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Aggregation and modeling using computational intelligence techniques

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Aggregation and modeling using computational intelligence techniques

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Content AGGREGATIONANDMODELINGUSINGCOMPUTATIONALINTELLIGENCE TECHNIQUES by MinshenHao ADissertationPresentedtothe FACULTYOFTHEUSCGRADUATESCHOOL UNIVERSITYOFSOUTHERNCALIFORNIA InPartialFulfillmentofthe RequirementsfortheDegree DOCTOROFPHILOSOPHY (ELECTRICALENGINEERING) May2014 Copyright 2014 MinshenHao Dedication To thememoryofmydad,DianpingHao,Imisshimeveryday mymom,YingDeng mygirlfriend,XiaochanNiu andallmyfamilyandfriends ii Acknowledgements First of all, I would like to express my deepest gratitude to my academic advisor and disser- tation committee chair, Prof. Jerry M. Mendel, for his guidance and support throughout my entirePh.D.study. Hisknowledge,experienceandencouragementhaveledmethroughmany difficulties I encountered during this long and tough process. And without a kind and patient advisorlikehim,myfiveyearshereatUSCwouldbemuchharder. AndIwouldalsoliketothank allmyfellowstudents herein Prof. Mendel’steamfor our fruitfuldiscussionsandfriendships,theyareDr. DaoyuanZhai,MohammadRezaRajati,and MohammadMehdiKorjani. I’despeciallyliketothankDr. DaoyuanZhaiforintroducingme toProf. Mendelandhisteam. I also want to extend my deepest thanks to the entire staff at the Center for Interactive Smart Oilfield Technologies (CiSoft) at USC, especially Director Dr. Iraj Ershaghi. Their hard work have created a stable source of financial and intellectual support for our projects, whichallowsmetowhole-heartedlyfocusontheresearchissuesinaworry-freecondition. I am also grateful to all my qualification and defense committee members: Prof. Iraj Ershaghi, Prof. Antonio Ortega, Prof. Keith Jenkins, and Prof. Fred Aminzadeh. I want to thankthemforsharingtheirvaluabletimewithmeandmakingalotofinsightfulsuggestions. Last but not least, my sincerest gratitude goes to all my family and friends. I will always rememberandbegrateful. iii Contents Dedication ii Acknowledgements iii Abstract xv Chapter1 Introduction 1 1.1 AggregatingInterwellConnectivities . . . . . . . . . . . . . . . . . . . . . . 2 1.2 SimilarityMeasureforGeneralType-2FuzzySets . . . . . . . . . . . . . . . 4 1.3 ModelingWordsbyNormalIntervalType-2FuzzySets . . . . . . . . . . . . 6 1.4 LinguisticWeightedStandardDeviation . . . . . . . . . . . . . . . . . . . . 8 1.5 PerceptualComputerApplicationinLearningOutcomeEvaluation . . . . . . 10 1.6 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.7 ThesisOutline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Chapter2 AggregatingPetroleumReservoirInterwellConnectivitiesUsingCom- putationalIntelligenceTechniques 14 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2.1 Liu-MendelModel(LMM) . . . . . . . . . . . . . . . . . . . . . . . 16 2.2.2 DistributedCapacitanceModel(DCM) . . . . . . . . . . . . . . . . 19 2.2.3 IteratedExtendedKalmanFilter(IEKF)andPredictor(EKP) . . . . . 20 2.2.4 ExtendedKalmanSmoother(EKS). . . . . . . . . . . . . . . . . . . 21 2.2.5 EllipseStrategyforChoosingtheContributingInjectors . . . . . . . 22 iv 2.2.6 FuzzyMeasuresandtheGeneralizedChoquetIntegral(GCI) . . . . . 23 2.2.7 QuantumParticleSwarmOptimization(QPSO) . . . . . . . . . . . . 24 2.3 AggregatingIPRestimatesobtainedfromdifferentIEKFs . . . . . . . . . . . 26 2.3.1 OverviewofDataProcessingUsingOnlytheIEKF . . . . . . . . . . 26 2.3.2 DetailsofOurAggregationApproach . . . . . . . . . . . . . . . . . 26 2.4 EvaluatingtheaggregatedIPRestimatesusingrealdata . . . . . . . . . . . . 32 2.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.5.1 AggregatingtheSRLMMandSRDCM . . . . . . . . . . . . . . . . 36 2.5.2 AggregatingtheNSRLMMandNSRDCM . . . . . . . . . . . . . . 36 2.5.3 AggregatingtheSRLMMandNSRLMM . . . . . . . . . . . . . . . 38 2.5.4 AggregatingtheSRDCMandNSRDCM . . . . . . . . . . . . . . . . 38 2.5.5 PartialConclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.5.6 AggregatingallFourIEKFs . . . . . . . . . . . . . . . . . . . . . . 40 2.5.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.6 ComparingtheWAandGCI . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Chapter3 Similarity Measures for General Type-2 Fuzzy Sets Based on the α- PlaneRepresentation 45 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.2.1 GT2FSs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.2.2 AnIT2FSSimilarityMeasure . . . . . . . . . . . . . . . . . . . . . 51 3.3 ANewSimilarityMeasureforGT2FSs . . . . . . . . . . . . . . . . . . . . 53 3.3.1 DesirablePropertiesforaGT2FSSimilarityMeasure. . . . . . . . . 53 3.3.2 Generalizationofsm J ( e A, e B)toGT2FSs . . . . . . . . . . . . . . . 53 3.4 PropertiesofTheNewSimilarityMeasureforTrapezoidalSecondaryMFs . . 56 3.5 NumericalExamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.5.1 Example1: DiscreteGT2FSs . . . . . . . . . . . . . . . . . . . . . 60 v 3.5.2 Example2: GT2FSswithSameFOUbutDifferentSecondaryMFs . 64 3.5.3 Example 3: GT2 FSs with Different FOUs and Triangular Secondary MFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.6 SimilarityPercentage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 Chapter4 EncodingWordsIntoNormalIntervalType-2FuzzySets 78 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.2 EncodingWordsIntoNormalIntervalType-2FuzzySets: HMMethod . . . . 78 4.2.1 DataPart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 4.2.2 FuzzySetPart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 4.4 ComparisonswithEIA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 Chapter5 LinguisticWeightedStandardDeviation 98 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 5.2 LinguisticWeightedPowerMeanandLinguisticWeightedStandardDeviation 99 5.2.1 LinguisticWeightedPowerMean . . . . . . . . . . . . . . . . . . . 99 5.2.2 Weighted Power Mean Enhanced Karnik-Mendel (WPMEKM) Algo- rithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 5.2.3 LinguisticWeightedStandardDeviation . . . . . . . . . . . . . . . . 102 5.3 NumericalExamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 5.3.1 RandomlySelectedWordFOUs . . . . . . . . . . . . . . . . . . . . 106 5.3.2 EvaluatingLocationChoiceApplication . . . . . . . . . . . . . . . . 108 5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Chapter6 PerceptualComputerApplicationinLearningOutcomeEvaluation 112 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 6.2 TraditionalLOEvaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 vi 6.3 DesignofPer-CforLOEvaluation . . . . . . . . . . . . . . . . . . . . . . . 114 6.3.1 Hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 6.3.2 Encoder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 6.3.3 Engine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 6.3.4 Decoder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 6.4 MandatoryRequirementsintheEvaluationProcess . . . . . . . . . . . . . . 120 6.5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 6.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 Chapter7 ConclusionsandFutureWorks 129 7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 7.2 FutureWorks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 7.2.1 ObtainingAMetricforDeterminingWhichWordModelIsBest . . . 131 7.2.2 ObtainingFOUsfromCyclicData . . . . . . . . . . . . . . . . . . . 131 7.2.3 ExtensionofWang-Mendel(WM)MethodstoIT2FSs . . . . . . . . 133 ReferenceList 135 AppendixA SquareRootIPRSVMsfora3-injector1-producerreservoir146 AppendixB Non-SquareRootIPRSVMs 149 AppendixC IEKFsfortheSquareRootandNon-SquareRootSVMs 151 C.1 GeneralEKFEquations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 C.2 IEKFEquations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 C.3 JacobianMatrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 C.4 Inequality-ConstrainedIEKFs(CIEKF) . . . . . . . . . . . . . . . . . . . . 154 AppendixD EKSsfortheLMMandDCM 156 AppendixE QPSOalgorithm 157 vii AppendixF Computing the centroid and standard deviation for e L in Fig. 4.2 158 viii ListofTables 2.1 AveragepredictionerrorsresultswhenaggregatingtheSRLMMandSRDCM, andtheNSRLMMandNSRLMMusingtheWA . . . . . . . . . . . . . . . . 37 2.2 AveragepredictionerrorsresultswhenaggregatingtheSRLMMandSRDCM, andtheNSRLMMandNSRDCMusingtheGCI. . . . . . . . . . . . . . . . 37 2.3 AveragepredictionerrorsresultswhenaggregatingtheSRLMMandNSRLMM, andSRDCMandNSRDCMusingtheWA . . . . . . . . . . . . . . . . . . . 39 2.4 AveragepredictionerrorsresultswhenaggregatingtheSRLMMandNSRLMM, andtheSRDCMandNSRDCMusingtheGCI . . . . . . . . . . . . . . . . . 39 2.5 SummarizedmeanandstdresultsofImp a l (l = 1,2,3,4)usingtheWA . . . 40 2.6 SummarizedmeanandstdresultsofImp a l (l = 1,2,3,4)usingtheGCI . . . 40 2.7 AveragepredictionerrorsresultswhenaggregatingallfourIEKFsusingtheWA 41 2.8 AveragepredictionerrorsresultswhenaggregatingallfourIEKFsusingtheGCI 41 3.1 CentroidsofT1similarityfunctionsforallpairsamongthe11generalT2FSs havingthesameFOUasFOU(little)forExample2 . . . . . . . . . . . . . 72 3.2 CentroidsofT1similarityfunctionsforallpairsamongthe11generalT2FSs havingthesameFOUasFOU(some)forExample2 . . . . . . . . . . . . . 72 3.3 CentroidsofT1similarityfunctionsforSM J ( e C i , e D j )(i,j = 1,...,5). . . . . 75 3.4 P( e A, e C j )andP( e A, e E j )(j = 1,...,5)resultsforlittle . . . . . . . . . . . . . 76 3.5 P( e B, e D j )andP( e B, e F j )(j = 1,...,5)resultsforsome . . . . . . . . . . . . . 76 4.1 Remainingdataintervalsandtheoverlapforleft-shoulderFOUs . . . . . . . 91 4.2 RemainingdataintervalsandtheoverlapforinteriorFOUs . . . . . . . . . . 91 4.3 Remainingdataintervalsandtheoverlapforright-shoulderFOUs . . . . . . 92 ix 4.4 Thecodebook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 4.5 Similarityresultsbetweenleft-shoulderFOUsgeneratedfromtheEIAandthe HMmethod. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.6 SimilarityresultsbetweeninteriorFOUsgeneratedfromtheEIAandtheHM method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 4.7 Similarity results between right-shoulder FOUs generated from the EIA and theHMmethod. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 6.1 LOsforCircuitsandSignals . . . . . . . . . . . . . . . . . . . . . . . . . . 113 6.2 AssessmentandLOmappings . . . . . . . . . . . . . . . . . . . . . . . . . 113 6.3 Jaccard’ssimilaritymeasuresmatrix . . . . . . . . . . . . . . . . . . . . . . 126 6.4 Jaccard’ssimilaritymeasuresmatrix . . . . . . . . . . . . . . . . . . . . . . 128 x ListofFigures 1.1 ConceptualstructureofthePer-C. . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2 FOUforwordSomefromtheEIA. . . . . . . . . . . . . . . . . . . . . . . . 8 1.3 RelationshipbetweencourseassessmentsandLOs. . . . . . . . . . . . . . . 10 2.1 Producer-centricreservoir. . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.2 AreservoirmodelwhoseproducerisacteduponbyN injectors. . . . . . . . 17 2.3 AtimingdiagramfortheIEKFappliedtoourrealdata. . . . . . . . . . . . . 26 2.4 Atimingdiagramforourproposedaggregationapproach. . . . . . . . . . . . 27 2.5 Flowchartforimplementingouraggregationapproach. . . . . . . . . . . . . 28 2.6 Aflowchartforevaluatingtheaggregationapproach. . . . . . . . . . . . . . 33 3.1 FOU(shaded),LMF(dashed)andUMF(solid)fortheIT2FS e A[74]. . . . . 48 3.2 2-1/2Dplotofα-planerepresentationofaGT2FS[72]. Theµ (x,u)direction appearstocomeoutofthepageandisthenewthirddimensionofaGT2FS. . 50 3.3 (a) and (c) illustrate two kinds of sm J ( e A α , e B α ) computed using (5.24) for allα∈ [0,1] (note that they are both functions ofα), wheresm J ( e A α , e B α ) in (a) is a monotonic function ofα, whereassm J ( e A α , e B α ) in (c) is not; (d) is a rotationof(c),wherethehorizontalaxesissm J ( e A α , e B α )andtheverticalaxes is α; and, (b) and (e) are the corresponding SM J ( e A, e B) (A T1 FS obtained using(3.22))for (a)and (c),respectively. . . . . . . . . . . . . . . . . . . . 56 3.4 AtrapezoidalsecondaryMFatx =x i . . . . . . . . . . . . . . . . . . . . . . 57 3.5 (a)µ e A (x,u)and(b)µ e B (x,u)inExample1. . . . . . . . . . . . . . . . . . . 60 3.6 sm J ( e A α , e B α )-αpairresults,whereα = 0.1,0.2,...,1. . . . . . . . . . . . . . 63 3.7 SM J ( e A, e B). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 xi 3.8 Theninepoints (a,b,c,d,e,f,g,i,h)torepresentanFOU. . . . . . . . . . . 64 3.9 FOUsforwords(a)littleand(b)some. . . . . . . . . . . . . . . . . . . . . . 65 3.10 (a) Five triangular secondary MFs and (b) five trapezoidal secondary MFs whenx = 6inFig. 3.9(b). . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.11 SM J ( e C i , e C j ), i,j = 1,...,5, where each SM J ( e C i , e C j ) has been computed using(3.22). ThewordislittleandthesecondaryMFsaretriangles. . . . . . 67 3.12 SM J ( e E i , e E j ), i,j = 1,...,5, where each SM J ( e E i , e E j ) has been computed using(3.22). ThewordislittleandthesecondaryMFsaretrapezoids. . . . . 68 3.13 (a) SM J ( e A, e C j ) and (b) SM J ( e A, e E j ) j = 1,...,5, where both SM J ( e A, e C j ) and SM J ( e A, e E j ) have been computed using (3.22). The word is little and in (a)the secondary MFs are triangles, where as in (b)the secondary MFs are trapezoids. Inboth(a)and(b) e AisanIT2FS. . . . . . . . . . . . . . . . . . 68 3.14 SM J ( e C i , e E j ), i,j = 1,...,5, where each SM J ( e C i , e E j ) has been computed using (3.22). The word is little and the row secondary MFs are triangles, whereasthecolumnMFsaretrapezoids. . . . . . . . . . . . . . . . . . . . . 69 3.15 SM J ( e D i , e D j ), i,j = 1,...,5, where each SM J ( e D i , e D j ) has been computed using(3.22). ThewordissomeandthesecondaryMFsaretriangles. . . . . . 69 3.16 SM J ( e F i , e F j ), i,j = 1,...,5, where each SM J ( e F i , e F j ) has been computed using(3.22). ThewordissomeandthesecondaryMFsaretrapezoids. . . . . 70 3.17 (a) SM J ( e B, e D j ) and (b) SM J ( e B, e F j ) j = 1,...,5, where both SM J ( e B, e D j ) and SM J ( e B, e F j ) have been computed using (3.22). The word is some and in (a)the secondary MFs are triangles, where as in (b)the secondary MFs are trapezoids. Inboth(a)and(b) e AisanIT2FS. . . . . . . . . . . . . . . . . . 70 3.18 SM J ( e D i , e F j ), i,j = 1,...,5, where each SM J ( e D i , e F j ) has been computed using (3.22). The word is some and the row secondary MFs are triangles, whereasthecolumnMFsaretrapezoids. . . . . . . . . . . . . . . . . . . . . 71 3.19 SM J ( e C i , e D j ), i,j = 1,...,5, where each SM J ( e C i , e D j ) has been computed using(3.22). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 xii 3.20 Results before supremum operation for (a) SM J ( e C 4 , e D 3 ) (b) SM J ( e C 4 , e D 5 ), and (c) SM J ( e C 5 , e D 4 ), where e C iα and e D jα denote the α-plane of e C i and e D j (i = 4,5andj = 3,4,5),respectively. . . . . . . . . . . . . . . . . . . . . . 74 4.1 Examplestoshowtherelationshipbetween(a)m a andξ ∗ ,and(b)σ a andξ ∗ . . 82 4.2 InteriorFOU. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.3 Left-shoulderFOU. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.4 Right-shoulderFOU. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 4.5 FOUsofall32wordsgeneratedusingtheHMmethod. . . . . . . . . . . . . 94 4.6 FOUs ofall32wordsusingtheEIA.Thewordshavebeenorderedaccording totheirincreasingaveragecentroid. . . . . . . . . . . . . . . . . . . . . . . 97 5.1 The(a)LWAand(b)LWSDofrandomlygeneratedIT2FSs. . . . . . . . . . 106 5.2 TheFOUsof e Y LWA ± e Σ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 5.3 TheFOUsoftheevaluationsof(a)economicgrowth;(b)cost;(c)government policies;and,(d)otherrisksforlocationAin[32]. . . . . . . . . . . . . . . . 109 5.4 TheFOUsof(a)thefinalperformance e Y F and(b)theLWSD e ΣoflocationA in[32]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 5.5 TheFOUsof e Y F − e Σand e Y F + e ΣforlocationAin[32]. . . . . . . . . . . . 110 5.6 TheFOUsoftruncated e Y F − e Σand e Y F + e ΣforlocationAin[32]. . . . . . . 110 6.1 HierarchyofPer-CforLOevaluationprocess. . . . . . . . . . . . . . . . . . 114 6.2 FOUsofthefivewordsforperformance. . . . . . . . . . . . . . . . . . . . . 116 6.3 FOUsforweights. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 6.4 Computingthefiringlevels. . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 6.5 Aconjunctivepartialabsorptionoperator. . . . . . . . . . . . . . . . . . . . 122 6.6 Aggregatingcollectionofmandatoryanddesiredrequirements,wherez isthe aggregatedoutputandthescaleofweightsis[0,1]. . . . . . . . . . . . . . . 123 6.7 Aggregatingcollectionofmandatoryanddesiredrequirements,wheretheweights arerepresentedbyIT2FSsandthescaleofweightsis[0,100]. . . . . . . . . 124 xiii 6.8 FOUsforLOsandthefinalperformanceusingLWPMwhenM A = 27,M L = 1andM E = 58. (a): e Y LO1 ; (b): e Y LO2 ; (c): e Y LO3 ; (d): e Y F . . . . . . . . . . . . 126 6.9 FOUsforLOsandthefinalperformanceusingLWAwhenM A = 27,M L = 1 andM E = 58. (a): e Y LO1 ; (b): e Y LO2 ; (c): e Y LO3 ; (d): e Y F . . . . . . . . . . . . . 127 7.1 Anexampleofdatafromacyclicprocess. . . . . . . . . . . . . . . . . . . . 132 7.2 ClustergradesobtainedfromtheFCMalgorithm. . . . . . . . . . . . . . . . 133 xiv Abstract In waterflood management, there exists several models to describe a petroleum reservoir for predicting the future production rates using scheduled injection rates. Most of them have the ability to estimate how much the injectors impact some specific producers, namely, the interwell connectivities between the injectors and the producers. Knowing these values not onlyreducesthecostofwaterinjection,butcanalsoincreasetheoilproduction. Inthefirstpartofthisthesis,weconstructfourdifferentmodelsfortheinteractionbetween a group of injectors and a producer and then dynamically estimate the parameters of these models, along with the interwell connectivities using an Iterated Extended Kalman Filter (IEKF)andSmoother(EKS).WethenusetheWeightedAverage(WA)andGeneralizedCho- quetIntegral(GCI)toaggregatetheestimatedinterwellconnectivities. Thesetwoaggregation functionsareoptimizedtominimizemean-squareerrorsinfutureforecastedproductionrates. This is done by using Quantum Particle Swarm Optimization (QPSO) to search for the opti- mal set of weights which are required by both aggregation methods. Several experiments are conductedtoshowtheimprovedaverageperformanceofourapproachonasetofdatafroma realreservoir,andtheperformancesoftheabovetwoaggregationmethodsarealsocompared andanalyzed. A similarity measure between fuzzy sets is a very important concept in fuzzy set theory. There have been a lot of different similarity measures proposed in the literature, for both T1 FSs and IT2 FSs. The second part of this thesis presents theoretical studies that were performed for the most advanced fuzzy logic sets that are currently under research-general type-2fuzzysets. Inourstudy,basedontheα-planerepresentationforageneraltype-2(GT2) xv FS, the similarity measure is generalized to such T2 FSs. Some examples that demonstrate howtocomputethesimilaritymeasuresfordifferentT2FSsaregiven. Next,thethirdpartofthisthesisproposesanewmethod−theHMmethod−tomodelwords by normal IT2 FSs, using data intervals that are collected from a group of subjects. The HM methodusesthesamebaddataprocessing,outlierprocessingandtolerancelimitprocessingto pre-processthedataintervals,asisusedintheEnhancedIntervalApproach(EIA);itthenuses anewconfidence-interval-basedreasonableintervaltesttokeeponlythose data intervalsthat share a common interval. In the Fuzzy Set Part, the common overlap is first determined for a group of data intervals; the IT2 FS model for a word is then determined from the remaining data intervals that exclude the overlap. The HM method has a new way to establish if a word shouldbemodeledasaleftshoulder,interiororrightshoulderIT2FS.TheresultingIT2FSs have both normal lower and upper membership functions (MFs), which makes them unique among IT2 FS word models. We also compare the IT2 model obtained from the HM method withthosefromtheEIAusingJaccard’ssimilaritymeasure. The fourth and fifth part of this dissertation, a generalization of the Linguistic Weigh-ted Average (LWA), the Linguistic Weighted Power Mean (LWPM), is studied. Based on the LWPM, a new type of fuzzy statistic, namely, the Linguistic Weighted Standard Deviation, is proposed. In classical statistics, the first- and second-order statistics, i.e., the mean and standarddeviation,arethemostimportantones. Inthisthesis,weextendthedefinitionofthe standarddeviationandmakesitpossibletocomputethestandarddeviationwhendatacontains notonlynumbers, butalsowords. ThegeneralizedstandarddeviationiscalledtheLinguistic Weighted Standard Deviation (LWSD). The LWSD is viewed as a special case of the LWPM when the parameter r in the LWPM is set to be 2. Two numerical examples that utilize the newLWSDarepresented: oneissyntheticwhereallthedataaregeneratedrandomly,andthe other is a practical decision making problem. These examples demonstrate that the LWSD canprovideextrainformationtoadecisionmakerwhenonlyuncertaininputdata(words)are available. We believe that the concept of the LWSD will certainly play an important role in manyfutureapplications. Inthefinalpartofthisdissertation,anIT2versionoftheLWPMis xvi defined,andanapplicationofthePerceptualComputer(Per-C)toevaluatelearningoutcomes usingtheIT2LWPMisalsopresented. The final part of this dissertation draws conclusions, and provides some suggestions for futureresearch. xvii Chapter1 Introduction Computational intelligence (CI) is a set of nature-inspired computational methodologies and approachestoaddresscomplexreal-worldproblemstowhichtraditionalapproaches,i.e.,first principles modeling or explicit statistical modeling, are ineffective or infeasible. The CI approaches primarily include artificial neural networks [94], evolutionary computation [22] and fuzzy logic [136]. In addition, CI also embraces biologically inspired algorithms such as swarmintelligence[4]andartificialimmunesystems,whichcanbeseenasapartofevolution- arycomputation,andincludesbroaderfieldssuchasimageprocessing[9],datamining[108], andnaturallanguageprocessing[6]. Furthermoreotherformalisms: theory,chaostheoryand many-valuedlogicareusedintheconstructionofcomputationalmodels. According to [114], the characteristic of ”intelligence” is usually attributed to humans. Morerecently,manyproductsanditemsalsoclaimtobe”intelligent”. Intelligenceisdirectly linked to the reasoning and decision making. Fuzzy logic was introduced in 1965 as a tool to formalise and represent the reasoning process and fuzzy logic systems which are based on fuzzy logic possess many characteristics attributed to intelligence. Fuzzy logic deals effec- tively with uncertainty that is common for human reasoning, perception and inference and, contrary to some misconceptions, has a very formal and strict mathematical backbone (’is quitedeterministicinitselfyetallowinguncertaintiestobeeffectivelyrepresentedandmanip- ulatedbyit’,sotospeak). Neuralnetworks,introducedin1940s(furtherdevelopedin1980s) mimicthehumanbrainandrepresentacomputationalmechanismbasedonasimplifiedmath- ematicalmodeloftheperceptrons(neurons)andsignalsthattheyprocess. Evolutionarycom- putation, introduced in the 1970s and more popular since the 1990s mimics the population- based sexual evolution through reproduction of generations. It also mimics genetics in so calledgeneticalgorithms. 1 This dissertation focuses on applying the CI techniques to solve practical problems and alsoextendingsomeoftheexistingCItechniquesintheComputingwithWords(CWW)field. The study philosophy employed herein has two separate pieces: 1) Design a methodology to aggregate the results from different models for a single problem, so that, without creat- ing a new model, an aggregated system is obtained which is capable of outperforming each of the single models; and, 2) Introduce some new concepts and approaches to model uncer- tainties and words and construct new word models using interval type-2 fuzzy sets that are unique among existing models. The dissertation also presents several important theoretical and practical results that were obtained using the approaches described above. These results willbecomekeyfoundationsforotherresearcherswhohavebeenworkingonCWWschemas anditsapplications. The rest of this chapter briefly goes through the background motivation and objectives of our studies using the CI techniques mentioned above, through which the applications and extensionsoftheCItechniquesarestudiedandexamined. 1.1 AggregatingInterwellConnectivities While primary production refers to oil that is recovered naturally from a producing well, Enhanced Oil Recovery (EOR) improves the amount of oil recovered from a well by using someformofadditionalengineeringtechnique. Waterinjection,alsoknownaswaterflood,is aformofthissecondaryEORproductionprocess. Used in onshore and offshore developments, water injection involves drilling injection wells into a reservoir and introducing water into that reservoir to encourage oil production. Whiletheinjectedwaterhelpstoincreasedepletedpressurewithinthereservoir,italsohelps tomovetheoilinplace. Whether water injection occurs after production has already been depleted or before pro- duction from the reservoir has been drained, waterflood sweeps remaining oil through the reservoirtoproductionwells,whereitcanberecovered. 2 Determining how well an injector communicates with a producer, interwell connectivity evaluation, provides a way to infer reservoir characteristics, identify locations for new wells, and manage waterflood performance. For instance, a large connectivity between a well pair could be an evidence for a channel or fracture between the wells. On the other hand, a small connectivity between two adjacent wells could mean that there is a low permeability zone or barrierintheinterwellregion. Iftheconnectivitymodelcanpredictproductionratesbasedon theinjectionrates,themodelcanoptimizeinjectionratestoimprovetheproductioneconomics [43]. In a waterflood, injection and production rate data are potentially a good source of infor- mation to estimate the interwell connectivity (referred as the “Injector-Producer-Relationship (IPR)”inthisproposal)betweeninjectorandproducerpairs. Inrecentyears,applyingsignal- processing-based techniques that use the most commonly available data in an oilfield (injec- tionandproductionrates)hasbecomeincreasinglypopularinwaterfloodmanagement. Heffer et al. [37] used Spearman rank correlations to relate injector-producer pairs and associated these relations with geomechanics. Panda and Chopra [85] used artificial neural networks to determine the interactions between injection and production rates. Albertoni and Lake [2] estimated the interwell connectivity based on a linear model using a multiple linear regres- sion(MLR)method. Yousefetal.[134,135]improvedthisworkbybuildingamorecomplex model,named“capacitancemodel,”todescribetherelationshipbetweeninjectionandproduc- tion rates. Lee [51] further generalized the capacitance model to the Distributed Capacitance Model(DCM)bytakingintoaccountthatthereservoirbetweensomeinjector-producerpairs is highly heterogeneous or includes some high permeability channels, fractures or faults. Liu andMendel[58]modeledthereservoirusingcontinuousimpulseresponseswhichwerechar- acterized as a two-parameter auto-regressive model between each single injector and a single producer. All of these approaches not only can estimate the IPR values using only measured injec- tion and production data, but can also predict future production rates given future scheduled injectionrates;however,differentapproachesfocusondifferentaspectsofmodelingthereser- voirandutilizedifferentestimationmethodstoestimatethemodel’sparameters. Forexample, 3 Lee’s DCM basically was derived from a total mass balance with compressibility (the same asthecapacitancemodel)andthenitsparameterswereestimatedusingconstrainedquadratic programming. Liu and Mendel’s model, referred to as the LMM in this thesis, was devel- opedbasedondomainexpertknowledge,anditsparameterswereestimatedusingtheIterated ExtendedKalmanFilter(IEKF). WearemotivatedbytheworksofKellerandhiscolleagues[3,44,100],whoshowedthat, instead of inventing another method for land-mine detection, they could outperform all exist- ing methods by aggregating the detection results obtained from a group of existing methods byusingtheGeneralizedChoquetIntegral(GCI).Morespecifically,wedevelopedamethod- ology that aggregates the IPS estimates from different models. We used both the WA and theGCIwithrespecttoλ-measure(whichisnon-additive)astheaggregatortoaggregatedif- ferent IPR estimates from LMM and DCM. One of our objectives is to learn which of these aggregationmethodsgivesthebestresults. In Chapter 2, we used the GCI with respect to λ-measure and the WA to aggregated IPR estimatesfromdifferentmodels;wealsousedanevolutionaryoptimization,namely,Quantum ParticleSwarmOptimization(QPSO),tooptimizetheweights associatedwitheachIPResti- mate in the GCI and WA. Our experiments suggested that our aggregated model outperforms eachsinglemodel. 1.2 SimilarityMeasureforGeneralType-2FuzzySets Zadeh[139]generalizedtheconceptoftype-1fuzzysets(T1FS)totype-2fuzzysets(T2FS): AT2FSisanextensionofaT1FSwherethemembershipfunction(MF)ofaT2FSisafuzzy set on the interval [0,1]. The MF of a T2 FS is three-dimensional, where the third dimension is the value of the membership function at each point on its two-dimensional domain that is calleditsfootprintofuncertainty(FOU).AT2FSletsusincorporateuncertaintyabouttheMF intofuzzysettheory,and,ifthereisnouncertainty,thenaT2FSreducestoaT1FS,whichis analogoustoprobabilityreducingtodeterminismwhenunpredictabilityvanishes.[77] 4 For an interval type-2 fuzzy set (IT2 FS) that third-dimension value is the same (e.g., 1) everywhere, which means that no new information is contained in the third dimension of an IT2FS.So,forsuchaset,thethirddimensionisignored,andonlytheFOUisusedtodescribe it. Most applications of T2 FSs use IT2 FSs (e.g., [8,20,29,30,39,53,55,74,105,127,128]). IT2 FSs have received the most attention because the mathematics that is needed for such sets−primarily interval arithmetic−is much simpler than the mathematics that is needed for generalT2FSs. However, recently, there has been a growing interest in using general T2 FSs (GT2 FSs) (e.g., [24,31,48,54,76,84,96,107,141]), because they have more design degrees of freedom then IT2 FSs, and, therefore, have the potential to outperform a system that uses IT2 FSs. Liu [54] and Mendel et al. [71,76] proposed a method that represents a GT2 FS, which is called an α-plane representation, and they demonstrated that this representation is useful for both theoretical and computational studies for GT2 FSs. Using the α-plane representation, set-theoretic operations and centroid computations become very simple, because they can be performedusingexistingalgorithmsthatareappliedtoeachα-plane,whichlendsthemselves tomassiveparallelprocessing. Also,in[106,107],WagnerandHagrasintroducedasystematic approach for developing general Type-2 Fuzzy Logic Systems (T2 FLSs) based on “zSlice Representation Theorem (RT)”. Mendel [71] and Mendel and Zhai [143] pointed out that a zSliceisthesameasanα-planeraisedtolevelα. Asimilaritymeasurebetweenfuzzysetsisaveryimportantconceptinfuzzysettheoryand has a lot of applications. In this dissertation, we proposed a new similarity measure for GT2 FSs which satisfies the desired properties for a “good” similarity measure using the α-plane representation. InChapter3, theα-planerepresentation, arecentlyproposedrepresentationforGT2FSs, was reviewed. Based the α-plane representation, we proposed a new similarity measure between two GT2 FSs, which is a very important concept in theoretical research of GT2 FSs andcanbeusedasadecoderinthePerceptualComputer(Per-C)thatutilizesGT2FSs. 5 Figure1.1:ConceptualstructureofthePer-C. 1.3 ModelingWordsbyNormalIntervalType-2FuzzySets Zadeh coined the phrase “computing with words” (CWW) [137,138], which is [138] “a methodology in which the objects of computation are words and propositions drawn from a natural language.” Words in the CWW paradigm may be modeled by type-1 fuzzy sets (T1 FSs)[136]ortheirextension,i.e.,intervaltype-2fuzzysets(IT2FSs)[65,139]. CWWusing T1 FSs has been studied by many researchers, e.g., [36,38]; however, since IT2 FSs can model both interpersonal and intrapersonal uncertainties [68,69], and T1 FSs are a special case of IT2 FSs and cannot model such uncertainties, we focus on CWW using IT2 FSs in this thesis [34,35,47,87,89]. By interpersonal uncertainties, we mean the variations in the understanding of a word between people, and by intrapersonal uncertainties, we mean the variation in one person’s understanding of a word. These linguistic uncertainties are at the heartoftheCWWparadigm[126]. Aspecificarchitecture,whichwasproposedin[66]andelaborateduponin[77]formaking subjectivejudgmentsbyCWW,isshowninFig. 1.1. ItiscalledaPerceptualComputer−Per- Cforshort. InFig. 1.1,theencodertransformslinguisticperceptionsintoIT2FSsthatactivate a CWW engine. The CWW engine performs operations on the IT2 FSs. The decoder maps theoutputoftheCWWengineintoarecommendation,whichcanbeaword,rank,orclass. 6 As shown in Fig. 1.1, the first step in the Per-C is to transform (model) words into IT2 FSs, i.e., the encoding problem. Liu and Mendel [56] proposed an Interval Approach (IA) to synthesize an IT2 FS model for a word, in which interval endpoint data about a word are collected from a group of subjects 1 ; each subject’s data interval is mapped into a T1 FS, and an IT2 FS mathematical model [represented by a footprint of uncertainty (FOU)] is obtained forthewordfromtheseT1FSs. As pointed out in [126], there are some limitations to the IA: the FOUs seem too fat and wide, and the LMFs of the interior FOUs usually have very small height. Wu et al. proposed an Enhanced Interval Approach (EIA) to overcome these limitations. The Data Part of the EIA has more steps and conditions than the IA to remove more data intervals, and its Fuzzy Set Part has an improvedprocedure to compute the lower membership function (LMF). They also demonstrated that the IT2 FS output from the EIA converges in a mean-squared sense to astablemodelasmoreandmoredataintervalsarecollected. Unfortunately, the EIA has its own limitations, as we explain next by using, .e.g, the EIA FOU of the word Some, which is depicted in Fig. 1.2. Observe that, although the EIA uses overlapping data intervals, it does not assign a membership function (MF) value of 1 to the common overlap. It seems to us that, if everyone is in an agreement about the overlap, there should not be any uncertainty about that overlap. In addition, the LMF of the word models fromtheEIAarenotnormal,whichleadstoextra(and,sometimes,tedious)carewhendealing withtheoreticalaswellascomputationalresults. In Chapter 4, we propose a new method−the HM method−that uses more information from the data intervals than the EIA, namely, the overlap shared by all those data intervals. TheHMmethodusesthesamebaddataprocessing,outlierprocessingandtolerancelimitpro- cessingtopre-process thedataintervals, as is usedinthe EIA; itthenuses anewconfidence- interval-based reasonable interval test to keep only those data intervals that share a common interval. In the Fuzzy Set Part, the common overlap is first determined for a group of data intervals, after which the IT2 FS model for a word is determined from those remaining data 1 Thesubjectsareasked: Onascaleof0-10,whataretheendpointsofanintervalthatyouassociatewiththe word ? 7 0 1 2 3 4 5 6 7 8 9 10 0 0.2 0.4 0.6 0.8 1 Some Figure1.2: FOUforwordSomefromtheEIA. intervals that exclude the overlap. The resulting IT2 FSs have both normal lower and upper membershipfunctions(MFs),whichmakesthemuniqueamongIT2FSwordmodels. 1.4 LinguisticWeightedStandardDeviation During the past two decades, many research studies have been performed in which a col- laboration of Fuzzy Set (FS) theory and statistics has been established for different pur- poses. According to [11], this has been done to: (1) introduce new data analysis problems inwhichtheobjectiveinvolveseitherfuzzyrelationshipsorfuzzyterms[13,83];(2)establish well-formalized models for elements combining randomness and fuzziness [19,52,133]; (3) developuni-andmultivariatestatisticalmethodologiestohandlefuzzy-valueddata[101,146]; and,(4)incorporatefuzzysetstohelpinsolvingtraditionalstatisticalproblemswithnumerical data[46,130]. Originally, imprecision and indeterminacy were considered to be random characteristics, and were taken into account by methods of probability theory [41]. In real-world situations, a frequent source of imprecision is not only the presence of random variables, but also the impossibility, in principle, of operating with numerical data as a result of the complexity of the system, where instead this can be handled by using fuzzy sets. According to [41], the motivation for the development of fuzzy statistics is its philosophical and conceptual relation 8 to subjective probability. In the subjective view, probability represents the degree of belief that a given person has in a given event on the basis of given evidence. There has been a lot of work done based on this philosophy, where it has been shown that better results can be obtainedwhensomefuzzystatisticsareused[95,130]. In classical statistics, the first- and second-order statistics, i.e., the mean and standard deviation 2 , are the most important ones, and, there are many algorithms and applications that usethemin,e.g.,theareasofimageprocessing,communicationandsignalprocessing[49,61, 102,132]. Moreover, with the help of α-cuts, there are also numerous applications that use fuzzymeansandfuzzystandarddeviations[19,95,130,146];however,allofthemusetype-1 fuzzysets(T1FSs). Recently, more and more researchers are focusing on interval type-2 fuzzy sets (IT2 FSs) [24,31,141], which contain more membership function (MF) uncertainties than T1 FSs and include T1 FSs as a special case. To compute the average of a collection of IT2 FSs, Wu andMendel[116,119]proposedtheLinguisticWeightedAverage(LWA),whosecomputation usesα-cutsandtheKM/EKMalgorithm;however,tothebestofourknowledge,therehasnot been any study about the standard deviation of IT2 FSs. Note that the standard deviation of IT2 FSs discussed in this thesis has nothing to do with the standard deviation introduced by WuandMendel[120],becausethestandarddeviationin[120]isanuncertaintymeasurefora singleIT2FS,whereasthestandarddeviationinthisthesisisdefinedforasetofIT2FSs. InChapter5,theLinguisticWeightedStandardDeviation(LWSD)isproposed. Itiscom- putedusingtheweightedpowermean(WMP).Analogoustothestandarddeviationinclassical statistics,theLWSDcomputestheweightedstandarddeviationofasetofIT2FSs. 2 Inthisthesis,weonlyconsiderthesamplearithmeticmeanandsamplestandarddeviation. 9 1.5 PerceptualComputerApplicationinLearningOutcome Evaluation Outcome-based education (OBE) is a model of education that rejects the traditional focus on what the school provides to students, in favor of making students demonstrate that they “knowandareabletodo”whatevertherequiredoutcomesare. Ithasbeenpracticedsinceits introductionbyWilliamSpadyin1988[97]. Foraspecificcourse,Fig. 1.3showsthegeneral relationshipbetweenLearningOutcomes(LOs)anditscourseassessments[10]. Although the proponents of OBE like the idea of measuring outputs (students’ perfor- mances)ratherthaninputs(theresourcesthatareavailabletostudents),therearesomemajor criticismsofthismodel: thevaguenessofOBE’sconceptionofa“measurableoutcome,”inap- propriateLOs(thestandardsoftheLOscanbesettoolow,toohighorbewronglyconceived), extraburdenoninstructorsandeducationalinstitutions,etc. Figure1.3: RelationshipbetweencourseassessmentsandLOs. In Chapter 6, a Perceptual Computer (Per-C) that implements the Linguistic Weighted Power Mean (LWPM) as the CWW engine is designed to evaluate the LOs and to then make afinalperformancerecommendationusingtheseLOevaluations. 1.6 Contributions This dissertation focuses on applying the CI techniques to solve practical problems and also extending some of the existing CI techniques in the Computing with Words (CWW) field. 10 In the first part of the dissertation, we proposed a new methodology to aggregated results from different models so that the aggregated model outperforms each single model. This methodology is very general so that it can be easily applied to other future applications. In the second part of the dissertation, we focused on applying and extending the current schema of a Per-C (see Fig. 1.1). For the Encoder, we proposed a new method to model words from intervaldata,whichleadstonormalIT2FSmodels. Thismethodisuniquetoourknowledge, andtheresultingwordmodelcansimplifynumericalcomputationsintheCWWapplications. For the CWW Engine, we used the LWPM as the engine and applied this Per-C to a practical application. From our simulation results, by changing the parameter in the LWPMoperation, it is possible to add more subjective requirements to a Per-C to enhance its performacen. For the Decoder, weproposedanewsimilaritymeasure, based ontheα-planerepresentation, for twoGT2FSs. ThenewsimilaritymeasurecanbeusedinaPer-CwhereGT2FSsareused. 1.7 ThesisOutline Therestofthethesisisorganizedasfollows: InChapter2,weconstructfourdifferentmodels for the interaction between a group of injectors and a producer and then dynamically esti- matetheparametersofthesemodels,alongwiththeinterwellconnectivitiesusinganIterated Extended Kalman Filter (IEKF) and Smoother (EKS). We then use the Weighted Average (WA) and Generalized Choquet Integral (GCI) to aggregate the estimated interwell connec- tivities. These two aggregation functions are optimized to minimize mean-square errors in futureforecastedproductionrates. ThisisdonebyusingQuantumParticleSwarmOptimiza- tion (QPSO) to search for the optimal set of weights which are required by both aggregation methods. Several experiments are conducted to show the improved average performance of our approach on a set of data from a real reservoir, and the performances of the above two aggregationmethodsarealsocomparedandanalyzed. In Chapter 3, a similarity measures for IT2 FSs, which is an extension of the Jaccard similarity measure for T1 FSs, is reviewed; then, based on the α-plane representation for a 11 generaltype-2(GT2)FS,thesimilaritymeasureisgeneralizedtosuchT2FSs. Someexamples thatdemonstratehowtocomputethesimilaritymeasuresfordifferentT2FSsaregiven. In Chapter 4, we propose a new method−the HM method−to model words by normal IT2 FSs, using data intervals that are collected from a group of subjects. The HM method uses the same bad data processing, outlier processing and tolerance limit processing to pre- process the data intervals, as is used in the Enhanced Interval Approach (EIA); it then uses anewconfidence-interval-basedreasonableintervaltesttokeeponlythose data intervalsthat share a common interval. In the Fuzzy Set Part, the common overlap is first determined for a group of data intervals; the IT2 FS model for a word is then determined from the remaining data intervals that exclude the overlap. The HM method has a new way to establish if a word shouldbemodeledasaleftshoulder,interiororrightshoulderIT2FS.TheresultingIT2FSs have both normal lower and upper membership functions (MFs), which makes them unique among IT2 FS word models. We also compare the IT2 model obtained from the HM method withthosefromtheEIAusingJaccard’ssimilaritymeasure. In Chapter 5, we extend the definition of the standard deviation and makes it possible to compute the standard deviation when data contains not only numbers, but also words. The generalizedstandarddeviationiscalledtheLinguisticWeightedStandardDeviation(LWSD). TheLinguisticWeightedPowerMean(LWPM)operationisalsoreviewedinthisChapter,and theLWSDisviewedasaspecialcaseoftheLWPMwhentheparameterr intheLWPMisset to be 2. Two numerical examples that utilize the new LWSD are presented: one is synthetic where all the data are generated randomly, and the other is a practical decision making prob- lem. TheseexamplesdemonstratethattheLWSDcanprovideextrainformationtoadecision maker when only uncertain input data (words) are available. We believe that the concept of theLWSDwillcertainlyplayanimportantroleinmanyfutureapplications. In Chapter 6, we describe an application of the Per-C to evaluate Learning Outcomes in an Outcome Based Education (OBE) system. The evaluation problem in this Chapter is viewed as a hierarchical decision making problem. In the first level of the hierarchy, the crisp score of each assessment is mapped into an IT2 FS for each student. The assessments are then aggregated to obtain the performance of each LO. The LWPM is used as the CWW 12 engine, so that some mandatory requirements can be implemented. Finally, by aggregating theperformanceofallLOs,thefinalperformanceisobtained. Wealsopresentanexampleof ourproposedPer-Cusingafictitiouscourse. InChapter7,wedrawconclusionsanddiscussseveralpossiblefutureresearchtopics. 13 Chapter2 AggregatingPetroleumReservoir InterwellConnectivitiesUsing ComputationalIntelligenceTechniques 2.1 Introduction Floodinganoilfieldwithextraneouswaterhasbeenawidelyacceptedmethodforincreas- ingareservoir’soilrecoverysincethe1950’s. Waterisinjectedintodedicatedinjectionwells strategically located throughout the reservoir, in order to displace the remaining oil towards the producing wells. If properly designed and operated, a waterflood can double the reservoir’soilrecovery. In almost all waterflood operations, measured injection and production rates are the most abundantavailabledata. Theyareconsideredtobecorrelatedtoeachotherinsomeverycom- plicatedway,andmanymethodshavebeenpreviouslyproposedtoinfertheinterwellconnec- tivities (referred as the “Injector-Producer-Relationship (IPR)” in this Chapter) between each producer and its surrounding contributing injectors using only these data. In all those works, thereservoirismodeledasadynamicalsysteminwhichtheinjectionratesactasthesystem’s inputsandtheproductionratesarethesystem’soutputs. As mentioned in Section 1.1, there has been a lot of work that uses signal-processing- based techniques to estimate the IPRs. In practice, the IPR values are estimated so that water can be allocated in an optimal manner. Water is a resource that can be expensive, and so its optimalallocationcanreducethecostsforextractingpetroleumfromareservoir;however,the IPR estimates are usually obtained only from one particular model, and it is very difficult to 14 tell which model should be used since injection rates are usually designed for one particular model, and not for all of the models. The key novelty of this work is to estimate and smooth theparametersinalldifferentmodelsusinganIEKFandExtendedKalmanSmoother(EKS), and to then aggregate different IPR estimates obtained for the different models, so that the aggregated IPR estimates produce less production rates prediction error for all the models. Note that we do not have to aggregate the IPR estimates in real time, because water is re- allocatedonadaily(orevenlessfrequent)basis. Aggregationfunctionsplayanimportantroleinmanyofthetechnologicaltasksscientists are faced with nowadays. They are specifically important in many problems related to the fusion of information [28]. The Weighted Average (WA) is one of the most commonly used linearaggregationmethods[21]. Also,theconceptsofnon-additivemeasuresandtheirrepre- sentationsasfuzzyintegralswereproposedinthe1970sandarewelldeveloped[26,59,113]. Many applications, such as information fusion, multiple regressions and classification have successfully employed non-additive measures as a data aggregation tool [23,27,50,79,99]. In this Chapter, we used both the WA and the GCI with respect toλ-measure (which is non- additive)to aggregate different IPR estimates from LMM and DCM. One of our objectives is tolearnwhichoftheseaggregationmethodsgivesthebestresults. 2.2 Background In order to aggregate the IPR estimates of LMM and DCM, the following techniques (two of whicharecomputationalintelligence(CI)techniques)wereused: 1. IteratedExtendedKalmanFilter(IEKF)andPredictor(EKP):UsedtoestimatetheIPR estimatesandpredictthefutureproductionratesforbothLMMandDCM. 2. ExtendedKalmanSmoother(EKS):UsedtoupdateandimprovetheIEKFestimatesby usingmoremeasurements. 3. Ellipse Strategy: Used to determine the contributing injectors to a specific producer in arealreservoir. 15 4. Weighted Average (WA) and Generalized Choquet Integral (GCI): Used to aggregate thedifferentIPRestimates. 5. Quantum Particle Swarm Optimization (QPSO): Used to find the optimal weights for eachaggregationapproach. The GCI and QPSO are CI techniques. The details of the LMM and DCM, and of all the techniques 1 just mentioned are described in this section. To begin, we explain the LMM and DCM. 2.2.1 Liu-MendelModel(LMM) Fig. 2.1 shows a portion of a reservoir consisting of 6 injectors and 3 producers. In this reservoir,theproductionrates ofproducerP 1 aredeterminedbyinjectorsI 1 ,I 2 ,I 5 andI 6 (as indicated by arrows); production rates of producer P 2 are determined by injectors I 2 , I 3 , I 4 andI 5 ; and production rates ofP 3 are determined byI 4 ,I 5 andI 6 . Injectors that determine a producer’s production rates are called the producer’s contributing injectors, i.e., injectors I 1 , I 2 ,I 5 andI 6 areP 1 ’s contributing injectors, etc. Models that view the whole reservoir in this manner are called producer-centric models, and producers are assumed to be independent of eachotherinthesemodels. Figure2.1: Producer-centricreservoir. 1 TheWAiswellknown,sothedetailsofitwillnotbedescribedinthissection. 16 In LMM, the reservoir is considered to be a system that can be modeled as a collection of continuous-time impulse responses that convert injection rates into a production rate. A reservoirinwhichaproducerisacteduponbyN injectorsisdepictedinFig. 2.2,whereI j (t), n j (t) and I m j (t) are the actual injection rates that flow into the reservoir, the corresponding measurementnoisesformeasuringinjectionrates,andthemeasuredinjectionratesforinjector j, respectively, and j = 1,2,...,N; P(t), n P (t) and P m (t) are the actual production rate, the corresponding measurement noise for measuring the production rate, and the measured production rate, respectively; P c j (t) is the channel production rate produced by injector j, which represents the amount of production rate in P(t) caused only by injector j. Note that thisreservoirmodelisanalogoustoadigitalcommunicationsystem,whereinjectionratesare the messages, h j (t)(j = 1,...,N) act as channels that distort the messages, and P(t) is the received signal which can only be observed with additive noise. Also note that the noise-free data,I j (t)(j = 1,2,...,N) andP(t) are not directly available. Their measured values,I m j (t) andP m (t),areusedforlaterdataprocessing. Figure2.2: AreservoirmodelwhoseproducerisacteduponbyN injectors. AsshowninFig. 2.2,eachinjector/producerpairisconsideredasanindependentsubsys- tem, and each subsystem is modeled as a continuous-time impulse response that converts the injection rate, I j (t), into production rate P c j (t). Using advice from petroleum engineers, Liu 17 and Mendel [58] chose the following two-parameter auto-regressive (AR) model to represent theimpulseresponsebetweenaproducerandinjectorj: h j (t) =f(r j ,k j )b j te −a j t (2.1) wheref(r j ,k j ) (j = 1,...,N) is a scale function that represents how much of each injection rate flows in the direction of a producer; it may be a linear or non-linear scalar function of the distance, r j , and the permeability, k j , between the producer and injector j; but, this is unimportant because f(r j ,k j ) and b j are absorbed into a single unknown parameter (γ ′ j in (2.2)). Only sampled injection and production rates are available for processing, hence, (2.1) isdiscretized. Liu and Mendel showed that a unit step change of an injection rate causes a step change of steady-state production rate, whose impact can be evaluated by the IPR value, where 0 ≤ IPR≤ 1. TheyshowedthatthenumericalIPRvaluebetweeninjectorj andaproduceristhe areaunderthediscretizedimpulseresponse,andobtainedthefollowingformulafortheIPR: IPR j = γ j f(r j ,k j ) (1−α j ) 2 = γ ′ j (1−α j ) 2 (2.2) whereα j =e −a j T ,γ j =b j α j T andT isthesampleperiod. Using the discretized impulse response and (2.2), the subsystem between a producer and injectorj canthenbemodeledasthefollowingsecond-orderfinite-differenceequation: P c j (k +1) = 2α j (k)P c j (k)−α 2 j (k)P c j (k−1) +γ ′ j (k)I j (k) (2.3) Note that in (2.3) we have made α j and γ j dependent on k, as is commonly done in system identification when an unknown parameter is modeled as a Markov process. In practice, the measurement noise of injection rate I j (k) is very small, so that it can be replaced by the measuredinjectionrate,I m j (k);hence,(2.3)isre-expressed,as: 18 P c j (k +1) = 2α j (k)P c j (k)−α 2 j (k)P c j (k−1) +γ ′ j (k)I m j (k) (2.4) orequivalently,using(2.2)toreplaceγ ′ j (k),as P c j (k +1) = 2α j (k)P c j (k)−α 2 j (k)P c j (k−1) +IPR j (k)[1−α j (k)] 2 I m j (k) (2.5) Notethat(2.5)istheLMMchannelequationusedinthisSection,wherebothIPR j (k)and α j (k) (or their square roots, to keepIPR j (k)≥ 0 andα j (k)≥ 0) are modeled as first-order Markovsequences,i.e., IPR j (k +1) =IPR j (k)+n IPR j (k) (2.6) α j (k +1) =α j (k)+n α j (k) (2.7) wheren IPR j (k)andn α j (k)arezero-meanadditivewhitenoises. (2.5)-(2.7)constituteanon- lineardiscrete-timestochasticsystem. 2.2.2 DistributedCapacitanceModel(DCM) Lee[51]developedanewreservoirmodel,calledtheDistributedCapacitanceModel(DCM), that can be viewed as a generalization of the capacitance model to include the case where there exists a high permeability channel or fracture that divides the reservoir into two parts. The DCM is also a producer-centric model in which the relationship between one producer and all its contributing injectors can also be represented by Fig. 2.2, but now the channel betweeninjectorj andaproducerisdescribedbyathree-parameterimpulseresponse,as: 19 h j (t) = λ j τ j1 −τ j2 (e −t τ j1 −e −t τ j2 ) (2.8) whereτ j1 andτ j2 arethe“timeconstants”ofthedrainagevolumeandλ j denotesthenormal- izedinterwellconnectivitybetweeninjectorj andtheproducer. Usingthediscretizedversionof(2.8)andtheareaunderthediscretizedh j (t)fortheIPR, itfollowsthattheIPR(0≤IPR≤ 1)intheDCMcanbecomputedas: IPR j = γ j (α j1 −α j2 ) 1−(α j1 +α j2 )+α j1 α j2 (2.9) whereα j1 =e −T/τ j1 ,α j2 =e −T/τ j2 ,γ j =λ j /(τ j1 −τ j2 )andT isthesamplingperiod. Usingthediscretizedversionof(2.8)andtheIPR j in(2.9),theDCMmodelisdescribed as: P c j (k +1) = [α j1 (k)+α j2 (k)]P c j (k) − α j1 (k)α j2 (k)P c j (k−1) + IPR j (k){1−[α j1 (k)+α j2 (k)] + α j1 (k)α j2 (k)}I m j (k) (2.10) As in (2.4) and (2.5), we have replaced I j (k) by its measurement I m j (k) (j = 1,...,N). (2.10) is the DCM channel equation used in this Section, whereIPR j (k),α j1 (k) andα j2 (k) (or their square roots) are modeled as first-order Markov sequences, analogous to (2.6) and (2.7)above. Thisisalsoanonlineardiscrete-timestochasticsystem. 2.2.3 IteratedExtendedKalmanFilter(IEKF)andPredictor(EKP) TheEKFandIEKFarewidelyusedasthestandardtechniquesforrecursivedynamicnonlinear estimation (e.g., [42,62,63]). They provide a first-order approximation of optimal nonlinear mean-squarestateestimationforanonlineardiscrete-timesystem. TheIEKFdiffersfromthe 20 EKFinthatititeratestheEKFcorrectionequationuntilastoppingcriterionismet;itprovides betterestimatesthantheEKF. The main reason for using the IEKF in our work is that we assume the IPRs and other channel parameters are not static. They may be affected by many factors, e.g., changing of bottom-holepressures,workovers,naturalorman-madegeomechanicaleffects,etc. TheIEKFandEKPequationsarebasedonthefollowingState-VariableModel(SVM):      x(k +1) = f[x(k),k]+n x (k) z(k +1) = h[x(k +1),k +1]+n z (k +1) (2.11) wheref(·)andh(·)arethestateandmeasurementequations,respectively,andn x (k)andn z (k) are additive zero-mean white noises for the state and measurement, with covariance matrix Q(k)andvariancer(k),respectively. TousetheIEKFandEKP,wefirstneedtoconstructanSVM.InthisChapter,fourdifferent SVMs are used: two for the LMM and two for the DCM. How to construct such SVMs is illustrated in Appendices A and B for a simple 3-injector 1-producer reservoir. It is easy to generalizetheseexamplestothemultiple-injectorcase,andthisislefttothereader. IEKFandEKPprocessingprovidefilteredandpredictedstateestimatesofx(k+1),b x(k+ 1|k+1)andb x(k+1|k),respectively,theformerusesallthemeasurementsuptoandincluding timek+1,andthelatterusesthemeasurementsuptotimek. DetailsoftheIEKFsandEKPs forthedifferentSVMsaregiveninAppendixC. 2.2.4 ExtendedKalmanSmoother(EKS) Using an SVM, one can also perform another kind of state estimation, called smoothing (or interpolation) [63,104]. A smoothed estimate ofx(k) not only uses measurements that occur up to and including k, but also uses measurements to the right (future) of k. Smoothed esti- matesofIPRarebetterthanfilteredestimatesofIPRbecausetheymakeuseofmoredata. Depending on how many future measurements are used and how they are used, there are three types of smoothers: fixed-interval, fixed point and fixed lag. In this Chapter, we only 21 use a fixed-interval smoother (that is called the EKS) b x(k|N),k = 0,1,...,N − 1, where N isafixedpositiveinteger. Thesituationisasfollows[63]: withanexperimentcompleted,we have measurements available over the fixed interval 1≤ k ≤ N. For each time point within this interval we obtain the optimal estimate of state vectorx(k), that is based on all available measurementdata{z(j),j = 1,2,...,N}. TheequationsfortheEKSaregiveninAppendixD. 2.2.5 EllipseStrategyforChoosingtheContributingInjectors The IEKF and EKS require an SVM in which the number of injectors that affect a producer mustbeknownapriori;but,inpractice,thisisnotknownaheadoftime. Thefollowingellipse strategy [144], suggested by petroleum engineers, is a procedure for estimating this number andidentifyingwhichinjectorsmustbeusedforeachproducer. 1. Choosealargeenoughinitialellipsecenteredoneachproducerandusealltheinjectors withinthisellipsetoconstructanSVM. 2. Shrinktheellipsebydifferentscalefactorsandusealltheinjectorswithintheseellipses toconstructdifferentSVMs. 3. For each of the SVMs, run an IEKF and compute average errors between the predicted andactualproductionratesforthelatest60daysofthedata. 4. Use the injectors that are contained in the ellipse that gives the minimum averaged predictionerrorasthecontributinginjectorsforaproducer. The reason an ellipse (and not a circle) was used is due to the geological fractures of the reservoir where our data came from. According to the petroleum engineers who are famil- iar with this reservoir, there are parallel fractures along each well that align 45 ◦ to NE, so it is believed that injectors located along this fracture alignment are more likely to affect a producer. For a different reservoir, one could use another geometric shape centered on each producer,wherethatshapeisprovidedbypetroleumengineerexperts. 22 2.2.6 FuzzyMeasuresandtheGeneralizedChoquetIntegral(GCI) In this section, we first review some basic concepts behind the theory of fuzzy measures, and thendefinetheGCI. Definition1 Let X = {x 1 ,...,x n } be any finite set. A discrete fuzzy measure on X is a functionofthepowersetofX,µ : 2 X → [0,1]withthefollowingproperties: 1. µ (∅)=0andµ (X)=1; 2. givenA,B∈ 2 X ,ifA⊂B thenµ (A)≤µ (B)(monotonocityproperty). The setX is considered to contain the names of sources of information (in our case,X is thesetofallmodels),andforasubsetA⊂X,µ (A)istheworthofA. TheSugenoλ-measureisaspecialclassoffuzzymeasures,denotedg. Definition2 Let X = {x 1 ,...,x n } be any finite set and let λ ∈ [−1,+∞]. A Sugeno λ- measure[98,112]isafunctiong(·)from 2 X to [0,1]withthefollowingproperties: 1. g(X) = 1; 2. ifA,B⊆X withA∩B =∅,then g(A∪B) =g(A)+g(B)+λg(A)g(B) (2.12) Themeasureofasingletonset{x i },denotedg i = g({x i }),iscalleda fuzzy densityofthe informationsourcex i ,andλsatisfiesthefollowingproperty: λ+1 = n Y i=1 (1+λg i ) (2.13) In[79],itisprovedthatthepolynomialequation(2.13)hasarealrootgreaterthan-1,and severalresearchershaveobservedthatthisequationisactuallyeasytosolve. From(2.12)and (2.13), computing a Sugeno λ-measure on a set X with n elements only requires specifying the n fuzzy densities, g i . Unfortunately, for our application we do not know the g i ahead of time. 23 Definition3 Let f be a function from X ={x 1 ,...,x n } toℜ. Let{x σ(1) ,...,x σ(n) } denote a reordering of the set X such that f(x σ(1) ) ≤ ... ≤ f(x σ(n) ), and let A (i) be a collection of subsets defined byA (i) ={x σ(i) ,...,x σ(n) }. The discrete GCI off with respect to the Sugeno λ-measureg onX is: C g (f) = n X i=1 g(A (i) )(f(x σ(i) )−f(x σ(i−1) )) = n X i=1 f(x σ(i) )(g(A (i) )−g(A (i+1) )) (2.14) wheref(x σ(0) )≡ 0andA (n+1) ≡∅,i.e.,g(A (n+1) )≡ 0. The function f is a particular instance of the partial support (evidence) supplied by each sourceofinformation. TheGCIfusesthisobjectivesupportaccordingtotheworthofvarious subsetsoftheinformationsources. Inpractice,(2.14)iscomputedasfollows: 1. Determinethef function. InthisChapter,f isanIPRestimateobtainedfromanEKS. 2. Compute λ using the given (or, as in our application, optimized by means of QPSO) fuzzydensitiesbysolving(2.13). 3. Computeg(A (i) ),i = 1,...,n,accordingto(2.12). 4. ComputeC g (f),withquantitiescomputedabove,using(2.14). 2.2.7 QuantumParticleSwarmOptimization(QPSO) QPSO [129,131] is a globally convergent search algorithm which outperforms the original PSO [45] in search ability and has fewer parameters to control. It is a population-based opti- mization technique that contains a set of different particles, where a population is called a swarm. Eachparticlerepresentsapossiblesolutiontoanoptimizationproblem(minimization problem in our case). During each iteration, the position of each particle is updated using its 24 most recent own best solution, best solutions found by all other particles, and the global best solutionfoundbyallparticlessofar. Let M denote the population size and n denote the number of dimensions of the search space. Each individual particle i (1 ≤ i ≤ M), at iteration t, has the following attributes: A current position in the search space X i (t) and a personal best (pbest) position P i (t) (the position giving the best fitness found by this particle). Also, the global best (gbest) position foundbyallparticlesduringiterationsuptot,P g (t),isdefinedas: P g (t) = argmin P i f(P i (t)),1≤i≤M (2.15) where the functionf is the fitness. Note that in this Chapter, each particle represents a set of weights that are used by the WA or GCI to compute the aggregated IPR estimates, and every elementinX i (t)(i = 1,...,M)isconstrainedtobebetween0and1forallt. Atiterationt,thepositionofparticlei,X i (t),isupdated,as: X i (t+1) = min{max{P i (t)±β|m(t)−X i (t)|ln(1/u),0},1} (2.16) where β, which controls the convergence speed of the algorithm, is called the contraction- expansioncoefficient;m(t)iscomputed,as m(t) = 1 M M X i=1 P i (t); (2.17) anduisarandomnumberuniformlydistributedin(0,1). InthisChapter,β decreaseslinearly from1to0.5asthenumberofiterationsincreases. ThereasonweuseQPSOisthatin[111], Wang et al. showedthat, fordeterminingfuzzy measures from data, PSO-based algorithms are feasible and havebetter performance than the existing genetic algorithms and gradient descent algorithms. Note that in our approach, the data used to optimize the weights are actual production rates, a time series, whereas the data usedin[111]totestitsalgorithmsarenottimeseries. ThedetailsoftheQPSOalgorithmaredescribedinAppendixE. 25 2.3 Aggregating IPR estimates obtained from different IEKFs 2.3.1 OverviewofDataProcessingUsingOnlytheIEKF In practice, measured injection and production rate data are not available in real time. Gen- erally speaking, our injection rates data are measured daily, but the production rates are only measured on a weekly or bi-weekly basis; hence, IEKF processing on real data can not be performedinrealtime. Fig. 2.3explainshowweappliedtheIEKFtotherealdata. Figure2.3: AtimingdiagramfortheIEKFappliedtoourrealdata. At the end of Day N 1 , the daily injection and production rates up to Day N 1 are both availabletous. AnIEKF(basedonanyoftheSVMs)isruntoDayN 1 ,sothatIPRestimatesat DayN 1 canbeestimatedfromthestatevectoroftheIEKF.Oncenewinjectionandproduction ratesdataaremeasureduptoDayN 2 ,insteadofre-estimatingeverythingfromthebeginning, the IEKF continues to run from Day N 1 to Day N 2 , so that IPR estimates at Day N 2 can be computed. In this way, new IPR estimates can be obtained at Days k = N 1 ,N 2 ,..., directly fromtheIEKF. 2.3.2 DetailsofOurAggregationApproach Fig. 2.4 provides a high-level description of how we aggregated different IPR estimates obtained from different IEKFs and EKSs. As can be seen, it involves filtering from, e.g., N 1 toN 2 ,smoothingfromN 2 backtoM 2 ,andaggregatingwithin[M 2 ,N 2 ]. Thesethreesteps arerepeatedfromdays 1toN 1 ,N 1 toN 2 ,...,etc. Thedetailsofouraggregationapproachfor 26 asystemofoneproducerandC contributinginjectorsareexplainedinthissection. Notethat the extension of this approach to the multi-injector multi-producer scenario is very straight- forward. Figure2.4: Atimingdiagramforourproposedaggregationapproach. Tobegin,somenotationsaredefined: 1. b x l (k|k): filteredestimatesatDayk froml-thIEKF(l = 1,...,L). 2. b x l (k|k ′ )(k ′ >k): smoothedestimatesatDayk usingallthemeasurementsuptoDayk ′ froml-thEKS. 3. b x l (k|k ′ ) (k ′ < k): predicted estimates at Day k using the measurements up to Day k ′ froml-thEKP. 4. d IPR l ≡ [ [ IPR 1 l ,..., [ IPR C l ] T : IPR estimates of all C injectors from l-th IEKF, where [ IPR j l denotestheIPRestimatebetweenInjectorj andtheproducer(l = 1,...,L). 5. d IPR a ≡ [ [ IPR 1 a ,..., [ IPR C a ] T : aggregatedIPRestimatesforallC injectors. 27 Figure2.5: Flowchartforimplementingouraggregationapproach. Our procedure for aggregating the IPR estimates is summarized in Fig. 2.5. Explanations oftheblocksinthisfigureareasfollows(inthefollowingsteps,j = 1,...,C,l = 1,...,Land w = 1,...,20, whereC is the number of injectors,L is the number of models andw refers to thenumberofparticlesthatareusedinQPSO): 28 1. Assignthemeasuredinjectionandproductionratestothedifferentestimatorsforfuture use. Note that the connections between this block and the IEKF, EKS, EKP and the RMSEblocksarenotshowninFig. 2.5,soasnottoclutterthatfigure. 2. Run the L IEKFs to obtain L filtered state estimates b x l (N t |N t ) (in our case L = 2 or L = 4). 3. Useb x l (N t |N t )andI j (k),P(k)(k =M t ,...,N t ),torunLEKSsbacktoDayM t ,tocom- pute L smoothed estimates b x l (M t |N t ). In this way, all of the available measurements areusedtoobtainimprovedestimatesofx l attheearliertimeM t . 4. In the “Inner loop,” the QPSO algorithm is used, where each particle in the swarm representsasetofweights,whereeachweightisassociatedwithanIPRestimate. 2 The goal is to find the weights that minimize a prediction error (described below) and to then use those weights to aggregate d IPR l (M t |N t ) in order to obtain d IPR a (M t |N t ). We used 20 particles and 50 generations. Also, since each weight represents the worth of its corresponding IPR estimate, it is constrained between 0 and 1. The details of how QPSOisapplied,are: (a) Randomize20particles,andletg w = [g w 1 ,...,g w CL ] T denotethecurrentpositionof anarbitraryparticlew,inwhichg w (j−1)L+1 ,...,g w jL areweightscorrespondingtothe IPRestimates [ IPR j l betweenInjectorj andtheproducerintheLIEKFs. (b) For every particle g w , the aggregated IPR estimate, [ IPR j,w a (M t |N t ), between Injectorj andtheproduceriscomputedas: [ IPR j,w a (M t |N t ) =F w [ [ IPR j 1 (M t |N t ),..., [ IPR j L (M t |N t ),g w (j−1)L+1 ,...,g w (j−1)L+L ] (2.18) 2 ThedimensionofeachparticleisCL,sincethereareC injectorsandoneweightisassignedtoeachInjector- Producerpair(IPRestimate)inallLIEKFs. 29 wherethefunctionF(·)istheaggregationfunction. InthisChapter,twoaggrega- tion function are used: the WA,F WA , and the GCI,F GCI . The formulas for these twofunctionsaregivenby: F WA [ [ IPR j 1 (M t |N t ),..., [ IPR j L (M t |N t ),g w (j−1)L+1 ,...,g w (j−1)L+L ] = P L l=1 IPR j l (M t |N t )g w (j−1)L+l P L l=1 g w (j−1)L+l (2.19) and F GCI [ [ IPR j 1 (M t |N t ),..., [ IPR j L (M t |N t ),g w (j−1)L+1 ,...,g w (j−1)L+L ] = L X l=1 [ IPR j σ(l) (M t |N t )[g(A σ(l) )−g(A σ(l+1) )] (2.20) In (2.20), {σ(1),...,σ(L)} denotes a reordering of {1,...,L} such that [ IPR j σ(1) (M t |N t ) ≤ ... ≤ [ IPR j σ(L) (M t |N t ); A σ(l) is defined by A σ(l) = {EKS σ(l) , EKS σ(l+1) ,..., EKS σ(L) }, where EKS σ(l) is the σ(l)-th EKS; and, g(A σ(l) )arerecursivelycomputedusing(2.13)and(2.12),as: λ j +1 = n Y l=1 (1+λ j g({EKS σ(l) })) (2.21) and g(A σ(l) ) =g(A σ(l+1) )+g({EKS σ(l) })+λ j g(A σ(l+1) )g({EKS σ(l) }) (2.22) where A σ(L+1) ≡ ∅, g(A σ(L+1) ) ≡ 0 and g({EKS σ(l) }) ≡ g w (j−1)L+σ(l) . For example, g(A σ(L) ) = g({EKS σ(L) }∪A σ(L+1) ) = g({EKS σ(L) }) +g(A σ(L+1) ) + λ j g({EKS σ(L) })g(A σ(L+1) ) =g({EKS σ(L) }) =g w (j−1)L+σ(L) ,g(A σ(L−1) ) 30 = g({EKS σ(L−1) } ∪ A σ(L) ) = g({EKS σ(L−1) }) + g(A σ(L) ) + λ j g({EKS σ(L−1) })g(A σ(L) ) = g w (j−1)L+σ(L−1) + g w (j−1)L+σ(L) + λ j g w (j−1)L+σ(L−1) g w (j−1)L+σ(L) ,andsoon. Denote the aggregated IPR estimates of all C injectors using particle w as d IPR w a (M t |N t ) ≡ [ [ IPR 1,w a (M t |N t ),..., [ IPR C,w a (M t |N t )], where [ IPR j,w a (M t |N t ) iscomputedusing(2.18). Notethatinthisstep, theIPRestimatesfromsmoothed stateestimatesb x l (M t |N t )havebeenusedtoobtaintheaggregatedIPRestimates. (c) Replace the IPR estimates in each of the L state vector estimates b x l (M t |N t ) with d IPR w a (M t |N t ). Run each of the L EKPs to obtain L production-rate predictions, denoted b P w l (k|N t ),wherek =M t +1,...,N t . (d) Use the actual production rates P(k) (k = M t + 1,...,N t ) and compute 20L weightedpredictionRMSEs: J w l = v u u t 1 N t −M t Nt X k=Mt+1 [ b P w l (k|N t )−P(k)] 2 b k (2.23) where the weights b k linearly increase between 0.5 and 1.5 from k = M t + 1 to k =N t . Then,computethefollowing20averageRMSEs: J(w) = 1 L L X l=1 J w l (2.24) Note that J(w) is the objective function minimized by QPSO for each of the 20 particles. (e) UseJ(w) to update the personal best weights (P w (t)) and the global best weights (P g (t)) that are defined in Section 2.2.7, and then generate the new positions for all 20 particles according to the QPSO algorithm. With the new positions, repeat Steps(b)-(e). 31 (f) Terminate the QPSO when Steps (b) - (e) are repeated 50 (maximum generation) times. Then output the winner of the QPSO, i.e., the particle (set of weights) that givestheglobalbestweights(P g (t)),denotedg ∗ . 5. Using g ∗ , compute the final aggregated IPR estimates, d IPR a (M t |N t ) ≡ [ [ IPR 1 a (M t |N t ),..., [ IPR C a (M t |N t )],as: [ IPR j a (M t |N t ) =F[ [ IPR j 1 (M t |N t ),..., [ IPR j L (M t |N t ),g ∗ (j−1)L+1 ,...,g ∗ (j−1)L+L ] (2.25) 6. Sett =t+1andrepeatSteps1-5untilalldataareused. 2.4 EvaluatingtheaggregatedIPRestimatesusingrealdata During the aggregation process described in the last section, EKPs are used to predict the production rates from Day M t + 1 to N t in order to optimize the weights for aggregating the IPR estimates. To evaluate the aggregated IPR estimates, EKPs are also used to predict the production rates from Day N t + 1 to N t + 30. These predicted production rates are then compared with the historical data. Fig. 2.6 summarizes how we evaluated the aggregation approach described in Section 2.3.2 for one producer. The data we used is welltest data from anactualreservoir,recordedfromJan. 2006toSep. 2010(atotalof1736days). 32 Figure2.6: Aflowchartforevaluatingtheaggregationapproach. During the evaluation process, two kinds of prediction errors from the IEKFs are com- puted: (1) using the aggregated IPR estimates from our aggregation approach, and (2) from theIEKFsdirectly. Theprocedureisdescribedasfollows: 1. SetM t andN t (t = 1,...,50)(seeFig. 2.4),asM t ≡ 195+30tandN t ≡ 205+30t,i.e., theaggregationapproachwasperformedevery30days,andforeveryt,N t −M t = 10 days of data were used to optimize the corresponding weights to obtain the aggregated IPRestimates. 2. UsetheEllipseStrategy(seeSection2.2.5)todeterminethecontributinginjectors. For eacht,wefirstchoseanellipsesizethatincludedallthepossiblecontributinginjectors forthisproducer;then,weshrankthesizeoftheellipsebyfactorsof0.95,0.9,0.85,0.8 and0.75;andthen,foreachsizeoftheellipse,weusedtheinjectorscontainedwithinit 33 asourcontributinginjectors,rantheLIEKFstoDayM t ,computedthepredictionerrors fortheperiodk = M t +1,...,N t ,andcomputedthesumofallLpredictionerrors. We found the ellipse size that gave the smallest sum of the prediction errors and used the injectorsinthatellipseasthefinalcontributinginjectors. Notethisstepwasdonebefore theaggregationapproachwasusedtoaggregatetheIPRestimates. 3. Two separate flows of computation are performed in this step. The first one is for our aggregationapproach: (a) ApplytheaggregationapproachdescribedinSection2.3.2,usingthecontributing injectors determined from Step 2, after whichL smoothed estimates are obtained at Day M t , from the L EKSs, b x l (M t |N t ) (l = 1,...,L), and the aggregated IPR estimate, d IPR a (M t |N t ),isobtainedfrom(2.25). (b) ReplaceonlytheIPRestimatesinb x l (M t |N t )(l = 1,...,L)with d IPR a (M t |N t ),and denotetheresultingvectorasb x a l (M t |N t ). (c) Use b x a l (M t |N t ) and the injection rates from M t + 1 to N t + 30, and run the L EKPs to obtain L predictions of the production rates, namely b P a l (k|N t ), where l = 1,...,Landk =M t +1,...,N t +30. (d) ComputethepredictionRootMeanSquareErrors(RMSE),E a l (t),as: E a l (t) = v u u t 1 30 Nt+30 X k=Nt+1 [ b P a l (k|N t )−P(k)] 2 (2.26) whereP(k)istheactualproductionratesandl = 1,...,L. ThesecondflowofcomputationisforusingjusttheIEKFs: (a) Run each of the L IEKFs to obtain b x l (N t |N t ) directly, and then run the L EKPs. ThepredictedproductionratesfromtheseLEKPsaredenotedas b P l (k|N t ),where l = 1,...,Landk =N t +1,...,N t +30. (b) ComputeeachoftheLIEKF’sRMSE,E l (t),as: 34 E l (t) = v u u t 1 30 Nt+30 X k=Nt+1 [ b P l (k|N t )−p(k)] 2 (2.27) 4. RepeatSteps2-8untilalldataareused,i.e.,untilt max = 50. 5. Compare E a l (t) and E l (t) (t = 1,...,50) by computing the following averaged predic- tionerrors(l = 1,...,L): E a l = 1 50 50 X t=1 E a l (t) (2.28) E l = 1 50 50 X t=1 E l (t) (2.29) and the averaged improvement percentage achieved by the aggregation approach, namely Imp a l = E l −E a l E l ×100%, (2.30) aswellasthemeanandstandarddeviationofImp a l (l = 1,...,L). 2.5 Results In order to test the aggregation approach described in Sections 2.3 and 2.4 and compare the results for the WA and GCI, we applied our aggregation approach to 10 producers. Four different IEKFs were constructed (See Appendices A-C): Square-Root LMM (SRLMM), Square-Root DCM (SRDCM), Non-Square-Root LMM (NSRLMM) and Non-Square-Root DCM (NSRDCM). We first show results when we aggregated IPRs from two out of four IEKFs 3 ,andthentheresultswhenweaggregatedallfourIPRs. 3 There are six possibilities, but we only focus on four of them in this Chapter. The other two possibilities, aggregatingSRLMMandNSRDCMandaggregatingSRDCMandNSRLMM,arenotshownsincetheirresults aresimilartoaggregatingSRLMMandNSRLMMand/oraggregatingSRDCMandNSRDCM. 35 2.5.1 AggregatingtheSRLMMandSRDCM Tables 2.1 and 2.2 (cols. 2-7) summarize the average prediction errors, (2.28) and (2.29) (l = 1,2),andtheimprovementpercentages(2.30),forF WA andF GCI andall10producers. Observefromthetables,that: 1. (Tables2.1and2.2)TheWAandtheGCIaggregationmethodsprovidemostlyidentical production-ratepredictionerrorresults. 2. (Tables 2.1 and 2.2) Both aggregation methods are able to decrease the standard devia- tionoftheaveragepredictionerrors,i.e.,E a 1 andE a 2 overthe10producers. 3. (Table 2.1) In the best case, the WA improved the SRLMM by 8.65% (for Producer 9), andtheSRDCMby10.09%(forProducer10);whileintheworstcase,theWAimproved theSRLMMby0.83%(forProducer2),andtheSRDCMby1.24%(forProducer4). 4. (Table2.2)TheGCIimprovedtheSRLMMby9.02%(forProducer9)inthebestcase, andtheSRDCMby9.82%(forProducer10);whileintheworstcase,theGCIimproved theSRLMMby1.02%(forProducer2),andtheSRDCMby1.24%(forProducer4). 5. (Tables 2.1 and 2.2) On average, aggregation improves the SRLMM by around 4.6% andtheSRDCMby5.0%,whichwefeelisnotlargeenoughtojustifytheaggregation. 2.5.2 AggregatingtheNSRLMMandNSRDCM TheaverageresultsfortheWAandtheGCIaresummarizedinTables2.1and2.2(cols. 8-13). Observethat: 1. (Tables 2.1 and 2.2) The production-rate prediction error results using the WA and the GCIarealsomostlyidenticaltoeachother,asinthepreviouscase. 2. (Tables2.1and2.2)Inthiscase,aggregationimprovesthetwomodelslessthan1%for mostofthe10producers. 36 3. (Table 2.1) In the best case, the WA improved the NSRLMM by 10.99% (for Producer 9), and the NSRDCM by 4.46% (for Producer 9); while in the worst case, the WA providednoimprovementforbothNSRLMMandNSRDCM. 4. (Table2.2) Inthebestcase, the GCIimprovedthe NSRLMM by11.06%(for Producer 9), and the NSRDCM by 4.87% (for Producer 9); while in the worst case, as the WA, theGCImadenoimprovementtobothNSRLMMandNSRDCM. 5. (Tables 2.1 and 2.2) On average, the two aggregation methods improve the NSRLMM byaround1.3%andtheNSRDCMby0.8%,hardlyworththeeffortofaggregation. Table2.1: AveragepredictionerrorsresultswhenaggregatingtheSRLMMandSRDCM,and theNSRLMMandNSRLMMusingtheWA SRLMM SRDCM NSRLMM NSRDCM Producer E 1 (b/d) E a 1 (b/d) Imp a 1 E 2 (b/d) E a 2 (b/d) Imp a 2 E 3 (b/d) E a 3 (b/d) Imp a 3 E 4 (b/d) E a 4 (b/d) Imp a 4 1 90.7439 87.0628 4.06% 90.7421 83.0458 8.48% 100.3351 99.4523 0.88% 101.2240 100.0205 1.19% 2 84.8228 84.1206 0.83% 87.5299 85.0149 2.87% 90.0289 89.4169 0.68% 90.9896 89.6256 1.50% 3 68.6777 63.3526 7.75% 68.6968 63.4625 7.62% 64.1131 64.0838 0.05% 64.1137 64.0949 0.03% 4 37.7788 37.3092 1.24% 37.7820 37.3134 1.24% 37.8807 37.8772 0.01% 37.8815 37.8778 0.01% 5 45.1636 43.0968 4.58% 44.8669 43.2503 3.60% 43.7291 43.5851 0.33% 43.7480 43.6225 0.29% 6 8.1366 7.9270 2.58% 8.1366 7.9270 2.58% 7.1405 7.1405 0% 7.1405 7.1405 0% 7 46.3569 43.9301 5.24% 46.3644 43.9449 5.22% 43.8837 43.8816 0% 43.8855 43.8837 0% 8 57.4151 55.5948 3.17% 57.4368 55.6425 3.12% 60.3661 60.3398 0% 60.3761 60.3471 0% 9 109.8968 100.3906 8.65% 98.8611 93.7772 5.14% 108.9694 96.9957 10.99% 104.6845 100.0204 4.46% 10 74.8505 68.8476 8.02% 78.0524 70.1751 10.09% 76.3100 76.2820 0.04% 76.3138 76.2892 0.03% Mean 62.3843 59.1632 4.61% 61.8469 58.3554 5.00% 63.2757 61.9055 1.30% 63.0357 62.2993 0.75% STD 29.6021 27.5236 2.80% 28.2558 26.1827 2.89% 31.4648 29.5313 3.42% 31.0102 20.0402 1.41% Table2.2: AveragepredictionerrorsresultswhenaggregatingtheSRLMMandSRDCM,and theNSRLMMandNSRDCMusingtheGCI SRLMM SRDCM NSRLMM NSRDCM Producer E 1 (b/d) E a 1 (b/d) Imp a 1 E 2 (b/d) E a 2 (b/d) Imp a 2 E 3 (b/d) E a 3 (b/d) Imp a 3 E 4 (b/d) E a 4 (b/d) Imp a 4 1 90.7439 87.0931 4.02% 90.7421 82.9338 8.60% 100.3351 99.4280 0.90% 101.2240 99.9962 1.21% 2 84.8228 83.9557 1.02% 87.5299 84.8568 3.05% 90.0289 89.3706 0.73% 90.9896 89.6026 1.52% 3 68.6777 63.3193 7.80% 68.6968 63.4298 7.67% 64.1131 64.0836 0.04% 64.1137 64.0947 0.03% 4 37.7788 37.3081 1.25% 37.7820 37.3123 1.24% 37.8807 37.8771 0.01% 37.8815 37.8777 0.01% 5 45.1636 43.1102 4.55% 44.8669 43.2647 3.57% 43.7291 43.5837 0.33% 43.7480 43.6212 0.29% 6 8.1366 7.9271 2.58% 8.1366 7.9271 2.58% 7.1405 7.1405 0% 7.1405 7.1405 0% 7 46.3569 43.9266 5.24% 46.3644 43.9414 5.23% 43.8837 43.8815 0% 43.8855 43.8837 0% 8 57.4151 55.5840 3.19% 57.4368 55.6316 3.14% 60.3661 60.3392 0% 60.3761 60.3465 0% 9 109.8968 99.9835 9.02% 98.8611 93.6420 5.28% 108.9694 96.9191 11.06% 104.6845 99.5911 4.87% 10 74.8505 68.9461 7.89% 78.0524 70.3875 9.82% 76.3100 76.2805 0.04% 76.3138 76.2877 0.03% Mean 62.3843 59.1154 4.66% 61.8469 58.3327 5.02% 63.2757 61.8904 1.31% 63.0357 62.2442 0.80% STD 29.6021 27.4458 2.82% 28.2558 26.1424 2.84% 31.4648 29.5130 3.44% 31.0102 29.9755 1.54% 37 2.5.3 AggregatingtheSRLMMandNSRLMM TheaverageresultsfortheWAandGCIarealsoshowninTables2.3and2.4(cols. 2-7). Observethat: 1. (Tables2.3and2.4)Onaverage,theWAprovidessomewhatbetterresultsthantheGCI. 2. (Table 2.3) In the best case, the WA improved the SRLMM by 19.11% (for Producer 9), and the NSRLMM by 17.71% (for Producer 1); while in the worst case, the WA improved the SRLMM by 1.95% (for Producer 4), and the NSRLMM by 0.27% (for Producer6). 3. (Table 2.4) In the best case, the GCI improved the SRLMM by 19.33% (for Producer 9), and the NSRLMM by 18.33% (for Producer 1); while in the worst case, the GCI improved the SRLMM by 0.59% (for Producer 4), but worsened the NSRLMM by 0.27%(forProducer6). 4. (Tables 2.3 and 2.4) On average, the WA improves the SRLMM by 9.61% and the NSRLMM by 9.87%, whereas the GCI improves the SRLMM by 9.21% and the NSRLMMby9.50%,bothofwhicharelargeenoughtojustifytheaggregation. 2.5.4 AggregatingtheSRDCMandNSRDCM The average results for the WA and GCI are also summarized in Tables 2.3 and 2.4 (cols. 8-13). Observethat: 1. (Tables2.3and2.4)Onaverage,theWAprovidesslightlybetterresultsthantheGCI. 2. (Table 2.3) In the best case, the WA improved the SRDCM by 18.14% (for Producer 9), and the NSRDCM by 22.07% (for Producer 9); while in the worst case, the WA improved the SRDCM by 1.68% (for Producer 2), and the NSRDCM by 0.32% (for Producer6). 38 3. (Table 2.4) In the best case, the GCI improved the SRDCM by 18.13% (for Producer 9), and the NSRDCM by 22.06% (for Producer 9); while in the worst case, the GCI improved the SRDCM by 1.01% (for Producers 2 and 4), and the NSRDCM by 0.36% (forProducer6). 4. (Tables 2.3 and 2.4) On average, the WA improves the SRDCM by 9.64% and the NSRDCM by 10.38%, where the GCI improves the SRDCM by 9.36% and the NSRDCMby10.12%,bothofwhicharelargeenoughtojustifytheaggregation. Table 2.3: Average prediction errors results when aggregating the SRLMM and NSRLMM, andSRDCMandNSRDCMusingtheWA SRLMM NSRLMM SRDCM NSRDCM Producer E 1 (b/d) E a 1 (b/d) Imp a 1 E 3 (b/d) E a 3 (b/d) Imp a 3 E 2 (b/d) E a 2 (b/d) Imp a 2 E 4 (b/d) E a 4 (b/d) Imp a 4 1 90.6776 86.7552 4.33% 101.1077 83.1974 17.71% 90.6671 82.1511 9.39% 100.6843 82.1941 18.36% 2 85.3595 83.2903 2.42% 92.5048 83.7167 9.50% 85.2244 83.7899 1.68% 90.9750 83.8385 7.84% 3 69.0398 58.5817 15.15% 64.5295 58.7755 8.92% 69.2235 58.7659 15.11% 64.6654 58.7701 9.12% 4 36.7043 35.9898 1.95% 38.3500 35.9921 6.15% 36.7142 36.0585 1.79% 38.3333 36.0589 5.93% 5 45.0895 40.1815 10.88% 44.4611 40.3551 9.24% 44.5388 40.7916 8.41% 44.4637 40.8070 8.22% 6 8.2059 7.4158 9.63% 7.4356 7.4158 0.27% 8.2059 7.4116 9.68% 7.4356 7.4116 0.32% 7 46.3213 41.0598 11.36% 44.1416 41.0665 6.97% 46.3255 41.3803 10.67% 44.1389 41.3812 6.25% 8 56.8262 51.8763 8.71% 59.9733 51.9193 13.43% 56.9555 52.0999 8.53% 60.1704 52.1036 13.41% 9 109.1295 88.2733 19.11% 108.5958 91.7039 15.55% 98.7538 80.8360 18.14% 103.7892 80.8809 22.07% 10 74.4225 65.9549 12.55% 76.5702 68.1644 10.98% 77.7238 67.6405 12.97% 77.1251 67.6578 12.28% Mean 62.1776 55.9379 9.61% 63.7670 56.2307 9.87% 61.4332 55.0925 9.64% 63.5771 55.1104 10.38% STD 29.5993 26.0111 5.51% 31.5979 26.2107 4.98% 28.1216 24.5552 5.19% 30.7288 24.5720 6.36% Table 2.4: Average prediction errors results when aggregating the SRLMM and NSRLMM, andtheSRDCMandNSRDCMusingtheGCI SRLMM NSRLMM SRDCM NSRDCM Producer E 1 (b/d) E a 1 (b/d) Imp a 1 E 3 (b/d) E a 3 (b/d) Imp a 3 E 2 (b/d) E a 2 (b/d) Imp a 2 E 4 (b/d) E a 4 (b/d) Imp a 4 1 90.6776 86.5245 4.58% 101.1077 82.5768 18.33% 90.6671 82.4140 9.10% 100.6843 82.4556 18.10% 2 85.3595 83.4127 2.28% 92.5048 83.8390 9.37% 85.2244 84.3618 1.01% 90.9750 84.4082 7.22% 3 69.0398 58.4137 15.39% 64.5295 58.6228 9.15% 69.2235 58.5704 15.39% 64.6654 58.5747 9.42% 4 36.7043 36.4874 0.59% 38.3500 36.4896 4.85% 36.7142 36.3439 1.01% 38.3333 36.3443 5.19% 5 45.0895 40.8471 9.41% 44.4611 41.0204 7.74% 44.5388 41.0998 7.72% 44.4637 41.1154 7.53% 6 8.2059 7.4524 9.18% 7.4356 7.4525 -0.23% 8.2059 7.4087 9.71% 7.4356 7.4087 0.36% 7 46.3213 41.6873 10.00% 44.1416 41.6941 5.54% 46.3255 41.6125 10.17% 44.1389 41.6134 5.72% 8 56.8262 51.8368 8.78% 59.9733 51.8781 13.50% 56.9555 52.1485 8.44% 60.1704 52.1521 13.33% 9 109.1295 88.0395 19.33% 108.5958 91.4554 15.78% 98.7538 80.8534 18.13% 103.7892 80.8965 22.06% 10 74.4225 65.9748 12.53% 76.5702 68.1582 10.99% 77.7238 67.6727 12.93% 77.1251 67.6906 12.23% Mean 62.1776 56.0676 9.21% 63.7670 56.3187 9.50% 61.4332 55.2486 9.36% 63.5771 55.2660 10.12% STD 29.5993 27.4458 5.74% 31.5979 25.9822 5.49% 28.1216 24.6042 5.48% 30.7288 24.6204 6.46% 39 2.5.5 PartialConclusions So as to see the forest from the trees, the mean and standard deviation results over the ten producersforImp a l (l = 1-4)aresummarizedinTables2.5and2.6. Comparingalltheresults inTables2.1-2.6,thefollowingareobserved: 1. Forallfouraggregations,theredoesnotappeartobeanadvantageofWAoverGCI,or vice-versa. 2. Using both WA and GCI, the aggregation approach did a much better job when aggre- gatingtheSRLMMandNSRLMMandaggregatingtheSRDCMandNSRDCMthanit didintheothertwocases,intermsofaveragepredictionerror. 3. Forthestandarddeviationsoftheaveragedpredictionerrorsoverthetenproducers,the WAandGCIgaveverysimilarresults. Table2.5: SummarizedmeanandstdresultsofImp a l (l = 1,2,3,4)usingtheWA SRLMMandSRDCM NSRLMMandNSRDCM SRLMMandNSRLMM SRDCMandNSRDCM Imp a 1 Imp a 2 Imp a 3 Imp a 4 Imp a 1 Imp a 3 Imp a 2 Imp a 4 Mean 4.61% 5.00% 1.30% 0.75% 9.61% 9.87% 9.64% 10.38% STD 2.90% 2.89% 3.42% 1.41% 5.51% 4.98% 5.19% 6.36% Table2.6: SummarizedmeanandstdresultsofImp a l (l = 1,2,3,4)usingtheGCI SRLMMandSRDCM NSRLMMandNSRDCM SRLMMandNSRLMM SRDCMandNSRDCM Imp a 1 Imp a 2 Imp a 3 Imp a 4 Imp a 1 Imp a 3 Imp a 2 Imp a 4 Mean 4.66% 5.02% 1.31% 0.80% 9.21% 9.50% 9.36% 10.12% STD 2.82% 2.84% 3.44% 1.54% 5.74% 5.49% 5.48% 6.46% 2.5.6 AggregatingallFourIEKFs Thesametenproducerswereusedasabove,andtheaveragedresultsaresummarizedinTables 2.7and2.8. ObservethatonaveragetheWAimprovedthepredictionperformanceformostproducers overthefourIEKFs,whereastheGCIactuallymadethepredictionperformanceworse. Also, comparing the improvement percentage results given in Tables 2.5 and 2.6 (cols. 6-9) with thoseinTables2.7and2.8(cols. 4,7,10and13),observethattheWAmethodforaggregating 40 SRLMM and NSRLMM, and for aggregating SRDCM and NSRDCM, are better than those ofaggregatingallfourIEKFs. Table2.7: AveragepredictionerrorsresultswhenaggregatingallfourIEKFsusingtheWA SRLMM SRDCM NSRLMM NSRDCM Producer E 1 (b/d) E a 1 (b/d) Imp a 1 E 2 (b/d) E a 2 (b/d) Imp a 2 E 3 (b/d) E a 3 (b/d) Imp a 3 E 4 (b/d) E a 4 (b/d) Imp a 4 1 90.3876 92.5818 -2.43% 91.3462 87.9688 3.70% 100.9915 87.8040 13.06% 101.4345 87.9911 13.25% 2 84.4201 84.8028 -0.45% 85.7162 85.9353 -0.26% 91.7624 85.7391 6.56% 91.9500 85.9602 6.51% 3 69.2126 62.3040 9.98% 69.2235 62.3979 9.86% 64.6659 62.3899 3.52% 64.6654 62.4024 3.50% 4 36.7130 37.6582 -2.57% 36.7142 37.6607 -2.58% 38.3325 37.6604 1.75% 38.3333 37.6611 1.75% 5 44.6058 40.5295 9.14% 44.2988 40.6937 8.14% 44.2061 40.6490 8.05% 44.2286 40.7163 7.94% 6 8.2059 7.4185 9.60% 8.2059 7.4185 9.59% 7.4356 7.4185 0.23% 7.4356 7.4186 0.23% 7 46.3213 41.1929 11.07% 46.3251 41.2017 11.06% 44.1416 41.2005 6.66% 44.1438 41.2028 6.66% 8 56.9204 57.0680 -0.26% 56.9555 57.1184 -0.29% 60.1589 57.1139 5.06% 60.1704 57.1221 5.07% 9 108.7192 90.9534 16.34% 98.5159 97.0256 1.51% 108.0766 89.7943 16.92% 104.5872 97.2108 7.05% 10 75.5370 70.4698 6.71% 78.3417 69.5994 11.16% 77.4711 69.5950 10.17% 77.4743 69.6189 10.14% Mean 62.1043 58.4979 5.71% 61.5643 58.7020 5.19% 63.7242 57.9365 7.20% 63.4423 58.7304 6.21% STD 29.5029 27.3508 6.64% 28.2687 27.6990 5.33% 31.4859 26.6219 5.12% 31.0321 27.7320 3.85% Table2.8: AveragepredictionerrorsresultswhenaggregatingallfourIEKFsusingtheGCI SRLMM SRDCM NSRLMM NSRDCM Producer E 1 (b/d) E a 1 (b/d) Imp a 1 E 2 (b/d) E a 2 (b/d) Imp a 2 E 3 (b/d) E a 3 (b/d) Imp a 3 E 4 (b/d) E a 4 (b/d) Imp a 4 1 90.3876 99.7175 -10.32% 91.3462 93.9237 -2.82% 100.9915 93.7530 7.17% 101.4345 93.9496 7.38% 2 84.4201 103.7370 -22.88% 85.7162 105.4892 -23.07% 91.7624 105.2790 -14.73% 91.9500 105.5134 -14.75% 3 69.2126 71.0721 -2.69% 69.2235 71.1844 -2.83% 64.6659 71.1758 -10.07% 64.6654 71.1894 -10.09% 4 36.7130 42.6009 -16.04% 36.7142 42.6034 -16.04% 38.3325 42.6032 -11.14% 38.3333 42.6038 -11.14% 5 44.6058 44.5094 0.22% 44.2988 44.6596 -0.81% 44.2061 44.6191 -0.93% 44.2286 44.6788 -1.02% 6 8.2059 7.6658 6.58% 8.2059 7.6658 6.58% 7.4356 7.6658 -3.10% 7.4356 7.6658 -3.10% 7 46.3213 49.4634 -6.78% 46.3251 49.4709 -6.79% 44.1416 49.4696 -12.07% 44.1438 49.4718 -12.07% 8 56.9204 73.9291 -29.88% 56.9555 73.9744 -29.88% 60.1589 73.9701 -22.96% 60.1704 73.9777 -22.95% 9 108.7192 100.7061 7.37% 98.5159 109.9309 -11.59% 108.0766 102.2063 5.43% 104.5872 110.0753 -5.25% 10 75.5370 80.6407 -6.76% 78.3417 79.5362 -1.52% 77.4711 79.5320 -2.66% 77.4743 79.5543 -2.68% Mean 62.1043 67.4042 -8.12% 61.5643 67.8439 -8.88% 63.7242 67.0274 -6.51% 63.4423 67.8680 -7.57% STD 29.5029 31.1527 12.10% 28.2687 31.9508 11.24% 31.4859 30.8536 9.37% 31.0321 31.9767 8.45% 2.5.7 Conclusion Comparing the averaged prediction errors for all five experiments, our results demonstrate that,usingtheWAtoaggregatetheSRDCMandNSRDCMgivesthebestresults. 2.6 ComparingtheWAandGCI Weweresurprisedbytheresultsfromouraggregationprocessing,namelythatwhenweaggre- gated two models, the results from the WA and GCI aggregation methods were very close to 41 each other, whereas when we aggregated four models, the WA provided much better results thantheGCI.Inthissection,weprovidesomepossibleexplanationsfortheseobservations. When we aggregate two models, L = 2 in (2.19) and (2.20), so that g(A (l) ) (l = 1,2) in (2.20)canbeexpressedasg(A (1) )≡ g(X) = 1,g(A (2) ) = g σ(2) andg(A (3) ) = 0(seeSection 2.2.6). Therefore,F GCI (IPR 1 ,IPR 2 ,g 1 ,g 2 )canbedirectlycomputedas: F GCI (IPR 1 ,IPR 2 ,g 1 ,g 2 ) =IPR σ(1) (1−g σ(2) )+IPR σ(2) g σ(2) (2.31) whereσ(1) andσ(2) are indices such thatIPR σ(1) ≤ IPR σ(2) . Note that we do not have to solvethenonlinearequation(2.13)inthiscase. ForF WA (·)in(2.19),wehave: F WA (IPR 1 ,IPR 2 ,g 1 ,g 2 ) = P 2 l=1 IPR l g l P 2 l=1 g l =IPR 1 g 1 g 1 +g 2 +IPR 2 g 2 g 1 +g 2 (2.32) Withoutlossofgenerality,assumethatIPR 1 ≤IPR 2 andletg ′ 2 ≡ g 2 g 1 +g 2 . Then(2.32)canbe re-expressedas: F WA (IPR 1 ,IPR 2 ,g 1 ,g 2 ) =IPR 1 (1−g ′ 2 )+IPR 2 g ′ 2 (2.33) Observe that (2.33) has exactly the same form as (2.31), so when the weights g 1 and g 2 are optimized using the WA or GCI, these two methods should give the same or very similar results, just as they did in our experiment. The very small differences between these two methods shown in last section are due to the random nature of QPSO, as well as the additive measurementnoise. When we aggregate four models, L = 4 in (2.19) and (2.20), and the GCI is computed usingthestepsdescribedattheendofSection2.2.6. From(2.21),λcanbeexpressedasafunc- tionofthefuzzydensities,i.e.,λ =λ(g 1 ,...,g 4 );then,g(A σ(l) )(l = 1,...,4)canbecomputed iteratively using (2.22), where g(A σ(5) ) ≡ 0. Let h l [λ(g 1 ,...,g 4 )] ≡ g(A σ(l) )−g(A σ(l+1) ), 42 whereA σ(l) is the same as defined in Section 2.3.2, Step 4. F GCI (IPR 1 ,...,IPR 4 ,g 1 ,...,g 4 ) canbethenexpressedas: F GCI (IPR 1 ,...,IPR 4 ,g 1 ,...,g 4 ) = 4 X l=1 IPR σ(l) h l [λ(g 1 ,...,g 4 )] (2.34) where{σ(1),...,σ(4)}denotesareorderingof{1,...,4}suchthatIPR σ(1) ≤...≤IPR σ(4) . ForF WA (·),wehave: F WA (IPR 1 ,...,IPR 4 ,g 1 ,...,g 4 ) = P 4 l=1 IPR l g l P 4 l=1 g l = 4 X l=1 IPR l g l P 4 i=1 g i (2.35) Letg ′ l ≡ g l P 4 i=1 g i (l = 1,...,4). Since{σ(1),...,σ(4)} is just a reordering of{1,...,4}, (2.35) canbere-expressedas: F WA (IPR 1 ,...,IPR 4 ,g 1 ,...,g 4 ) = 4 X l=1 IPR σ(l) g ′ σ(l) (2.36) Notice that P 4 l=1 h l [λ(g 1 ,...,g 4 )]≡ 1, and P 4 l=1 g ′ σ(l) ≡ 1, i.e., the weights in (2.34) and (2.36) are both normalized. (2.34) and (2.36) have the same form; however, unlike g ′ l which is a simple non-linear function of the weights, h l [λ(g 1 ,...,g 4 )] is a complicated non-linear function of the weights, g l (l = 1,..,4). It may be that, when a random search algorithm is used (i.e., QPSO), the complicated nonlinearity of the GCI makes such an algorithm more difficulttolocatetheoptimumthanfortheWA,asdemonstratedbyourexperiments. 4 2.7 Summary In this Chapter, we have shown how the optimized WA and GCI can be used to aggregate differentIPRestimatesfromdifferentIEKFs,andthattheaggregatedIPRestimatesreducethe averagedfutureproductionpredictionerrorsforallIEKFs. Ourresultssuggestthatthenumber 4 We also conducted the same experiment for QPSO using 40 particles and 100 generations, and the results weresimilartowhatweshowedinthisChapter. 43 ofinformationsourcestobeaggregatedisanimportantfactorthataffectstheperformanceof the aggregation approach, and should be chosen carefully in practice. In fact, sometimes it is better to aggregate fewer than the maximum number of possible IPR estimates. Also, from our experiments, when aggregating two IEKFS, the optimized WA and GCI methods aresimilartoeachother,butwhenaggregatingfourIEKFS,theoptimizedWAismuchbetter than the optimized GCI with the same computational cost (same population size and number of generations in QPSO). Because it is much easier to understand the optimized WA, we recommendthatitshouldbeusedforthisapplication. We believe that the GCI is a better method than WA for obtaining improved predictions forapplicationsthatdonotrequirethosepredictionstobemadeinreal-time. 44 Chapter3 SimilarityMeasuresforGeneralType-2 FuzzySetsBasedontheα-Plane Representation 3.1 Introduction Asimilaritymeasurebetweenfuzzysetsisaveryimportantconceptinfuzzysettheory. There havebeenalotofdifferentsimilaritymeasuresproposedintheliterature,forbothT1FSsand IT2 FSs [25,81,121,123,140,145]. Many applications have been made use of these similar- ity measures, e.g., Buckley and Hayashi [5] and Turksen and Zhong [103] used a similarity measure between fuzzy sets for rule matching; Candan et al. [7] applied a similarity measure in multimedia data base query; Meng et al. [80] used a fuzzy set-valued similarity measure for a pattern recognition problem; Mohamed et al. [82] applied a new similarity measure to clustering problem, etc. In this Chapter, we propose a new similarity measure between GT2 FSsbasedontheα-planerepresentation. 3.2 Background This section reviews some important concepts in T2 FS theory and describes the α-plane representation of a GT2 FS developed by Liu [54] and Mendel et al. [71,76]. It also reviews WuandMendel’s[77,121,123]similaritymeasureforIT2FSs. 45 3.2.1 GT2FSs Definition4 A GT2 FS, denoted e A, is a bivariate function [1] on the Cartesian product µ : X×[0,1] into [0,1], whereX is the universe for theprimaryvariable of e A,x. The 3D MF of e Aisusuallydenotedµ e A (x,u),wherex∈X andu∈U = [0,1],i.e., e A = (x,u),µ e A (x,u) |∀x∈X,∀u∈ [0,1] (3.1) inwhich 0≤µ e A (x,u)≤ 1. e Acanalsobeexpressedas e A = Z x∈X Z u∈[0,1] µ e A (x,u)/(x,u) (3.2) where R R denotes union 1 over all admissiblex andu. For discrete universes of discourse R is replaced by P , and X and U by X d and U d . In (3.1) and (3.2) u is called the secondary variable,andhasdomainU = [0,1]ateachx∈X. In Definition 4, the first restriction that∀u ∈ [0,1] is consistent with the T1 constraint that 0 ≤ µ A (x) ≤ 1, i.e., when uncertainty disappears a T2 MF must reduce to a T1 MF, in which case the variable u equals 2 µ A (x) and 0 ≤ µ A (x) ≤ 1. The second restriction that 0≤ µ e A (x,u)≤ 1 is consistent with the fact that the amplitudes of a MF should lie between orbeequalto0and1. Definition5 Whenµ e A (x,u) = 1 for∀x∈ X and∀u∈ U then e A is called an intervalT2 FS (IT2FS). Definition6 At each value of x, say x = x ′ , the 2D plane whose axes are u and µ e A (x ′ ,u) is called a vertical slice of µ e A (x,u). A secondary MF is a vertical slice of µ e A (x,u). It is µ e A (x =x ′ ,u)forx ′ ∈X and∀u∈ [0,1],i.e., 1 RecallthattheunionoftwosetsAandBisbydefinitionanothersetthatcontainstheelementsineitherAor B.WhenwevieweachelementofaT2FSasasubset,thentheunionsin(3.2)conformtotheclassicaldefinition ofunion,sinceeachelementofthatsetisdistinct. Ataspecificvalueofxanduonlyonetermisactivatedinthe union. 2 Inthiscase,thethirddimensiondisappears. 46 µ e A (x =x ′ ,u)≡µ e A (x ′ ) = Z u∈J u x ′ ⊆[0,1] f x (u)/u (3.3) J u x ′ is the subset of U that is the support of µ e A (x ′ ), and is called the primary membership of e A. The amplitude of the secondary MF,f x (u), is called the secondary grade. The secondary grades of an IT2 FS are all equal to 1. Because∀x ′ ∈ X, we drop the prime notation on µ e A (x ′ ), and refer to µ e A (x) as 3 a secondary MF; it is a T1 FS, which we also refer to as a secondaryset. Based on the concept of secondary sets, a GT2 FS can be reinterpreted as the union of all secondarysets,i.e. e A = (x,µ e A (x))|∀x∈X (3.4) or,alternativelyas e A = Z ∀x∈X µ e A (x)/x = Z ∀x∈X Z ∀u∈J u x f x (u)/u /x (3.5) Definition7 UncertaintyintheprimarymembershipsofanIT2FS, e A,consistsofabounded region that is called the footprint of uncertainty (FOU). It is the two-dimensional support of e A,i.e.,[1] FOU( e A) = (x,u)∈X×U|µ e A (x,u)> 0 (3.6) FOU( e A)canalsobeexpressedastheunionofallprimarymemberships,i.e., FOU( e A) = [ x∈X J u x (3.7) This is a vertical-slice representation of the FOU, because each of the primary memberships isaverticalslice. 3 μ e A (x)isactuallyafunctionofsecondaryvariableu;hence,abetternotationforitisμ e A (u|x). Becausethe notationμ e A (x)isalreadywidelyusedbytheT2FScommunity,itisnotchangedhere. 47 Definition8 Theuppermembershipfunction(UMF)and lowermembershipfunction(LMF) of FOU( e A) are two T1 MFs that bound the FOU. The UMF is associated with the upper bound ofFOU( e A) andis denotedUMF FOU( e A) (x),∀x∈ X,and theLMF is associatedwith thelowerboundofFOU( e A)andisdenotedLMF FOU( e A) (x)∀x∈X,i.e.,[1] UMF FOU( e A) (x) = sup{u|u∈ [0,1],µ e A (x,u)> 0} ∀x∈X (3.8) LMF FOU( e A) (x) = inf{u|u∈ [0,1],µ e A (x,u)> 0} ∀x∈X (3.9) Note,also,thatJ u x canbeexpressed,as: J u x = n (x,u) :∀u∈ h LMF FOU( e A) (x),UMF FOU( e A) (x) io ⊆ [0,1] (3.10) For an IT2 FS, UMF FOU( e A) (x) ≡ UMF( e A) and LMF FOU( e A) (x) ≡ LMF( e A) (as in Fig. 3.1),andµ e A (x)andµ e A (x)areusuallyusedtodenotetheseT1upperandlowerMFs. Figure3.1: FOU(shaded),LMF(dashed)andUMF(solid)fortheIT2FS e A[74]. Liu [54] introduced the horizontal-slice decomposition (representation) of a GT2 FS. Becauseahorizontalsliceisanalogoustoanα-cut(raisedtolevelα)ofaT1FS,itiscalledthe 48 α-planerepresentationofaGT2FS[76],or,whenx,yandzareusedforthethreecoordinates ofaGT2FS,azSlicerepresentation[106,107]. Definition9 Anα-plane for the GT2 FS e A, denoted e A α , is the union of all primary member- shipsof e Awhosesecondarygradesaregreaterthanorequaltoα (0≤α≤ 1),i.e., e A α = (x,u),µ e A (x,u)≥α|∀x∈X,∀u∈ [0,1] = Z ∀x∈X Z ∀u∈[0,1] {(x,u)|f x (u)≥α} (3.11) Notethat FOU( e A) = e A 0 (3.12) and that, just as an α-cut of a T1 FS resides on its 1D domain axis X, e A α resides on its 2D domainaxesX×U. Each e A α canbeconvertedtoaspecialIT2FSR e Aα [71]where R e Aα (x,u) =α/ e A α ∀x∈X,u∈ [0,1] (3.13) ObserveR e Aα (x,u)raises e A α tolevel-αsothatR e Aα isanIT2FSallofwhosesecondaryMFs equal α (rather than 1 as would be the case for the usual IT2 FS). R e Aα is called an “α-level (zSlice)T2FS;”itisalsodesignatedas e A(α)[54]. AnexampleisprovidedinFig. 3.2[72]. Theorem1 Theα-plane(zSlice)representationforaGT2FSis: e A = [ α∈[0,1] R e Aα = [ α∈[0,1] α/ e A α = sup α∈[0,1] h α/ e A α i (3.14) Finally, we have the following connection between an α-plane and α-cuts of the secondary MFs: Definition10 Theα-cutofasecondaryMF,f x (u),isS e A (x|α)where: 49 Figure 3.2: 2-1/2D plot of α-plane representation of a GT2 FS [72]. The µ (x,u) direction appearstocomeoutofthepageandisthenewthirddimensionofaGT2FS. 50 S e A (x|α) = [s L (x|α),s R (x|α)] (3.15) S e A (x|α)isaverticalsliceof e A α . An example of S e A (x|α) that is raised to level α is depicted in Fig. 3.2. Note that for the orientationofthisfigures L (x|α)istheright-endofS e A (x|α),whereass R (x|α)istheleft-end ofS e A (x|α). This RT is very useful, because, since each e A α can be viewed as the FOU of an α-level T2 FS−a special IT2 FS−operations involving T2 FSs can be performed by using existing techniquesthathavealreadybeendevelopedforIT2FSs. 3.2.2 AnIT2FSSimilarityMeasure InthisChapter,usingtheα-planeRT,anewsimilaritymeasureforGT2FSsisproposed;itis an extension of a similarity measure for IT2 FSs suggested by Wu and Mendel [77,121]. In thissection,wefirstprovidefouraxiomaticpropertiesthatareconsideredverydesirablefora similaritymeasure,andthentheWu-MendelsimilaritymeasureforIT2FSs. Letsm( e A, e B)bethesimilaritymeasurebetweentwoIT2FSs e Aand e B. Thefollowingfouraxiomaticproperties[123]areconsidereddesirableforanIT2FSsim- ilaritymeasure: 1. Reflexivity: sm( e A, e B) = 1⇔ e A = e B. 2. Symmetry: sm( e A, e B) =sm( e B, e A). 3. Transitivity 4 : If e C ≤ e A≤ e B,thensm( e C, e A)≥sm( e C, e B). 4. Overlapping 5 : If e A∩ e B6=∅,thensm( e A, e B)> 0;otherwise,sm( e A, e B) = 0. 4 e A≤ e B ifμ e A (x)≤μ e B (x)andμ e A (x)≤μ e B (x)for∀x∈X. 5 Two IT2 FSs e A and e B overlap, i.e., e A∩ e B 6=∅, if∃x such that min(μ e A (x),μ e B (x)) > 0. e A and e B do not overlap,thatis, e A∩ e B =∅,if min(μ e A (x),μ e B (x)) = min(μ e A (x),μ e B (x)) = 0for∀x. 51 Jaccard’ssimilaritymeasure[40]fortwoT1FSs,AandB,sm J (A,B),is: sm J (A,B) = p(A∩B) p(A∪B) = R X µ A∩B (x)dx R X µ A∪B (x)dx (3.16) where p(A∩B) and p(A∪B) are the cardinalities of A∩B and A∪B, respectively, and, µ A∩B (x)andµ A∪B (x)aretheMFsofA∩B andA∪B,respectively. Wu and Mendel [121,123] proposed the following new crisp similarity measure between IT2FSs e Aand e B,sm J ( e A, e B),thatisbasedon(3.16): sm J ( e A, e B) ≡ AC( e A∩ e B) AC( e A∪ e B) ≡ [p(µ e A∩ e B (x))+p(µ e A∩ e B (x))]/2 [p(µ e A∪ e B (x))+p(µ e A∪ e B (x))]/2 = R X min(µ e A (x),µ e B (x))dx+ R X min(µ e A (x),µ e B (x))dx R X max(µ e A (x),µ e B (x))dx+ R X max(µ e A (x),µ e B (x))dx (3.17) whereAC( e A∩ e B) andAC( e A∪ e B) are the average cardinalities [77] ofA∩B andA∪B, respectively; µ e A∩ e B (x) and µ e A∩ e B (x) are the UMF and LMF of e A∩ e B, respectively; and, µ e A∪ e B (x) andµ e A∪ e B (x) arethe UMFandLMFof e A∪ e B, respectively. Notethateachintegral in (3.17) is an area, e.g., R X min(µ e A (x),µ e B (x))dx is the area under the minimum of µ e A (x) andµ e B (x). Closed-formsolutionsusuallycannotbefoundfortheseintegrals,sothefollowing discreteversionof(3.17)isused: sm J ( e A, e B) = P N i=1 min(µ e A (x i ),µ e B (x i ))+ P N i=1 min(µ e A (x i ),µ e B (x i )) P N i=1 max(µ e A (x i ),µ e B (x i ))+ P N i=1 max(µ e A (x i ),µ e B (x i )) (3.18) wherex i (i = 1,...,N)areequallyspacedinthesupportofA∪B. WuandMendel[77,121]provedthatsm J ( e A, e B) satisfiestheabovefouraxiomaticprop- erties. 52 3.3 ANewSimilarityMeasureforGT2FSs In this section, some desirable properties for a GT2 similarity measure are introduced; then, a new similarity measure for GT2 FSs using the α-plane RT is proposed; and, finally, some theorems are proved, that demonstrate the new similarity measure satisfies all the desirable properties when trapezoidal secondary MFs are used. Trapezoidal secondary MFs are very useful and include triangular secondary MFs as a special case; and, in this Chapter, only this kindofsecondaryMFisconsidered. 3.3.1 DesirablePropertiesforaGT2FSSimilarityMeasure Letsm( e A, e B)beacrispsimilaritymeasurebetweentwoGT2FSs e Aand e B. Definition11 e A≤ e Bifµ e Aα (x)≤µ e Bα (x)andµ e Aα (x)≤µ e Bα (x)for∀x∈X and∀α∈ [0,1]. Definition12 e A and e B overlap, i.e., e A ∩ e B 6= ∅, if ∃x,α ∈ [0,1] such that min(µ e Aα (x),µ e Bα (x)) > 0. e A and e B do not overlap, that is, e A ∩ e B = ∅, if min(µ e Aα (x),µ e Bα (x)) = min(µ e Aα (x),µ e Bα (x)) = 0for∀xandforallα∈ [0,1]. SimilartotheIT2FScase(seeSection3.2.2),thefollowingfouraxiomaticpropertiesare considereddesirableforasimilaritymeasureinvolvingGT2FSs: 1. Reflexivity: sm( e A, e B) = 1⇔ e A = e B. 2. Symmetry: sm( e A, e B) =sm( e B, e A). 3. Transitivity: If e A≤ e B≤ e C,thensm( e A, e B)≥sm( e A, e C). 4. Overlapping: If e A∩ e B6=∅,thensm( e A, e B)> 0;otherwise,sm( e A, e B) = 0. 3.3.2 Generalizationofsm J ( e A, e B)toGT2FSs Inthissection,wegeneralizesm J ( e A, e B)in(5.24)toGT2FSs. Letum( e A α )denoteanuncer- tainty measure for e A α , and um( e A) denote an uncertainty measure for GT2 FS, e A. Zhai and Mendel[142]provedthat: 53 um( e A) = [ α∈[0,1] α/um( e A α ) (3.19) Similarto(3.19),weproposeanewsimilarityforGT2FSs,as: SM J ( e A, e B) = [ α∈[0,1] α/sm J ( e A α , e B α ) (3.20) where e Aand e B aretwoGT2FSs,and e A α and e B α areα-planesof e Aand e B,respectively. For two GT2 FSs, e A and e B, the way to compute the similarity measure, sm J ( e A, e B), betweenthesetwoFSsis: 1. Decideonhowmanyα-planeswillbeused,whereα∈ [0,1]. CallthatnumberM,and itschoicewilldependontheaccuracythatisrequired. RegardlessofM,α = 1mustbe alwaysincluded. 2. Foreachα j (j = 1,...,M),compute e A α j and e B α j . 3. Computesm J ( e A α j , e B α j )using(5.24),as: sm J ( e A α j , e B α j ) = P N i=1 min(µ e Aα j (x i ),µ e Bα j (x i ))+ P N i=1 min(µ e Aα j (x i ),µ e Bα j (x i )) P N i=1 max(µ e Aα j (x i ),µ e Bα j (x i ))+ P N i=1 max(µ e Aα j (x i ),µ e Bα j (x i )) (3.21) Note that the accuracy of this result will depend on the discretization of the primary variable. 4. Repeatsteps2)and3)fortheM differentα j choseninStep1. 5. Collectallsm J ( e A α j , e B α j )(j = 1,...,M)andconstructaT1FSSM J ( e A, e B),as SM J ( e A, e B) = [ ∀α j α j /sm J ( e A α j , e B α j ) (3.22) 54 Note that this operation is analogous to how an uncertainty measure is computed in [142], and that SM J ( e A, e B) is a T1 FS. In (3.22), the union denotes the supremum (maximum)operationwithrespecttoα j . 6. ComputethecentroidofSM J ( e A, e B)toobtainacrispsimilaritymeasure,sm J ( e A, e B). If sm J ( e A α , e B α ) is a monotonic function of α, the supremum operation does not remove anyα-sm J ( e A α , e B α )pairobtainedinStep3,asshowninFigs. 3.3(a)and(b). Inthiscase,the centroidofSM J ( e A, e B),SM J ( e A, e B),iscomputedas: SM J ( e A, e B) = P M j=1 α j sm J ( e A α j , e B α j ) P M j=1 α j (3.23) Note that unlike the conventional centroid calculation, in which the primary variable (sm J ( e A α j , e B α j )) is equally discretized, (3.23) equally discretizes the α-axis. The two kinds of discretizations lead to the same results whenM is large enough [143]. In this Chapter, we usedM = 100. Ontheotherhand,ifsm J ( e A α , e B α )isnotamonotonicfunctionofα,thesupremumoper- ation will remove some of the α-sm J ( e A α , e B α ) pairs obtained in Step 3. For example, Fig. 3.3 (c) shows an enumeration of sm J ( e A α , e B α ) as a function of α, and Fig. 3.3 (e) depicts the corresponding T1 FSSM J ( e A α , e B α ). In this case, instead of using (3.23), SM J ( e A, e B) is computedusingthepointsremainingafterthesupremumoperationhasbeenperformed. LetS be the subset ofα j ’s remaining after the supremum operation, thenSM J ( e A, e B) is computed as: SM J ( e A, e B) = P ∀α i ∈S α j sm J ( e A α j , e B α j ) P ∀α i ∈S α j (3.24) 55 Figure 3.3: (a) and (c) illustrate two kinds of sm J ( e A α , e B α ) computed using (5.24) for all α∈ [0,1] (note that they are both functions ofα), wheresm J ( e A α , e B α ) in (a) is a monotonic function of α, whereas sm J ( e A α , e B α ) in (c) is not; (d) is a rotation of (c), where the hori- zontal axes issm J ( e A α , e B α ) and the vertical axes isα; and, (b) and (e) are the corresponding SM J ( e A, e B)(AT1FSobtainedusing(3.22))for (a)and (c),respectively. 3.4 Properties of The New Similarity Measure for Trape- zoidalSecondaryMFs Inthissection,twotheoremsareproved. Thefirstprovidesapreliminaryresultthatisusedin the proof of the second which in turn demonstrates that our new crisp similarity measure for GT2FSs,in(3.23),satisfiesallfouraxiomaticpropertiesdefinedinSection3.3.1. Theorem2 For two GT2 FSs, e A and e B, with trapezoidal secondary MFs, if e A ≤ e B, sm J ( e A α , e B α )isamonotonicfunctionofα (α∈ [0,1]). Proof: 56 Figure3.4: AtrapezoidalsecondaryMFatx =x i . A trapezoidal secondary MF for a GT2 FS e A (at x = x i ) is shown in Fig. 3.4. In this case, µ e A α ′ (x i ) and µ e A α ′ (x i ) can be expressed using µ e A 0 (x i ), µ e A 0 (x i ), µ e A 1 (x i ) and µ e A 1 (x i ), respectively,as(α ′ ∈ [0,1]): µ e A α ′ (x i ) =α ′ µ e A 1 (x i )+(1−α ′ )µ e A 0 (x i ) (3.25) and µ e A α ′ (x i ) =α ′ µ e A 1 (x i )+(1−α ′ )µ e A 0 (x i ) (3.26) If e A≤ e B,itfollows,fromDefinition11,thatµ e Aα (x)≤µ e Bα (x)andµ e Aα (x)≤µ e Bα (x)for ∀x∈X and∀α∈ [0,1]. Thesimilaritymeasurebetween e A α and e B α ,sm J ( e A α , e B α ),canthen beexpressed,from(5.24),andthen(3.25)and(3.26),as: 57 sm J ( e A α , e B α ) = P N i=1 µ e Aα (x i )+ P N i=1 µ e Aα (x i ) P N i=1 µ e Bα (x i )+ P N i=1 µ e Bα (x i ) = P N i=1 [αµ e A 1 (x i )+(1−α)µ e A 0 (x i )]+ P N i=1 [αµ e A 1 (x i )+(1−α)µ e A 0 (x i )] P N i=1 [αµ e B 1 (x i )+(1−α)µ e B 0 (x i )]+ P N i=1 [αµ e B 1 (x i )+(1−α)µ e B 0 (x i )] = α P N i=1 [µ e A 1 (x i )+µ e A 1 (x i )]+(1−α) P N i=1 [µ e A 0 (x i )+µ e A 0 (x i )] α P N i=1 [µ e B 1 (x i )+µ e B 1 (x i )]+(1−α) P N i=1 [µ e B 0 (x i )+µ e B 0 (x i )] (3.27) Let U A ≡ P N i=1 [µ e A 1 (x i ) + µ e A 1 (x i )], V A ≡ P N i=1 [µ e A 0 (x i ) + µ e A 0 (x i )], U B ≡ P N i=1 [µ e B 1 (x i )+µ e B 1 (x i )] andV B ≡ P N i=1 [µ e B 0 (x i )+µ e B 0 (x i )]; then, (3.27) can be re-written as: sm J ( e A α , e B α ) = αU A +(1−α)V A αU B +(1−α)V B (3.28) Computingthederivativeof(3.28)withrespecttoα,weobtain: dsm J ( e A α , e B α ) dα = [U A −V A ][αU B +(1−α)V B ]−[U B −V B ][αU A +(1−α)V A ] [αU B +(1−α)V B ] 2 = U A V B −V A U B [αU B +(1−α)V B ] 2 (3.29) For any given non-empty e A and e B, the denominator of (3.29) is always positive and the numerator is independent of α and is a constant value; hence, sm J ( e A α , e B α ) is a monotonic functionofα(α∈ [0,1]). Theorem3 ThecrispsimilaritymeasureSM J ( e A, e B)in(3.23),betweentwoGT2FSs e Aand e B, both of whichhave trapezoidalsecondary MFs, satisfies all four axiomatic propertiesof a similaritymeasurethataredescribedinSection3.3.1. Proof: WuandMendel[77,121]provedthatsm J ( e A α , e B α )(∀α∈ [0,1])satisfiesreflexiv- ity,symmetry,transitivityandoverlapping. Wewillusethesefactsdirectlyinourproof. 58 1. Reflexivity: Consider first the necessity, that is, SM J ( e A, e B) = 1 ⇒ e A = e B. SM J ( e A, e B) = 1 indicates that ∀α j (j = 1,...,M) e A α j = e B α j , from which it fol- lows that e A = e B. Consider next the sufficiency, that is, e A = e B ⇒ SM J ( e A, e B) = 1. When e A = e B, e A α j = e B α j for allα j ; hence, it follows thatsm J ( e A α j , e B α j ) = 1 for all α j . After the supremum operation in (3.22) is performed, SM J ( e A, e B) only contains a singletonvalueat1. Therefore,thecentroid,SM J ( e A, e B),isalsoequalto1. 2. Symmetry: We know that for all α j , sm J ( e A α j , e B α j ) = sm J ( e B α j , e A α j ); hence, from (3.23),SM J ( e A, e B) =SM J ( e B, e A)isobtained. 3. Transitivity: If e A ≤ e B ≤ e C (see Definition 11), when we apply Theorem 2 to e A and e B,andto e Aand e C,itfollowsthatSM J ( e A, e B)andSM J ( e A, e C)canbecomputedusing (3.23),as SM J ( e A, e B) = P M j=1 α j sm J ( e A α j , e B α j ) P M j=1 α j (3.30) and SM J ( e A, e C) = P M j=1 α j sm J ( e A α j , e C α j ) P M j=1 α j (3.31) In [77,121], is is proved that sm J ( e A α j , e B α j ) ≥ sm J ( e A α j , e C α j ) for all j = 1,...,M; hence,itfollowsthat SM J ( e A, e B)≥SM J ( e A, e C). 4. Overlapping: If e A ∩ e B 6= ∅ (see Definition 12), it follows that ∃α j such that sm J ( e A α j , e B α j ) > 0, from which it then follows that SM J ( e A, e B) > 0. If e A and e B donotoverlap,thatis, e A∩ e B =∅,wehaveforallj,sm J ( e A α j , e B α j ) = 0;consequently, SM J ( e A, e B) = 0. 3.5 NumericalExamples In this section, some examples are provided that use the new similarity measure described in theprevioussections. WestartwithanexampleusingtwodiscreteGT2FSs. 59 3.5.1 Example1: DiscreteGT2FSs ConsiderthefollowingtwoGT2FSswiththesameuniverseX ={1,2,3}: e A ={(0.3/0.6+0.9/0.8+0.7/0.9)/1,(0.6/0.4+0.9/0.7)/2,(0.5/0.3+1.0/0.4+0.4/0.6)/3} (3.32) and e B ={(0.3/0.5+0.9/0.6+0.4/0.8)/1,(0.7/0.5+0.3/0.6)/2,(0.6/0.2+1.0/0.3+0.7/0.6)/3} (3.33) Fig. 3.5depicts(a)µ e A (x,u)and(b)µ e B (x,u),andtheα-planewhenα = 0.5. Figure3.5: (a)µ e A (x,u)and(b)µ e B (x,u)inExample1. Inthisexample,tenα-planesareused(i.e.,M = 10asdescribedinSection3.3.2),where α = 0.1,0.2,...,1. Consider α = 1 first. According to Definition 9, it is straightforward to see thatµ e A 1 (3) = µ e A 1 (3) = 0.4 andµ e B 1 (3) = µ e B 1 (3) = 0.3, and for bothx = 1 andx = 2, µ e A 1 (x) =µ e A 1 (x) =µ e B 1 (x) =µ e B 1 (x) = 0. Using(5.24),weobtain: 60 sm J ( e A 1 , e B 1 ) = 0.3+0.3 0.4+0.4 = 0.75 (3.34) Next,considerα = 0.9. Inthiscase,for e A,wehaveµ e A 0.9 (1) =µ e A 0.9 (1) = 0.8;µ e A 0.9 (2) = µ e A 0.9 (2) = 0.7; and, µ e A 0.9 (3) = µ e A 0.9 (3) = 0.4. For e B, we haveµ e B 0.9 (1) = µ e B 0.9 (1) = 0.6; and,µ e B 0.9 (3) =µ e B 0.9 (3) = 0.3. Therefore, sm J ( e A 0.9 , e B 0.9 ) = 0.6+0+0.3+0.6+0+0.3 0.8+0.7+0.4+0.8+0.7+0.4 = 0.47 (3.35) Similarly,wehaveµ e A 0.8 (1) = µ e A 0.8 (1) = 0.8;µ e A 0.8 (2) = µ e A 0.8 (2) = 0.7;and,µ e A 0.8 (3) = µ e A 0.8 (3) = 0.4. For e B, we haveµ e B 0.8 (1) = µ e B 0.8 (1) = 0.6; and, µ e B 0.8 (3) = µ e B 0.8 (3) = 0.3. Therefore,wehave sm J ( e A 0.8 , e B 0.8 ) = 0.6+0+0.3+0.6+0+0.3 0.8+0.7+0.4+0.8+0.7+0.4 = 0.47 (3.36) whichisthesameasthecasewhereα = 0.9. Thisisbecausethereisnox,upairthathasMF valueµ x,u = 0.8. For α = 0.7, we have µ e A 0.7 (1) = 0.9; µ e A 0.7 (1) = 0.8; µ e A 0.7 (2) = µ e A 0.7 (2) = 0.7; and, µ e A 0.7 (3) = µ e A 0.7 (3) = 0.4. For e B,wehaveµ e B 0.7 (1) = µ e B 0.7 (1) = 0.6;µ e B 0.7 (2) = µ e B 0.7 (2) = 0.5;and,µ e B 0.7 (3) = 0.6;µ e B 0.7 (3) = 0.3. sm J ( e A 0.7 , e B 0.7 ) = 0.6+0.5+0.4+0.6+0.5+0.3 0.9+0.7+0.6+0.8+0.7+0.4 = 0.71 (3.37) Forα = 0.6,wehaveµ e A 0.6 (1) = 0.9;µ e A 0.6 (1) = 0.8;µ e A 0.6 (2) = 0.7;µ e A 0.6 (2) = 0.4;and, µ e A 0.6 (3) = µ e A 0.6 (3) = 0.4. For e B,wehaveµ e B 0.6 (1) = µ e B 0.6 (1) = 0.6;µ e B 0.6 (2) = µ e B 0.6 (2) = 0.5;and,µ e B 0.6 (3) = 0.6;µ e B 0.6 (3) = 0.2. sm J ( e A 0.6 , e B 0.6 ) = 0.6+0.5+0.4+0.6+0.4+0.2 0.9+0.7+0.6+0.8+0.5+0.4 = 0.69 (3.38) 61 For α = 0.5, we have µ e A 0.5 (1) = 0.9; µ e A 0.5 (1) = 0.8; µ e A 0.5 (2) = 0.7; µ e A 0.5 (2) = 0.4; and, µ e A 0.5 (3) = 0.4; µ e A 0.5 (3) = 0.3. For e B, we haveµ e B 0.5 (1) = µ e B 0.5 (1) = 0.6; µ e B 0.5 (2) = µ e B 0.5 (2) = 0.5;and,µ e B 0.5 (3) = 0.6;µ e B 0.5 (3) = 0.2. sm J ( e A 0.5 , e B 0.5 ) = 0.6+0.5+0.4+0.6+0.4+0.2 0.9+0.7+0.6+0.8+0.5+0.3 = 0.71 (3.39) For α = 0.4, we have µ e A 0.4 (1) = 0.9; µ e A 0.4 (1) = 0.8; µ e A 0.4 (2) = 0.7; µ e A 0.4 (2) = 0.4; and, µ e A 0.4 (3) = 0.6; µ e A 0.4 (3) = 0.3. For e B, we have µ e B 0.4 (1) = 0.8; µ e B 0.4 (1) = 0.6; µ e B 0.4 (2) =µ e B 0.4 (2) = 0.5;and,µ e B 0.4 (3) = 0.6;µ e B 0.4 (3) = 0.2. sm J ( e A 0.4 , e B 0.4 ) = 0.8+0.5+0.6+0.6+0.4+0.2 0.9+0.7+0.6+0.8+0.5+0.3 = 0.82 (3.40) For α = 0.3, we have µ e A 0.3 (1) = 0.9; µ e A 0.3 (1) = 0.6; µ e A 0.3 (2) = 0.7; µ e A 0.3 (2) = 0.4; and, µ e A 0.3 (3) = 0.6; µ e A 0.3 (3) = 0.3. For e B, we have µ e B 0.3 (1) = 0.8; µ e B 0.3 (1) = 0.5; µ e B 0.3 (2) = 0.6;µ e B 0.3 (2) = 0.5;and,µ e B 0.3 (3) = 0.6;µ e B 0.3 (3) = 0.2. sm J ( e A 0.3 , e B 0.3 ) = 0.8+0.6+0.6+0.5+0.4+0.2 0.9+0.7+0.6+0.6+0.5+0.3 = 0.86 (3.41) For α = 0.1 and α = 0.2, since there is no (x,u) pair that has µ (x,u) ≤ 0.2, we have sm J ( e A 0.2 , e B 0.2 ) =sm J ( e A 0.1 , e B 0.1 ) =sm J ( e A 0.3 , e B 0.3 ) = 0.86. All ten sm J ( e A α , e B α )-α pairs are depicted in Fig. 3.6. In order to obtain the T1 result SM J ( e A, e B), a fuzzy union operation needs to be performed (see Section 3.3.2). Fig. 3.7 depicts the MF of SM J ( e A, e B). In this example, SM J ( e A, e B) is also discrete, and can be expressedas:{0.3/0.86,0.4/0.82,0.6/0.69,0.7/0.71,0.9/0.47, 1.0/0.75}. 62 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 α smJ( e Aα, e Bα) Figure3.6: sm J ( e A α , e B α )-αpairresults,whereα = 0.1,0.2,...,1. 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 α smJ( e Aα, e Bα) Figure3.7: SM J ( e A, e B). Finally,thecentroidofSM J ( e A, e B),SM J ( e A, e B),iscomputedas: 63 SM J ( e A, e B) = 0.86×0.3+0.82×0.4+0.69×0.6+0.71×0.7+0.47×0.9+0.75×1 0.3+0.4+0.6+0.7+0.9+1.0 = 0.6846 (3.42) 3.5.2 Example 2: GT2 FSs with Same FOU but Different Secondary MFs TorepresentanFOU,weusethesamenine-parameternotationasin[77](seeFig. 3.8),where thefirstfourparametersdefinetheUMF,thelastfiveparametersdefinetheLMF,andtheninth parameteristheheightoftheLMF. Figure3.8: Theninepoints (a,b,c,d,e,f,g,i,h)torepresentanFOU. TheGT2FSsusedinthissectionwereselectedasfollows: 1. The FOUs of the words little and some were selected from the 32-word codebook in [56], and are depicted in Fig. 3.9. The nine parameters [77] of little and some are (0.38,1.58,3.50,5.62,1.79,2.20, 2.20,2.40,0.24)and (1.28,3.50,5.50,7.83,3.79,4.41,4.41,4.91,0.36),respectively. 64 0 2 4 6 8 10 0 0.25 0.5 0.75 1 x u 0 2 4 6 8 10 0 0.25 0.5 0.75 1 x u Little Some (a) (b) Figure3.9: FOUsforwords(a)littleand(b)some. 2. ThreeclassesofsecondaryMFswereused,namely,interval,triangularandtrapezoidal: (a) Interval secondary MFs satisfyf x (u) = 1 for allu∈ J x ,∀x∈ X. e A denotes our GT2 FS with little FOU and interval secondary MF; e B denotes our GT2 FS with someFOUandintervalsecondaryMFs. (b) TriangularsecondaryMFshavebaseequaltos R (x|0)−s L (x|0)andapexlocation Apex(x)parameterizedas[76](w = 0,0.25,0.5,0.75,1): Apex(x) =s L (x|0)+w[s R (x|0)−s L (x|0)] (3.43) An example of the triangular secondary MFs for the word some is shown in Fig. 3.10 (a) for all five w values. e C 1 - e C 5 denote the five GT2 FSs with little FOU and triangular secondary MFs whenw = 0,...,1, respectively; e D 1 - e D 5 denote the five GT2 FSs with some FOU and triangular secondary MFs when w = 0,...,1, respectively. 65 0 0.2 0.4 0.6 0.8 1 0 0.5 1 f x (u) u 0 0.2 0.4 0.6 0.8 1 0 0.5 1 f x (u) u (a) (b) w=0 w=0.25 w=0.5 w=0.75 w=1 Figure 3.10: (a) Five triangular secondary MFs and (b) five trapezoidal secondary MFs when x = 6inFig. 3.9(b). (c) Trapezoidal secondary MFs also have base equal tos R (x|0)−s L (x|0), and a top defined by the locations of the left and right end points,Ep l (x) andEp r (x), both ofwhichareparameterizedas[76](w=0,0.25,0.5,0.75,1): Ep l (x) =s L (x|0)+0.6w[s R (x|0)−s L (x|0)] (3.44) Ep l (x) =s R (x|0)−0.6(1−w)[s R (x|0)−s L (x|0)] (3.45) Anexampleof the trapezoidal secondary MFs for the word some is shownin Fig. 3.10 (b) for all five w values. e E 1 - e E 5 denote the five GT2 FSs with little FOU and trapezoidal secondary MFs whenw = 0,...,1, respectively; e F 1 - e F 5 denote the five GT2 FSs with some FOU and trapezoidal secondary MFs whenw = 0,...,1, respectively. In summary, for each FOU (little or some), there is one GT2 FS with interval secondary MF( e Aor e B),fiveGT2FSswithtriangularMFs( e C 1 - e C 5 or e D 1 - e D 5 )andanotherfiveGT2FSs withtrapezoidalsecondaryMFs( e E 1 - e E 5 or e F 1 - e F 5 )foratotalof22GT2FSs(11perword). In this section, similarity measure (3.22) is computed between each pair of the GT2 FSs withthe sameFOU,i.e., e A, e C 1 - e C 5 and e E 1 - e E 5 ;and, e B, e D 1 - e D 5 and e F 1 - e F 5 . Fortheword little: 66 Fig. 3.11 depicts SM J ( e C i , e C j ), i,j = 1,...,5, where the plot in the i th row and j th column correspondstoSM J ( e C i , e C j ). Fig. 3.12depictsSM J ( e E i , e E j ),i,j = 1,...,5,wheretheplotin thei th row andj th column corresponds toSM J ( e E i , e E j ). Fig. 3.13 depicts bothSM J ( e A, e C j ) and SM J ( e A, e E j ), j = 1,...,5, where the j th plot in (a) corresponds to SM J ( e A, e C j ), and the j th plot in (b) corresponds to SM J ( e A, e E j ). Fig. 3.14 depicts SM J ( e C i , e E j ), i,j = 1,...,5, wheretheplotinthei th rowandj th columncorrespondstoSM J ( e C i , e E j ). Forthewordsome, Fig. 3.15 depicts SM J ( e D i , e D j ), i,j = 1,...,5, where the plot in the i th row and j th column correspondstoSM J ( e D i , e D j ). Fig. 3.16depictsSM J ( e F i , e F j ),i,j = 1,...,5,wheretheplotin thei th row andj th column corresponds toSM J ( e F i , e F j ). Fig. 3.17 depicts bothSM J ( e B, e D j ) andSM J ( e B, e F j ), j = 1,...,5, where thej th plot in (a) corresponds toSM J ( e A, e D j ), and the j th plot in (b) corresponds to SM J ( e A, e F j ). Fig. 3.18 depicts SM J ( e D i , e F j ), i,j = 1,...,5, where the plot in the i th row and j th column corresponds to SM J ( e D i , e F j ). We remind the readerthatinallofthesefigures,thehorizontalaxisisSM J andtheverticalaxisisα. 2 0 0.2 0.4 0.6 0.8 1 0 0.5 1 0 0.2 0.4 0.6 0.8 1 0 0.5 1 0 0.2 0.4 0.6 0.8 1 0 0.5 1 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1 0 0.5 1 1.5 2 0 0.5 1 0.4 0.6 0.8 1 0 0.5 1 0.2 0.4 0.6 0.8 1 0 0.5 1 0.2 0.4 0.6 0.8 1 0 0.5 1 1 0.4 0.6 0.8 1 0 0.5 1 0 0.5 1 1.5 2 0 0.5 1 0.7 0.8 0.9 1 0 0.5 1 0.4 0.6 0.8 1 0 0.5 1 1 0.2 0.4 0.6 0.8 1 0 0.5 1 0.7 0.8 0.9 1 0 0.5 1 0 0.5 1 1.5 2 0 0.5 1 0.7 0.8 0.9 1 0 0.5 1 1 0.2 0.4 0.6 0.8 1 0 0.5 1 0.4 0.6 0.8 1 0 0.5 1 0.7 0.8 0.9 1 0 0.5 1 0 0.5 1 1.5 2 0 0.5 1 e C2 e C3 e C4 e C5 Figure3.11: SM J ( e C i , e C j ),i,j = 1,...,5,whereeachSM J ( e C i , e C j )hasbeencomputedusing (3.22). ThewordislittleandthesecondaryMFsaretriangles. 67 2 0.4 0.6 0.8 1 0 0.5 1 0.4 0.6 0.8 1 0 0.5 1 0.2 0.4 0.6 0.8 1 0 0.5 1 0.2 0.4 0.6 0.8 1 0 0.5 1 1 0 0.5 1 1.5 2 0 0.5 1 0.7 0.8 0.9 1 0 0.5 1 0.4 0.6 0.8 1 0 0.5 1 0.4 0.6 0.8 1 0 0.5 1 1 0.7 0.8 0.9 1 0 0.5 1 0 0.5 1 1.5 2 0 0.5 1 0.7 0.8 0.9 1 0 0.5 1 0.7 0.8 0.9 1 0 0.5 1 1 0.4 0.6 0.8 1 0 0.5 1 0.7 0.8 0.9 1 0 0.5 1 0 0.5 1 1.5 2 0 0.5 1 0.8 0.85 0.9 0.95 1 0 0.5 1 1 0.4 0.6 0.8 1 0 0.5 1 0.7 0.8 0.9 1 0 0.5 1 0.8 0.85 0.9 0.95 1 0 0.5 1 0 0.5 1 1.5 2 0 0.5 1 e E2 e E3 e E4 e E5 Figure3.12: SM J ( e E i , e E j ),i,j = 1,...,5,whereeachSM J ( e E i , e E j )hasbeencomputedusing (3.22). ThewordislittleandthesecondaryMFsaretrapezoids. 1 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 e C2 e C3 e C4 e C5 e E2 e E3 e E4 e E5 (a) (b) Figure 3.13: (a)SM J ( e A, e C j ) and (b)SM J ( e A, e E j )j = 1,...,5, where bothSM J ( e A, e C j ) and SM J ( e A, e E j )havebeencomputedusing(3.22). Thewordislittleandin(a)thesecondaryMFs aretriangles,whereasin(b)thesecondaryMFsaretrapezoids. Inboth(a)and(b) e AisanIT2 FS. 68 1 0 0.2 0.4 0.6 0.8 1 0 0.5 1 0 0.2 0.4 0.6 0.8 1 0 0.5 1 0 0.2 0.4 0.6 0.8 1 0 0.5 1 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1 0.4 0.6 0.8 1 0 0.5 1 0.4 0.6 0.8 1 0 0.5 1 0.4 0.6 0.8 1 0 0.5 1 0.2 0.4 0.6 0.8 1 0 0.5 1 1 0.7 0.8 0.9 1 0 0.5 1 0.7 0.8 0.9 1 0 0.5 1 0.7 0.8 0.9 1 0 0.5 1 0.7 0.8 0.9 1 0 0.5 1 1 0.4 0.6 0.8 1 0 0.5 1 0.7 0.8 0.9 1 0 0.5 1 0.7 0.8 0.9 1 0 0.5 1 0.7 0.8 0.9 1 0 0.5 1 1 0.2 0.4 0.6 0.8 1 0 0.5 1 0.4 0.6 0.8 1 0 0.5 1 0.7 0.8 0.9 1 0 0.5 1 0.8 0.85 0.9 0.95 1 0 0.5 1 e E2 e E3 e E4 e E5 Figure3.14: SM J ( e C i , e E j ),i,j = 1,...,5,whereeachSM J ( e C i , e E j )hasbeencomputedusing (3.22). The word is little and the row secondary MFs are triangles, whereas the column MFs aretrapezoids. 2 0 0.2 0.4 0.6 0.8 1 0 0.5 1 0 0.2 0.4 0.6 0.8 1 0 0.5 1 0 0.2 0.4 0.6 0.8 1 0 0.5 1 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1 0 0.5 1 1.5 2 0 0.5 1 0.4 0.6 0.8 1 0 0.5 1 0.2 0.4 0.6 0.8 1 0 0.5 1 0.2 0.4 0.6 0.8 1 0 0.5 1 1 0.4 0.6 0.8 1 0 0.5 1 0 0.5 1 1.5 2 0 0.5 1 0.7 0.8 0.9 1 0 0.5 1 0.4 0.6 0.8 1 0 0.5 1 1 0.2 0.4 0.6 0.8 1 0 0.5 1 0.7 0.8 0.9 1 0 0.5 1 0 0.5 1 1.5 2 0 0.5 1 0.7 0.8 0.9 1 0 0.5 1 1 0.2 0.4 0.6 0.8 1 0 0.5 1 0.4 0.6 0.8 1 0 0.5 1 0.7 0.8 0.9 1 0 0.5 1 0 0.5 1 1.5 2 0 0.5 1 e D2 e D3 e D4 e D5 Figure3.15: SM J ( e D i , e D j ),i,j = 1,...,5,whereeachSM J ( e D i , e D j )hasbeencomputedusing (3.22). ThewordissomeandthesecondaryMFsaretriangles. 69 2 0.7 0.8 0.9 1 0 0.5 1 0.4 0.6 0.8 1 0 0.5 1 0.2 0.4 0.6 0.8 1 0 0.5 1 0.2 0.4 0.6 0.8 1 0 0.5 1 1 0 0.5 1 1.5 2 0 0.5 1 0.7 0.8 0.9 1 0 0.5 1 0.4 0.6 0.8 1 0 0.5 1 0.4 0.6 0.8 1 0 0.5 1 1 0.7 0.8 0.9 1 0 0.5 1 0 0.5 1 1.5 2 0 0.5 1 0.7 0.8 0.9 1 0 0.5 1 0.7 0.8 0.9 1 0 0.5 1 1 0.4 0.6 0.8 1 0 0.5 1 0.7 0.8 0.9 1 0 0.5 1 0 0.5 1 1.5 2 0 0.5 1 0.8 0.85 0.9 0.95 1 0 0.5 1 1 0.4 0.6 0.8 1 0 0.5 1 0.7 0.8 0.9 1 0 0.5 1 0.8 0.85 0.9 0.95 1 0 0.5 1 0 0.5 1 1.5 2 0 0.5 1 e F2 e F3 e F4 e F5 Figure 3.16: SM J ( e F i , e F j ),i,j = 1,...,5, where eachSM J ( e F i , e F j ) has been computed using (3.22). ThewordissomeandthesecondaryMFsaretrapezoids. 1 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 e D2 e D3 e D4 e D5 e F2 e F3 e F4 e F5 (a) (b) Figure3.17: (a)SM J ( e B, e D j ) and (b)SM J ( e B, e F j )j = 1,...,5, wherebothSM J ( e B, e D j ) and SM J ( e B, e F j ) have been computed using (3.22). The word is some and in (a)the secondary MFs are triangles, where as in (b)the secondary MFs are trapezoids. In both (a) and (b) e A is anIT2FS. 70 1 0 0.2 0.4 0.6 0.8 1 0 0.5 1 0 0.2 0.4 0.6 0.8 1 0 0.5 1 0 0.2 0.4 0.6 0.8 1 0 0.5 1 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1 0.4 0.6 0.8 1 0 0.5 1 0.4 0.6 0.8 1 0 0.5 1 0.4 0.6 0.8 1 0 0.5 1 0.2 0.4 0.6 0.8 1 0 0.5 1 1 0.7 0.8 0.9 1 0 0.5 1 0.7 0.8 0.9 1 0 0.5 1 0.7 0.8 0.9 1 0 0.5 1 0.7 0.8 0.9 1 0 0.5 1 1 0.4 0.6 0.8 1 0 0.5 1 0.7 0.8 0.9 1 0 0.5 1 0.7 0.8 0.9 1 0 0.5 1 0.7 0.8 0.9 1 0 0.5 1 1 0.2 0.4 0.6 0.8 1 0 0.5 1 0.4 0.6 0.8 1 0 0.5 1 0.7 0.8 0.9 1 0 0.5 1 0.8 0.85 0.9 0.95 1 0 0.5 1 e F2 e F3 e F4 e F5 Figure3.18: SM J ( e D i , e F j ),i,j = 1,...,5,whereeachSM J ( e D i , e F j )hasbeencomputedusing (3.22). The word is some and the row secondary MFs are triangles, whereas the column MFs aretrapezoids. Tables 3.1 and 3.2 summarize the centroid of each of the T1 similarity functions shown in Figs. 3.11-3.14 (for little) and Figs. 3.15-3.18 (for some), respectively, computed using (3.23). By studying Figs. 3.11-3.18 and Tables 3.1 and 3.2, it is possible to observe the followingabouttheaxiomaticpropertiesofsimilarityforGT2FSs: 1. The entries of Tables 3.1 and 3.2 are symmetric with respect to its diagonal (symmetry axiom),andonlytheelementsalongthemaindiagonalareequalto1(reflexivityaxiom). 2. From the secondary MFs in Fig. 3.10, observe that e C 1 < e C 2 < ... < e C 5 , and e E 1 < e E 2 < ... < e E 5 . Accordingly, from, e.g., the second and seventh rows of Table 3.1, observe that SM J ( e C 1 , e C 2 ) > SM J ( e C 1 , e C 3 ) > SM J ( e C 1 , e C 4 ) > SM J ( e C 1 , e C 5 ), and,SM J ( e E 1 , e E 2 ) > SM J ( e E 1 , e E 3 ) > SM J ( e E 1 , e E 4 ) > SM J ( e E 1 , e E 5 ) (transitivity axiom). Similarly, e D 1 < e D 2 < ... < e D 5 , and e F 1 < e F 2 < ... < e F 5 , and we observe from the second and seventh rows of Table 3.2, that SM J ( e D 1 , e D 2 ) > SM J ( e D 1 , e D 3 ) > SM J ( e D 1 , e D 4 ) > SM J ( e D 1 , e D 5 ), and, SM J ( e F 1 , e F 2 )>SM J ( e F 1 , e F 3 )>SM J ( e F 1 , e F 4 )>SM J ( e F 1 , e F 5 )(transitivityaxiom). 71 3. AllGT2FSsusedinExample2sharethesameFOU,soallentriesinTables3.1and3.2 aregreaterthanzero(overlappingaxiom). 4. Comparing Tables 3.1 and 3.2, observe that the values of the corresponding entries are veryclose. BecausethesetwotablesaregeneratedusingtwodifferentFOUs,thismeans thesecondaryMFshavesimilareffectsondifferentFOUs. Table 3.1: Centroids of T1 similarity functions for all pairs among the 11 general T2 FSs havingthesameFOUasFOU(little)forExample2 e A e C 1 e C 2 e C 3 e C 4 e C 5 e E 1 e E 2 e E 3 e E 4 e E 5 e A 1 0.3567 0.4544 0.5252 0.5792 0.6218 0.6140 0.6516 0.6822 0.7076 0.7291 e C 1 0.3567 1 0.4860 0.3567 0.2879 0.2436 0.5288 0.4220 0.3567 0.3114 0.2776 e C 2 0.4544 0.4860 1 0.6784 0.5252 0.4327 0.6529 0.6989 0.6784 0.5762 0.5033 e C 3 0.5252 0.3567 0.6784 1 0.7626 0.6218 0.6140 0.7517 0.7750 0.7941 0.7291 e C 4 0.5792 0.2879 0.5252 0.7626 1 0.8109 0.4778 0.6202 0.7626 0.8194 0.8317 e C 5 0.6218 0.2436 0.4327 0.6218 0.8109 1 0.3949 0.5083 0.6218 0.7353 0.8487 e E 1 0.6140 0.5288 0.6529 0.6140 0.4778 0.3949 1 0.7523 0.6140 0.5233 0.4582 e E 2 0.6516 0.4220 0.6986 0.7517 0.6202 0.5083 0.7523 1 0.8070 0.6822 0.5936 e E 3 0.6822 0.3567 0.6784 0.7750 0.7626 0.6218 0.6140 0.8070 1 0.8411 0.7291 e E 4 0.7076 0.3114 0.5762 0.7941 0.8194 0.7353 0.5233 0.6822 0.8411 1 0.8645 e E 5 0.7291 0.2776 0.5033 0.7291 0.8317 0.8487 0.4582 0.5936 0.7291 0.8645 1 Table 3.2: Centroids of T1 similarity functions for all pairs among the 11 general T2 FSs havingthesameFOUasFOU(some)forExample2 e B e D 1 e D 2 e D 3 e D 4 e D 5 e F 1 e F 2 e F 3 e F 4 e F 5 e B 1 0.3902 0.4787 0.5438 0.5940 0.6338 0.6341 0.6681 0.6960 0.7193 0.7392 e D 1 0.3902 1 0.5281 0.3902 0.3158 0.2676 0.5730 0.4603 0.3902 0.3413 0.3046 e D 2 0.4787 0.5281 1 0.6951 0.5438 0.4507 0.6770 0.7169 0.6951 0.5946 0.5219 e D 3 0.5438 0.3902 0.6951 1 0.7719 0.6338 0.6341 0.7642 0.7853 0.8028 0.7392 e D 4 0.5940 0.3158 0.5438 0.7719 1 0.8169 0.4982 0.6351 0.7719 0.8261 0.8376 e D 5 0.6338 0.2676 0.4507 0.6338 0.8169 1 0.4141 0.5239 0.6338 0.7437 0.8535 e F 1 0.6341 0.5730 0.6770 0.6341 0.4982 0.4141 1 0.7687 0.6341 0.5440 0.4784 e F 2 0.6681 0.4603 0.7169 0.7642 0.6351 0.5239 0.7687 1 0.8171 0.6960 0.6088 e F 3 0.6960 0.3902 0.6951 0.7853 0.7719 0.6338 0.6341 0.8171 1 0.8480 0.7392 e F 4 0.7193 0.3413 0.5946 0.8028 0.8261 0.7437 0.5440 0.6960 0.8480 1 0.8696 e F 5 0.7392 0.3046 0.5219 0.7392 0.8376 0.8535 0.4784 0.6088 0.7392 0.8696 1 72 3.5.3 Example 3: GT2 FSs with Different FOUs and Triangular Sec- ondaryMFs Inthisexample,weusethesamenotationsasinExample2. Inordertoexaminetheeffectsof secondaryMFswhendifferentFOUsareused,weshowthesimilaritymeasureSM J ( e C i , e D j ) for all i,j = 1,...,5, in Fig. 3.19. Recall that as we move either from left to right in each row, or from top to bottom in each column, the triangular secondary MFs (see Fig. 3.10 (a)) move from a left-hand right triangle to a right-hand right triangle. In Fig. 3.19, as we explained in Section 3.3.2, three T1 results changed after the union operation in (3.22), namely: SM J ( e C 4 , e D 3 ), SM J ( e C 4 , e D 5 ), and SM J ( e C 5 , e D 4 ). The similarity and α pairs com- putedbeforetheunionoperationaredepictedinFig. 3.20. 0.4 0 0.1 0.2 0.3 0.4 0 0.5 1 0 0.1 0.2 0.3 0.4 0 0.5 1 0 0.1 0.2 0.3 0.4 0 0.5 1 0 0.1 0.2 0.3 0.4 0 0.5 1 0.5 0.33 0.34 0.35 0.36 0.37 0 0.5 1 0.25 0.3 0.35 0.4 0.45 0 0.5 1 0.2 0.25 0.3 0.35 0.4 0 0.5 1 0.1 0.2 0.3 0.4 0.5 0 0.5 1 0.5 0.3 0.32 0.34 0.36 0.38 0.4 0 0.5 1 0.3647 0.3647 0.3647 0.3647 0 0.5 1 0.33 0.34 0.35 0.36 0.37 0 0.5 1 0.25 0.3 0.35 0.4 0.45 0 0.5 1 0.4 0.25 0.3 0.35 0.4 0.45 0 0.5 1 0.34 0.35 0.36 0.37 0 0.5 1 0.36 0.37 0.38 0.39 0 0.5 1 0.355 0.36 0.365 0.37 0 0.5 1 0.4 0.2 0.25 0.3 0.35 0.4 0 0.5 1 0.3 0.32 0.34 0.36 0.38 0.4 0 0.5 1 0.364 0.366 0.368 0.37 0.372 0 0.5 1 e D2 e D3 e D4 e D5 0.36 0.37 0.38 0.39 0.4 0 0.5 1 Figure3.19: SM J ( e C i , e D j ),i,j = 1,...,5,whereeachSM J ( e C i , e D j )hasbeencomputedusing (3.22). 73 0.356 0.358 0.36 0.362 0.364 0.366 0.368 smJ( e C4α, e D3α) smJ( e C4α, e D5α) α smJ( e C5α, e D4α) (a) (b) (c) 0.357 0.358 0.359 0.36 0.361 0.362 0.363 0.364 0.365 0.366 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.3655 0.366 0.3665 0.367 0.3675 0.368 0.3685 0.369 Figure3.20: Resultsbeforesupremumoperationfor(a)SM J ( e C 4 , e D 3 )(b)SM J ( e C 4 , e D 5 ),and (c) SM J ( e C 5 , e D 4 ), where e C iα and e D jα denote the α-plane of e C i and e D j (i = 4,5 and j = 3,4,5),respectively. Table 3.3 summarizes the centroid of each of the T1 similarity functions shown in Fig. 3.19,ascomputedusing(3.23). BystudyingFig. 3.19andTable3.3,oneisabletomakethefollowingobservations: 1. From Fig. 3.19, observe that when sm J ( e C α , e D α ) is a monotonic function of α (ver- tical axis), it is not always an increasing [SM J ( e C 4 , e D 4 )] or a decreasing function [SM J ( e C 4 , e D 3 )]. 2. SM J ( e C 5 , e D 4 ) is a very interesting example that demonstrates the effects of the union operation used in (3.22). To ComputeSM J ( e C 5 , e D 4 ), according to the supremum oper- ation, we only keep theα-sm J ( e C α , e D α ) pairs having the largestα value; this results in the T1 similarity function shown in the 5 th row and 4 th column in Fig. 3.19. Note that because of the values of the α-sm J ( e C α , e D α ) pairs (see Fig. 3.20 (c)), the resulting T1 similarityfunctionSM J ( e C 5 , e D 4 )isdiscontinuous. 3. Comparing Table 3.3 with the 2-3 and 3-2 blocks of Tables 3.1 and 3.2, observe that the values of the entries in Table 3.3 are generally smaller than those in Tables 3.1 and 74 3.2. AllGT2FSsusedinTables3.1and3.2havethesameFOU,whereastwodifferent FOUs areused inTable3.3. It seems obvious therefore thatGT2FSswithsimilar(or thesame)FOUsaremoresimilarthanGT2FSswithlesssimilarFOUs. Table3.3: CentroidsofT1similarityfunctionsforSM J ( e C i , e D j )(i,j = 1,...,5). e D 1 e D 2 e D 3 e D 4 e D 5 e C 1 0.2663 0.2180 0.1845 0.1600 0.1413 e C 2 0.2663 0.3473 0.3136 0.2738 0.2441 e C 3 0.2403 0.3348 0.3647 0.3461 0.3153 e C 4 0.2121 0.3125 0.3560 0.3731 0.3611 e C 5 0.1873 0.2868 0.3376 0.3673 0.3782 SimilarresultswereobtainedforlittleandsomeandtrapezoidalsecondaryMFs. 3.6 SimilarityPercentage As mentioned in the Introduction, GT2 FSs are becoming more and more popular. Of great interest to us is learning whether or not there is much difference between a GT2 FS and an IT2 FS. In this section, we use the similarity results between a GT2 FS with triangular ( e C’s and e E’s) or trapezoidal ( e D’s and e F’s) secondary MFs and the IT2 FS with the same FOU ( e A and e B)toanswertheabovequestion. Tobemoreprecise,weusetheresultsinthefirstrowof Tables3.1and3.2. Of course, it is the variability of the secondary MFs of a GT2 FS that makes a GT2 FS different from an IT2 FS, when that GT2 FS has the same FOU as the IT2 FS. So, in order to obtain a better understanding of the effects of the secondary MFs, we define the following similaritypercentageP( e A, e B)fortwoGT2FSs e Aand e B,as: P( e A, e B) = |SM J ( e A, e B)−sm J ( e A 0 , e B 0 )| sm J ( e A 0 , e B 0 ) ×100% (3.46) 75 ThelargerP( e A, e B)is,themoreeffectsthesecondaryMFshaveonthesimilaritybetween e Aand e B. The e A and e B used in this section always have the same FOU, i.e., sm J ( e A 0 , e B 0 ) ≡ 1. In thiscase,(3.46)canbere-writtenas: P( e A, e B) =|SM J ( e A, e B)−1|×100% (3.47) P( e A, e C j ) and P( e A, e E j ) (j = 1,...,5) are consolidated from Table 3.1 in Table 3.4; and, P( e B, e D j )andP( e B, e F j )(j = 1,...,5)areconsolidatedfromTable3.2in3.5. Table3.4: P( e A, e C j )andP( e A, e E j )(j = 1,...,5)resultsforlittle e C 1 e C 2 e C 3 e C 4 e C 5 e E 1 e E 2 e E 3 e E 4 e E 5 e A 64.30 54.56 47.48 42.08 37.82 38.60 34.84 31.78 29.24 27.09 Table3.5: P( e B, e D j )andP( e B, e F j )(j = 1,...,5)resultsforsome e D 1 e D 2 e D 3 e D 4 e D 5 e F 1 e F 2 e F 3 e F 4 e F 5 e B 60.98 52.13 45.62 40.60 36.62 36.59 33.19 30.40 28.07 26.08 FromTables3.4and3.5,observethat: Almostallofthesimilaritypercentagesaregreater than 30%, which means (in our opinion) secondary MFs can make GT2 FSs very different fromtheirFOUs,i.e.,anIT2FS. Hence, our answer to the question: “Is there much difference between a GT2 FS and an IT2FS?”isyes. ThissupportsresearcherswhoareveryactivelyresearchingGT2FSs. 3.7 Summary InthisChapter,anewsimilaritymeasureforGT2FSsisproposedusingtheα-planerepresen- tation;itisanextensionofWuandMendel’ssimilaritymeasureforIT2FSs. Someproperties ofthenewsimilaritymeasureareproved;numericalexamplesaregiven,andtheeffectsofthe secondary MFS in different scenarios are discussed. Our overall conclusion about “Is there 76 much of a difference between a GT2 FS and an IT2 FS” can be answered by using our new similarity measure. From our simulations, we showed that the secondary MFs can make a GT2FSverydifferentfromitsFOU(anIT2FS). WehaveonlyusedtriangularandtrapezoidalsecondaryMFsinthisChapter(becausethey are, in our opinion, the most reasonable ones to use). We leave the extension of our work to otherkindsofsecondaryMFsasanopenproblem. 77 Chapter4 EncodingWordsIntoNormalInterval Type-2FuzzySets 4.1 Introduction In this Chapter, we propose a new method−the HM method−to further improve the perfor- mance of the Enhanced Interval Approach (EIA) by using more information from the data intervals, namely, the overlap shared by all those data intervals. The HM method uses the samebaddataprocessing,outlierprocessingandtolerancelimitprocessingtopre-processthe data intervals, as is used in the EIA; it then uses a new confidence-interval-based reasonable interval test to keep only those data intervals that share a common interval. In the Fuzzy Set Part, the common overlap is first determined for a group of data intervals; the IT2 FS model forawordisthendeterminedfromtheremainingdataintervalsthatexcludetheoverlap. The resulting IT2 FSs have both normal lower and upper membership functions (MFs), which makesthemuniqueamongIT2FSwordmodels. 4.2 Encoding Words Into Normal Interval Type-2 Fuzzy Sets: HMMethod The HM method consists of two parts (as in the EIA): the Data Part and the Fuzzy Set Part. For each word, it is assumed thatn data intervals, [a (i) ,b (i) ],i = 1,...,n, have been collected fromagroupofnsubjects. 78 4.2.1 DataPart TheDataPartconsistsofthefollowingthreesteps: 1. PerformtheEIA[126]baddataprocessing(reducingthenumberofdataintervalsfrom n to n ′ ), outlier processing (reducing the number of data intervals from n ′ to m ′ ) and tolerancelimitprocessing(reducingthenumberofdataintervalsfromm ′ tom ′′ ). 2. Modify the EIA reasonable interval processing (ensures overlapping intervals and that those intervals are not overly long) to account for uncertainties that are associated with using sample mean and sample standard deviation, using a three-step procedure that is explainednext. The EIA assumes that both the left and right data interval end points are normally distributed, and that each reasonable interval must contain the point that best separates thetwonormaldistributionswhichoccursattheirinnerintersection. Todothis,onehas tocomputetwovaluesofξ ∗ ,givenby ξ ∗ ={(m b σ a −m a σ b )±σ a σ b [(m a −m b ) 2 +2(σ 2 a −σ 2 b )ln(σ b /σ a )] 1/2 }/(σ 2 a −σ 2 b ) (4.1) andthenchoosetheuniqueonevalueforwhich m a ≤ξ ∗ ≤m b (4.2) In(4.1)and(4.2)m a andσ a arethemeanandstandarddeviationofleftendpointsa (i) , respectively,andm b andσ b arethemeanandstandarddeviationofrightendpointsb (i) . 79 In the EIA, m a , σ a , m b and σ b are estimated using sample mean and standard devi- ations, i.e., ˆ m a , ˆ σ a , ˆ m b and ˆ σ b , respectively. Then, the EIA only keeps data intervals [a (i) ,b (i) ]suchthat 1 2ˆ m a − ˆ ξ ∗ ≤a (i) < ˆ ξ ∗ <b (i) ≤ 2ˆ m b − ˆ ξ ∗ (4.3) Because ˆ ξ, ˆ m a and ˆ m b are used in (4.3) instead of ξ, m a and m b , there are lots of uncertainties associated with (4.3). The EIA does not account for the uncertainties, but theHMdoes. In the HM method, instead of a single point, we find an interval of ˆ ξ ∗ , [ ˆ ξ ∗ , ˆ ξ ∗ ], so that only data intervals that cover this interval are kept. To do this, rather than using the sample mean and standard deviation to estimate the four parameters, we compute confidenceintervals(CI)fortheseparameters. Bychoosingdifferentvaluesofthemean and standard deviation within their CIs, we obtain an interval of ˆ ξ ∗ , [ ˆ ξ ∗ ˆ ξ ∗ ], using (4.1) and(4.2),asexplainednext. (a) Compute 100(1−α)% CIs of mean and standard deviation of left end pointsa (i) (i = 1,...,m ′′ ),[ˆ m l , ˆ m l ]and[ˆ σ l ,ˆ σ l ],respectively. Similarly,Compute100(1−α)% CIsofmeanandstandarddeviationofrightendpointsb (i) (i = 1,...,m ′′ ),[ˆ m r , ˆ m r ] and [ˆ σ r ,ˆ σ r ],respectively. Recall [12] that 100(1−α)% CIs of the meanµ and standard deviationσ of a randomsampleofsizenfromanormaldistributionarecomputedas: [µ ,µ ] = [µ −z α/2 σ/ √ n,µ +z α/2 σ/ √ n] (4.4) and [σ,σ] = [ s (n−1)σ 2 χ α/2,n−1 , s (n−1)σ 2 χ 1−α/2,n−1 ] (4.5) 1 TheGaussiandistributionsfora (i) andb (i) havethreeintersectionsthatoccurat2m a −ξ ∗ ,ξ ∗ and2m b −ξ ∗ . 80 wherez α/2 is the upper 100α/2 percentage point of the standard normal distribu- tion,χ p,n−1 isthecritical valueoftheχ 2 distributionwith p-valueequalstop and n−1 degrees of freedom. In (4.5), only twop values are used,α/2 and 1−α/2. Different values of α can be used; however, in this paper, we use α = 0.05, i.e., wecomputethe 95%CIsofthedistributionparameters. (b) Computetheinterval[ ˆ ξ ∗ , ˆ ξ ∗ ]byconsideringthecaseswherethemeanandstandard deviation of the two Gaussian distributions are chosen from one of the end points of their CI (i.e., m a = ˆ m l or ˆ m l , σ a = ˆ σ l or ˆ σ l , m b = ˆ m r or ˆ m r , σ b = ˆ σ r or ˆ σ r ). There are 16 combinations of these parameters. 2 For each of the combinations, a value ofξ ∗ j (j = 1,...,16) is computed using (4.1) and (4.2). The interval [ ˆ ξ ∗ , ˆ ξ ∗ ] isthencomputedas: [ ˆ ξ ∗ , ˆ ξ ∗ ] = [min j ˆ ξ ∗ j ,max j ˆ ξ ∗ j ] (4.6) Fig. 4.1 depicts two examples to show the relationship between one single parameter(m a orσ a )andξ ∗ . Recallthat[56]ξ ∗ occursattheinnerintersectionof p(a (i) ) and p(b (i) ). In Fig. 4.1 (a), ξ ′ corresponds to the ξ ∗ computed using (4.1) and (4.2) whenm a = ˆ m l , andξ ′′ corresponds to theξ ∗ computed using (4.1) and (4.2)whenm a = ˆ m l . Noticethatwhenm a ∈ [ˆ m l , ˆ m r ],ξ ∗ willalwaysbebetween interval [ξ ′ ,ξ ′′ ], which means, when all other parameters (i.e., σ a , m b and σ b ) are fixed,inordertoobtainarangeoftheξ ∗ foranuncertainmeanm a ∈ [ˆ m l , ˆ m l ],we onlyneedtocomputetwoξ’s,oneform a = ˆ m l andoneform a = ˆ m l . Similarlyin Fig. 4.1(b),ξ ′ correspondstotheξ ∗ computedusing(4.1)and(4.2)whenσ a = ˆ σ l , andξ ′′ corresponds to theξ ∗ computed using (4.1) and (4.2) whenσ a = ˆ σ l . From Fig. 4.1 (b), when m a , m b and σ b are fixed, the range of the ξ ∗ for an uncertain σ a ∈ [ˆ σ l ,ˆ σ l ] is computed as [ξ ′ ,ξ ′′ ]. Similar results can also be obtained for m b andσ b . 2 Wehavefourparametersandeachofthemhavetwooptions,sothetotalnumberofcombinationsis2 4 = 16. 81 Figure4.1:Examplestoshowtherelationshipbetween(a)m a andξ ∗ ,and(b)σ a andξ ∗ . 82 (c) Remove data intervals that do not contain the interval [ ˆ ξ ∗ , ˆ ξ ∗ ], and remove data intervals that are overly long as in the EIA by removing those that do not satisfy (4.3)when ˆ ξ ∗ = ˆ ξ ∗ and ˆ ξ ∗ = ˆ ξ ∗ ,respectively. 3 As a result of the new reasonable interval test, some of the m ′′ data intervals maybediscardedandtherewillfinallybemremainingdataintervals. 3. EstablishthenatureoftheFOUusingtoleranceintervalsforeachword. Thisisdoneby first recomputing the tolerance interval, as in Tolerance Limit Processing [77], for end pointsa (i) andb (i) (i = 1,...,m),namely, [a,a]and [b,b],respectively,as: [a,a] = [ˆ m l −kˆ σ l , ˆ m l +kˆ σ l ] (4.7) and [b,b] = [ˆ m r −kˆ σ r , ˆ m r +kˆ σ r ] (4.8) where ˆ m l and ˆ σ l are the sample mean and standard deviation ofa (i) , ˆ m r and ˆ σ r are the samplemeanandstandarddeviationofb (i) ,k isthetolerancefactorandk = 2.549was used so that one can assert with 95% confidence that the given limits contain at least 95%oftheendpoints. Next, a word is classified as follows: when a ≤ 0, this indicates uncertainty to the left of 0, and the word is classified as a left-shoulder FOU; when b ≥ 10, this indicates uncertainty to the right of 10, and the word is classified as a right-shoulder FOU;otherwise,thewordisclassifiedasaninteriorFOU. 4.2.2 FuzzySetPart TheFuzzySetPartconsistsofthefollowingtwosteps: 3 Notethat,for ˆ ξ ∗ = ˆ ξ ∗ , ˆ m a and ˆ m b usedin(4.3)arethoseusedtocompute ˆ ξ ∗ in(4.1). Similarly,for ˆ ξ ∗ = ˆ ξ ∗ , ˆ m a and ˆ m b usedin(4.3)arethoseusedtocompute ˆ ξ ∗ in(4.1). 83 Figure4.2: InteriorFOU. 1. Determine the overlap [o l ,o r ] for each word. For an interior FOU, this is computed as [o l ,o r ] = [max i a (i) ,min i b (i) ] (i = 1,...,m). For a left-shoulder FOU, since there is no uncertainty to the left, the overlap is computed as [o l ,o r ] = [0,min i b (i) ]. For a right-shoulder, there is no uncertainty to the right, the overlap is computed as [o l ,o r ] = [max i a (i) ,10]. 2. Determine the FOU using the means and standard deviations of the remaining subsets ofintervals. (a) For an interior FOU, after the overlap interval [o l ,o r ] is determined, the FOU is constructed as shown in Fig. 4.2, where the left-part FOU, e L, corresponds to the uncertainty of the remaining intervals [a (i) ,o l ] (i = 1,...,m) and the right- part FOU, e R, corresponds to the uncertainty of the remaining intervals [o r ,b (i) ] (i = 1,...,m). To complete the description of an interior FOU, we need to compute a l , a r , b l andb r . Howtodothisisexplainednext. 84 IntheHMmethod,asintheEIA,alldataintervalsareassumedtobeuniformly distributed;hence,foraninterval [α,β],itsmeanandstandarddeviationarecom- putedas[77]: m = α+β 2 (4.9) s = β−α √ 12 (4.10) The two corresponding uncertainty measures of an FOU are its centroid and standarddeviation[120]. RecallthatthecentroidofanIT2FS e Aisdefinedas: C e A = [ ∀Ae c(A e ) = [c l ( e A),c r ( e A)] (4.11) where S istheunionoperation,and c l ( e A) = min ∀Ae c(A e ) (4.12) c r ( e A) = max ∀Ae c(A e ) (4.13) whereA e isanembeddedT1FSof e A,andc(A e )isthecentroidofA e . Recall,also,thatthevariance[120]ofanIT2FSisdefinedas: V e A = [ ∀Ae v e A (A e ) = [v l ( e A),v r ( e A)] (4.14) wherev l ( e A)andv r ( e A)aretheminimumandmaximumrelativevarianceofallA e , respectively,i.e., v l ( e A) = min ∀Ae v e A (A e ) (4.15) 85 v r ( e A) = max ∀Ae v e A (A e ) (4.16) Finally,thestandarddeviationofanIT2FSisdefinedas[120]: STD( e A) = q v l ( e A), q v r ( e A) (4.17) For the left-part FOU e L in Fig. 4.2, because all its embedded T1 FSs share the samepoint 4 o l ,wedonotneedtoperformanyiterativecomputationstodetermine itscentroidandstandarddeviation;theycanbeobtainedbydirectlycomputingthe centroidandstandarddeviationoftheLMFandUMF,as(seeAppendixF): C e L = [c(UMF( e L)),c(LMF( e L))] (4.18) and STD e L = [s(LMF( e L)),s(UMF( e L))] = q v e L (LMF( e L)), q v e L (UMF( e L)) (4.19) whereLMF( e L)andUMF( e L)aretheLMFandUMFof e L,respectively. Observe thatLMF( e L)isthelinesegmenta r o l andUMF( e L)isthelinesegmenta l o l . Using results from Table 3.3 in [77], the formulas to compute the mean and standarddeviationfortheLMFandUMFof e Lare: c(LMF( e L)) = 2o l +a r 3 (4.20) 4 MendelandJohn’sRepresentationtheorem[70,73,75]forIT2FSsstatesthattheFOUistheunionofallits embeddedT1FSs. Wu[115]hasshownthatcertainFOUscanalsobecoveredbyasubsetofembeddedT1FSs. Inthispaper,weusesuchasubset,namely,allstraightlinesthatpassthrougho l (o r ). 86 c(UMF( e L)) = 2o l +a l 3 (4.21) s(LMF( e L)) = " 1 6 (o l +a r ) 2 +2o 2 l − h c(LMF( e L)) i 2 # 1/2 (4.22) s(UMF( e L)) = " 1 6 (o l +a l ) 2 +2o 2 l − h c(UMF( e L)) i 2 # 1/2 (4.23) The HM method requiresFOU( e L) to contain the same average uncertainty as the data intervals [a (i) ,o l ] (i = 1,...,m). To accomplish this, we first compute the means of the mean and standard deviation of the m data intervals, and then, we equate these two quantities to the centers of the centroid and standard deviation of e L, respectively. In this way, since we use the mean (first-order statistic) of the two uncertainty measures of the data intervals, we are using both the individual andgroupuncertaintyoveralldataintervals. Thecentersofthemeanandstandard deviationof e Lare: c( e L)≡ c(LMF( e L))+c(UMF( e L)) 2 (4.24) s( e L)≡ s(LMF( e L))+s(UMF( e L)) 2 (4.25) 87 Letthemeanandstandarddeviationofthedataintervals [a (i) ,o l ](i = 1,...,m) bem i ands i ,respectively. Theyarecomputedusing(4.9)and(4.10),respectively. Define ˆ m a and ˆ s a asthesamplemeanofm i ands i (i = 1,...,m),i.e.: ˆ m a ≡ P m i=1 m i m (4.26) and ˆ s a ≡ P m i=1 s i m (4.27) For e L,a l anda r areobtainedbysolvingtheequations      c( e L) = ˆ m a s( e L) = ˆ s a (4.28) Thesolutionsare: 5      a l = 3ˆ m a −2o l −3 √ 2ˆ s a a r = 3ˆ m a −2o l +3 √ 2ˆ s a (4.29) Proceeding in a similar manner, for the right-part FOU e R in Fig. 4.2, suppose the means of the mean and standard deviation of the data intervals [o r ,b (i) ] (i = 1,...,m)are ˆ m b and ˆ s b ,respectively. Because e Risanalogousto e L,theformulasto computetheparametersb l andb r are 6 : 5 Therearetwosetsofsolutionsfor(4.28); wekeeponlytheonewherea r >a l . Also, becauseourscalefor FOUsis [0,10],weseta l tobe0ifa l from(4.29)islessthan0. 6 BecauseourscaleforFOUsis [0,10],wesetb r tobe10ifb r from(4.30)isgreaterthan10. 88      b l = 3ˆ m b −2o r −3 √ 2ˆ s b b r = 3ˆ m b −2o r +3 √ 2ˆ s b (4.30) (b) For a left-shoulder (Fig. 4.3), the procedure to compute b l and b r is similar to the right-part FOU e R of an interior FOU. We compute the means of the mean and standard deviation of the intervals [o r ,b (i) ](i = 1,...,m), ˆ m b and ˆ s b , and then computeb l andb r using(4.30). Figure4.3: Left-shoulderFOU. (c) For a right-shoulder (Fig. 4.4), the procedure to compute a l and a r is similar to the left-part FOU e L of an interior FOU. We compute the means of the mean and standard deviation of the intervals [a (i) ,o l ](i = 1,...,m), ˆ m a and ˆ s a , and then computea l anda r using(4.29). 89 Figure4.4: Right-shoulderFOU. 4.3 Examples In this section, we apply the HM method to the same set of data intervals collected from 174 peoplefor32words,asusedin[126]. Tables4.1-4.3summarizehowmanydataintervalsareleftaftereachprocessingstagefor left-shoulder,interiorandright-shoulderFOUs,respectively;theyalsogivetheoverlap[o l ,o r ] computed from the data intervals. Note that for word Quite a bit, we actually got no data intervalsleftafterthereasonableintervaltest,sothereisnoFOUcorrespondingtoit. Quite a bit is a very confusing word because “a bit” has the connotation of small where “quite” has the connotation of large. Table II in [126] reveals that very few interval survive EIA’s preprossing stages. Our recommendation is to remove “Quite a bit” from the 32 word vocabulary. 90 Table4.1: Remainingdataintervalsandtheoverlapforleft-shoulderFOUs Words DataPart Overlap n n ′ m ′ m ′′ m o l o r Quiteabit 175 163 149 132 0 N/A N/A Teeny-weeny 174 161 111 92 45 0 0.28 Nonetoverylittle 174 162 112 95 84 0 0.10 Tiny 174 165 117 96 59 0 0.27 Verysmall 174 167 110 93 66 0 0.25 VeryLittle 174 167 128 107 71 0 0.30 Asmidgen 174 152 107 91 30 0 0.58 Abit 174 164 120 112 42 0 1.02 Little 174 166 130 120 53 0 1.27 Lowamount 174 164 128 113 87 0 1.27 Small 174 167 137 123 83 0 1.27 Table4.2: RemainingdataintervalsandtheoverlapforinteriorFOUs Words DataPart Overlap n n ′ m ′ m ′′ m o l o r Somewhatsmall 174 167 141 127 43 1.68 2.00 Some 174 162 145 136 34 3.00 3.70 Modestamount 174 168 161 155 31 4.08 4.73 Sometomoderate 174 164 152 144 76 4.07 4.87 Medium 174 165 132 124 118 4.90 5.17 Moderateamount 174 162 118 107 75 4.83 5.23 Fairamount 174 165 159 152 59 5.03 5.73 Goodamount 174 162 156 135 25 6.93 7.48 Considerableamount 174 163 150 132 28 7.05 7.85 91 Table4.3: Remainingdataintervalsandtheoverlapforright-shoulderFOUs Words DataPart Overlap n n ′ m ′ m ′′ m o l o r Sizable 174 152 146 132 16 6.27 10 Substantialamount 175 158 144 128 64 7.53 10 Verysizable 174 155 125 112 51 8.08 10 Large 174 150 121 110 84 8.62 10 Alot 174 164 126 114 87 8.65 10 Highamount 174 163 130 119 61 8.98 10 Verylarge 174 154 115 100 81 9.75 10 Veryhighamount 174 161 115 111 94 9.90 10 Hugeamount 174 155 106 99 83 9.90 10 Humongousamount 175 148 102 85 76 9.90 10 Extremeamount 174 147 107 90 70 9.90 10 Maximumamount 174 148 118 89 77 9.90 10 Table4.4isthecodebookforthe31words(excludingQuiteabit). Itprovidesthecoordi- nates(code)fortheLMFandUMFofeachFOU.Fig. 4.5depictstheFOUsforallwordsthat arerankedusingthecenterofcentroidgiveninthelastcolumnofTable4.4. 4.4 ComparisonswithEIA In this section, we compare the IT2 FS models generated from the HM method and the EIA using Jaccard’s similarity. Recall that the Jaccard’s similarity [121,123] between IT2 FSs e A and e B,sm J ( e A, e B),isdefinedas: sm J ( e A, e B) = P N i=1 min(µ e A (x i ),µ e B (x i ))+ P N i=1 min(µ e A (x i ),µ e B (x i )) P N i=1 max(µ e A (x i ),µ e B (x i ))+ P N i=1 max(µ e A (x i ),µ e B (x i )) (4.31) 92 Table4.4: Thecodebook Centerof Words UMF LMF Centroid centroid Teeny-weeny [0,0,0.28,0.82] [0,0,0.28,0.34,1] [0.16,0.30] 0.23 Nonetoverylittle [0,0,0.10,1.24] [0,0,0.10,0.22,1] [0.09,0.49] 0.29 Tiny [0,0,0.27,1.44] [0,0,0.27,0.39,1] [0.17,0.52] 0.34 Verysmall [0,0,0.25,1.45] [0,0,0.25,0.37,1] [0.16,0.53] 0.34 Verylittle [0,0,0.30,1.66] [0,0,0.30,0.44,1] [0.19,0.60] 0.39 Asmidgen [0,0,0.58,1.33] [0,0,0.58,0.66,1] [0.31,0.51] 0.41 Abit [0,0,1.02,1.85] [0,0,1.02,1.10,1] [0.53,0.74] 0.64 Little [0,0,1.27,3.44] [0,0,1.27,1.49,1] [0.70,1.27] 0.98 Lowamount [0,0,1.27,4.17] [0,0,1.27,1.56,1] [0.72,1.50] 1.11 Small [0,0,1.27,4.40] [0,0,1.27,1.58,1] [0.73,1.58] 1.15 Somewhatsmall [0,1.68,2.00,4.28] [1.50,1.68,2.00,2.23,1] [1.37,2.56] 1.96 Some [0,3.00,3.70,5.50] [2.69,3.00,3.70,3.88,1] [2.45,3.80] 3.12 Modestamount [1.92,4.08,4.73,7.16] [3.86,4.08,4.73,4.98,1] [3.82,5.10] 4.46 Sometomoderate [0.71,4.07,4.87,8.37] [3.73,4.07,4.87,5.22,1] [3.51,5.48] 4.50 Medium [1.72,4.90,5.17,8.40] [4.58,4.90,5.17,5.49,1] [3.99,6.10] 5.05 Moderateamount [2.04,4.83,5.23,8.24] [4.55,4.83,5.23,5.54,1] [4.19,5.97] 5.08 Fairamount [1.79,5.03,5.73,8.87] [4.71,5.03,5.73,8.87,1] [4.33,6.28] 5.36 Goodamount [4.69,6.93,7.48,10] [6.71,6.93,7.48,7.74,1] [6.58,7.94] 7.26 Considerableamount [5.05,7.05,7.85,10] [6.85,7.05,7.85,8.07,1] [6.92,8.04] 7.48 Sizable [4.20,6.27,10,10] [6.06,6.27,10,10,1] [7.57,8.07] 7.82 Substantialamount [4.91,7.53,10,10] [7.27,7.53,10,10,1] [8.02,8.69] 8.35 Verysizable [6.16,8.08,10,10] [7.89,8.08,10,10,1] [8.50,8.98] 8.74 Large [5.33,8.62,10,10] [8.28,8.62,10,10,1] [8.32,9.21] 8.77 Alot [5.61,8.65,10,10] [8.34,8.65,10,10,1] [8.42,9.23] 8.83 Highamount [6.59,8.98,10,10] [8.74,8.98,10,10,1] [8.77,9.42] 9.10 Verylarge [6.65,9.75,10,10] [9.44,9.75,10,10,1] [8.66,9.78] 9.22 Veryhighamount [7.63,9.90,10,10] [9.67,0.90,10,10,1] [8.98,9.88] 9.43 Hugeamount [7.91,9.90,10,10] [9.70,9.00,10,10,1] [9.11,9.89] 9.50 Humongousamount [8.70,9.90,10,10] [9.78,9.90,10,10,1] [9.48,9.91] 9.70 Extremeamount [9.09,9.90,10,10] [9.82,9.90,10,10,1] [9.66,9.93] 9.79 Maximumamount [9.87,9.90,10,10] [9.90,9.90,10,10,1] [9.94,9.95] 9.95 93 Quite a bit Teeny−weeny None to very little Tiny Very small Very little A smidgen A bit Little Low amount Small Somewhat small Some Modest amount Some to moderate Medium Moderate amount Fair amount Good amount Considerable amount Sizeable Substantial amount Very sizeable Large A lot High amount Very large Very high amount Huge amount Humongous amount Extreme amount Maximum amount Figure4.5: FOUsofall32wordsgeneratedusingtheHMmethod. Forcomparisonpurposes,theFOUsgeneratedfromtheEIAaredepictedinFig. 4.6,using thesamedatasetusedinSection4.3. The similarity results for left-shoulder, interior and right-shoulder FOUs are summarized inTables4.5-4.7,respectively. Observethat: 1. Although the HM method utilizes a different FOU classification procedure from the EIA, the nature of all FOUs for these two approaches are the same, i.e., if a word is modeled as a left-shoulder by the EIA, it is also modeled as a left-shoulder by the HM method(seeFigs. 4.5and4.6). 2. The FOUs from the HM method are generally visually “thinner” than those from the EIA, because more information is used from the data intervals in the HM method, i.e., theoverlap,whichinturnreducestheremaininguncertainties(seeFigs. 4.5and4.6). 94 3. All similarity measures shown in Tables 4.5-4.7 are greater than 0.5, which means the HM method and EIA FOUs are more similar than not. On the other hand, most of the similarities are less than 0.8, which indicates the choice of which method to use may makeadifferenceinsomeCWWapplications. Table4.5: Similarityresultsbetweenleft-shoulderFOUsgeneratedfromtheEIAandtheHM method. words sm J Teeny-weeny 0.74 Nonetoverylittle 0.75 Tiny 0.83 Verysmall 0.83 Verylittle 0.73 Asmidgen 0.74 Abit 0.69 Little 0.78 Lowamount 0.80 Small 0.83 95 Table 4.6: Similarity results between interior FOUs generated from the EIA and the HM method. words sm J Somewhatsmall 0.72 Some 0.64 Modestamount 0.52 Sometomoderate 0.62 Medium 0.76 Moderateamount 0.68 Fairamount 0.66 Goodamount 0.62 Considerableamount 0.60 Table 4.7: Similarity results between right-shoulder FOUs generated from the EIA and the HMmethod. words sm J Sizable 0.70 Substantialamount 0.74 Verysizable 0.71 Large 0.81 Alot 0.80 Highamount 0.78 Verylarge 0.85 Veryhighamount 0.75 Hugeamount 0.72 Humongousamount 0.77 Extremeamount 0.66 Maximumamount 0.55 96 Teeny−weeny Tiny Very small None to very little A smidgen Very little A bit Little Small Low amount Somewhat small Some Quite a bit Modest amount Some to moderate Medium Moderate amount Fair amount Good amount Considerable amount Sizeable Substantial amount Very sizeable Large A lot High amount Very large Very high amount Huge amount Humongous amount Extreme amount Maximum amount Figure 4.6: FOUs of all 32 words using the EIA. The words have been ordered according to theirincreasingaveragecentroid. 4.5 Summary This Chapter has presented a new method-the HM method-for constructing normal IT2 FS models for words from interval data that are collected from a group of subjects. The HM method uses more information about the overlap of these data intervals than does the EIA. It leads,forthefirsttime,tofullynormalFOUs,i.e.,FOUswhoseLMFandUMFarenormalT1 FSs. Such fully normal word FOUs may be quite useful in advanced computing with words applicationsbecausetheycanbeusedtosimplifynumericalcomputations[88]. Wehavealso compared the FOUs from the HM method with those from the EIA, using similarities. Our experiments suggest that a metric is needed to determine which model is better for a specific problem. 97 Chapter5 LinguisticWeightedStandardDeviation 5.1 Introduction Originally, imprecision and indeterminacy were considered to be random characteristics, and were taken into account by methods of probability theory [41]. In real-world situations, a frequent source of imprecision is not only the presence of random variables, but also the impossibility, in principle, of operating with numerical data as a result of the complexity of the system, where instead this can be handled by using fuzzy sets. According to [41], the motivation for the development of fuzzy statistics is its philosophical and conceptual relation to subjective probability. In the subjective view, probability represents the degree of belief that a given person has in a given event on the basis of given evidence. There has been a lot of work done based on this philosophy, where it has been shown that better results can be obtainedwhensomefuzzystatisticsareused[95,130]. In classical statistics, the first- and second-order statistics, i.e., the mean and standard deviation 1 , are the most important ones, and, there are many algorithms and applications that usethemin,e.g.,theareasofimageprocessing,communicationandsignalprocessing[49,61, 102,132]. Moreover, with the help of α-cuts, there are also numerous applications that use fuzzymeansandfuzzystandarddeviations[19,95,130,146];however,allofthemusetype-1 fuzzysets(T1FSs). Recently, more and more researchers are focusing on interval type-2 fuzzy sets (IT2 FSs) [24,31,141], which contain more membership function (MF) uncertainties than T1 FSs and include T1 FSs as a special case. To compute the average of a collection of IT2 FSs, Wu andMendel[116,119]proposedtheLinguisticWeightedAverage(LWA),whosecomputation 1 InthisChapter,weonlyconsiderthesamplearithmeticmeanandsamplestandarddeviation. 98 usesα-cutsandtheKM/EKMalgorithm;however,tothebestofourknowledge,therehasnot been any study about the standard deviation of IT2 FSs. Note that the standard deviation of IT2FSsdiscussedinthisChapterhasnothingtodowiththestandarddeviationintroducedby WuandMendel[120],becausethestandarddeviationin[120]isanuncertaintymeasurefora singleIT2FS,whereasthestandarddeviationinthisChapterisdefinedforasetofIT2FSs. In this Section, the Linguistic Weighted Standard Deviation (LWSD) is proposed to com- putetheweightedstandarddeviationofacollectionofIT2FSs. Weviewthisasanadjunctto theLWAmuchastheclassicalstandarddeviationisanadjuncttotheclassicalmean. 5.2 Linguistic Weighted Power Mean and Linguistic WeightedStandardDeviation In [93], Rickard et al introduced the Linguistic Weighted Power Mean (LWPM) and its com- putation via a generalization of the Karnik-Mendel (KM) algorithm, namely, the Weighted Power Mean Enhanced Karnik-Mendel (WPMEKM) algorithm. The LWPM admits interval type-2fuzzymembershipfunctionsforbothitsinputsanditsweightstoaccountforimprecise knowledgeofthesequantities. Inthis section, theLWPMandWPMEKM are reviewed, after whichtheLinguisticWeightedStandardDeviation(LWSD)isproposed. 5.2.1 LinguisticWeightedPowerMean Recently, Rickard et al. [90,93] have generalized the LWA to the linguistic weighted power mean(LWPM),i.e.,to(usingtheirnotation): L r ( e X, e W), lim q→r ! P n i=1 f W i e X q i P n i=1 f W i % 1/q (5.1) where f W i and e X i areIT2FSs 2 foralliandthepowerr isallowedtovaryovertheentirereal line. 2 Inthispaper,wetreatthemasIT2fuzzynumbers. 99 As stated in [93], for T1 FSs, “... as r ranges over the real line, L r ranges from logical conjunction x 1 ∧x 2 ∧...∧x n of the inputs (in the limit as r → −∞) to logical disjunction x 1 ∨x 2 ∨...∨x n of the inputs (in the limit as r → ∞). L r is thus an orand operator.” Of course,whenr = 1,L r istheFuzzyWeightedAverage(FWA)forT1FSsortheLWAforIT2 FSs. The $64 question is “How is (5.1) computed?” Rickard, et al. [90] have explained how to dothisbymeansofWeightedPowerMean(WPM)EKMalgorithms. Becausetheirapproach forobtainingtheWPMEKMalgorithmsisquitegeneralandseemsapplicabletoothermodi- ficationsoftheLWA,someofitsmoreimportantdetailsarecoverednext. 5.2.2 Weighted Power Mean Enhanced Karnik-Mendel (WPMEKM) Algorithm Asin[90,93],inthissection,wefocusonthesimplestcaseswhereall f W i and e X i in(5.1)are justintervals,givenas x i ∈ [x i ,x i ] and w i = [w i ,w i ],i = 1,...,N (5.2) wherex i ,x i ,w i ,w i ≥ 0. 3 Inthiscase,(5.1)canbeexpressedas 4 : L r (x,w) = P n i=1 w i x r i P n i=1 w i 1/r (5.3) TheWPMEKMalgorithm[90,93]isanextensionoftheEKMalgorithm,andisdesigned to compute the WPM when the inputs are both intervals. Note that in this case, L r (x,w) is also an interval, and we denote it as [L r (x,w),L r (x,w)]. To provide the readers a more comprehensive picture, we briefly review the WPMEKM algorithm and the intuition behind it. 3 Inreal-worldapplications,itisalmostalwaystruethatboththescores(x i )andweights(w i )arenon-negative. 4 When r > 0 (r < 0) we have partial disjunction (conjunction), i.e. L r is more similar to disjunction (conjunction) than to conjunction (disjunction). For these values ofr the limit in (5.1) is not needed; however, whenr = 0thelimitisneeded,andthereadersshouldconsult[90]and[93]fordiscussionsaboutthiscase. 100 First of all, the extreme cases when r = +∞ and r = −∞ must be taken care of. In these two cases, L r (x,w) is independent ofw i , for alli. The resulting interval endpoints are determinedjustbytheendpointsofx i ,as[93] L −∞ (x,w) = [L −∞ (x,w),L −∞ (x,w)] = [min i:w i >0 x i , min i:w i >0 x i ] (5.4) and L ∞ (x,w) = [L ∞ (x,w),L ∞ (x,w)] = [max i:w i >0 x i , max i:w i >0 x i ] (5.5) NexttheWPMwithfinitepowerr isconsidered. Accordingto[93],anauxiliaryfunction h r (z)needstobeused,whichisdefinedas: h r (z) =      z 1/r r6= 0 e z r = 0 (5.6) Hence,theinversefunctionofh r (z),h −1 r (z),isgivenby: h −1 r (z) =      z r r6= 0 lnz r = 0 (5.7) Let y r (x,w) = P n i=1 w i h −1 r (x i ) P n i=1 w i (5.8) (5.3)isthenrewrittenas: 101 L r (x,w) = h r ( n X i=1 w i h −1 r (x i ) . n X i=1 w i ) = h r (y r (x,w)) (5.9) Notice that h r (z) and h −1 r (z) are strictly increasing on non-negative real numbers when r≥ 0andarestrictlydecreasingwhenr < 0;therefore,from(5.9),wehave L r (x,w) = L r ,L r =            h h r (y r ),h r (y r ) i r≥ 0 h h r (y r ),h r (y r ) i r < 0 (5.10) where [y r ,y r ] = min P n i=1 w i h −1 r (x i ) P n i=1 w i ,max P n i=1 w i h −1 r (x i ) P n i=1 w i (5.11) wheretheminandmaxfunctionsaretakenoverx i ∈ [x i ,x i ]andw i ∈ [w i ,w i ],foralli. To compute (5.11), h −1 r (x i ) (i = 1,...,n) needs to be computed first. When we only considernon-negativerealnumbers,theycanbecomputedas: h −1 r (x i ) =            [h −1 r (x i ),h −1 r (x i )] r≥ 0 [h −1 r (x i ),h −1 r (x i )] r < 0 (5.12) Then,(5.11)canbecomputedusingtheregularintervalweightedaverage(IWA)[72,116,119] withinputsh −1 r (x i )andw i ,afterwhichL r (x,w)canthenbeobtainedusing(5.10). 5.2.3 LinguisticWeightedStandardDeviation Inclassicalstatistics,givenanon-emptysetofdata{x 1 ,x 2 ,...,x n },withnon-negativeweights {w 1 ,w 2 ,...,w n },theweightedmeaniscomputedas: 102 x = P n i=1 w i x i P n i=1 w i (5.13) Thecorrespondingweightedsamplevarianceisthencomputedas: σ 2 = P n i=1 w i (x i −x) 2 P n i=1 w i (5.14) Supposewearegivenanon-emptysetofdatamodeledbyIT2FSs{ e X 1 , e X 2 ,..., e X n },with non-negativeweights{ f W 1 , f W 2 ,..., f W n } 5 . Similarto(5.13),theLWAiscomputedas: e Y LWA = P n i=1 f W i e X i P n i=1 f W i (5.15) From (5.14), it is straightforward to see that the weighted variance of the given data set is computed,as: e Σ 2 = P n i=1 f W i ( e X i − e Y LWA ) 2 P n i=1 f W i (5.16) Similarly, the weighted standard deviation of the given data is defined as the square root oftheweightedvariance,as: e Σ = " P n i=1 f W i ( e X i − e Y LWA ) 2 P n i=1 f W i # 1/2 (5.17) (5.17) is exactly the same as the LWPM when r = 2, which means we can use the WPMEKM algorithm to compute it. Because all the quantities in (5.17) can be IT2 FSs, wecallittheLinguisticWeightedStandardDeviation(LWSD). Note that (5.15)-(5.17) are expressive equations, i.e., they are not performed by multiply- ing,addinganddividingIT2FSs. Howtheyareactuallycomputedisexplainedbelow. Like the LWA, computing the LWSD is equivalent to computing two WPMs of T1 FSs [77]. To begin, the LWA e Y LWA and e S i = e X i − e Y LWA (i = 1,...,n) are computed. Let Σ and 5 InthisChapter,non-negativeIT2weightsrefertoIT2fuzzynumbersthathaveanon-negativeleftendpoint foreveryα-cut. 103 ΣbetheUpperMembershipFunction(UMF)andLowerMembershipFunction(LMF)of e Σ, respectively. Thentocompute Σ,we: 1. Selectappropriatemα-cutsfor Σ(e.g.,divide [0,1] 6 intom−1intervalsandsetα j = (j−1)/(m−1),j = 1,2,...,m.) 2. Let S i and W i be the UMFs of e S i and f W i (i = 1,...,n), respectively. Find the corre- sponding α-cut on S i and W i for each α j ; denote the end points of the α-cuts S i (α j ) and W i (α j ) as [a il (α j ),b ir (α j )] and [c il (α j ),d ir (α j )], respectively. Note that S i (α j ) arecomputedusingintervalarithmetic,i.e.,X i (α j )−Y LWA (α j ) 7 ,sotheymaycontain negativevalues. 3. Use a WPMEKM algorithm to compute the α-cut of Σ, Σ(α j ) = [Σ Ll (α j ),Σ Rr (α j )], where: Σ Ll (α j ) = min ! P n i=1 W i (α j )S 2 i (α j ) P n i=1 W i (α j ) % 1/2 (5.18) Σ Rr (α j ) = max ! P n i=1 W i (α j )S 2 i (α j ) P n i=1 W i (α j ) % 1/2 (5.19) where the min and max functions are taken over S i (α j ) ∈ [a il (α j ),b ir (α j )] and W i (α j ) ∈ [c il (α j ),d ir (α j )]. Notice that, because S i (α j ) may contain negative num- bers,S 2 i (α j )cannotbecomputedusing(5.12)directly. Instead,itiscomputedas: S 2 i (α j ) =            [a 2 il (α j ),b 2 ir (α j )] a il (α j )≥ 0 [0,max{a 2 il (α j ),b 2 ir (α j )}] a il (α j )< 0 (5.20) 4. RepeatSteps(3)and(4)foreveryα j (j = 1,...,m). 6 InthisChapter,weassumethatallUMFsarenormalT1FSsasin[77]. 7 X i (α j )andY LWA (α j )aretheα-cutsofX i andY LWA atlevelα j ,respectively. 104 5. Connect all left coordinates (Σ Ll (α j ), α j ) and all right coordinates (Σ Rr (α j ), α j ) to formtheUMF Σ. Tocompute Σ: 1. Select appropriate p α-cuts for Σ (e.g., divide [0,h min ] 8 into p− 1 intervals and set α j =h min (j−1)/(p−1),j = 1,2,...,p.) 2. Let S i and W i be the LMFs of e S i and f W i (i = 1,...,n), respectively. Find the corre- spondingα-cut [a ir (α j ),b il (α j )]and [c ir (α j ),d il (α j )]onS i andW i . 3. Use a WPMEKM algorithm to compute the α-cut of Σ, Σ(α j ) = [Σ Lr (α j ),Σ Rl (α j )], where: Σ Lr (α j ) = min P n i=1 W i (α j )S 2 i (α j ) P n i=1 W i (α j ) 1/2 (5.21) Σ Rl (α j ) = max P n i=1 W i (α j )S 2 i (α j ) P n i=1 W i (α j ) 1/2 (5.22) where the min and max functions are taken over S i (α j ) ∈ [a ir (α j ),b il (α j )] and W i (α j )∈ [c ir (α j ),d il (α j )]. S 2 i in(5.21)and(5.22)iscomputedas: S 2 i (α j ) =            [a 2 ir (α j ),b 2 il (α j )] a ir (α j )≥ 0 [0,max{a 2 ir (α j ),b 2 il (α j )}] a ir (α j )< 0 (5.23) 4. RepeatSteps(3)and(4)foreveryα j (j = 1,...,p). 5. Connect all left coordinates (Σ Lr (α j ), α j ) and all right coordinates (Σ Rl (α j ), α j ) to formtheLMF Σ. 8 The LMFs might not be normal T1 FSs, hence we let h min = min{min ∀i {h X i },min ∀i {h W i },h Y LWA }, whereh X i ,h W i andh Y LWA aretheheightsoftheLMFsof e X i , f W i and e Y LWA ,respectively. 105 5.3 NumericalExamples In this section, two numerical examples are provided that demonstrate the LWSD introduced in Section 5.2.C. To represent an FOU, we use the same nine-parameter notation as in [77] (seeFig. 3.8). 5.3.1 RandomlySelectedWordFOUs We randomly selected 1000 word FOUs from the 32-word codebook in [56], and then com- putedtheLWAandtheLWSDofthem. Formally, we indexed the word FOUs from none to very little to maximum amount with integers from 1 to 32, respectively. 1000 numbers were then drawn from a random number generator, which generated an integer from 1 to 32 uniformly. For the i-th integer U i , we let e X i betheU i -thwordFOUinthecodebook. Similarly,wedrewanother1000randomnumbers U j (j = 1,...,1000)andlet f W j bethej-thcorrespondingwordFOUinthecodebook. The LWA e Y LWA defined in (5.15) for the e X i with weights f W i (i = 1,...,1000) generated above is shown in Fig. 5.1 (a). The LWSD e Σ defined in (5.17) is depicted in Fig. 5.1 (b). Their nine-parameter representations are (3.34,4.87,5.94,7.25,5.00,5.21,5.71,5.96,0.24) and (1.83,2.21,2.44,2.69,2.24,2.26,2.42,2.44,0.24),respectively. 0 5 10 0 0.5 1 0 5 10 0 0.5 1 (a) (b) e Y LWA e Σ Figure5.1: The(a)LWAand(b)LWSDofrandomlygeneratedIT2FSs. Observethat: 1. The support sets of the FOUs in the codebook are inside the interval [0,10], and the FOU of the LWA e Y LWA lies around the middle of [0,10]. Recall that all the words and 106 weights are selected according to a uniformly distributed random number generator, so thisresultmakesintuitivesense. 2. The FOU of e Σ looks “thinner” than the FOU of e Y LWA , which means in this example, theLWSD e Σcontainslessuncertaintythan e Y LWA . Asinstatistics,usingtheLWSD e Σ,wecancompute,e.g.,theinterval[ e Y LWA − e Σ, e Y LWA + e Σ] to provide a “spread” of the linguistic data. The interval [ e Y LWA − e Σ, e Y LWA + e Σ] for this example is shown in Fig. 5.2. The nine-parameter representation of e Y LWA − e Σ is (0.58,2.37,3.69,5.38,2.51,2.75,3.39,3.67,0.24), and the nine-parameter representation of e Y LWA + e Σis (5.07,6.99,8.31,9.87,7.16,7.40,8.04,8.32,0.24). 0 5 10 0 0.5 1 e Y LWA + e Σ e Y LWA − e Σ y Figure5.2: TheFOUsof e Y LWA ± e Σ. In perceptual computing, words are modeled using IT2 FSs; hence, once we know the interval [ e Y LWA − e Σ, e Y LWA + e Σ],wecandecodeitsendpointFOUsbackintowords. Mendel and Wu [77] extended Jaccard’s similarity measure for T1 FSs to IT2 FSs, and suggested Jaccard’s similarity measure should be used in perceptual computing applications. Therefore, in this Chapter, the Jaccard’s similarity measure [defined in (5.24)] is used as the decoder,andthesimilaritybetween e Y LWA ± e Σandallthewordsinthecodebookarecomputed; then,thewordwithhighestsimilarityisusedasthedecodedwordforthetwoendpointFOUs. TheJaccard’ssimilaritymeasurebetweentwoIT2FSs e Aand e B isdefinedas: sm J ( e A, e B) = P N i=1 min(µ e A (x i ),µ e B (x i ))+ P N i=1 min(µ e A (x i ),µ e B (x i )) P N i=1 max(µ e A (x i ),µ e B (x i ))+ P N i=1 max(µ e A (x i ),µ e B (x i )) (5.24) 107 where µ and µ are the UMF and LMF of an IT2 FS, respectively, and x i (i = 1,...,N) are equallyspacedinthesupportof e A∪ e B. According to (5.24), e Y LWA is decoded as modest amount 9 ; and, the interval [ e Y LWA − e Σ, e Y LWA + e Σ] is decoded as [Little, Considerable amount] 10 . Note that [Little, Considerable amount] covers a relatively wide range, because, as explained above, both the words and weightsaredrawnaccordingtoauniformrandomvariable. 5.3.2 EvaluatingLocationChoiceApplication Perceptual Computer (Per-C) [64,77,78] is a very useful tool when dealing with words and uncertainties,especiallyfordecisionmakingproblems[32]. In[32,33],HanandMendelshowedhowtoapplyaPer-Ctothelocationchoiceproblem. This problem is one of hierarchical multi-criteria decision making where a decision has to be made about which of three locations (A, B and C) should be chosen based on four criteria: economicgrowth,cost,governmentpoliciesandotherrisks. Foreachcriterion,therearealso several sub-criteria that have to be evaluated. All the evaluation information for this problem issummarizedinTableIIin[32]. HanandMendelusedtheLWAtoaggregatealloftheinformation(includingbothnumbers andwords)abouteachlocationandsuggestedafinaldecisionfortheproblem. UsingtheLWSDweareabletoobtainevenmoreinformationwithoutcollectinganymore data. As an example, we will only compute the LWSD for the four criteria of location A (see TableIIin[32]forallrequisiteFOUs). The aggregated evaluation of the above four criteria for location A is depicted in Fig. 5.3, where e Y EG , e Y C , e Y GP and e Y OR are performances for economic growth, cost, government policiesandotherrisks,respectively. 9 Thenine-parameterrepresentationofmodestamount is (3.59,4.75,6.00,7.41,4.79,5.30,5.30,5.71,0.42). 10 The nine-parameter representations of Little and Considerable amount are (0.38,1.58,3.50,5.62,1.79,2.20,2.20,2.40,0.24) and (4.38,6.50,8.25,9.62,7.19,7.58,7.58,8.21,0.37), respectively. 108 e Σ = " f W EG ( e Y EG − e Y F ) 2 + f W C ( e Y C − e Y F ) 2 + f W GP ( e Y GP − e Y F ) 2 + f W OR ( e Y OR − e Y F ) 2 f W EG + f W C + f W GP + f W OR # 1/2 (5.26) 0 5 10 0 0.5 1 e Y EG (a) 0 5 10 0 0.5 1 e Y C (b) 0 5 10 0 0.5 1 e Y GP (c) 0 5 10 0 0.5 1 e Y OR (d) Figure 5.3: The FOUs of the evaluations of (a) economic growth; (b) cost; (c) government policies;and,(d)otherrisksforlocationAin[32]. The aggregated performance over the four criteria e Y F and their LWSD, e Σ, are given by (5.25)and 11 (5.26),respectively. e Y F = f W EG e Y EG + f W C e Y C + f W GP e Y GP + f W OR e Y OR f W EG + f W C + f W GP + f W OR (5.25) TheFOUsof e Y F and e ΣaredepictedinFig. 5.4. Thecentroidof e Y F is[2.77,6.11],andthe centerofthecentroidof e Y F is4.44[32]. 11 (5.26)isatthetopofthenextpage. 109 0 5 10 0 0.5 1 0 5 10 0 0.5 1 (a) (b) e Y F e Σ Figure 5.4: The FOUs of (a) the final performance e Y F and (b) the LWSD e Σ of location A in[32]. Asinourlastexample,wealsocomputedtheinterval h e Y F − e Σ, e Y F + e Σ i ,whichisshown inFig. 5.5. −5 0 5 10 15 0 0.5 1 e Y F + e Σ e Y F − e Σ y Figure5.5: TheFOUsof e Y F − e Σand e Y F + e ΣforlocationAin[32]. Observe the FOUs in Fig. 5.5 exceed the [0,10] range, so we “truncated” them (in a linguistically suitable way) so as to make them lie inside the [0,10] interval, as shown in Fig. 5.6. 0 5 10 0 0.5 1 e Y F + e Σ e Y F − e Σ y Figure5.6: TheFOUsoftruncated e Y F − e Σand e Y F + e ΣforlocationAin[32]. 110 Using the Jaccard’s similarity measure as a decoder, e Y F is decoded as some to moderate, andthetruncatedinterval h e Y F − e Σ, e Y F + e Σ i isdecodedas[little,goodamount]. Thecentroidsofthetruncated e Y F − e Σand e Y F + e Σare[0.88,4.72]and[4.49,8.90],respec- tively; and, their centers are 2.80 and 6.70. Therefore, using the LWSD, another interval, [2.80,6.70]isobtained,whichaccountsfortheuncertaintiesoverthefourcriteria. Noticethat the interval [2.80,6.70] is different from the centroid of e Y F , [2.77,6.11], which is obtained onlyfromtheFOUof e Y F . Observe, also, from Fig. 5.5 that the interval h e Y F − e Σ, e Y F + e Σ i covers the entire [0,10] range,whichindicatesalotof“uncertainty”associatedwiththeperformanceoflocationA 12 . Alltheaboveinformation, whichis morethanin[32,33], canbetakenintoconsideration bythedecisionmakerinordertomakethefinaldecision. 5.4 Summary In this Chapter, the Linguistic Weighted Standard Deviation (LWSD) is proposed. It is com- puted using the weighted power mean and the WMPEKM algorithm. Analogous to the stan- dard deviation in classical statistics, the LWSD computes the weighted standard deviation of asetofIT2FSs. TwonumericalexamplesaregiventodemonstratetheuseoftheLWSDand thepossibilitiesforpotentialapplicationindecisionmakingproblems. 12 Thisisnotsurprisingsince e Y F hasalargesupport. 111 Chapter6 PerceptualComputerApplicationin LearningOutcomeEvaluation 6.1 Introduction Outcome-based education (OBE) is a model of education that rejects the traditional focus on what the school provides to students, in favor of making students demonstrate that they “knowandareabletodo”whatevertherequiredoutcomesare. Ithasbeenpracticedsinceits introductionbyWilliamSpadyin1988[97]. Foraspecificcourse,Fig. 1.3showsthegeneral relationshipbetweenLearningOutcomes(LOs)anditscourseassessments[10]. In this Chapter, a Perceptual Computer (Per-C) [67,77,125] is proposed to evaluate the LOsandfinalperformanceinanOBEsystem. 6.2 TraditionalLOEvaluation Tokeeptheproblemformulationsimple,weusethesameexampleasin[10]toexplainthetra- ditional LO evaluation process, and our proposed Per-C. Suppose we have a fictitious course called “Circuits and Signals,” and we assume that there are three LOs associated with this course,andthreeformsofassessment,namely,assignments,laboratoryexperimentsandwrit- ten exams (see Table 6.1). Table 6.2 shows the corresponding points (out of 100) assigned to theassessmentsandLOs. 112 Table6.1: LOsforCircuitsandSignals LO Description Assessments LO1 Performmathematicalmodeling Assignments(A) ofsignalsinelectriccircuits WrittenExams(E) LO2 Performfrequencydomain Assignments(A) analysisofelectriccircuitsusing Labs(L) FourierandLaplacetransforms WrittenExams(E) LO3 Synthesizeandanalyzeoneport Labs(L) andtwoportnetworks WrittenExams(E) Table6.2: AssessmentandLOmappings LO Assignment Lab WrittenExam Total LO1 15 0 20 35 LO2 15 5 20 40 LO3 0 5 20 25 Total 30 10 60 100 In the traditional LO evaluation method, the following equation is used to evaluate the LOs:         ELO 1 ELO 2 ELO 3         =         15/30 0 20/60 15/30 5/10 20/60 0 5/10 20/60         ×         M A M L M E         (6.1) whereM A ,M L andM E arescoresobtainedbyastudentforassignment,labandexam,respec- tively,andELO i (i = 1,2,3)istheevaluationofi-thLOachievement. In[10],Chuaetal. discussedthepossibilityofemployingfuzzyLOevaluationbyassign- ing some linguistic terms, that are modeled by type-1 fuzzy sets (T1 FSs), to the assessments aswellastotheLOs. Theyshowedthat,usingfuzzyrules(describedbythelinguisticterms) 113 that can be tuned by the instructor, their methods have the capacity to adjust the input-output relationshipoftheevaluationsystem. However,wordsmeandifferentthingstodifferentpeo- ple,andsoareuncertain. MendelandWu[77]illustratedthatanintervaltype-2fuzzyset(IT2 FS) should be used to model a word, since it is characterized by its footprint of uncertainty (FOU)and,therefore,hasthepotentialtocaptureworduncertainties. 6.3 DesignofPer-CforLOEvaluation Inspired by the work of Chua and his colleagues, a Per-C for LO evaluation using IT2 FSs is introduced in this section. The hierarchy of the Per-C is depicted first, and then the encoder, CWWengineanddecoderareexplained,respectively. 6.3.1 Hierarchy Figure6.1: HierarchyofPer-CforLOevaluationprocess. ThehierarchyoftheLOevaluationprocessfor“CircuitsandSignals”isillustratedinFig. 6.1. The inputs to nodes in Layer 1 are crisp scores a student obtains for the three assessments, denoted M A , M L and M E . The nodes in this layer map the input crisp scores into three 114 FOUs, denoted e Y A , e Y L and e Y E , respectively. Unlike the crisp scores, these FOUs take some uncertaintyintoconsiderationduringtheassessmentprocedure. Then,theseFOUsarepassed to nodes in Layer 2 where they are aggregated. The resulting FOUs can be viewed as the performance of different LOs, denoted e Y LO1 , e Y LO2 and e Y LO3 , respectively. Finally, another aggregation is performed using the LO FOUs obtained from Layer 2, so that the FOU for the final performance, e Y F , is computed. This FOU is then mapped into a recommendation. Note that all FOUs in the above procedure can be mapped into some words like Poor, Good, Excellent,etc,asexplainedlater. The details of the nodes in the different layers of Fig. 6.1 are described in following sections. 6.3.2 Encoder There are two codebooks used, one for the words that are used to describe the assessments, LOsandfinalperformance,andtheotherforweights,asexplainednext. CodebookUsedforPerformance Thiscodebookcontainsfivewords(Poor,Marginal,Adequate,Good andExcellent)andtheir FOUscanbeusedtodescribetheLOandfinalperformance. FOUsofthesewordsaredepicted in Fig. 6.2. These FOUs are the same as those used by Wu and Mendel in [77,125] in their Social Judgment Advisor (SJA). The scale for these words are from 0-10; therefore, the crisp scores must first be normalized into the interval [0,10]. One thing to note is that these FOUs were not obtained by collecting data from a group of subjects, i.e., they are synthetic; however,theuserofthisPer-Cmaymodifythemaccordingtosomespecificrequirements,as by collecting data from a group of subjects and then using the intervalapproach (IA) [57,77] tosynthesizethem. 115 Figure6.2: FOUsofthefivewordsforperformance. CodebookUsedforWeights The weightsfordifferentassessmentsand LOs aredeterminedbytheinstructor, accordingto Table6.2,butinpracticeitisverydifficultfortheinstructortodecideexactlyhowmanypoints ineachassessmentarecontributingtothethreeLOs. Instead,itismucheasiertodescribethis assessment-LOrelationshipusingsentenceslike“around15pointsintheassignmentrepresent the performance of LO1”, or “about 5 points in the lab represent the performance of LO2,” etc. In order to do this, we model all the weights using IT2 FSs. How to represent aroundw ′ points using an FOU ( f W(w ′ )) is depicted in Fig. 6.3. Note that our weight FOUs are valid only ifw ′ ≥ 5, which is in agreement with the “Circuits and Signals” example. Other FOUs mayalsobeusedfordifferentapplications. 116 Figure6.3: FOUsforweights. 6.3.3 Engine Layer1 InanIT2fuzzylogicsystem(FLS)[65],anantecedent(IT2FS)canbefiredbyacrispnumber (or an interval), and how strong this antecedent is fired is represented by an interval, i.e., the firing interval. In order to map a normalized crisp score of an assessment into an IT2 FS that resemblesFOUsinthecodebook[77],wefirstcomputethe“firingintervals”oftheinputscore for all five word FOUs, and then use these firing intervals as weights to obtain the FOU for theassessment. How we obtain the firing intervals is illustrated in Fig. 6.4. The five FOUs are words for performance as shown in Fig. 6.2, e.g., in Fig. 6.4,x = x ′ only “fires” two FOUs, Adequate and Good. The firing intervals for words e X 3 = Adequate and e X 4 = Good are denoted as f 3 (x ′ ) = [f 3 (x ′ ),f 3 (x ′ )] and f 4 (x ′ ) = [f 4 (x ′ ),f 4 (x ′ )], respectively. Note that in Fig. 6.4, f i (x ′ ) = 0for e X 1 , e X 2 and e X 5 (i = 1,2,3). 117 Figure6.4: Computingthefiringlevels. Let e X≡ ( e X 1 ,..., e X 5 )andF(M)≡ (f 1 (M),...,f 5 (M))(M isacrispscore);then, e Y A , e Y L and e Y E canthenbecomputedas: e Y A =H A [ e X,F(M A )] (6.2) e Y L =H L [ e X,F(M L )] (6.3) and e Y E =H E [ e X,F(M E )] (6.4) whereH A (·),H L (·)andH E (·)areaggregationfunctionsthatareexplainedinSection6.4. Layer2 ThenodesinthislayeraggregatethejustcomputedassessmentFOUs( e Y A , e Y L and e Y E )accord- ingtotheirfuzzyweights(Table6.2),as: e Y LO1 =H LO1 [ e Y AE , e W AE ] (6.5) e Y LO2 =H LO2 [ e Y ALE , e W ALE ] (6.6) 118 and e Y LO3 =H LO3 [ e Y LE , e W LE ] (6.7) where e Y AE ≡ [ e Y A , e Y E ], e W AE ≡ [ f W(15), f W(20)], e Y ALE ≡ [ e Y A , e Y L , e Y E ], e W ALE ≡ [ f W(15), f W(5), f W(20)], e Y LE ≡ [ e Y L , e Y E ], e W LE ≡ [ f W(5), f W(20)], and H LO1 (·), H LO2 (·) and H LO3 (·) are aggregation functions that compute the three LO FOUs, respectively. Note that different aggregation functions can be used so as to satisfy some requirements proposed bytheinstructor,asexplainedinSection6.4. Layer3 Similarly,thefinalperformanceFOU, e Y F ,canthenbecomputedbyaggregatingthethreeLO performanceFOUs,as: e Y F =H F [ e Y LO , e W LO ] (6.8) where e Y LO ≡ [ e Y LO1 , e Y LO2 , e Y LO3 ], e W LO ≡ [ f W(30), f W(10), f W(60)]andH F (·)isanaggrega- tionfunctionexplainedinSection6.4,usedtocomputethefinalperformanceFOU. 6.3.4 Decoder After computing the LO and final performance FOUs, a Jaccard similarity measure [63, Ch. 4], sm J , is computed between each of these FOUs and the five words in the codebook, e X 1 , ..., e X 5 , namely, sm J ( e Y LOi , e X j ) andsm J ( e Y F , e X j ) (i = 1,2,3 andj = 1,...,5). For each LO and the final performance, the word with maximum similarity measure is assigned to it as the linguistic description. For example, if argmax j {sm J ( e Y LO3 , e X j )} = 4, the performance of LO3isdecodedasGood (whichcorrespondsto e X 4 inFig. 6.2). 119 6.4 MandatoryRequirementsintheEvaluationProcess Dujmovic[15]proposed16importantpropertiesofhumanreasoningintheprocessofmaking evaluationdecisions,oneofwhichisofspecialinteresttous: mandatoryanddesired require- ments. According to [15], an example of an evaluation process with mandatory and desired requirementscanbedescribedasfollows: acarbuyerevaluatescarsusingsafetyandcomfort amongothercriteria,andrequiressafetyasamandatoryrequirementandcomfortasadesired requirement. In this case, if a car does not satisfy minimum safety features, the car will be rejected directly. On the other hand, a car with better comfort features will have higher score and vice versa, but even the least comfortable car will not be rejected purely because of its comfortfeatures. In system evaluation, if one input is mandatory (with score m) and the other is desired (withscored),thentheaggregatedscoreforthissystem,z =A m/d ,mustsatisfythefollowing conditions: A m/d (0,d) = 0, 0≤d≤ 10 0<A m/d (m,0)<m, 0<m≤ 10 m<A m/d (m,1)< 10, 0<m< 10 (6.9) Note that the range for all preference scores in this Chapter is [0,10]. If the mandatory requirementisnotsatisfied(m = 0),theaggregatedscorewillbe0;ifthedesiredrequirement is not satisfied, it reduces the aggregated score without rejecting such a system; if the desired requirementisfullysatisfied,itincreasestheaggregatedscore. Dujmovic suggested (in a type-1 setting) that the weighted powermean (WPM) performs thebestforageneralclassofparameterizedaggregationoperators. WPMisdefinedas: L r (X,W) = lim q→r P n i=1 w i x q i P n i=1 w i 1/q (6.10) 120 wherer ∈ℜ, X≡ (x 1 ,...,x n ) and W≡ (w 1 ,...,w n ). Asr ranges from−∞ to +∞, WPM rangescontinuouslybetweenpureconjunctionandpuredisjunction. Rickard et al. [92] proposed a new family of aggregation operators that admits IT2 FSs for both the inputs and weights, namely, the linguistic weighted power mean (LWPM). The LWPMincludestheLinguisticWeightedAverage(LWA)[117,118,122]asaspecialcase,and isconsideredtobeamoreflexibleaggregationoperatorthantheLWA.LWPMisdefinedas: L r ( e X, e W) = lim q→r ! P n i=1 f W i e X q i P n i=1 f W i % 1/q (6.11) wherer ∈ℜ, e X≡ ( e X 1 ,..., e X n ) is the set of input FOUs and e W≡ ( f W 1 ,..., f W n ) is the set of associatedweightFOUs. Notethatwhenallinputsarecrisp,theLWPMreducestotheWPM operation. Like the WPM, as r ranges over the real line, L r (·) ranges from logical conjunction (as r→−∞)oftheinputs, e X 1 ∧···∧ e X n ,tologicaldisjunction(asr→ +∞), e X 1 ∨···∨ e X n . L r (·) is thus an orand operator. Other special cases of the LWPM include the LWA (r = 1), geometric mean (r = 0), harmonic mean (r = −1) and Euclidean mean (r = 2). How to chooser inourapplicationwillbeexplainedinSection6.5. In order to compute the LWPM, Rickard et al. [91] have generalized the EKM algorithm [124]totheWeightedPowerMeanEKMalgorithm(WPMEKM). When parameterized aggregation operators, e.g., WPM and LWPM, range between con- junctionanddisjunction,andnessisdefinedasameasureofsimilaritybetweentheaggregation operatorandthefullconjunction,andornessisdefinedasameasureofsimilaritybetweenthe aggregation operator and the full disjunction. For the WPM (inputs are all crisp), the global andness(α g )andtheglobalorness(ω g )aredefinedas[14,15,92]: α (r) g (n) = n−(n+1)L r (X,W) n−1 = 1−ω (r) g (n) (6.12) and 121 ω (r) g (n) = (n+1)L r (X,W) n−1 = 1−α (r) g (n) (6.13) where L r (X,W) is defined in (6.10) and the overbar denotes averaging over the hypercube [0,10] n inwhichXlies. WhenL r (X,W) is the WPM with fixed weights and input scores,L r (X,W) is a continu- ous and increasing function of r. Therefore, from (6.12), the andness α (r) g (n) is a monotoni- cally decreasing function ofr. When−∞ < r < 1,L r (X,W) represents partial conjunction (α (r) g (n) > 1/2); for 1 < r < +∞,L r (X,W) represents partial disjunction. When the inputs are IT2 FSs (using the LWPM), the qualitative aspects of partial disjunction/conjunction are stillpreserved. Figure6.5: Aconjunctivepartialabsorptionoperator. A partial absorption (PA) operator combines a mandatory requirement and a desired requirement. A conjunctive PA operator is depicted in Fig. 6.5 [15]. W 1 and W 2 are two adjustable parameters, “A” is an aggregation operator (e.g., LWPM) with ω (r) g (2) = 1/2, “Δ” is an aggregation operator (also using the LWPM) with α (r) g (2) ≥ 2/3, and t is the output of “A”. In Fig. 6.5, t = L r ([x,y],[w 1 ,w 2 ]), where w 1 = W 1 and w 2 = 1− W 1 (0≤W 1 ,W 2 ≤ 1),andz =L r ([x,t],[w 1 ,w 2 ]),wherew 1 =W 2 andw 2 = 1−W 2 . 1 Observe,inFig. 6.5,thattheoutputz isaffectedbyboththevalueofthemandatoryinput x and the value of the desired input y. If the mandatory input is zero, the output is zero, regardless of the value of the desired input y; however, if the mandatory input is partially 1 Notethatin(6.10)and(6.11),theresultsoftheWPMandLWPMarenormalizedbytheinputweights,that is,thescaleoftheweightsdoesnotaffecttheoutput. Herethescaleweusedis[0,1],asin[15];however,weuse [0,100]laterfortheIT2FScase. 122 satisfied, then there are two possibilities. If y < x, then the output z is x decremented by a penalty function, P(x,y) ≡ x−z, and, if y > x, z is x incremented by a reward function, R(x,y)≡z−x,where 0≤x,y≤ 10andP(x,y),R(x,y)≥ 0. AccordingtoDujmovic,inpractice,theweightsW 1 andW 2 areobtainedfromthedesired mean values of penalty and reward function, defined as P = 1 10 2 R 10 0 R 10 0 P(x,y)dxdy and R = 1 10 2 R 10 0 R 10 0 R(x,y)dxdy, respectively. 2 There are two ways to determine these two weights. Thefirstistousetablesintroducedin[17]. Thesecondwayistoprovideatrainingset of desired{x,y,z} triplets, and determine the weights using the preferential neuron training techniquedescribedin[18]. Forsimplicity,weusedthetablein[17]todetermineW 1 andW 2 . Theglobalandness(α g )andtheglobalorness(ω g )canbesatisfiedbychoosingpropervalues ofr intheLWPM. Based on the conjunctive partial absorption operation shown in Fig. 6.5, a nested aggre- gator[15,16]thatcombinesseveralmandatoryanddesiredinputscanbegeneralizeddirectly, as in Fig. 6.6. Comparing Fig. 6.6 to Fig. 6.5, observe that all mandatory inputs and desired inputs are first aggregated using the “Δ” and “A” aggregators, respectively, so that their mandatory/desired properties are preserved. The resulting outputs are then aggregated as in Fig. 6.5, where, again, W 1 and W 2 can be determined according to the user-defined average penaltyandrewardfunctions(0≤W 1 ,W 2 ≤ 1),asin[17]. Figure 6.6: Aggregating collection of mandatory and desired requirements, where z is the aggregatedoutputandthescaleofweightsis[0,1]. 2 The scale of x, y, P(x,y) and R(x,y) in [17] are all [0,1]. In order to use the table in [17], we have to normalizeaveragepenaltyandrewardfunctionssothatitwillstillbeinthatrange. 123 Fig. 6.6 is also applicable to LWPM operations, where inputs and output are IT2 FSs. In this Chapter, the weights are IT2 FSs and the scale of the weights is [0,100], so Fig. 6.6 is re-drawnasinFig. 6.7. In Fig. 6.7, e Y 1 and e Y 2 are obtained by aggregating the inputs using their respective fuzzy weights,accordingtoTable6.2andFig. 6.3. Figure6.7: Aggregatingcollectionofmandatoryanddesiredrequirements,wheretheweights arerepresentedbyIT2FSsandthescaleofweightsis[0,100]. 6.5 Example Inthissection,anexampleofourproposedPer-Cisdescribed,thatimplementsLWPMasthe CWWengine. In this example, the instructor who is teaching “Circuits and Signals” includes the fol- lowingtwosubjectivemandatoryrequirementsintheevaluationprocess: (1)Ifastudentgets a very low score on lab (M L is very low), no matter how well she does in assignment and exam,theperformanceofLO2(Table6.1)shouldalsoberelativelylow;and(2)Ifastudent’s performance on LO2 is very low, no matter how well she does in LO1 and LO3, the final performance should be relatively low. By using the LWPM, the above requirements can be implementedautomaticallybythePer-C. Dujmovic[14]computedseveralvaluesofα (r) g andω (r) g fortheWPM.Accordingto[14], α (−1) g (2) = 0.77(≥ 2/3),soweusedr =−1forthe“Δ”operator(seeFigs. 6.5and6.6). For the“A”operator,weusedr = 1,asexplainedabove. WealsochoseP = 20%andR = 10%. Usingthetablefrom[17],wechoseW 1 = W 2 = 0.6(Fig. 6.6). Thescaleofweightsusedin 124 ourexampleis[0,100],sotheweightsarefirstscaledtoW ′ 1 = W ′ 2 = 60andthenmappedto aFOUusingFig. 6.3,i.e., f W 1 = f W 2 = f W(60). According to our assumptions, assignment and exam are desired requirements and lab is amandatoryrequirement. FromFig. 6.7,themandatoryrequirementonLO2isimplemented as: e Y 1 = e Y L e Y 2 =L 1 ( e Y AE , e W AE ) e Y 3 =L 1 ([ e Y 1 , e Y 2 ],[ f W(60), f W(40)]) e Y LO2 =L −1 ([ e Y 1 , e Y 3 ],[ f W(60), f W(40)]) (6.14) where e Y AE , e W AE aredefinedjustafter(6.7). There are no special requirements on LO1 and LO3, so they are computed directly using theLWA(i.e.,theLWPMwhenr = 1),as: e Y LO1 =L 1 [ e Y AE , e W AE ] (6.15) e Y LO3 =L 1 [ e Y LE , e W LE ] (6.16) where e Y LE and e W LE arethesameasdefinedafter(6.7). Similarto(6.14), e Y F iscomputedas: e Y 1 = e Y LO2 e Y 2 =L 1 ([ e Y LO1 , e Y LO3 ],[ f W(35), f W(25)]) e Y 3 =L 1 ([ e Y 1 , e Y 2 ],[ f W(60), f W(40)]) e Y F =L −1 ([ e Y 1 , e Y 3 ],[ f W(60), f W(40)]) (6.17) wheretheweightsforLO1andLO3areobtainedfromTable6.2. 125 Suppose,forexample,thescoresofonestudentareM A = 27,M L = 1andM E = 58,i.e., shedidreallywellforassignmentandexambutonlygot1pointforthelab. TheFOUsofthe threeLOsandthefinalperformancecomputedusing(6.14)-(6.17)areshowninFig. 6.8. Figure 6.8: FOUs for LOs and the final performance using LWPM whenM A = 27,M L = 1 andM E = 58. (a): e Y LO1 ; (b): e Y LO2 ; (c): e Y LO3 ; (d): e Y F . Jaccard’s similarity measures were computed between these FOUs and those of the five wordsinthecodebook(Fig. 6.2),andtheresultsareshowninTable6.3. 3 Table6.3: Jaccard’ssimilaritymeasuresmatrix e Y e X 1 e X 2 e X 3 e X 4 e X 5 e Y LO1 0 0 0.03 0.22 0.86 e Y LO2 0.62 0.40 0.07 0 0 e Y LO3 0 0.03 0.19 0.66 0.31 e Y F 0.51 0.55 0.13 0.02 0 3 Notice that some of the similarity measures are very small, due to the fact that some of the LO FOUs are verynarrowcomparingtotheperformanceFOUsinthecodebook. 126 Although the total raw scores for this student are 27 + 1 + 58 = 86, the performance of LO1, LO2, LO3 and final performance for her are Excellent, Poor, Good and Marginal, respectively. The final performance of Poor is because of the mandatory requirements since sheperformedbadlyforlab. Forcomparativepurposes,wealsocomputed e Y LO2 andthen e Y F usingonlytheLWA,as e Y LO2 =L 1 [ e Y ALE , e W ALE ] (6.18) and e Y F =L 1 [ e Y LO , e W LO ] (6.19) The FOUs of the three LOs and the final performance using the LWA are shown in Fig. 6.9. TheJaccard’ssimilaritymeasureswerecomputedandtheresultsareshowninTable6.4. Observe that the performance of LO1, LO2, LO3 and final performance for this student are Excellent, Good, Good and Excellent, respectively. The final performance does not fulfill the mandatory requirements described at the beginning of this section; hence, this illustrates that theLWPMisabetterchoicefortheaggregationfunctionthanistheLWA. Figure6.9: FOUsforLOsandthefinalperformanceusingLWAwhenM A = 27,M L = 1and M E = 58. (a): e Y LO1 ; (b): e Y LO2 ; (c): e Y LO3 ; (d): e Y F . 127 Table6.4: Jaccard’ssimilaritymeasuresmatrix e Y e X 1 e X 2 e X 3 e X 4 e X 5 e Y LO1 0 0 0.03 0.22 0.86 e Y LO2 0 0.01 0.12 0.56 0.42 e Y LO3 0 0.03 0.19 0.66 0.31 e Y F 0 0.01 0.09 0.48 0.49 6.6 Summary InthisChapter,aPer-CisdesignedtoevaluatetheLOsandtothenmakeafinalperformance recommendationusingtheseLOevaluations. ThereareseveraladvantagestousingaPer-Cin theevaluationprocess: 1. The use of IT2 FSs “fuzzifies” the standards of the LOs so that they will never be too concrete. 2. ByusingtheLWPM,theCWWenginecanaccomplishseveralcompoundlogicaloper- ations,whichletstheinstructorincludesubjectivegradingpoliciesintotheevaluationsystem. 3. Once the instructor provides some subjective inputs to the Per-C, the entire evaluation process can be done automatically, which relieves the instructor’s burden to track and report separateLOs. 128 Chapter7 ConclusionsandFutureWorks 7.1 Conclusions This thesis consists of four major parts: 1) aggregating the Injector Producer Relationship (IPR) estimates from different models; 2) a new similarity measure for general type-2 fuzzy sets(GT2FSs);3)anewmethodtomodelwordsusingnormalintervaltype-2fuzzysets(IT2 MFs);4)extensionandapplicationofthelinguisticweightedpowermean(LWPM)operation. In Chapter 2, we constructed four different models for the interaction between a group of injectors and a producer and then dynamically estimated the parameters of these mod- els, along with the interwell connectivities using an Iterated Extended Kalman Filter (IEKF) and Smoother (EKS). In order to obtain even better production forecasting results, we then used both Weighted Average (WA) and Generalized Choquet Integral (GCI) to aggregated IPR estimates from different models. These two aggregation functions were optimized to minimize mean-square errors in future forecasted production rates. This was done by using QuantumParticleSwarmOptimization(QPSO)tosearchfortheoptimalsetofweightswhich are required by both aggregation methods. Several experiments were conducted to show the improved average performance of our approach on a set of data from a real reservoir, and the performancesoftheabovetwoaggregationmethodswerealsocomparedandanalyzed. Ourresultssuggestedthatthenumberofinformationsourcestobeaggregatedisanimpor- tant factor that affects the performance of the aggregation approach, and should be chosen carefully in practice. In fact, sometimes it is better to aggregate fewer than the maximum numberofpossibleIPRestimates. Also,fromourexperiments,whenaggregatingtwoIEKFS, the optimized WA and GCI methods were very similar to each other, but when aggregating four IEKFS, the optimized WA was much better than the optimized GCI with the same com- putational cost (same population size and number of generations in QPSO). Because it is 129 much easier to understand the optimized WA, we recommend that it should be used for this application. In Chapter 3, a new similarity measure for GT2 FSs was proposed using the α-plane representation; it is an extension of Wu and Mendel’s similarity measure for IT2 FSs. Some properties of the new similarity measure were proved; numerical examples were given, and theeffectsofthesecondaryMFSindifferentscenarioswerediscussed. Ouroverallconclusion about “Is there much of a difference between a GT2 FS and an IT2 FS” can be answered by usingournewsimilaritymeasure. Fromoursimulations,weshowedthatthesecondaryMFs canmakeaGT2FSverydifferentfromitsFOU(anIT2FS). InChapter4,anewmethod−theHMmethod−forconstructingnormalIT2FSmodelsfor words from interval data that are collected from a group of subjects was presented. The HM method uses more information about the overlap of these data intervals than does the EIA. It leads, for the first time, to fully normal FOUs, i.e., FOUs whose LMF and UMF are normal T1 FSs. Such fully normal word FOUs may be quite useful in advanced computing with words applications because they can be used to simplify numerical computations. We have also compared the FOUs from the HM method with those from the EIA, using similarities. Our experiments suggested that a metric is needed to determine which model is better for a specificproblem. In Chapter 5, a new fuzzy statistic, the LWSD was proposed. It was computed using the weighted power mean and the WMPEKM algorithm. Analogous to the standard deviation in classical statistics, the LWSD computes the weighted standard deviation of a set of IT2 FSs. Two numerical examples are given to demonstrate the use of the LWSD and the possibilities forpotentialapplicationindecisionmakingproblems. InChapter6,asanapplicationoftheLWPMoperationonIT2FSs,aPerceptualComputer (Per-C)isdesignedtoevaluatetheLOsandtothenmakeafinalperformancerecommendation usingtheseLOevaluations. 130 7.2 FutureWorks Inthissection,somefutureresearchtopicsaredescribed. 7.2.1 ObtainingAMetricforDeterminingWhichWordModelIsBest InChapter4,anewmethod,namely,theHMmethodisproposedtomodelwordsusingnormal IT2FSs,anditiscomparedwiththeEIAusingsimilarities. TheFOUsgeneratedfromtheHM method is compared with those from the EIA. Our experiments suggested that, on one hand, theFOUgeneratedfromtheHMmethodandEIAFOUsaremoresimilarthannot,sincemost ofthesimilaritiesaregreaterthan0.5;ontheotherhand,mostofthesimilaritiesarelessthan 0.8,whichindicatesthechoiceofwhichmethodtousemaymakeadifferenceinsomeCWW applications. Therefore, for different applications, which modeling method would one use needstobedeterminedaheadoftime. Unfortunately,tothebestofourknowledge,thereisno systematicwaytodothis. In both the EIA and HM method, we try to use IT2 FSs to model the randomness and uncertainties associated with the data intervals collected from a group of subjects. There has beenseveralinformativemeasuresdevelopedforfuzzysets,e.g.,theentropy. Onefuturetopic of research is to define a metric of information, so that the modeling method that keeps the mostinformationfromthedataintervalscanbedeterminedforaspecificapplication. 7.2.2 ObtainingFOUsfromCyclicData In real life, there are lots of processes that contain a cyclic nature, e.g., the Carnot cycle, the Clapeyroncycle,andtheClausius-Rankinecycle. Theyaresomeofthethermodynamiccycles thathaveplayedanimportantroleintheevolutionofthegeneralprinciplesofthermodynamics and in the development of engineering applications of thermodynamics. In these processes, dataarealsomeasuredperiodically. Fig. 7.1demonstratesanexampleofsuchdataset. 131 0 1 2 3 4 5 6 7 x 10 4 0 200 400 600 800 1000 1200 Pressure Sample Figure7.1: Anexampleofdatafromacyclicprocess. AsshowninFig. 7.1,notonlyistherenoise(uncertainties)ineachcycle,buttherearealso uncertainties among cycles. In order to capture these uncertainties, fuzzy clustering methods areextremelyusefulinprocessing. Oneofthemostwidelyusedfuzzyclusteringalgorithmsis the Fuzzy C-Means (FCM) Algorithm [60]. The FCM algorithm partitions a finite collection of n elements X = {x 1 ,...,x n } into a collection of c clusters with respect to some given criterion. Givenafinitesetofdata,theFCMAlgorithmreturnsalistofcclustercentersC = {c 1 ,c 2 ,...,c c } and a partition matrix W = w i,j ∈ [0,1], where i = 1,...,n and j = 1,...,c. Eachw i,j isthedegreetowhichelementx i belongstoclusterc j . Using w i,j , one is able to construct type-1 membership functions (MFs) for each cluster. Anexamplethatclustersasetofdatainto3clustersusingtheFCMalgorithmisshowninFig. 7.2. 132 200 250 300 350 400 450 500 550 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 c Low c Medium c High Low Medium High Figure7.2: ClustergradesobtainedfromtheFCMalgorithm. Inouropinion,however,theFCMalgorithmisnotenoughtoalsomodeltheuncertainties among cycles. To do that, we need a model that captures both kinds of uncertainties, and we believe this can be done by using an interval type-2 (IT2) fuzzy set. Therefore, in the future, wewilldevelopamethodologytoaggregatetheT1MFsobtainedfromtheFCMalgorithmso thatanIT2modelcanthenbeobtained. 7.2.3 ExtensionofWang-Mendel(WM)MethodstoIT2FSs In1992, WangandMendel [110]proposed anewapproachto generatefuzzy IF-THENrules from numerical data pairs, now known as the Wang-Mendel (WM) method. The generated fuzzy rule base can be combined with experts’ knowledge to construct a combined system of rules. They proved that the resulting fuzzy system is capable of approximating any nonlinear continuousfunctiononacompactsettoarbitraryaccuracy. In 2003, Wang [109] introduced two other rule-extracting methods that are based on the original WM method. These two new methods are variants of the WM method and are used tosolvedifferentproblemsfordifferentpurposes. To the best of our knowledge, there has not been any work that extends the WM methods to IT2 FSs. As mentioned in Section 7.2.2, an IT2 clustering is capable of capturing more uncertainties from the data. Therefore, we will extend all three WM methods to IT2 FSs so 133 that they can make use of the IT2 version of the clustering results. Experiments will also be performedtoseewhetherwecanobtainbetterresultsusingIT2FSs. 134 ReferenceList [1] J. Aisbett, J. T. Rickard, and D. G. Morgenthaler. Type-2 fuzzy sets as functions on spaces. IEEETransactionsonFuzzySystems,18:841–844,August2010. [2] A. Albertoni and L. W. Lake. 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Usingsuch states,thestatevectorsfortheLMMandDCMareconstructedasfollows. For the LMM, (2.5) is described by a 4 × 1 state vector x j (k), where x j (k) ≡ [ p IPR j (k), p α j (k),P c j (k − 1),P c j (k)] T . Note that for this model, it is p IPR j (k) and p α j (k)thataregivenbythemodelin(2.1)and(2.2). Itisstraightforwardtoshow[144],that theSVMforP c j is(j = 1,2,3):                                      x j (k +1) =                x j1 (k) x j2 (k) x j4 (k) 2x 2 j2 (k)x j4 (k)−x 4 j2 (k)x j3 (k) +x 2 j1 (k)[1−x 2 j2 (k)] 2 I m j (k)                +n x j (k) p c j (k +1) = h 0 0 0 1 i x j (k +1)+n p c j (k +1) (A.1) wheren x j (k) = [n √ IPR j (k),n √ α j (k),0,n P c j (k)] T ,eachofwhosenon-zerocomponentsisan additivezero-meanwhiteGaussiannoiseandthecovariancematrix,Q nx j ,isgivenby: 146 Q nx j =diag[r √ IPR j ,r √ α j ,0,r P c j ] (A.2) wherer √ IPR j ,r √ α j andr P c j arevariancesofn √ IPR j (k),n √ α j (k)andn P c j (k),respectively. Let x(k + 1) ≡ [x 1 (k + 1),x 2 (k + 1),x 3 (k + 1)] T , then the complete SVM for a simple 3-injector1-producersystemisgivenby:                      x(k +1) =        x 1 (k +1) x 2 (k +1) x 3 (k +1)        +n x j (k) P m (k +1) = h H 1 H 2 H 3 i x(k +1)+n P (k +1) (A.3) where H j = h 0 0 0 1 i (j = 1,2,3) (A.4) Similarly for the DCM, (2.10) is described by a 5× 1 state vector x j (k), where x j (k) ≡ [ p IPR j (k), p α j1 (k), p α j2 (k),P c j (k− 1),P c j (k)] T , for which p IPR j (k), p α j1 (k) and p α j2 (k) are given by models like (2.8) and (2.9). The DCM SVM for P c j (j = 1,2,3) is givenby:                                                    x j (k +1) =                        x j1 (k) x j2 (k) x j3 (k) x j5 (k) [x 2 j2 (k)+x 2 j3 (k)]x j5 (k) −x 2 j2 (k)x 2 j3 (k)x j4 (k) +x 2 j1 (k)[1−x 2 j2 (k)][1−x 2 j3 (k)]I m j (k)                        +n x j (k) p c j (k +1) = h 0 0 0 0 1 i x j (k +1)+n p c j (k +1) (A.5) 147 where n x j (k) = [n √ IPR j (k),n √ α j1 (k),n √ α j2 (k),0,n P c j (k)] T , each of whose non-zero com- ponentsisanadditivezero-meanwhiteGaussiannoiseand Q nx j =diag[r √ IPR j ,r √ α j1 ,r √ α j2 ,0,r P c j ] (A.6) ThecompleteDCMSVMfora3-injector1-producersystemisalsogivenby(A.3),where inthiscase,x j isgivenby(A.5)and H j = h 0 0 0 0 1 i (j = 1,2,3) (A.7) Inthispaper,EKFs(orIEKFs)basedontheaboveSVMsarecalledthe Square Root (SR) EKFs(orIEKFs)(seeAppendixC). 148 AppendixB Non-SquareRootIPRSVMs For the LMM, (2.5) is now described by a new 4× 1 state vector x j (k), where x j (k) ≡ [IPR j (k), p α j (k),P c j (k − 1),P c j (k)] T . Similar to Appendix A, the SVM for P c j is now (j = 1,2,3):                                      x j (k +1) =                x j1 (k) x j2 (k) x j4 (k) 2x 2 j2 (k)x j4 (k)−x 4 j2 (k)x j3 (k) +x j1 (k)[1−x 2 j2 (k)] 2 I m j (k)                +n x j (k) P c j (k +1) = h 0 0 0 1 i x j (k +1)+n p c j (k +1) (B.1) wheren x j (k) is the same noise vector that was defined in Appendix A (below (A.1)), but the firstelementisnown IPR j (k)ratherthann √ IPR j (k). For the DCM, (2.10) is now described by a new 5× 1 state vector x j (k), where x j (k)≡ [IPR j (k), p α j1 (k), p α j2 (k),P c j (k− 1),P c j (k)] T . The DCM SVM for P c j (j = 1,2,3) is givenby: 149                                                    x j (k +1) =                        x j1 (k) x j2 (k) x j3 (k) x j5 (k) [x 2 j2 (k)+x 2 j3 (k)]x j5 (k) −x 2 j2 (k)x 2 j3 (k)x j4 (k) +x j1 (k)[1−x 2 j2 (k)][1−x 2 j3 (k)]I m j (k)                        +n x j (k) P c j (k +1) = h 0 0 0 0 1 i x j (k +1)+n p c j (k +1) (B.2) wheren x j (k) is the same noise vector that was defined in Appendix A (below (A.5)), but the firstelementisnown IPR j (k)ratherthann √ IPR j (k). The EKFs (or IEKFs) based on the above SVMs are called the non-square root (NSR) EKFs(orIEKFs)(seeAppendixC). 150 AppendixC IEKFsfortheSquareRootand Non-SquareRootSVMs C.1 GeneralEKFEquations The EKF has two stages, predictor and corrector. The predictor computes the predicted esti- mate of x(k + 1) based on the measurements up to time k, denoted as b x(k + 1|k); and, the correctorcomputesthefilteredestimateofx(k+1)usingthemeasurementsuptotimek+1, denotedasb x(k +1|k +1). IntheEKF,thestateequationislinearizedaboutb x(k|k),andthemeasurementequationis linearizedaboutb x(k +1|k),i.e.,                    x(k +1)≈ f[b x(k|k),k]+F x [b x(k|k),k] ×[x(k)−b x(k|k)]+n x (k) y(k +1)≈ h[b x(k +1|k),k +1]+H x [b x(k +1|k),k +1] ×[x(k +1)−b x(k +1|k)]+n y (k +1) (C.1) where b x(k +1|k) = f[b x(k|k),k],F x = ∂f[x(k),k]/∂x(k),andH x = ∂h[x(k),k]/∂x(k). Note thatF x andH x areJacobianmatricesforaspecificSVM,thataregivenbelowinSectionC. TheEKFfor(2.11)is: 1. InitializetheEKFwithb x(0|0),P(0|0),Q k andr(k). 2. Predictor(EKP)(k = 0,1,...): b x(k +1|k) = f[b x(k|k),k] (C.2) 151 P(k +1|k) = F x [b x(k|k),k]P(k|k)F ′ x [b x(k|k),k]+Q k (C.3) 3. Corrector(k = 0,1,...): b x(k +1|k +1) = b x(k +1|k)+K(k +1) ×{y(k +1)−h[b x(k +1|k),k +1]} (C.4) K(k +1) = P(k +1|k)H ′ x H x P(k +1|k)H ′ x +r(k +1) (C.5) P(k +1|k +1) = [I−K(k +1)H x ]P(k +1|k) (C.6) NotethattheequationfortheEKPisgivenby(C.2). C.2 IEKFEquations TheIEKFiteratesthecorrectionequationin(C.4)usingthemostrecentfilteredestimate. The equationsforl-thcorrector(l = 2,...)is: b x l (k +1|k +1) = b x l−1 (k +1|k +1)+K l (k +1) ×{y(k +1)−h[b x l−1 (k +1|k +1),k +1]} (C.7) K l (k +1) = P l−1 (k +1|k +1)H ′ x H x P l−1 (k +1|k +1)H ′ x +r(k +1) (C.8) P l (k +1|k +1) = [I−K l (k +1)H x ]P l−1 (k +1|k +1) (C.9) 152 wherethesubscriptldenotestheiterationnumber,andb x 1 (k+1|k+1),K 1 (k+1)andP 1 (k+ 1|k +1) are initialized by (C.4), (C.5) and (C.6), respectively. Note that the iterative process stops either whenl = L max whereL max is the maximum number of iterations, or when there isverylittleimprovementbetweentwoconsecutiveiterations,i.e.,k b x l (k+1|k+1)−b x l−1 (k+ 1|k +1)k≤ǫ. Inthispaper,wechoseL max = 20andǫ = 10 −3 . C.3 JacobianMatrices ForthesquarerootSVMbasedontheLMM,inthe3-injector1-producercase,iffollowsfrom (A.3),that F x =        A 1 0 0 0 A 2 0 0 0 A 3        (C.10) and H x = h H 1 H 2 H 3 i (C.11) where0isa 4×4zeromatrix,H j aregivenby(A.4)andA j isgivenby(C.12) (j = 1,2,3). A j =            1 0 0 0 0 1 0 0 0 0 0 1 2x j1 (k)(1−x 2 j2 (k)) 2 I m j (k) 4x j2 (k)x j4 (k)−4x 3 j2 (k)x j3 (k)−4x j2 (k)(1−x 2 j2 (k))x 2 j1 I m j (k) −x 4 j2 (k) 2x 2 j2 (k)            (C.12) For the square root SVM based on the DCM, F x and H x are also given by (C.10) and (C.11). Now, 0 is a 5× 5 zero matrix, H j are given by (A.7) and A j is given by (C.13) (j = 1,2,3). 153 A j =                     1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 2x j1 (k)(1−x 2 j2 (k))(1 2x j2 (k)[x j5 (k)−x 2 j3 (k)x j4 (k) 2x j3 (k)[x j5 (k)−x 2 j2 (k)x j4 (k) −x 2 j2 (k)x 2 j3 (k) x 2 j2 (k) −x 2 j3 (k))I m j (k) −(1−x 2 j3 (k))x 2 j1 I m j (k)] −(1−x 2 j2 (k))x 2 j1 I m j (k)] +x 2 j3 (k)                     (C.13) Similarly,forthenon-squarerootSVMbasedontheLMM,A j isgivenby(C.14)andfor thenon-squarerootSVMbasedontheDCM,A j isgivenby(C.15), (j = 1,2,3). A j =            1 0 0 0 0 1 0 0 0 0 0 1 (1−x 2 j2 (k)) 2 I m j (k) 4x j2 (k)x j4 (k)−4x 3 j2 (k)x j3 (k)−4x j2 (k)(1−x 2 j2 (k))x j1 I m j (k) −x 4 j2 (k) 2x 2 j2 (k)            (C.14) A j =                     1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 (1−x 2 j2 (k))(1 2x j2 (k)[x j5 (k)−x 2 j3 (k)x j4 (k) 2x j3 (k)[x j5 (k)−x 2 j2 (k)x j4 (k) −x 2 j2 (k)x 2 j3 (k) x 2 j2 (k) −x 2 j3 (k))I m j (k) −(1−x 2 j3 (k))x j1 I m j (k)] −(1−x 2 j2 (k))x j1 I m j (k)] +x 2 j3 (k)                     (C.15) C.4 Inequality-ConstrainedIEKFs(CIEKF) As mentioned in Appendix A, there is a physical constraint that the IPR in any model should begreaterthanorequalto0. Inordertosatisfythisconstraint,aCIEKFwasusedinourIEKF processingforthenon-squarerootIPRSVMs. 154 ToimplementaCIEKF,weadoptedtheMean-SquareMethoddescribedin[86]. Consider the non-square root SVM based on the LMM described in (B.1) whose complete state vector x is given by (A.3). After we obtained an unconstrained estimate b x from the IEKF, we found aconstrainedestimate,e x,suchthat: De x≤ d (C.16) whereD≡diag[−1,0,0,0,−1,0,0,0,−1,0,0,0]andd≡ 0. Qianetal.[86]showedthate xcanbefoundbysolvingthefollowingproblem: min e x {(e x−b x) T (e x−b x)}, suchthatDe x≤ d. (C.17) This is done at each time point using quadratic programming in our processing. Note that, from(C.17),ifb xsatisfies(C.16),i.e.,Db x≤ d,wehavee x = b x. Note that the constrained estimate using the non-square root SVM based on the DCM can also be computed using (C.17), where in this case, D ≡ diag[−1,0,0,0,0,−1,0,0,0,0,−1,0,0,0,0]. 155 AppendixD EKSsfortheLMMandDCM Thefixed-intervalsmoothedestimatorofx(k),b x(k|N),wherel = 1,2,...,isgivenby(Lesson 21in[63]): b x(k|N) = b x(k|k−1)+P(k|k−1)r(k|N) (D.1) wherek =N−1,N−2,...,1,andn×1vectorrsatisfiestheback-recursiveequation r(j|N) = Φ ′ p (j +1,1)r(j +1|N) +H ′ x (j)[H x (j)P(j|j−1)H ′ x (j)+r(j)] −1 [y(j)−b x(j|j−1)] (D.2) wherej =N,N−1,...,1andr(N +1|N) = 0. Matrix Φ p isdefinedas: Φ p (k +1,k) = F ′ x (k)[I−K(k)H x (k)] (D.3) NotethatF x (k),H x (k),K(k)andP(k|k−1)areallgiveninAppendixC. During EKS computations, r(N|N) and r(N − 1|N) are first computed iteratively using (D.2), then b x(N − 1|N) is computed, after which, r(N − 2|N), b x(N − 2|N), ..., etc are computed. This is done until the time point M j is reached at which the smoothed estimate b x(M j |N j )iscomputed. 156 AppendixE QPSOalgorithm TheQPSOalgorithmusedinthispaperissummarizedasfollows: Initializepopulation: randomlygenerateX i (0)andsetP i (0) = X i (0)(i = 1,...,M); fort = 0toMaxGenerationdo computem(t)using(2.17); fori = 1toM do iff(X i (t))<f(P i (t))then P i (t) = X i (t); endif g = argmin(f(P i (t))); η =rand(0,1); p i (t+1) =ηP i (t)+(1−η)P g (t); u =rand(0,1); ifrand(0,1)> 0.5then X i (t+1) = p i (t+1)−β|m(t)−X i (t)|ln(1/u); else X i (t+1) = p i (t+1)+β|m(t)−X i (t)|ln(1/u); endif endfor endfor 157 AppendixF Computingthecentroidandstandard deviationfor e LinFig. 4.2 Thissectionexplainshow(4.18)and(4.19)areobtainedforcomputingthecentroidandstan- dard deviation for e L. The mean of a “straight-line” T1 FS shown in Fig. 4.2 (e.g.,o l −a ′ ) is takenfromTableIin[126]: m MF = 2o l +a ′ 3 (F.1) Thestandarddeviationiscomputedas: s MF = (o l −a ′ ) 3 √ 2 (F.2) Note that∀a ′ ∈ [a l ,a r ],m MF in (F.1) is an increasing function ofa ′ ; and,s MF in (F.2) is a decreasing function ofa ′ . Therefore, the lower and upper bounds of the mean of “straight- line” T1 FS in e L (see Fig. 4.2) can be obtained whena ′ = a l anda ′ = a r , respectively; and, the lower and upper bounds of the standard deviation of “straight-line” T1 FS in e L can be obtainedwhena ′ =a r anda ′ =a l ,respectively. 158

Abstract (if available)

Abstract In waterflood management, there exists several models to describe a petroleum reservoir for predicting the future production rates using scheduled injection rates. Most of them have the ability to estimate how much the injectors impact some specific producers, namely, the interwell connectivities between the injectors and the producers. Knowing these values not only reduces the cost of water injection, but can also increase the oil production. ❧ In the first part of this thesis, we construct four different models for the interaction between a group of injectors and a producer and then dynamically estimate the parameters of these models, along with the interwell connectivities using an Iterated Extended Kalman Filter (IEKF) and Smoother (EKS). We then use the Weighted Average (WA) and Generalized Choquet Integral (GCI) to aggregate the estimated interwell connectivities. These two aggregation functions are optimized to minimize mean-square errors in future forecasted production rates. This is done by using Quantum Particle Swarm Optimization (QPSO) to search for the optimal set of weights which are required by both aggregation methods. Several experiments are conducted to show the improved average performance of our approach on a set of data from a real reservoir, and the performances of the above two aggregation methods are also compared and analyzed. ❧ A similarity measure between fuzzy sets is a very important concept in fuzzy set theory. There have been a lot of different similarity measures proposed in the literature, for both T1 FSs and IT2 FSs. The second part of this thesis presents theoretical studies that were performed for the most advanced fuzzy logic sets that are currently under research‐general type‐2 fuzzy sets. In our study, based on the α-plane representation for a general type‐2 (GT2) FS, the similarity measure is generalized to such T2 FSs. Some examples that demonstrate how to compute the similarity measures for different T2 FSs are given. ❧ Next, the third part of this thesis proposes a new method—the HM method—to model words by normal IT2 FSs, using data intervals that are collected from a group of subjects. The HM method uses the same bad data processing, outlier processing and tolerance limit processing to pre‐process the data intervals, as is used in the Enhanced Interval Approach (EIA)

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Creator Hao, Minshen (author)

Core Title Aggregation and modeling using computational intelligence techniques

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Electrical Engineering

Publication Date 11/12/2014

Defense Date 03/14/2014

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag aggregation,computational intelligence techniques,computing with words,general type‐2 fuzzy sets,interval type‐2 fuzzy sets,interval type‐2 word models,OAI-PMH Harvest,similarity measure,uncertainty modeling

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Mendel, Jerry M. (committee chair), Ershaghi, Iraj (committee member), Ortega, Antonio K. (committee member)

Creator Email donkeybo@gmail.com,minshenhao@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c3-413110

Unique identifier UC11296327

Identifier etd-HaoMinshen-2508.pdf (filename),usctheses-c3-413110 (legacy record id)

Legacy Identifier etd-HaoMinshen-2508.pdf

Dmrecord 413110

Document Type Dissertation

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Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

Repository Name University of Southern California Digital Library

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