第三章系統(tǒng)的描繪與模型成立Chapter3.DescriptionofSystemsandModeling3.6不確立性描繪(二)–模糊性3.6DescriptionofUncertainty(2)-Randomness一.精準(zhǔn)與模糊1.Precisionandfuzziness統(tǒng)計剖析隨機(jī)性描繪主元剖析-不確立性描繪因子剖析模糊性描繪DescriptionofuncertaintystatisticanalysisDescriptionofrandomnessPCA,PrincipleComponentAnalysisfactoranalysisDescriptionoffuzziness-

精準(zhǔn)思想及方法在科學(xué)技術(shù)發(fā)展中日趨獲得成功.往?！鼻уN百煉”被以為是科學(xué)工作者的美德.Thinkingandperformingwithprecisionhavebeensuccessfulindevelopmentofscienceandtechnology,and“tobeaccurateaspossible”hasbeenconsideredasavirtueofascientificresearcher.-精準(zhǔn)方法研究的對象是無生命的機(jī)械系統(tǒng),界線分明的機(jī)械事務(wù).Theobjectsbeinginvestigatedwithaccuracyarenormallynon-lifemechanicalsystems,whichhaveclear-cutboundaries.相關(guān)生命現(xiàn)象,社會現(xiàn)象,心理要素的科學(xué),因?yàn)樗芯康膶ο蟠蠖嗍菦]有明確界限的模糊事物,很難進(jìn)行精準(zhǔn)的丈量,所以很難使用精準(zhǔn)的定量方法.Phenomenaregardinglives,socialaffairs,psychology,etc,duetotheobjectsbeinginvestigatedaremostlyfuzzyproblemswithoutclearboundaries,itisdifficulttoconducttheaccuratemeasurement,sohardtoquantifywithhighaccuracy.即便對于無生命的系統(tǒng),一旦系統(tǒng)宏大,若進(jìn)行精準(zhǔn)地丈量,仍很困難.-Eventothosenon-lifesystems,onceitgrowsenormous,itisstilldifficulttomeasurewithprecision.二.模糊系統(tǒng)理論Fuzzysystemtheory語言變量定義及相關(guān)觀點(diǎn):Languagevariableandsomerelatedconcepts:用自然語言中的詞句表示.每一個語言變量值均對應(yīng)著一個論域(Discourseuniverse).Expressedwithawordorsentenceofnaturallanguage,andeachandeverylanguagevariablecorrespondstoaDiscourseUniverse.例:人的年紀(jì)的語言值可為:“特別年青,年青,較年青,不年青,較老,老,很老”,等等.Ex.Thelanguagevalueofageofhumanmaybe:“veryyoung,young,somewhatyoung,notyoung,somewhatold,old,everyold”etc.現(xiàn)設(shè)論域U=[0,150],則以上每一個值都對應(yīng)著該論域的一個模糊子集可用隸屬函數(shù)(或稱類屬函數(shù),資格函數(shù),成員函數(shù)等)來表示.

.

模糊子集SupposethediscourseuniverseU=[0,150],inwhicheveryvalueshouldcorrespondtoafuzzyset.Thefuzzysetmaybedescribedusingmembershipfunction.例:給定一個一般會合A,某一元素屬a于A,則隸屬度為Ex.GivenanormalsetA,andacertainelementabelongingtoA,thenthedegreeofbelongingisA(a)1若不屬于A,則IncaseitdoesnotbelongtoA，thenA(a)0對于一模糊子集A~,某一元素x除了A~(x)1外,有可能在必定程度上隸屬0于A~,0<A~(x)<1.ForafuzzysubsetA~,foracertainelementxotherthanA~(x)1,itmaytoa0certainextentbelongtoA~,0<A~(x)<1.例:對于小于10的自然數(shù),小于5的數(shù)的會合為A={1,2,3,4}對這四個數(shù),均有Ex.Forthosenaturalnumberslessthan10,asetofwhichallelementsarelessthan5isA={1,2,3,4}.Forthesefournumbers,itholdsA(a)1a=1,,4.而對于A={5,6,7,8,9},均有ForA={5,6,7,8,9},ontheotherhand,itholdsA(a)0a=5,,9但對于小于10的自然數(shù)(論域),“比較小的數(shù)”則組成了一個模糊子集A~,對于這個子集,存在以下關(guān)系:Butinregardtothenumberslessthan10(universeofdiscourse),“relativesmallnumber”formsafuzzysubsetA~,andforthissubset,itholdsthefollowingrelationship142必定屬于A~,很有可能屬于A~,356不大可能屬于A~8必定不屬于A~79142certanlybelongtoA~,likelybelongtoA~,356notlikelybelongtoA~8certainlynotbelongtoA~79用模糊數(shù)學(xué)表示為: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.語言變量系統(tǒng)包含5個方面:Thelanguagevariablesinvolve5aspects語言變量x(上例中,自然數(shù))Languagevariablex(thenaturalnumber,intheaboveexample)語言值會合T(上例中,“必定”,”很有可能”,等)論域U(上例中,“小于10的自然數(shù)”)TheuniverseofdiscourseU(“Thenaturalnumberslessthan10”)語法例則GGrammaticalrulesG語義規(guī)則M.ImplicationrulesM.例:Ex.年紀(jì)語言變量語法例則很年青年青老語言值語義變量基本變量1020306070論域Age

LanguagevariableGrammaticalrulesVeryyoung

youngold

LanguagevalueImplicationvar.Basicvariable1020306070Univofdiscourse-模糊數(shù):對應(yīng)于實(shí)數(shù)軸上某一個凸?fàn)畹哪：蛹?往常是三角形,鐘形,矩形等)Fuzzynumber:foracertainconvex-shapedfuzzysubset(normallytriangle,bell,rectangle,etc.)模糊關(guān)系:不清楚不完整確立的關(guān)系.可由隸屬函數(shù)描繪的模糊子集,隸屬函數(shù)值代表的親密程度.Fuzzyrelation:unclearanduncertainrelation.Itmayberepresentedbyassociationdegreetoafuzzysubsetdescribedbyamembershipfunctionorfunctionvalue.三.模糊性與隨機(jī)性的差別(續(xù)見以下筆錄)Thedifferencebetweenfuzzinessandrandomness(continuewiththenotesbelow)INTRODUCTIONTOFUZZYSETS模糊集的介紹Twovaluedlogic:-blackandwhiteoddandeven兩個邏輯值：黑和白奇數(shù)和偶數(shù)Situationsthattwovaluedlogicmaynotbesuitable:Tallman,SmallerrorSignificanterror,etc兩個邏輯值可能不般配的狀況：高大的男人小的錯誤存心義的錯誤等等。Pileofseedexample:Oneseedisnotapile,Twoseedsdon’tconstituteapileCanwesaythat121078seedsdonotconstituteapilebut121079seedsconstitutedo?100x106seedsconstituteapile.Conclusion:definitionofapileissomewhatfuzzy.打個比方：多少量量的種子才算是一堆種子一粒種子不是一堆種子。兩粒種子也不可以算是一堆種子。那我們能說：121078粒種子不可以算是一堆種子，但121079粒種子是一堆種子嗎？結(jié)論：“一堆”的定義有些模糊。Setbybelongingtoaset（設(shè)置屬于某個會合）SetObjectIntwovaluedlogic,anobjecteitherbelongstosetordoesnotbelongtoset.在兩個邏輯值中，一個對象可能屬于這個會合也可能不屬于這個會合。比如,，EvennumbersetObject1=O1=3,Object2=O2=2AO1AO2∈AInfuzzysets,gradesofbelongingchangesbetweencompletelybelongs(1)completelyexcluded(2)在模糊集中，隸書度介于(a)完整屬于(b)部分屬于比如:Degreeofbelongingtoconcept“pile”1numberofseeds比如:：Towvaluedlogicofhightemperature（高溫的兩個邏輯值）Degreeofbelongingtoconcept“hightemperature”Temperature*TFuzzydefinitionofhightemperature（高溫的模糊定義）Degreeofbelonging(=membership)toconcept“hightemperature”temperature比如:Degreeofmembershiptoconcept“young”（屬于“年青”觀點(diǎn)的隸屬度）Young1functiontodescribethemembershipold（描繪老的程度)

yearsx502μold(x)=[1]-1550<=x<=100（x表示old）old(x)0.5x<=5SmallerrorLargeerrorMediumerrorErrorFORMALDEFINITIONOFFUZZYSET模糊集正式的定義Aisconceptsuchasyoung,mediumerror,largetemperature,etc.表示如年青，均勻偏差，高溫等觀點(diǎn)。Xisauniverseofdiscoursesuchastemperature,numberofseeds,years,error,etc.定義如溫度，種子的數(shù)目，年數(shù)，偏差等A(X)isanexpressionexpressingtheextendtowhichXfulfilsthecategoryspecifiedbyA.SometimeswewilluseμA(x)toindicatethemembershipofxtoA.IfthereisnoconfusionwewillsimplyuseA(x)orsometimesμA(x).有時，我們用μA(x)顯示x與A的關(guān)系。若是那不存在模糊，我們將簡單的用A(x)或μA(x).來表示。Normalfuzzyset.一般模糊集SupxμA(x).=1Sup:supremum（或運(yùn)算？？）比如：A:aconceptofyoung表示youngX:auniverseofdiscourse,years表示一個域，yearsx:avalueintheuniverseofdiscourse表示域上的值A(chǔ)(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補(bǔ)集:A(x)1A(x),x∈XA(x)A(x)：notlargeerror.A(x)XNegation:A(x)1A(x),x∈XA(x)：notlargeerror.SOMEEXAMPLESONTHEPROPERTIESOFFUZZYSETS幾個對于模糊集特征的例子Wehaveadiscreteuniverseofdiscoursehere:（這里有個失散域的推理過程）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:：若是是二值邏輯集

SetA1Note:ifitisthetwo-valuelogic.A=(1111)Ac=(0000)A∩Ac=(0.00.00.00.0)A∪Ac=(1.01.01.01.0)1234我們注意到A∩Ac不是零值.，A∪Ac不是獨(dú)一的。這對嚴(yán)格定義的模糊集都是合用的。對照這個例子，可知二值邏輯不是模糊邏輯。NoticethattheoverlapmembershipA∩Acisnotzero.AlsonotethattheunderlapmembershipA∪Acisnotunity.Thisholdsforallproperfuzzysets.Comparethisexamplewiththeonepresentedinthesubsetabovewherewehave2-valuedlogic,i.e.,nofuzzylogic.FUZZINESSANDRANDOMNESS:HANDLINGUNCERTAINTY任意性和模糊性：操作的不確立X:afiniteuniverseofdiscourse,x∈X（一個有限的域）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].模糊關(guān)系FUZZYRELATIONSR:X

Y→[0,1]

xX,y

Y對于每對x,y對應(yīng)于一個屬于(0,1)的值。ToeverypairxXandyYandmemberfromarange[0,1]isassigned..R表示序偶<x,y>的隸屬程度，也描繪了<u,v>間擁有模糊關(guān)系R的量級。Rexpressesthestrengthoftiesbetweenthem.模糊關(guān)系EXAMINGONFUZZYRELATIONS例1R(x,y):x相像y.的程度函數(shù)EX1R(x,y):xissimilartoy.1R(x,y)=41(xy)R(x,y)xy例2例1的失散域的論域.EX2DiscreteuniverseofdiscoursexR(x,)yR=

1.00.70.0y1y2y3yx1x2x我們怎么來確立模糊關(guān)系HOWDOWEDETERMINEFUZZYRELATIONS定義一個標(biāo)準(zhǔn)模糊關(guān)系t:Definitionoft-norm:t:[0,1]x[0,1]→[0,1]它知足以下特征：Ithasthefollowingproperties非遞減性:non-decreasing:xtwytzforxyandwz(注意：t代表標(biāo)準(zhǔn)t域，不是一個變量)比如：xy是一個標(biāo)準(zhǔn)t域,也就是.,簡單相乘是一個標(biāo)準(zhǔn)t域，所以，xw<yzify>xandz>w(ii)知足互換律：commutative:xty=ytxEX.xy=yx(iii)知足聯(lián)合律：associative:(xty)tz=xt(ytz)EX.(xy)z=x(yz)(iv)知足限制條件：satisfytheboundingconditionxt0=0EX.x0=0xt1=xEX.x1=x一些對于標(biāo)準(zhǔn)t域的其余例子：SOMEOTHEREXAMPLESOFt-nor