chapter 9 uncertaintybased information9章基于信息不确定性.docx
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chapter9uncertaintybasedinformation9章基于信息不确定性
chapter9uncertainty-basedinformation:
9章基于信息不确定性
chapter9uncertainty-basedinformation:
9章基于信息不确定性Chapter9Uncertainty-BasedInformation9.1InformationandUncertainty•Uncertainty:
何謂uncertainty?
所謂的uncertainty即是意指資訊量的不足.suchas,incomplete,imprecise,fragmentary,vague,contradictory,unreliable•一般而言,資訊量增加相當於不確定性(uncertainty)的減少(Uncertainty-BasedInformation)suchas,newfact,measurement,experience,knowledge•接著我們嘗試利用數學理論去建造一個數學model來判斷一個系統的不確定性(uncertainty)程度如何,其流程如下面此範例e.q.settheoryprobabilistictheoryinformationtheorymeasureifprobabilisticuncertainty(ClaudeShannon)generalizedinformationtheorye.q.FuzzysettheoryFuzzymeasuretheoryPossibilitytheoryEvidencetheoryUncertainty-Basedinformation•不確定性(uncertainty)通常有三種種類,分別如下:
1.nonspecificity(orimprecision)-cardinalities2.fuzziness(vagueness)-imprecisionboundaries3.strife(discord)-conflictions9.2NonspecificityofCrispSets•首先我們要量測一個系統的uncertainty,必須先著重在settheory,定義如下式其中:
cardinalityofsetAb>1,c>0:
determinetheunitofuncertaintye.q.b=2,c=1:
Hortlyfunctionuncertaintyismeasureinbitsonebitofuncertainty=thetotaluncertaintyregardingthetruthonfalsityofoneproposition.•Formally,U:
-•然而所謂的不確定性(uncertainty)完全是根據不同的setA而有不同的解釋e.q.predicativeuncertainty:
Aisasetofpredictedstatesofavariablediagnosticuncertainty:
Aisasetofpossiblediseasesofapatientretrodictiveuncertainty:
Aisasetofpossibleanswertoaquestionprescriptiveuncertainty:
Aisasetofpossiblepolicies•任何一個集合都存在著潛在的nonspecificity如這兩種情形:
Largesetslessspecific,Singletonsfullspecificity•若把集合A縮減成集合B則theamountofuncertainty-basedinformationI(A,B)=theamountofreduceduncertaintyU(A)-U(B)i.e.,I(A,B)=U(A)-U(B)==where|B|=1I(A,1)==U(A)U(A)mustalsobeconceivedastheamountofinformationcharacterizeoneelementofsetA•Let,:
universalsets:
arelation:
domainofR:
rangeofRLet,,U()=,U()=simpleuncertainty:
單純的兩個集合,其各自的uncertainty.U(,)=Jointuncertainty:
而其中uncertainty是具有結合性質,可將兩集合合併,依舊可以得知此新的集合其uncertainty的程度.•ConditionaluncertaintydefinedbyU(|)=:
theaveragenumberifelementsofgivenanelementofU(|):
theaveragenonspecificityofgiven(p.s此定義可以配合Example9.1就可以清楚了解)U(|)=•U(|)==-=U(|)-U()---(9.8)U(|)=U(|)-U()---(9.9)(9.8)-(9.9)U()-U()=U(|)-U(|)(而其中可分為Noninterative和Iterative)•,,:
NoninterativeifU(|)=U(),U(|)=U()U(,)=U()+U()IterativeIfU(,)0:
iterative(同樣的可分為兩種情形)By(9.8),(9.9)T(,)=U()+U()-U()-U(,)=U()-U(,)T(,)=U()-U(,)•其通常表示成下列:
=totalinformationjointinformation•Example9.1:
={low,median,high},={1,2,3,4},U()===1.6U()===2U(,)===2.8U(|)=U(|)-U()=2.8–2=0.8U(|)=U(|)-U()=2.8–1.6=1.2T(,)=U()+U()-U(,)=1.6+2–2.8=0.8•(9.1):
applicabletofinitesetsForinfinitesetsU(A)=log[1+(A)]Where(A):
themeasureofAdefinedbytheLebesgueintegralofthecharacteristicfunctionofAe.q.A=[a,b]onR(A)=b–aU([a,b])=log[1+b–a]•由投影片上可清楚發現Riemannintegral和lebesgueintegral之間主要的差別9.3NonspecificityofFuzzysets首先我們先定義一些數學方程式以方便之後用來決定Fuzzysets的nonspecificity程度為何A:
fuzzyset---(9.22):
thecardinalityofthe-cutofAh(A):
theheightofAU(A):
aweightedaverageofvaluesoftheHartleyfunctionforall-cutsofA(X)/h(A);Eachweightisadifferencebetweenthevaluesofofagiven-cutandtheimmediatelypreceding-cut•IfA(X)/h(A)=B(X)/h(B)U(A)=U(B)i.e.根據Example9.2可以發現同樣的normalizedfuzzysets經由U的計算後,會得到相同的nonspecificity•(9.22):
applicabletofiniteuniversalsetForinfinitesetswhere:
memsurableandlebesgueintegrable:
themeasureofdefinedbytheLebesgueintegralofthecharacteristicfunctionof¤U-uncertaintyU:
whereR:
thesetofallfiniteandorderedpossibilitydistributions※Eachpossibilitydistributions.t.U-uncertaintyofU()=---(9.25)Where=0IfrepresentsanormalfuzzysetA,U()=U(A),where>0andirepresentswith=žalternativeof(9.25)U()==……………………=Futhermore,U(m)=Wherem=():
basicprobabilityassignmentcorrespondingtožExample9.4:
Givenpossibilitydistribution=(1,1,0.8,0.7,0.7,0.7,0.4,0.3,0.2,0.2)CalculateU()=(0,0.2,0.1,0,0.3,0.1,0.1,0,0.2)U()===2.18=U()žLet:
jointpossibilitydistributiondefinedonU():
jointU-uncertainlyU(),U():
simpleU-uncertainly,:
marginalpossibilitydistributionU()=U()=U(,)=:
setoffocalelementsbyžConditionalU-uncertaintyU(|)=U(|)=whererepresents,theaveragenumberofelementsofAthatremainpossiblealternativegivenanelementofžFornormalfuzzysets,by(9.22)U(|)===U(,)-U()Thisisageneralizationof(9.8)(9.10)Similarlyfor(9.9)~(9.16)validforU-uncertainty(9.17)canbedefinedforU-uncertainty(9.18),(9.19)validforU-uncertainty9.4Fuzziness(Vagueness)žmeasureoffuzzinessf:
:
fuzzypowersetf(A):
thedegreetowhichtheboundaryoffuzzysetAisnotsharpžRequirements1.f(A)=0iffA:
acrispset(當Aisacrispset,很明顯其一點都不Fuzzy)2.A(x)=0.5ifff(A0):
maximum(而一般而言,當其membership值=0.5時,最難決定其狀態,也因此理所當然的,此時的fuzziness最大)3.f(A)f(B):
AsharperthanBi.e.A(x)B(x)whenB(x)0.5A(x)B(x)whenB(x)0.5(同樣地,若A比B更尖銳,即可發現A會比B不fuzzy)由上訴幾點可以簡單得知,若一個membershipfunction越平滑,其fuzziness越高.ž測量一個set的fuzzinessways的方法有二:
1.測量此set的membershipfunction與最接近此set的crispset’scharacteristicfunction之間的差異2.測量此set與其complement之間的lack※若此set與其complement之間的差距越小,就表示此set越fuzzier.t※ifparticularfuzzycomplementwereused,thevalue0.5inrequirements2and3wouldhavetobereplacedwiththeequilibriumsofthefuzzycomplementžUsestandardfuzzycomplementHammingdistanceThesumofabsolutevaluesofdifferencež1,the,localdistinctionofsetAanditscomplement|A(x)-(1-A(x)|=|2A(x)-1|2,thelackofalocaldistinction1-|2A(x)-1|3.Themeasureoffuzziness=0iffA:
crispwhenA(x)=0.5žContinuouscasežExample9.5i,f()=4-==]=4–0.5-0.5-0.5-0.5=2ii,=4-1-0.5-0.5-1=1iii,=4-1-0.5-0.5-1=1žAsetanditsnormalcounterpart即使擁有同樣的nonspecificity,但卻不會有相同的fuzziness,如下例Example:
Letwithh()=0.4h()=0.625i.e.,(x-1)=(3-x)0otherwisek=4,5==4-1-0.6-0.6-1=0.8==4-1-0.425-0.425-1=1.15※theshadedareaindicatethedifferencebetweenmembershipgradesofsetanditscomplement.thedegreeoffuzzinessisthedeficiencyofthedifferencew.r.t.1andismeasuredbythetotalsizeoftheunshadedareasž由上便可以得知nonspecificity和fuzziness是不同類型的uncertainty,而且它們是彼此獨立的!
theyareindependentofeachotheržnonspecificityreducedgainininformationfuzzinessreducedgainininformationdependsontheaccompaniedchangeinnonspecificityU(A)=2.99,f(A)=6U(B)=3,f(B)=0由上面例子可清楚的看出U:
increasedwhilef:
decreased9.5uncertaintyinEvidenceTheorygeneralizationofnonspecificity:
classicalsettheoryfuzzysettheorypossibilitytheoryevidencetheory¤NonspecificityžN-uncertainty:
bodyofevidencei,Nisauniquemeasureofnonspecificityii,whenfocalelementsinarenested,NUiii,NisaweightedaverageoftheHartleyfunctionfocalelementAweights:
valuesofthebasicprobabilityassignmentm(A)m(A):
thedegreeofevidencefocusingonA:
thelackofspecificityofthisevidenceclaimiv,m(A)evidenceA=specificityv,therangeofN:
[0,]N(m)=0whenx,m({x})=0.(nouncertain)N(m)=whenm()=1(totalignorance)vi,(9.8)-(9.19)validforNžfocalelementsinprobabilitymeasuresaresingletons,i.e.|A|=1==0N(m)=0probabilitymeasure¤ShannonEntropyH---(9.37)--measurestheaverageuncertaintyassociatedwiththepredictionofoutcomesinarandomexperiment--range[0,]H(m)=0whenx,s.t.m({x})=1H(m)=,whenx,m({x})=i.e.uniformprobabilitydistributionon--(9.8)~(9.19)validforHž,m({x})=1-(9.37)H(m)=LetCon({x})=--thetotalevidentialclaimpertainingtofocalelementsthataredifferentwiththefocalelement[x]--thesumofallevidentialclaimsthatfullyconflictwiththeonefocusingon{x}H(m):
theexpectedvalueoftheconflictamongevidentialclaimswithinagivenprobabilisticbodyofevidencežShannonCross-entropy(directeddivergence)D(f(x),g(x)|)=Wheref(x),g(x):
probabilitydensityfunctiondefinedon[a,b]DiscretecaseD(p(x),q(x)|)=-continuousinformationtransimission=wheref(x,y):
jointprobabilitydistributionon:
densityfunctionofmarginaldistributiononand⊙Entropy-likeMeasureinEvidenceTheoryA,dissonanceFrom(7.11)Let-ThetotalevidentialclaimpertainingtofocalelementsthataredisjointwithA-ThesumofallevidentialclaimsthatfullyconflictwithA.→E(m)i.Theexpectedvalueoftheconflictamongevidentialclaimswithinagivenbodyofevidence(F,m)ii.Measureconflictiii.Range:
iv.Notfullysatisfactoryasameasureconflictswithm(A),whenandB,ConfusionFrom(7.10)Let-thesumofallevidentialclaimwithAaccordingtom(B)conflictswithm(A)wheneveri.Theexpectedvalueoftheconflictamongevidentialclaimswithinagivenbodyofevidence(F,m)ii.Notfullysatisfactoryasameasureofconflict.doesnotproperlyscaleeachparticularconflictofm(B),w.r.tm(A)accordingtothedegreeofviolationofthesubsethoodrelation.※Themorethissubsethoodrelationisviolated,thegreatertheconflictiii.C,DiscordLetconflict-thesumofindividualconflictsofevidentialclaimsw.r.t.A-eachconflicthasbescaledbythedegreetowhichthesubsethoodisviolated-range:
[0,1]※A,Measureofthemeanconflictamongevidentialclaimswithineach(F,m)※※‧DefectofD(m)i.LetAccordingtofunctionContheclaimm(B)istakentobeinconflictwiththeclaimm(A)tothedegree--------(a)
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