Mathematical measures of the likelihood of an event being true include probability theory, capacity, fuzzy logic, possibility, and credibility, as well as uncertainty.
Axiom 3. (Countable Subadditivity Axiom) For every countable sequence of events , we have
.
Axiom 4. (Product Measure Axiom) Let be uncertainty spaces for . Then the product uncertain measure is an uncertain measure on the product σ-algebra satisfying
.
Principle. (Maximum Uncertainty Principle) For any event, if there are multiple reasonable values that an uncertain measure may take, then the value as close to 0.5 as possible is assigned to the event.
Uncertainty distribution is inducted to describe uncertain variables.
Definition: The uncertainty distribution of an uncertain variable ξ is defined by .
Theorem (Peng and Iwamura, Sufficient and Necessary Condition for Uncertainty Distribution): A function is an uncertain distribution if and only if it is an increasing function except and .
Independenceedit
Definition: The uncertain variables are said to be independent if
for any Borel sets of real numbers.
Theorem 1: The uncertain variables are independent if
for any Borel sets of real numbers.
Theorem 2: Let be independent uncertain variables, and measurable functions. Then are independent uncertain variables.
Theorem 3: Let be uncertainty distributions of independent uncertain variables respectively, and the joint uncertainty distribution of uncertain vector . If are independent, then we have
for any real numbers .
Operational lawedit
Theorem: Let be independent uncertain variables, and a measurable function. Then is an uncertain variable such that
where are Borel sets, and means for any.
Expected Valueedit
Definition: Let be an uncertain variable. Then the expected value of is defined by
provided that at least one of the two integrals is finite.
Theorem 1: Let be an uncertain variable with uncertainty distribution . If the expected value exists, then
Theorem 2: Let be an uncertain variable with regular uncertainty distribution . If the expected value exists, then
Theorem 3: Let and be independent uncertain variables with finite expected values. Then for any real numbers and , we have
Varianceedit
Definition: Let be an uncertain variable with finite expected value . Then the variance of is defined by
Theorem: If be an uncertain variable with finite expected value, and are real numbers, then
Critical valueedit
Definition: Let be an uncertain variable, and . Then
Theorem 1: Let be an uncertain variable with regular uncertainty distribution . Then its α-optimistic value and α-pessimistic value are
,
.
Theorem 2: Let be an uncertain variable, and . Then we have
if , then ;
if , then .
Theorem 3: Suppose that and are independent uncertain variables, and . Then we have
,
,
,
,
,
.
Entropyedit
Definition: Let be an uncertain variable with uncertainty distribution . Then its entropy is defined by
where .
Theorem 1(Dai and Chen): Let be an uncertain variable with regular uncertainty distribution . Then
Theorem 2: Let and be independent uncertain variables. Then for any real numbers and , we have
Theorem 3: Let be an uncertain variable whose uncertainty distribution is arbitrary but the expected value and variance . Then
Inequalitiesedit
Theorem 1(Liu, Markov Inequality): Let be an uncertain variable. Then for any given numbers and , we have
Theorem 2 (Liu, Chebyshev Inequality) Let be an uncertain variable whose variance exists. Then for any given number , we have
Theorem 3 (Liu, Holder's Inequality) Let and be positive numbers with , and let and be independent uncertain variables with and . Then we have
Theorem 4:(Liu [127], Minkowski Inequality) Let be a real number with , and let and be independent uncertain variables with and . Then we have
Convergence conceptedit
Definition 1: Suppose that are uncertain variables defined on the uncertainty space . The sequence is said to be convergent a.s. to if there exists an event with such that
for every . In that case we write ,a.s.
Definition 2: Suppose that are uncertain variables. We say that the sequence converges in measure to if
for every .
Definition 3: Suppose that are uncertain variables with finite expected values. We say that the sequence converges in mean to if
.
Definition 4: Suppose that are uncertainty distributions of uncertain variables , respectively. We say that the sequence converges in distribution to if at any continuity point of .
Theorem 1: Convergence in Mean Convergence in Measure Convergence in Distribution. However, Convergence in Mean Convergence Almost Surely Convergence in Distribution.
Conditional uncertaintyedit
Definition 1: Let be an uncertainty space, and . Then the conditional uncertain measure of A given B is defined by
Theorem 1: Let be an uncertainty space, and B an event with . Then M{·|B} defined by Definition 1 is an uncertain measure, and is an uncertainty space.
Definition 2: Let be an uncertain variable on . A conditional uncertain variable of given B is a measurable function from the conditional uncertainty space to the set of real numbers such that
.
Definition 3: The conditional uncertainty distribution of an uncertain variable given B is defined by
provided that .
Theorem 2: Let be an uncertain variable with regular uncertainty distribution , and a real number with . Then the conditional uncertainty distribution of given is
Theorem 3: Let be an uncertain variable with regular uncertainty distribution , and a real number with . Then the conditional uncertainty distribution of given is
Definition 4: Let be an uncertain variable. Then the conditional expected value of given B is defined by
provided that at least one of the two integrals is finite.
Referencesedit
Wikimedia Commons has media related to Uncertainty Theory.
^Liu, Baoding (2015). Uncertainty theory: an introduction to its axiomatic foundations. Springer uncertainty research (4th ed.). Berlin: Springer. ISBN978-3-662-44354-5.
Cuilian You, Some Convergence Theorems of Uncertain Sequences, Mathematical and Computer Modelling, Vol.49, Nos.3-4, 482-487, 2009.
Yuhan Liu, How to Generate Uncertain Measures, Proceedings of Tenth National Youth Conference on Information and Management Sciences, August 3–7, 2008, Luoyang, pp. 23–26.
Baoding Liu, Some Research Problems in Uncertainty Theory, Journal of Uncertain Systems, Vol.3, No.1, 3-10, 2009.
Yang Zuo, Xiaoyu Ji, Theoretical Foundation of Uncertain Dominance, Proceedings of the Eighth International Conference on Information and Management Sciences, Kunming, China, July 20–28, 2009, pp. 827–832.
Yuhan Liu and Minghu Ha, Expected Value of Function of Uncertain Variables, Proceedings of the Eighth International Conference on Information and Management Sciences, Kunming, China, July 20–28, 2009, pp. 779–781.
Zhongfeng Qin, On Lognormal Uncertain Variable, Proceedings of the Eighth International Conference on Information and Management Sciences, Kunming, China, July 20–28, 2009, pp. 753–755.
Jin Peng, Value at Risk and Tail Value at Risk in Uncertain Environment, Proceedings of the Eighth International Conference on Information and Management Sciences, Kunming, China, July 20–28, 2009, pp. 787–793.
Yi Peng, U-Curve and U-Coefficient in Uncertain Environment, Proceedings of the Eighth International Conference on Information and Management Sciences, Kunming, China, July 20–28, 2009, pp. 815–820.
Wei Liu, Jiuping Xu, Some Properties on Expected Value Operator for Uncertain Variables, Proceedings of the Eighth International Conference on Information and Management Sciences, Kunming, China, July 20–28, 2009, pp. 808–811.
Xiaohu Yang, Moments and Tails Inequality within the Framework of Uncertainty Theory, Proceedings of the Eighth International Conference on Information and Management Sciences, Kunming, China, July 20–28, 2009, pp. 812–814.
Yuan Gao, Analysis of k-out-of-n System with Uncertain Lifetimes, Proceedings of the Eighth International Conference on Information and Management Sciences, Kunming, China, July 20–28, 2009, pp. 794–797.
Xin Gao, Shuzhen Sun, Variance Formula for Trapezoidal Uncertain Variables, Proceedings of the Eighth International Conference on Information and Management Sciences, Kunming, China, July 20–28, 2009, pp. 853–855.
Zixiong Peng, A Sufficient and Necessary Condition of Product Uncertain Null Set, Proceedings of the Eighth International Conference on Information and Management Sciences, Kunming, China, July 20–28, 2009, pp. 798–801.
January 01, 1970
uncertainty, theory, confused, with, uncertainty, principle, this, article, multiple, issues, please, help, improve, discuss, these, issues, talk, page, learn, when, remove, these, template, messages, this, article, includes, list, general, references, lacks, . Not to be confused with Uncertainty principle This article has multiple issues Please help improve it or discuss these issues on the talk page Learn how and when to remove these template messages This article includes a list of general references but it lacks sufficient corresponding inline citations Please help to improve this article by introducing more precise citations May 2018 Learn how and when to remove this message This article may be too technical for most readers to understand Please help improve it to make it understandable to non experts without removing the technical details December 2009 Learn how and when to remove this message This article provides insufficient context for those unfamiliar with the subject Please help improve the article by providing more context for the reader July 2020 Learn how and when to remove this message Learn how and when to remove this message The uncertainty theory invented by Baoding Liu 1 is a branch of mathematics based on normality monotonicity self duality countable subadditivity and product measure axioms clarification needed Mathematical measures of the likelihood of an event being true include probability theory capacity fuzzy logic possibility and credibility as well as uncertainty Contents 1 Four axioms 2 Uncertain variables 3 Uncertainty distribution 4 Independence 5 Operational law 6 Expected Value 7 Variance 8 Critical value 9 Entropy 10 Inequalities 11 Convergence concept 12 Conditional uncertainty 13 References 14 SourcesFour axioms editAxiom 1 Normality Axiom M G 1 for the universal set G displaystyle mathcal M Gamma 1 text for the universal set Gamma nbsp Axiom 2 Self Duality Axiom M L M L c 1 for any event L displaystyle mathcal M Lambda mathcal M Lambda c 1 text for any event Lambda nbsp Axiom 3 Countable Subadditivity Axiom For every countable sequence of events L 1 L 2 displaystyle Lambda 1 Lambda 2 ldots nbsp we have M i 1 L i i 1 M L i displaystyle mathcal M left bigcup i 1 infty Lambda i right leq sum i 1 infty mathcal M Lambda i nbsp dd Axiom 4 Product Measure Axiom Let G k L k M k displaystyle Gamma k mathcal L k mathcal M k nbsp be uncertainty spaces for k 1 2 n displaystyle k 1 2 ldots n nbsp Then the product uncertain measure M displaystyle mathcal M nbsp is an uncertain measure on the product s algebra satisfying M i 1 n L i min 1 i n M i L i displaystyle mathcal M left prod i 1 n Lambda i right underset 1 leq i leq n operatorname min mathcal M i Lambda i nbsp dd Principle Maximum Uncertainty Principle For any event if there are multiple reasonable values that an uncertain measure may take then the value as close to 0 5 as possible is assigned to the event Uncertain variables editAn uncertain variable is a measurable function 3 from an uncertainty space G L M displaystyle Gamma L M nbsp to the set of real numbers i e for any Borel set B of real numbers the set 3 B g G 3 g B displaystyle xi in B gamma in Gamma mid xi gamma in B nbsp is an event Uncertainty distribution editUncertainty distribution is inducted to describe uncertain variables Definition The uncertainty distribution F x R 0 1 displaystyle Phi x R rightarrow 0 1 nbsp of an uncertain variable 3 is defined by F x M 3 x displaystyle Phi x M xi leq x nbsp Theorem Peng and Iwamura Sufficient and Necessary Condition for Uncertainty Distribution A function F x R 0 1 displaystyle Phi x R rightarrow 0 1 nbsp is an uncertain distribution if and only if it is an increasing function except F x 0 displaystyle Phi x equiv 0 nbsp and F x 1 displaystyle Phi x equiv 1 nbsp Independence editDefinition The uncertain variables 3 1 3 2 3 m displaystyle xi 1 xi 2 ldots xi m nbsp are said to be independent if M i 1 m 3 B i min 1 i m M 3 i B i displaystyle M cap i 1 m xi in B i mbox min 1 leq i leq m M xi i in B i nbsp for any Borel sets B 1 B 2 B m displaystyle B 1 B 2 ldots B m nbsp of real numbers Theorem 1 The uncertain variables 3 1 3 2 3 m displaystyle xi 1 xi 2 ldots xi m nbsp are independent if M i 1 m 3 B i max 1 i m M 3 i B i displaystyle M cup i 1 m xi in B i mbox max 1 leq i leq m M xi i in B i nbsp for any Borel sets B 1 B 2 B m displaystyle B 1 B 2 ldots B m nbsp of real numbers Theorem 2 Let 3 1 3 2 3 m displaystyle xi 1 xi 2 ldots xi m nbsp be independent uncertain variables and f 1 f 2 f m displaystyle f 1 f 2 ldots f m nbsp measurable functions Then f 1 3 1 f 2 3 2 f m 3 m displaystyle f 1 xi 1 f 2 xi 2 ldots f m xi m nbsp are independent uncertain variables Theorem 3 Let F i displaystyle Phi i nbsp be uncertainty distributions of independent uncertain variables 3 i i 1 2 m displaystyle xi i quad i 1 2 ldots m nbsp respectively and F displaystyle Phi nbsp the joint uncertainty distribution of uncertain vector 3 1 3 2 3 m displaystyle xi 1 xi 2 ldots xi m nbsp If 3 1 3 2 3 m displaystyle xi 1 xi 2 ldots xi m nbsp are independent then we have F x 1 x 2 x m min 1 i m F i x i displaystyle Phi x 1 x 2 ldots x m mbox min 1 leq i leq m Phi i x i nbsp for any real numbers x 1 x 2 x m displaystyle x 1 x 2 ldots x m nbsp Operational law editTheorem Let 3 1 3 2 3 m displaystyle xi 1 xi 2 ldots xi m nbsp be independent uncertain variables and f R n R displaystyle f R n rightarrow R nbsp a measurable function Then 3 f 3 1 3 2 3 m displaystyle xi f xi 1 xi 2 ldots xi m nbsp is an uncertain variable such that M 3 B sup f B 1 B 2 B n B min 1 k n M k 3 k B k if sup f B 1 B 2 B n B min 1 k n M k 3 k B k gt 0 5 1 sup f B 1 B 2 B n B c min 1 k n M k 3 k B k if sup f B 1 B 2 B n B c min 1 k n M k 3 k B k gt 0 5 0 5 otherwise displaystyle mathcal M xi in B begin cases underset f B 1 B 2 cdots B n subset B sup underset 1 leq k leq n min mathcal M k xi k in B k amp text if underset f B 1 B 2 cdots B n subset B sup underset 1 leq k leq n min mathcal M k xi k in B k gt 0 5 1 underset f B 1 B 2 cdots B n subset B c sup underset 1 leq k leq n min mathcal M k xi k in B k amp text if underset f B 1 B 2 cdots B n subset B c sup underset 1 leq k leq n min mathcal M k xi k in B k gt 0 5 0 5 amp text otherwise end cases nbsp dd where B B 1 B 2 B m displaystyle B B 1 B 2 ldots B m nbsp are Borel sets and f B 1 B 2 B m B displaystyle f B 1 B 2 ldots B m subset B nbsp means f x 1 x 2 x m B displaystyle f x 1 x 2 ldots x m in B nbsp for anyx 1 B 1 x 2 B 2 x m B m displaystyle x 1 in B 1 x 2 in B 2 ldots x m in B m nbsp Expected Value editDefinition Let 3 displaystyle xi nbsp be an uncertain variable Then the expected value of 3 displaystyle xi nbsp is defined by E 3 0 M 3 r d r 0 M 3 r d r displaystyle E xi int 0 infty M xi geq r dr int infty 0 M xi leq r dr nbsp provided that at least one of the two integrals is finite Theorem 1 Let 3 displaystyle xi nbsp be an uncertain variable with uncertainty distribution F displaystyle Phi nbsp If the expected value exists then E 3 0 1 F x d x 0 F x d x displaystyle E xi int 0 infty 1 Phi x dx int infty 0 Phi x dx nbsp nbsp Theorem 2 Let 3 displaystyle xi nbsp be an uncertain variable with regular uncertainty distribution F displaystyle Phi nbsp If the expected value exists then E 3 0 1 F 1 a d a displaystyle E xi int 0 1 Phi 1 alpha d alpha nbsp Theorem 3 Let 3 displaystyle xi nbsp and h displaystyle eta nbsp be independent uncertain variables with finite expected values Then for any real numbers a displaystyle a nbsp and b displaystyle b nbsp we have E a 3 b h a E 3 b h displaystyle E a xi b eta aE xi b eta nbsp Variance editDefinition Let 3 displaystyle xi nbsp be an uncertain variable with finite expected value e displaystyle e nbsp Then the variance of 3 displaystyle xi nbsp is defined by V 3 E 3 e 2 displaystyle V xi E xi e 2 nbsp Theorem If 3 displaystyle xi nbsp be an uncertain variable with finite expected value a displaystyle a nbsp and b displaystyle b nbsp are real numbers then V a 3 b a 2 V 3 displaystyle V a xi b a 2 V xi nbsp Critical value editDefinition Let 3 displaystyle xi nbsp be an uncertain variable and a 0 1 displaystyle alpha in 0 1 nbsp Then 3 s u p a sup r M 3 r a displaystyle xi sup alpha sup r mid M xi geq r geq alpha nbsp is called the a optimistic value to 3 displaystyle xi nbsp and 3 i n f a inf r M 3 r a displaystyle xi inf alpha inf r mid M xi leq r geq alpha nbsp is called the a pessimistic value to 3 displaystyle xi nbsp Theorem 1 Let 3 displaystyle xi nbsp be an uncertain variable with regular uncertainty distribution F displaystyle Phi nbsp Then its a optimistic value and a pessimistic value are 3 s u p a F 1 1 a displaystyle xi sup alpha Phi 1 1 alpha nbsp 3 i n f a F 1 a displaystyle xi inf alpha Phi 1 alpha nbsp dd Theorem 2 Let 3 displaystyle xi nbsp be an uncertain variable and a 0 1 displaystyle alpha in 0 1 nbsp Then we have if a gt 0 5 displaystyle alpha gt 0 5 nbsp then 3 i n f a 3 s u p a displaystyle xi inf alpha geq xi sup alpha nbsp if a 0 5 displaystyle alpha leq 0 5 nbsp then 3 i n f a 3 s u p a displaystyle xi inf alpha leq xi sup alpha nbsp Theorem 3 Suppose that 3 displaystyle xi nbsp and h displaystyle eta nbsp are independent uncertain variables and a 0 1 displaystyle alpha in 0 1 nbsp Then we have 3 h s u p a 3 s u p a h s u p a displaystyle xi eta sup alpha xi sup alpha eta sup alpha nbsp 3 h i n f a 3 i n f a h i n f a displaystyle xi eta inf alpha xi inf alpha eta inf alpha nbsp 3 h s u p a 3 s u p a h s u p a displaystyle xi vee eta sup alpha xi sup alpha vee eta sup alpha nbsp 3 h i n f a 3 i n f a h i n f a displaystyle xi vee eta inf alpha xi inf alpha vee eta inf alpha nbsp 3 h s u p a 3 s u p a h s u p a displaystyle xi wedge eta sup alpha xi sup alpha wedge eta sup alpha nbsp 3 h i n f a 3 i n f a h i n f a displaystyle xi wedge eta inf alpha xi inf alpha wedge eta inf alpha nbsp Entropy editDefinition Let 3 displaystyle xi nbsp be an uncertain variable with uncertainty distribution F displaystyle Phi nbsp Then its entropy is defined by H 3 S F x d x displaystyle H xi int infty infty S Phi x dx nbsp where S x t ln t 1 t ln 1 t displaystyle S x t ln t 1 t ln 1 t nbsp Theorem 1 Dai and Chen Let 3 displaystyle xi nbsp be an uncertain variable with regular uncertainty distribution F displaystyle Phi nbsp Then H 3 0 1 F 1 a ln a 1 a d a displaystyle H xi int 0 1 Phi 1 alpha ln frac alpha 1 alpha d alpha nbsp dd Theorem 2 Let 3 displaystyle xi nbsp and h displaystyle eta nbsp be independent uncertain variables Then for any real numbers a displaystyle a nbsp and b displaystyle b nbsp we have H a 3 b h a E 3 b E h displaystyle H a xi b eta a E xi b E eta nbsp dd Theorem 3 Let 3 displaystyle xi nbsp be an uncertain variable whose uncertainty distribution is arbitrary but the expected value e displaystyle e nbsp and variance s 2 displaystyle sigma 2 nbsp Then H 3 p s 3 displaystyle H xi leq frac pi sigma sqrt 3 nbsp dd Inequalities editTheorem 1 Liu Markov Inequality Let 3 displaystyle xi nbsp be an uncertain variable Then for any given numbers t gt 0 displaystyle t gt 0 nbsp and p gt 0 displaystyle p gt 0 nbsp we have M 3 t E 3 p t p displaystyle M xi geq t leq frac E xi p t p nbsp dd Theorem 2 Liu Chebyshev Inequality Let 3 displaystyle xi nbsp be an uncertain variable whose variance V 3 displaystyle V xi nbsp exists Then for any given number t gt 0 displaystyle t gt 0 nbsp we have M 3 E 3 t V 3 t 2 displaystyle M xi E xi geq t leq frac V xi t 2 nbsp dd Theorem 3 Liu Holder s Inequality Let p displaystyle p nbsp and q displaystyle q nbsp be positive numbers with 1 p 1 q 1 displaystyle 1 p 1 q 1 nbsp and let 3 displaystyle xi nbsp and h displaystyle eta nbsp be independent uncertain variables with E 3 p lt displaystyle E xi p lt infty nbsp and E h q lt displaystyle E eta q lt infty nbsp Then we have E 3 h E 3 p p E h p p displaystyle E xi eta leq sqrt p E xi p sqrt p E eta p nbsp dd Theorem 4 Liu 127 Minkowski Inequality Let p displaystyle p nbsp be a real number with p 1 displaystyle p leq 1 nbsp and let 3 displaystyle xi nbsp and h displaystyle eta nbsp be independent uncertain variables with E 3 p lt displaystyle E xi p lt infty nbsp and E h q lt displaystyle E eta q lt infty nbsp Then we have E 3 h p p E 3 p p E h p p displaystyle sqrt p E xi eta p leq sqrt p E xi p sqrt p E eta p nbsp dd Convergence concept editDefinition 1 Suppose that 3 3 1 3 2 displaystyle xi xi 1 xi 2 ldots nbsp are uncertain variables defined on the uncertainty space G L M displaystyle Gamma L M nbsp The sequence 3 i displaystyle xi i nbsp is said to be convergent a s to 3 displaystyle xi nbsp if there exists an event L displaystyle Lambda nbsp with M L 1 displaystyle M Lambda 1 nbsp such that lim i 3 i g 3 g 0 displaystyle lim i to infty xi i gamma xi gamma 0 nbsp dd for every g L displaystyle gamma in Lambda nbsp In that case we write 3 i 3 displaystyle xi i to xi nbsp a s Definition 2 Suppose that 3 3 1 3 2 displaystyle xi xi 1 xi 2 ldots nbsp are uncertain variables We say that the sequence 3 i displaystyle xi i nbsp converges in measure to 3 displaystyle xi nbsp if lim i M 3 i 3 e 0 displaystyle lim i to infty M xi i xi leq varepsilon 0 nbsp dd for every e gt 0 displaystyle varepsilon gt 0 nbsp Definition 3 Suppose that 3 3 1 3 2 displaystyle xi xi 1 xi 2 ldots nbsp are uncertain variables with finite expected values We say that the sequence 3 i displaystyle xi i nbsp converges in mean to 3 displaystyle xi nbsp if lim i E 3 i 3 0 displaystyle lim i to infty E xi i xi 0 nbsp dd Definition 4 Suppose that F ϕ 1 F 2 displaystyle Phi phi 1 Phi 2 ldots nbsp are uncertainty distributions of uncertain variables 3 3 1 3 2 displaystyle xi xi 1 xi 2 ldots nbsp respectively We say that the sequence 3 i displaystyle xi i nbsp converges in distribution to 3 displaystyle xi nbsp if F i F displaystyle Phi i rightarrow Phi nbsp at any continuity point of F displaystyle Phi nbsp Theorem 1 Convergence in Mean displaystyle Rightarrow nbsp Convergence in Measure displaystyle Rightarrow nbsp Convergence in Distribution However Convergence in Mean displaystyle nLeftrightarrow nbsp Convergence Almost Surely displaystyle nLeftrightarrow nbsp Convergence in Distribution Conditional uncertainty editDefinition 1 Let G L M displaystyle Gamma L M nbsp be an uncertainty space and A B L displaystyle A B in L nbsp Then the conditional uncertain measure of A given B is defined by M A B M A B M B if M A B M B lt 0 5 1 M A c B M B if M A c B M B lt 0 5 0 5 otherwise displaystyle mathcal M A vert B begin cases displaystyle frac mathcal M A cap B mathcal M B amp displaystyle text if frac mathcal M A cap B mathcal M B lt 0 5 displaystyle 1 frac mathcal M A c cap B mathcal M B amp displaystyle text if frac mathcal M A c cap B mathcal M B lt 0 5 0 5 amp text otherwise end cases nbsp provided that M B gt 0 displaystyle text provided that mathcal M B gt 0 nbsp dd Theorem 1 Let G L M displaystyle Gamma L M nbsp be an uncertainty space and B an event with M B gt 0 displaystyle M B gt 0 nbsp Then M B defined by Definition 1 is an uncertain measure and G L M B displaystyle Gamma L M mbox B nbsp is an uncertainty space Definition 2 Let 3 displaystyle xi nbsp be an uncertain variable on G L M displaystyle Gamma L M nbsp A conditional uncertain variable of 3 displaystyle xi nbsp given B is a measurable function 3 B displaystyle xi B nbsp from the conditional uncertainty space G L M B displaystyle Gamma L M mbox B nbsp to the set of real numbers such that 3 B g 3 g g G displaystyle xi B gamma xi gamma forall gamma in Gamma nbsp dd Definition 3 The conditional uncertainty distribution F 0 1 displaystyle Phi rightarrow 0 1 nbsp of an uncertain variable 3 displaystyle xi nbsp given B is defined by F x B M 3 x B displaystyle Phi x B M xi leq x B nbsp dd provided that M B gt 0 displaystyle M B gt 0 nbsp Theorem 2 Let 3 displaystyle xi nbsp be an uncertain variable with regular uncertainty distribution F x displaystyle Phi x nbsp and t displaystyle t nbsp a real number with F t lt 1 displaystyle Phi t lt 1 nbsp Then the conditional uncertainty distribution of 3 displaystyle xi nbsp given 3 gt t displaystyle xi gt t nbsp is F x t 0 if F x F t F x 1 F t 0 5 if F t lt F x 1 F t 2 F x F t 1 F t if 1 F t 2 F x displaystyle Phi x vert t infty begin cases 0 amp text if Phi x leq Phi t displaystyle frac Phi x 1 Phi t land 0 5 amp text if Phi t lt Phi x leq 1 Phi t 2 displaystyle frac Phi x Phi t 1 Phi t amp text if 1 Phi t 2 leq Phi x end cases nbsp dd Theorem 3 Let 3 displaystyle xi nbsp be an uncertain variable with regular uncertainty distribution F x displaystyle Phi x nbsp and t displaystyle t nbsp a real number with F t gt 0 displaystyle Phi t gt 0 nbsp Then the conditional uncertainty distribution of 3 displaystyle xi nbsp given 3 t displaystyle xi leq t nbsp is F x t F x F t if F x F t 2 F x F t 1 F t 0 5 if F t 2 F x lt F t 1 if F t F x displaystyle Phi x vert infty t begin cases displaystyle frac Phi x Phi t amp text if Phi x leq Phi t 2 displaystyle frac Phi x Phi t 1 Phi t lor 0 5 amp text if Phi t 2 leq Phi x lt Phi t 1 amp text if Phi t leq Phi x end cases nbsp dd Definition 4 Let 3 displaystyle xi nbsp be an uncertain variable Then the conditional expected value of 3 displaystyle xi nbsp given B is defined by E 3 B 0 M 3 r B d r 0 M 3 r B d r displaystyle E xi B int 0 infty M xi geq r B dr int infty 0 M xi leq r B dr nbsp dd provided that at least one of the two integrals is finite References edit nbsp Wikimedia Commons has media related to Uncertainty Theory Liu Baoding 2015 Uncertainty theory an introduction to its axiomatic foundations Springer uncertainty research 4th ed Berlin Springer ISBN 978 3 662 44354 5 Sources editXin Gao Some Properties of Continuous Uncertain Measure International Journal of Uncertainty Fuzziness and Knowledge Based Systems Vol 17 No 3 419 426 2009 Cuilian You Some Convergence Theorems of Uncertain Sequences Mathematical and Computer Modelling Vol 49 Nos 3 4 482 487 2009 Yuhan Liu How to Generate Uncertain Measures Proceedings of Tenth National Youth Conference on Information and Management Sciences August 3 7 2008 Luoyang pp 23 26 Baoding Liu Uncertainty Theory 4th ed Springer Verlag Berlin 1 2009 Baoding Liu Some Research Problems in Uncertainty Theory Journal of Uncertain Systems Vol 3 No 1 3 10 2009 Yang Zuo Xiaoyu Ji Theoretical Foundation of Uncertain Dominance Proceedings of the Eighth International Conference on Information and Management Sciences Kunming China July 20 28 2009 pp 827 832 Yuhan Liu and Minghu Ha Expected Value of Function of Uncertain Variables Proceedings of the Eighth International Conference on Information and Management Sciences Kunming China July 20 28 2009 pp 779 781 Zhongfeng Qin On Lognormal Uncertain Variable Proceedings of the Eighth International Conference on Information and Management Sciences Kunming China July 20 28 2009 pp 753 755 Jin Peng Value at Risk and Tail Value at Risk in Uncertain Environment Proceedings of the Eighth International Conference on Information and Management Sciences Kunming China July 20 28 2009 pp 787 793 Yi Peng U Curve and U Coefficient in Uncertain Environment Proceedings of the Eighth International Conference on Information and Management Sciences Kunming China July 20 28 2009 pp 815 820 Wei Liu Jiuping Xu Some Properties on Expected Value Operator for Uncertain Variables Proceedings of the Eighth International Conference on Information and Management Sciences Kunming China July 20 28 2009 pp 808 811 Xiaohu Yang Moments and Tails Inequality within the Framework of Uncertainty Theory Proceedings of the Eighth International Conference on Information and Management Sciences Kunming China July 20 28 2009 pp 812 814 Yuan Gao Analysis of k out of n System with Uncertain Lifetimes Proceedings of the Eighth International Conference on Information and Management Sciences Kunming China July 20 28 2009 pp 794 797 Xin Gao Shuzhen Sun Variance Formula for Trapezoidal Uncertain Variables Proceedings of the Eighth International Conference on Information and Management Sciences Kunming China July 20 28 2009 pp 853 855 Zixiong Peng A Sufficient and Necessary Condition of Product Uncertain Null Set Proceedings of the Eighth International Conference on Information and Management Sciences Kunming China July 20 28 2009 pp 798 801 Retrieved from https en wikipedia org w index php title Uncertainty theory Liu amp oldid 1219231271, wikipedia, wiki, book, books, library,