第三章系统描述与模型建立

第三章系统的描绘与模型成立Chapter3.DescriptionofSystemsandModeling3.6不确立性描绘(二)–模糊性3.6DescriptionofUncertainty(2)-Randomness一.精准与模糊1.Precisionandfuzziness统计剖析随机性描绘主元剖析-不确立性描绘因子剖析模糊性描绘DescriptionofuncertaintystatisticanalysisDescriptionofrandomnessPCA,PrincipleComponentAnalysisfactoranalysisDescriptionoffuzziness-

精准思想及方法在科学技术发展中日趋获得成功.往常”千锤百炼”被以为是科学工作者的美德.Thinkingandperformingwithprecisionhavebeensuccessfulindevelopmentofscienceandtechnology,and“tobeaccurateaspossible”hasbeenconsideredasavirtueofascientificresearcher.-精准方法研究的对象是无生命的机械系统,界线分明的机械事务.Theobjectsbeinginvestigatedwithaccuracyarenormallynon-lifemechanicalsystems,whichhaveclear-cutboundaries.相关生命现象,社会现象,心理要素的科学,因为所研究的对象大多是没有明确界限的模糊事物,很难进行精准的丈量,所以很难使用精准的定量方法.Phenomenaregardinglives,socialaffairs,psychology,etc,duetotheobjectsbeinginvestigatedaremostlyfuzzyproblemswithoutclearboundaries,itisdifficulttoconducttheaccuratemeasurement,sohardtoquantifywithhighaccuracy.即便对于无生命的系统,一旦系统宏大,若进行精准地丈量,仍很困难.-Eventothosenon-lifesystems,onceitgrowsenormous,itisstilldifficulttomeasurewithprecision.二.模糊系统理论Fuzzysystemtheory语言变量定义及相关观点:Languagevariableandsomerelatedconcepts:用自然语言中的词句表示.每一个语言变量值均对应着一个论域(Discourseuniverse).Expressedwithawordorsentenceofnaturallanguage,andeachandeverylanguagevariablecorrespondstoaDiscourseUniverse.例:人的年纪的语言值可为:“特别年青,年青,较年青,不年青,较老,老,很老”,等等.Ex.Thelanguagevalueofageofhumanmaybe:“veryyoung,young,somewhatyoung,notyoung,somewhatold,old,everyold”etc.现设论域U=[0,150],则以上每一个值都对应着该论域的一个模糊子集可用隶属函数(或称类属函数,资格函数,成员函数等)来表示.

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模糊子集SupposethediscourseuniverseU=[0,150],inwhicheveryvalueshouldcorrespondtoafuzzyset.Thefuzzysetmaybedescribedusingmembershipfunction.例:给定一个一般会合A,某一元素属a于A,则隶属度为Ex.GivenanormalsetA,andacertainelementabelongingtoA,thenthedegreeofbelongingisA(a)1若不属于A,则IncaseitdoesnotbelongtoA，thenA(a)0对于一模糊子集A~,某一元素x除了A~(x)1外,有可能在必定程度上隶属0于A~,0<A~(x)<1.ForafuzzysubsetA~,foracertainelementxotherthanA~(x)1,itmaytoa0certainextentbelongtoA~,0<A~(x)<1.例:对于小于10的自然数,小于5的数的会合为A={1,2,3,4}对这四个数,均有Ex.Forthosenaturalnumberslessthan10,asetofwhichallelementsarelessthan5isA={1,2,3,4}.Forthesefournumbers,itholdsA(a)1a=1,,4.而对于A={5,6,7,8,9},均有ForA={5,6,7,8,9},ontheotherhand,itholdsA(a)0a=5,,9但对于小于10的自然数(论域),“比较小的数”则组成了一个模糊子集A~,对于这个子集,存在以下关系:Butinregardtothenumberslessthan10(universeofdiscourse),“relativesmallnumber”formsafuzzysubsetA~,andforthissubset,itholdsthefollowingrelationship142必定属于A~,很有可能属于A~,356不大可能属于A~8必定不属于A~79142certanlybelongtoA~,likelybelongtoA~,356notlikelybelongtoA~8certainlynotbelongtoA~79用模糊数学表示为:Expressedinfuzzymathematicsasfollows:A~1.01.01.00.10.00.01234567891.5Membershipfunctionvalue10.50123456789100NaturalNumber上式中,4到7的界限是模糊的.Intheequationabove,theboundaryof4to7isfuzzy.语言变量系统包含5个方面:Thelanguagevariablesinvolve5aspects语言变量x(上例中,自然数)Languagevariablex(thenaturalnumber,intheaboveexample)语言值会合T(上例中,“必定”,”很有可能”,等)论域U(上例中,“小于10的自然数”)TheuniverseofdiscourseU(“Thenaturalnumberslessthan10”)语法例则GGrammaticalrulesG语义规则M.ImplicationrulesM.例:Ex.年纪语言变量语法例则很年青年青老语言值语义变量基本变量1020306070论域Age

LanguagevariableGrammaticalrulesVeryyoung

youngold

LanguagevalueImplicationvar.Basicvariable1020306070Univofdiscourse-模糊数:对应于实数轴上某一个凸状的模糊子集(往常是三角形,钟形,矩形等)Fuzzynumber:foracertainconvex-shapedfuzzysubset(normallytriangle,bell,rectangle,etc.)模糊关系:不清楚不完整确立的关系.可由隶属函数描绘的模糊子集,隶属函数值代表的亲密程度.Fuzzyrelation:unclearanduncertainrelation.Itmayberepresentedbyassociationdegreetoafuzzysubsetdescribedbyamembershipfunctionorfunctionvalue.三.模糊性与随机性的差别(续见以下笔录)Thedifferencebetweenfuzzinessandrandomness(continuewiththenotesbelow)INTRODUCTIONTOFUZZYSETS模糊集的介绍Twovaluedlogic:-blackandwhiteoddandeven两个逻辑值：黑和白奇数和偶数Situationsthattwovaluedlogicmaynotbesuitable:Tallman,SmallerrorSignificanterror,etc两个逻辑值可能不般配的状况：高大的男人小的错误存心义的错误等等。Pileofseedexample:Oneseedisnotapile,Twoseedsdon’tconstituteapileCanwesaythat121078seedsdonotconstituteapilebut121079seedsconstitutedo?100x106seedsconstituteapile.Conclusion:definitionofapileissomewhatfuzzy.打个比方：多少量量的种子才算是一堆种子一粒种子不是一堆种子。两粒种子也不可以算是一堆种子。那我们能说：121078粒种子不可以算是一堆种子，但121079粒种子是一堆种子吗？结论：“一堆”的定义有些模糊。Setbybelongingtoaset（设置属于某个会合）SetObjectIntwovaluedlogic,anobjecteitherbelongstosetordoesnotbelongtoset.在两个逻辑值中，一个对象可能属于这个会合也可能不属于这个会合。比如,，EvennumbersetObject1=O1=3,Object2=O2=2AO1AO2∈AInfuzzysets,gradesofbelongingchangesbetweencompletelybelongs(1)completelyexcluded(2)在模糊集中，隶书度介于(a)完整属于(b)部分属于比如:Degreeofbelongingtoconcept“pile”1numberofseeds比如:：Towvaluedlogicofhightemperature（高温的两个逻辑值）Degreeofbelongingtoconcept“hightemperature”Temperature*TFuzzydefinitionofhightemperature（高温的模糊定义）Degreeofbelonging(=membership)toconcept“hightemperature”temperature比如:Degreeofmembershiptoconcept“young”（属于“年青”观点的隶属度）Young1functiontodescribethemembershipold（描绘老的程度)

yearsx502μold(x)=[1]-1550<=x<=100（x表示old）old(x)0.5x<=5SmallerrorLargeerrorMediumerrorErrorFORMALDEFINITIONOFFUZZYSET模糊集正式的定义Aisconceptsuchasyoung,mediumerror,largetemperature,etc.表示如年青，均匀偏差，高温等观点。Xisauniverseofdiscoursesuchastemperature,numberofseeds,years,error,etc.定义如温度，种子的数目，年数，偏差等A(X)isanexpressionexpressingtheextendtowhichXfulfilsthecategoryspecifiedbyA.SometimeswewilluseμA(x)toindicatethemembershipofxtoA.IfthereisnoconfusionwewillsimplyuseA(x)orsometimesμA(x).有时，我们用μA(x)显示x与A的关系。若是那不存在模糊，我们将简单的用A(x)或μA(x).来表示。Normalfuzzyset.一般模糊集SupxμA(x).=1Sup:supremum（或运算？？）比如：A:aconceptofyoung表示youngX:auniverseofdiscourse,years表示一个域，yearsx:avalueintheuniverseofdiscourse表示域上的值A(x)=μA(x)1A(x)xX模糊集的特征SOMEPROPERTIESOFFUZZYSETS并集:(A∪B)(x)=max(A(x),B(x))=max(μA(x),μB(x)),x∈XUnion:(A∪B)(x)=max(A(X),B(X))=max(μA(x),μB(x)),x∈X(A∪B)(x)A:largeerror1B:mediumerror(A∪B)(x):mediumorlargeB(x)errorA(x)Xx交集:(A∩B)(x)=min(A(x),B(x))=min(μA(x),μB(x)),Intersection:(A∩B)(x)=min(A(X),B(X))=min(μA(x),μB(x)),B(x)(A∩B)(x):mediumandlargeerrorA(x)(A∩B)(x)

Xx补集:A(x)1A(x),x∈XA(x)A(x)：notlargeerror.A(x)XNegation:A(x)1A(x),x∈XA(x)：notlargeerror.SOMEEXAMPLESONTHEPROPERTIESOFFUZZYSETS几个对于模糊集特征的例子Wehaveadiscreteuniverseofdiscoursehere:（这里有个失散域的推理过程）A=(1.0)11234B=(0.7)11234A∪B=(1.0)A∩B=(0.5)Ac=(0.0)A∩Ac=(0.0)A∪Ac=(1.0)注意:：若是是二值逻辑集

SetA1Note:ifitisthetwo-valuelogic.A=(1111)Ac=(0000)A∩Ac=(0.00.00.00.0)A∪Ac=(1.01.01.01.0)1234我们注意到A∩Ac不是零值.，A∪Ac不是独一的。这对严格定义的模糊集都是合用的。对照这个例子，可知二值逻辑不是模糊逻辑。NoticethattheoverlapmembershipA∩Acisnotzero.AlsonotethattheunderlapmembershipA∪Acisnotunity.Thisholdsforallproperfuzzysets.Comparethisexamplewiththeonepresentedinthesubsetabovewherewehave2-valuedlogic,i.e.,nofuzzylogic.FUZZINESSANDRANDOMNESS:HANDLINGUNCERTAINTY任意性和模糊性：操作的不确立X:afiniteuniverseofdiscourse,x∈X（一个有限的域）A(x)=a:avalueofmembershipfunctionjofAforcertainelementofXisequalto“a”.P(x∈A)=aprobabilityofxbelongingtoAisequalto“a”.Wenowperformanexperimentinwhichwepickupxandobservetheoutcome:BeforeexperimentAfterexperimentFuzziness:A(x)=aA(x)=aRandomnessP(x∈A)=a1ifx∈A0ifxAConsiderthefollowingsituation:xistheoutcomeofadiethrowingexperiment,Aisanevennumber.Uponobservationofx,theaprioriprobability:P(x∈A)=abecomesaposterior,i.e.,1ifx∈A,0otherwise.AtthesametimeA(x),beingameasureoftheextenttowhichxbelongstoA,remainthesame.RANDOMNESS:statisticalinexactnessduetorandomevents.FUZZINESS:inexactnessduetoperceptionprocessofhumanbeing.Bothconceptsdescribeuncertaintywithnumbersintheinterval[0,1].模糊关系FUZZYRELATIONSR:X

Y→[0,1]

xX,y

Y对于每对x,y对应于一个属于(0,1)的值。ToeverypairxXandyYandmemberfromarange[0,1]isassigned..R表示序偶<x,y>的隶属程度，也描绘了<u,v>间拥有模糊关系R的量级。Rexpressesthestrengthoftiesbetweenthem.模糊关系EXAMINGONFUZZYRELATIONS例1R(x,y):x相像y.的程度函数EX1R(x,y):xissimilartoy.1R(x,y)=41(xy)R(x,y)xy例2例1的失散域的论域.EX2DiscreteuniverseofdiscoursexR(x,)yR=

1.00.70.0y1y2y3yx1x2x我们怎么来确立模糊关系HOWDOWEDETERMINEFUZZYRELATIONS定义一个标准模糊关系t:Definitionoft-norm:t:[0,1]x[0,1]→[0,1]它知足以下特征：Ithasthefollowingproperties非递减性:non-decreasing:xtwytzforxyandwz(注意：t代表标准t域，不是一个变量)比如：xy是一个标准t域,也就是.,简单相乘是一个标准t域，所以，xw<yzify>xandz>w(ii)知足互换律：commutative:xty=ytxEX.xy=yx(iii)知足联合律：associative:(xty)tz=xt(ytz)EX.(xy)z=x(yz)(iv)知足限制条件：satisfytheboundingconditionxt0=0EX.x0=0xt1=xEX.x1=x一些对于标准t域的其余例子：SOMEOTHEREXAMPLESOFt-nor