershipforeach transactionas Count(A)= i=t ∑ i=1 µ A (x) where µ A is TA, IA or FA
• calculatesupportforeachlinguisticterm s = Count(A) No oftrnsactions
Step6 Theaboveprocedurehasbeenrepeatedforeveryquantitativeattributeinthedatabase. Inordertoshowtheworkingprocedureoftheapproach,weconsiderthetemperatureasan attributeandtheterms“verycold”,“cold”,“cool”,“warm”,and“hot”astheirlinguisticterms torepresentthetemperatureofanobject.Then,followingthestepsoftheproposedapproach, constructtheirmembershipfunctionasbelow:
Step1 Theattributetemperature’hassetthelinguisticterms“verycold”,“cold”,“cool”,“warm”, and“hot”,andtheirrangesaredefinedinTable 4
Table4. Linguistictermsranges.
LinguisticTermCoreRangeLeftBoundaryRangeRightBoundaryRange VeryCold ∞–0N/A0–5 Cold5–100–510–15 Cool15–2010–1520–25 Warm25–3020–2530–35 Hot35–∞ 30–35N/A Step2 Basedontheselinguistictermranges,thetruth-membershipfunctionsofeachlinguistic variablearedefined,asfollows: Tvery cold (x)=
1; forx ≤ 0 (5 x)/5; for 0 < x < 5 0; forx ≥ 5 Tcold (x)=
1; for 5 ≤ x ≤ 10 (15 x)/5; for 10 < x < 15 x/5; for 0 < x < 5 0; forx ≥ 15 orx ≤ 0 Tcool (x)=
1; for 15 ≤ x ≤ 20 (25 x)/5; for 20 < x < 25 (x 10)/5; for 10 < x < 15 0; otherwise Twarm (x)=
1; for 25 ≤ x ≤ 30 (35 x)/5; for 30 < x < 35 (x 20)/5; for 20 < x < 25 0; otherwise
Thot (x)=
1; forx ≥ 35 (x 30)/5; for 30 < x < 35 0; otherwise
Thefalsity-membershipfunctionsofeachlinguisticvariablearedefinedasfollows:
Fvery cold (x)=
0; forx ≤ 0 x/5; for 0 < x < 5 1; forx ≥ 5 ;
Fwarm (x)=
0; for 25 ≤ x ≤ 30 (x 30)/5; for 30 < x < 35 (25 x)/5; for 20 < x < 25 1; otherwise
Fhot (x)=
0; forx ≥ 35 (35 x)/5; for 30 < x < 35 1; otherwise
Ivery cold (x)=
0; forx ≤−2.5 (x + 2.5)/5; for 2.5 ≤ x ≤ 2.5 (7.5 x)/5; for 2.5 ≤ x ≤ 7.5 0; forx ≥ 7.5
(x + 2.5)/5; for 2.5 ≤ x ≤ 2.5 (7.5 x)/5; for 2.5 ≤ x ≤ 7.5 (x 7.5)/5; for 7.5 ≤ x ≤ 12.5 (17.5 x)/5; for 12.5 ≤ x ≤ 17.5 0; otherwise Icool (x)=
Iwarm (x)=
(x 17.5)/5; for 17.5 ≤ x ≤ 22.5 (27.5 x)/5; for 22.5 ≤ x ≤ 27.5 (x 27.5)/5; for 27.5 ≤ x ≤ 32.5 (37.5 x)/5; for 32.5 ≤ x ≤ 37.5 0; otherwise
Ihot (x)= (x 27.5)/5; for 27.5 ≤ x ≤ 32.5 (37.5 x)/5; for 32.5 ≤ x ≤ 37.5 0; otherwise
ThegraphicalmembershipdegreesofthesevariablesaresummarizedinFigure 3.Thegraphical falsitydegreesofthesevariablesaresummarizedinFigure 4.Also,thegraphicalindeterminacy degreesofthesevariablesaresummarizedinFigure 5.Ontheotherhand,foraparticularlinguistic term,‘Cool’inthetemperatureattribute,theirneutrosophicmembershipfunctionsarerepresentedin Figure 6.
Figure3. Truth-membershipfunctionoftemperatureattribute.
Figure4. Falsity-membershipfunctionoftemperatureattribute.
Figure5. Indeterminacy-membershipfunctionoftemperatureattribute.
Figure6. Cool(T,I,F)fortemperatureattribute.
Step3 Basedonthemembershipgrades,differenttransactionhasbeensetupbytakingdifferent setsofthetemperatures.Themembershipgradesintermsoftheneutrosophicsetsofthese transactionsaresummarizedinTable 5
Table5. MembershipfunctionfordatabaseTransactions. TransactionTemp.MembershipDegree
T118
Very-cold<0,0,1> cold<0,0,1> cool<1,0.1,0> warm<0,0.1,1> hot<0,0,1>
T213
Verycold<0,0,1> cold<0.4,0.9,0.6> cool<0.6,0.9,0.4> warm<0,0,1> hot<0,0,1>
T312
Verycold<0,0,1> cold<0.6,0.9,0.4> cool<0.4,0.9,0.6> warm<0,0,1> hot<0,0,1>
T433
Verycold<0,0,1> cold<0,0,1> cool<0,0,1> warm<0.4,0.9,0.6> hot<0.6,0.9,0.4>
T521
T625
Verycold<0,0,1> cold<0,0,1> cool<0.8,0.7,0.2> warm<0.2,0.7,0.8> hot<0,0,1>
Verycold<0,0,1> cold<0,0,1> cool<0,0,1> warm<1,0.5,0> hot<0,0,1>
Step4 Now,wecountthesetoflinguisticterms{verycold,cold,cool,warm,hot}foreveryelement intransactions.Sincethetruth,falsity,andindeterminacy-membershipsareindependent functions,thesetoflinguistictermscanbeextendedto Tvery cold, Tcold, Tcool , Twarm, Thot
Fvery cold, Fcold, Fcool , Fwarm, Fhot Ivery cold, Icold, Icool , Iwarm, Ihot where Fwarm meansnotworm and Iwarm meansnotsureofwarmness.Thisenhancesdealingwithnegativeassociationrules, whichishandledaspositiveruleswithoutextracalculations.
Step5 ByusingthemembershipdegreesthataregiveninTable 5 forcandidate1-itemset,thecount andsupporthasbeencalculated,respectively.Thecorrespondingresultsaresummarizedin Table 6
Table6. Supportforcandidate1-itemsetneutrosophicset.
1-itemsetCountSupport
Tverycold 00
TCold 10.17
TCool 2.80.47
TWarm 1.60.27
THot 0.60.1
Iverycold 00
ICold 1.80.3
ICool 2.60.43
IWarm 2.20.37
IHot 0.90.15
Fverycold 61
FCold 50.83
FCool 3.20.53
FWarm 4.40.73 FHot 5.40.9
Similarly,thetwo-itemsetsupportisillustratedinTable 7 andtherestofitemsetgeneration (k-itemsetfor k = 3,4 8) areobtainedsimilarly.Thecountfor k-itemsetindatabaserecordisdefined byminimumcountofeachone-itemsetexists.
Forexample:{TCold, TCool}countis0.8 BecausetheyexistsinbothT2andT3.
InT2: TCold =0.4and TCool =0.6so,countfor{TCold, TCool}inT2=0.4
InT3: TCold =0.6and TCool =0.4so,countfor{TCold, TCool}inT2=0.4
Thus,countof{TCold, TCool}in(Database)DBis0.8.
Table7. Supportforcandidate2-itemsetneutrosophicset. 2-itemsetCountSupport2-itemsetCountSupport {TCold, TCool}0.80.13{ICold, ICool}1.80.30
{TCold, ICold}10.17 {ICold, Fverycold} 1.80.30 {TCold, ICool}10.17{ICold, FCold}10.17 {TCold, Fverycold} 10.17{ICold, FCool}10.17
{TCold, FCold}0.80.13{ICold, FWarm}1.80.30
{TCold, FCool}10.17{ICold, FHot}1.80.30
{TCold, FWarm}10.17{ICool, IWarm}0.80.13
{TCold, FHot}10.17 {ICool, Fverycold} 2.60.43
{TCool, TWarm}0.20.03{ICool, FCold}1.80.30
{TCool, ICold}10.17{ICool, FCool}1.20.20
{TCool, FCool}1.80.30{ICool, FWarm}2.60.43
{TCool, IWarm}0.80.13{ICool, FHot}2.60.43
{TCool, Fverycold} 2.80.47{IWarm, IHot}0.90.15
Table7. Cont.
{TCool, FCold}2.80.47
{IWarm, Fverycold} 2.20.37
{TCool, FCool}10.17{IWarm, FCold}2.20.37
{TCool, FWarm}2.80.47{IWarm, FCool}1.60.27
{TCool, FHot}2.80.47{IWarm, FWarm}1.40.23
{TWarm, THot}0.40.07{IWarm, FHot}1.70.28
{TWarm, ICool}0.20.03
{IHot, Fverycold} 0.90.15
{TWarm, IWarm}1.10.18{IHot, FCold}0.90.15
{TWarm, IHot}0.40.07{IHot, FCool}0.90.15
{TWarm, Fverycold} 1.60.27{IHot, FWarm}0.60.10
{TWarm, FCold}1.60.27{IHot, FHot}0.40.07
{TWarm, FCool}1.60.27
{TWarm, FWarm}0.60.10
{TWarm, FHot}1.60.27
{Fverycold, FCold} 50.83
{Fverycold, FCool} 3.20.53
{Fverycold, FWarm} 4.40.73
{THot, IWarm}0.60.10 {Fverycold, FHot} 5.40.90
{THot, IHot}0.60.10{FCold, FCool}30.50
{THot, Fverycold} 0.60.10{FCold, FWarm}3.40.57
{THot, FCold}0.60.10{FCold, FHot}4.40.73
{THot, FCool}0.60.10{FCool, FWarm}1.80.30
{THot, FWarm}0.60.10{FCool, FHot}2.60.43
{THot, FHot}0.40.07{FWarm, FHot}4.20.70
5.CaseStudy
Inthissection,thecaseofTelecomEgyptCompanystockrecordshasbeenstudied.Egyptianstock markethasmanycompanies.Oneofthemajorquestionsforstockmarketusersiswhentobuyor tosellaspecificstock.Egyptianstockmarkethasthreeindicators,EGX30,EGX70,andEGX100. Eachindicatorgivesareflectionofthestockmarket.Also,theseindicatorshaveanimportantimpact onthestockmarketusers,affectingtheirdecisionsofbuyingorsellingstocks.Wefocusinourstudy ontherelationbetweenthestockandthethreeindicators.Also,weconsiderthemonthandquarter oftheyeartobeanotherdimensioninourstudy,whilethesell/buyvolumeofthestockperdayis consideredtobethethirddimension.
Inthisstudy,thehistoricaldatahasbeentakenfromtheEgyptianstockmarketprogram(Mist) duringtheprogramSeptember2012untilSeptember2017.Foreverystock/indicator,Mistkeepsa dailytrackofnumberofvalues(openingprice,closingprice,highpricereached,lowpricereached, andvolume).ThecollecteddataofTelecomEgyptStockaresummarizedinFigure 7
Inthisstudy,weusetheopenpriceandclosepricevaluestogetpricechangerate,whichare definedasfollows: pricechangerate = closeprice openprice openprice × 100 andchangethevolumetobeapercentageoftotalvolumeofthestockwiththefollowingrelation: percentageofvolume = volume totalvolume × 100
Thesamewasperformedforthestockmarketindicators.Now,wetaketheattributesas“quarter”, “month”,“stockchangerate”,“volumepercentage”,and“indicatorschangerate”.Table 8 illustrates thesegmentofresulteddataafterpreparation.
Table8. Segmentofdataafterpreparation.
Ts_DateMonthQuarterChangeVolumeChange30Change70Change100
13September2012September30.640.03 1.110.01 0.43
16September2012September30.070.022.824.503.67 17September2012September33.470.121.270.760.81
18September2012September31.380.03 0.08 0.48 0.43
19September2012September3 1.480.020.35 1.10 0.64
20September2012September30.470.05 1.41 1.64 1.55
23September2012September33.640.02 0.211.000.41
24September2012September3 0.470.050.27 0.090.03
25September2012September3 2.770.152.151.791.85
26September2012September31.960.040.220.960.57
27September2012September30.900.05 1.38 0.88 0.92
30September2012September3 0.140.00 1.11 0.79 0.75
1October2012October4 1.600.02 2.95 4.00 3.51
Basedontheselinguisticterms,definetherangesundertheSVNSsenvironment.Forthis, correspondingtotheattributein“changerate”and“volume”,thetruth-membershipfunctionsby definingtheirlinguistictermsas{“highup”,“highlow”,“nochange”,“lowdown”,“highdown”} correspondingtoattribute“changerate”,whilefortheattribute“volume”,thelinguisticterms are(low,medium,andhigh)andtheirrangesaresummarizedinFigures 8 and 9,respectively. Thefalsity-membershipfunctionandindeterminacy-membershipfunctionhavebeencalculatedand appliedaswellforchangerateattribute.
Figure8. Changerateattributetruth-membershipfunction.
Figure9. Volumeattributetruth-membershipfunction.
6.ExperimentalResults
Weproceededtoacomparisonbetweenfuzzyminingandneutrosophicminingalgorithms, andwefoundoutthatthenumberofgeneratedassociationrulesincreasedinneutrosophicmining. AprogramhasbeendevelopedtogeneratelargeitemsetsforTelecomEgypthistoricaldata. VB.nethasbeenusedincreatingthisprogram.Theobtaineddatahavebeenstoredinanaccess database.Thecomparisondependsonthenumberofgeneratedassociationrulesinadifferent min-supportthreshold.Itshouldbenotedthattheperformancecannotbepartofthecomparison becauseofthenumberofitems(attributes)thataredifferentinfuzzyvs.neutrosophicassociation rulesmining.Infuzzymining,thenumberofitemswas14,whileinneutrosophicminingitis 34.Thishappensbecausethenumberofattributesincreased.Spreadingeachlinguisticterminto three(True,False,Indeterminacy)termsmakethegeneratedrulesincrease.Thefalsity-generated associationrulescanbeconsideredanegativeassociationrules.Aspointedoutin[36],theconviction ofarule conv(X → Y) isdefinedastheratiooftheexpectedfrequencythat X happenedwithout Y falsity-associationrulestobeusedtogeneratenegativeassociationrulesif T(x)+ F(x)= 1.InTable 9, thenumberofgeneratedfuzzyrulesineach k itemsetusingdifferentmin-supportthresholdare reported,whilethetotalgeneratedfuzzyassociationruleispresentedinFigure 10
Figure10. No.offuzzyassociationruleswithdifferentmin-supportthreshold.
Table9. No.ofresultedfuzzyruleswithdifferentmin-support.
Min-Support0.020.030.040.05 1-itemset10101010 2-itemset37363633 3-itemset55291510 4-itemset32420
Ascomparedtothefuzzyapproach,byapplyingthesamemin-supportthreshold,wegetahuge setofneutrosophicassociationrules.Table 10 illustratestheboomingthathappenedtogenerated neutrosophicassociationrules.Westopgeneratingitemsetsatiteration4duetothenotedexpansion intheresultsshowninFigure 11,whichshowsthenumberofneutrosophicassociationrules.
Table10. No.ofneutrosophicruleswithdifferentmin-supportthreshold.
Min-Support0.020.030.040.05 1-itemset26262626 2-itemset313311309300 3-itemset2293216420301907 4-itemset11,233968985237768
Figure11. No.ofneutrosophicassociationruleswithdifferentmin-supportthreshold.
Experimenthasbeenre-runusingdifferentmin-supportthresholdvaluesandtheresulted neutrosophicassociationrulescountshasbeennotedandlistedinTable 11.Notethehighvaluesthat areusedformin-supportthreshold.Figure 12 illustratesthegeneratedneutrosophicassociationrules formin-supportthresholdfrom0.5to0.9.
Table11. No.ofneutrosophicruleswithdifferentmin-supportthreshold.
Min-Support0.50.60.70.80.9 1-itemset119965 2-itemset5033301110 3-itemset12264501010 4-itemset175714555 5-itemset151452111 6-itemset8838800
Figure12. No.ofneutrosophicrulesformin-supportthresholdfrom0.5to0.9.
Usingtheneutrosophicminingapproachmakesassociationrulesexistformostofthemin-support thresholddomain,whichmaybesometimesmisleading.Wefoundthatusingtheneutrosophic approachisusefulingeneratingnegativeassociationrulesbesidepositiveassociationrules minings.Hugegeneratedassociationrulesprovoketheneedtore-minegeneratedrules(mining ofminingassociationrules).Usingsuitablehighmin-supportvaluesmayhelpintheneutrosophic miningprocess.
7.ConclusionsandFutureWork
Bigdataanalysiswillcontinuetogrowinthenextyears.Inordertoefficientlyandeffectively dealwithbigdata,weintroducedinthisresearchanewalgorithmforminingbigdatausing neutrosophicassociationrules.Convertingquantitativeattributesisthemainkeyforgenerating suchrules.Previously,itwasperformedbyemployingthefuzzysets.However,duetofuzzy drawbacks,whichwediscussedintheintroductorysection,wepreferredtouseneutrosophicsets. Experimentalresultsshowedthattheproposedapproachgeneratedanincreaseinthenumberof rules.Inaddition,theindeterminacy-membershipfunctionhasbeenusedtopreventlosingrules fromboundariesproblems.Theproposedmodelismoreeffectiveinprocessingnegativeassociation rules.Bycomparingitwiththefuzzyassociationrulesminingapproaches,weconcludethatthe proposedmodelgeneratesalargernumberofpositiveandnegativeassociationrules,thusensuring theconstructionofarealandefficientdecision-makingsystem.Inthefuture,weplantoextendthe comparisonbetweentheneutrosophicassociationruleminingandotherintervalfuzzyassociation ruleminings.Furthermore,weseizedthefalsity-membershipfunctioncapacitytogeneratenegative associationrules.Conjointly,weavailedoftheindeterminacy-membershipfunctiontopreventlosing rulesfromboundariesproblems.Manyapplicationscanemergebyadaptionsoftruth-membership function,indeterminacy-membershipfunction,andfalsity-membershipfunction.Futureworkwill benefitfromtheproposedmodelingeneratingnegativeassociationrules,orinincreasingthequality ofthegeneratedassociationrulesbyusingmultiplesupportthresholdsandmultipleconfidence thresholdsforeachmembershipfunction.Theproposedmodelcanbedevelopedtomixpositive associationrules(representedinthetruth-membershipfunction)andnegativeassociationrules (representedinthefalsity-membershipfunction)inordertodiscovernewassociationrules,andthe indeterminacy-membershipfunctioncanbeputforthtohelpintheautomaticadoptionofsupport thresholdsandconfidencethresholds.Finally,yetimportantly,weprojecttoapplytheproposedmodel inthemedicalfield,duetoitscapabilityineffectivediagnosesthroughdiscoveringbothpositive andnegativesymptomsofadisease.Allfuturebigdatachallengescouldbehandledbycombining neutrosophicsetswithvarioustechniques.
AuthorContributions: Allauthorshavecontributedequallytothispaper.Theindividualresponsibilities andcontributionofallauthorscanbedescribedasfollows:theideaofthiswholepaperwasputforward byMohamedAbdel-BassetandMaiMohamed,VictorChangcompletedthepreparatoryworkofthepaper. FlorentinSmarandacheanalyzedtheexistingwork.Therevisionandsubmissionofthispaperwascompletedby MohamedAbdel-Basset.
ConflictsofInterest: Theauthorsdeclarenoconflictofinterest.
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