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| {{distinguish|Uncertainty principle}}
| | If you present photography effectively, it helps you look much more properly at the globe around you. Also, you may want to opt for a more professioanl theme if you are planning on showing your site off to a high volume of potential customers each day. * A community forum for debate of the product together with some other customers in the comments spot. They found out all the possible information about bringing up your baby and save money at the same time. You can customize the appearance with PSD to Word - Press conversion ''. <br><br>Luckily, for Word - Press users, WP Touch plugin transforms your site into an IPhone style theme. The higher your blog ranks on search engines, the more likely people will find your online marketing site. We also help to integrate various plug-ins to expand the functionalities of the web application. These four plugins will make this effort easier and the sites run effectively as well as make other widgets added to a site easier to configure. W3C compliant HTML and a good open source powered by Word - Press CMS site is regarded as the prime minister. <br><br>It is also popular because willing surrogates,as well as egg and sperm donors,are plentiful. s cutthroat competition prevailing in the online space won. If Gandhi was empowered with a blogging system, every event in his life would have been minutely documented so that it could be recounted to the future generations. If you treasured this article so you would like to acquire more info concerning [http://snipitfor.me/backup_plugin_553209 backup plugin] i implore you to visit our own website. Thousands of plugins are available in Word - Press plugin's library which makes the task of selecting right set of plugins for your website a very tedious task. " Thus working with a Word - Press powered web application, making any changes in the website design or website content is really easy and self explanatory. <br><br>It is the convenient service through which professionals either improve the position or keep the ranking intact. Quttera - Quttera describes itself as a 'Saa - S [Software as a Service] web-malware monitoring and alerting solution for websites of any size and complexity. When we talk about functional suitability, Word - Press proves itself as one of the strongest contestant among its other rivals. IVF ,fertility,infertility expert,surrogacy specialist in India at Rotundaivf. It does take time to come up having a website that gives you the much needed results hence the web developer must be ready to help you along the route. <br><br>Website security has become a major concern among individuals all over the world. Mahatma Gandhi is known as one of the most prominent personalities and symbols of peace, non-violence and freedom. It's not a secret that a lion share of activity on the internet is takes place on the Facebook. If this is not possible you still have the choice of the default theme that is Word - Press 3. I have never seen a plugin with such a massive array of features, this does everything that platinum SEO and All In One SEO, also throws in the functionality found within SEO Smart Links and a number of other plugins it is essentially the swiss army knife of Word - Press plugins. |
| {{multiple issues|
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| {{one source|date=November 2009}}
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| {{jargon|date=December 2009}}
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| {{howto|date=December 2009}}
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| {{citation style|date=December 2009}}
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| {{misleading|date=December 2009}}
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| '''Uncertainty theory''' is a branch of [[mathematics]] based on normality, monotonicity, self-duality, countable subadditivity, and product measure axioms.{{Clarify|date=December 2009}} It was founded by Baoding Liu <ref>Baoding Liu, Uncertainty Theory, 2nd ed., Springer-Verlag, Berlin, 2007.</ref> in 2007 and refined in 2009.<ref>Baoding Liu, Uncertainty Theory, 4th ed., http://orsc.edu.cn/liu/ut.pdf.</ref>
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| Mathematical measures of the likelihood of an event being true include [[probability theory]], capacity, [[fuzzy logic]], possibility, and credibility, as well as uncertainty.
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| ==Five axioms==
| |
| '''Axiom 1.''' (Normality Axiom) <math>\mathcal{M}\{\Gamma\}=1\text{ for the universal set }\Gamma</math>.
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| | |
| '''Axiom 2.''' (Monotonicity Axiom) <math>\mathcal{M}\{\Lambda_1\}\le\mathcal{M}\{\Lambda_2\}\text{ whenever }\Lambda_1\subset\Lambda_2</math>.
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| | |
| '''Axiom 3.''' (Self-Duality Axiom) <Math>\mathcal{M}\{\Lambda\}+\mathcal{M}\{\Lambda^c\}=1\text{ for any event }\Lambda</math>.
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| | |
| '''Axiom 4.''' (Countable Subadditivity Axiom) For every countable sequence of events Λ<sub>1</sub>, Λ<sub>2</sub>, ..., we have | |
| ::<math>\mathcal{M}\left\{\bigcup_{i=1}^\infty\Lambda_i\right\}\le\sum_{i=1}^\infty\mathcal{M}\{\Lambda_i\}</math>.
| |
| | |
| '''Axiom 5.''' (Product Measure Axiom) Let <math>(\Gamma_k,\mathcal{L}_k,\mathcal{M}_k)</math> be uncertainty spaces for <math>k=1,2,\cdots,n</math>. Then the product uncertain measure <math>\mathcal{M}</math> is an uncertain measure on the product σ-algebra satisfying
| |
| ::<math>\mathcal{M}\left\{\prod_{i=1}^n\Lambda_i\right\}=\underset{1\le i\le n}{\operatorname{min} }\mathcal{M}_i\{\Lambda_i\}</math>.
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| | |
| '''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.
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| ==Uncertain variables==
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| An uncertain variable is a [[measurable function]] ξ from an uncertainty space <math>(\Gamma,L,M)</math> to the [[set (mathematics)|set]] of [[real numbers]], i.e., for any [[Borel set]] '''B''' of [[real numbers]], the set
| |
| <math>\{\xi\in B\}=\{\gamma \in \Gamma|\xi(\gamma)\in B\}</math> is an event.
| |
| | |
| ==Uncertainty distribution==
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| Uncertainty distribution is inducted to describe uncertain variables.
| |
| | |
| '''Definition''':The '''uncertainty distribution''' <math>\Phi(x):R \rightarrow [0,1]</math> of an uncertain variable ξ is defined by <math>\Phi(x)=M\{\xi\leq x\}</math>.
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| | |
| '''Theorem'''(Peng and Iwamura, ''Sufficient and Necessary Condition for Uncertainty Distribution'') A function <math>\Phi(x):R \rightarrow [0,1]</math> is an uncertain distribution if and only if it is an increasing function except <math>\Phi (x) \equiv 0</math> and <math>\Phi (x)\equiv 1</math>.
| |
| | |
| ==Independence==
| |
| '''Definition''': The uncertain variables <math>\xi_1,\xi_2,\ldots,\xi_m</math> are said to be independent if
| |
| :<math>M\{\cap_{i=1}^m(\xi \in B_i)\}=\mbox{min}_{1\leq i \leq m}M\{\xi_i \in B_i\} </math>
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| for any Borel sets <math>B_1,B_2,\ldots,B_m</math> of real numbers.
| |
| | |
| '''Theorem 1''': The uncertain variables <math>\xi_1,\xi_2,\ldots,\xi_m</math> are independent if
| |
| :<math>M\{\cup_{i=1}^m(\xi \in B_i)\}=\mbox{max}_{1\leq i \leq m}M\{\xi_i \in B_i\} </math>
| |
| for any Borel sets <math>B_1,B_2,\ldots,B_m</math> of real numbers.
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| | |
| '''Theorem 2''': Let <math>\xi_1,\xi_2,\ldots,\xi_m</math> be independent uncertain variables, and <math>f_1,f_2,\ldots,f_m</math> measurable functions. Then <math>f_1(\xi_1),f_2(\xi_2),\ldots,f_m(\xi_m)</math> are independent uncertain variables.
| |
| | |
| '''Theorem 3''': Let <math>\Phi_i</math> be uncertainty distributions of independent uncertain variables <math>\xi_i,\quad i=1,2,\ldots,m</math> respectively, and <math>\Phi</math> the joint uncertainty distribution of uncertain vector <math>(\xi_1,\xi_2,\ldots,\xi_m)</math>. If <math>\xi_1,\xi_2,\ldots,\xi_m</math> are independent, then we have
| |
| :<math>\Phi(x_1, x_2, \ldots, x_m)=\mbox{min}_{1\leq i \leq m}\Phi_i(x_i)</math>
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| for any real numbers <math>x_1, x_2, \ldots, x_m</math>.
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| ==Operational law==
| |
| '''Theorem''': Let <math>\xi_1,\xi_2,\ldots,\xi_m</math> be independent uncertain variables, and <math>f: R^n \rightarrow R</math> a measurable function. Then <math>\xi=f(\xi_1,\xi_2,\ldots,\xi_m)</math> is an uncertain variable such that
| |
| ::<math>\mathcal{M}\{\xi\in B\}=\begin{cases} \underset{f(B_1,B_2,\cdots,B_n)\subset B}{\operatorname{sup} }\;\underset{1\le k\le n}{\operatorname{min} }\mathcal{M}_k\{\xi_k\in B_k\}, & \text{if } \underset{f(B_1,B_2,\cdots,B_n)\subset B}{\operatorname{sup} }\;\underset{1\le k\le n}{\operatorname{min} }\mathcal{M}_k\{\xi_k\in B_k\} > 0.5 \\ 1-\underset{f(B_1,B_2,\cdots,B_n)\subset B^c}{\operatorname{sup} }\;\underset{1\le k\le n}{\operatorname{min} }\mathcal{M}_k\{\xi_k\in B_k\}, & \text{if } \underset{f(B_1,B_2,\cdots,B_n)\subset B^c}{\operatorname{sup} }\;\underset{1\le k\le n}{\operatorname{min} }\mathcal{M}_k\{\xi_k\in B_k\} > 0.5 \\ 0.5, & \text{otherwise} \end{cases}</math>
| |
| where <math>B, B_1, B_2, \ldots, B_m</math> are Borel sets, and <math>f( B_1, B_2, \ldots, B_m)\subset B</math> means<math>f(x_1, x_2, \ldots, x_m) \in B</math> for any<math>x_1 \in B_1, x_2 \in B_2, \ldots,x_m \in B_m</math>.
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| ==Expected Value==
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| '''Definition''': Let <math>\xi</math> be an uncertain variable. Then the expected value of <math>\xi</math> is defined by
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| :::<math>E[\xi]=\int_0^{+\infty}M\{\xi\geq r\}dr-\int_{-\infty}^0M\{\xi\leq r\}dr</math>
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| provided that at least one of the two integrals is finite.
| |
| | |
| '''Theorem 1''': Let <math>\xi</math> be an uncertain variable with uncertainty distribution <math>\Phi</math>. If the expected value exists, then
| |
| :::<math>E[\xi]=\int_0^{+\infty}(1-\Phi(x))dx-\int_{-\infty}^0\Phi(x)dx</math>.
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| [[File:Uncertain expected value.jpg|300px|center]]
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| '''Theorem 2''': Let <math>\xi</math> be an uncertain variable with regular uncertainty distribution <math>\Phi</math>. If the expected value exists, then
| |
| :::<math>E[\xi]=\int_0^1\Phi^{-1}(\alpha)d\alpha</math>.
| |
| | |
| '''Theorem 3''': Let <math>\xi</math> and <math>\eta</math> be independent uncertain variables with finite expected values. Then for any real numbers <math>a</math> and <math>b</math>, we have
| |
| :::<math>E[a\xi+b\eta]=aE[\xi]+b[\eta]</math>.
| |
| | |
| ==Variance==
| |
| '''Definition''': Let <math>\xi</math> be an uncertain variable with finite expected value <math>e</math>. Then the variance of <math>\xi</math> is defined by | |
| :::<math>V[\xi]=E[(\xi-e)^2]</math>.
| |
| | |
| '''Theorem''': If <math>\xi</math> be an uncertain variable with finite expected value, <math>a</math> and <math>b</math> are real numbers, then
| |
| :::<math>V[a\xi+b]=a^2V[\xi]</math>.
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| ==Critical value==
| |
| '''Definition''': Let <math>\xi</math> be an uncertain variable, and <math>\alpha\in(0,1]</math>. Then
| |
| :<math>\xi_{sup}(\alpha)=\mbox{sup}\{r|M\{\xi\geq r\}\geq\alpha\}</math>
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| is called the α-[[optimistic]] value to <math>\xi</math>, and | |
| :<math>\xi_{inf}(\alpha)=\mbox{inf}\{r|M\{\xi\leq r\}\geq\alpha\}</math>
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| is called the α-[[pessimistic]] value to <math>\xi</math>.
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| | |
| '''Theorem 1''': Let <math>\xi</math> be an uncertain variable with regular uncertainty distribution <math>\Phi</math>. Then its α-[[optimistic]] value and α-[[pessimistic]] value are
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| ::<math>\xi_{sup}(\alpha)=\Phi^{-1}(1-\alpha)</math>,
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| ::<math>\xi_{inf}(\alpha)=\Phi^{-1}(\alpha)</math>.
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| '''Theorem 2''': Let <math>\xi</math> be an uncertain variable, and <math>\alpha\in(0,1]</math>. Then we have
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| * if <math>\alpha>0.5</math>, then <math>\xi_{inf}(\alpha)\geq \xi_{sup}(\alpha)</math>;
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| * if <math>\alpha\leq 0.5</math>, then <math>\xi_{inf}(\alpha)\leq \xi_{sup}(\alpha)</math>.
| |
| | |
| '''Theorem 3''': Suppose that <math>\xi</math> and <math>\eta</math> are independent uncertain variables, and <math>\alpha\in(0,1]</math>. Then we have | |
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| <math>(\xi + \eta)_{sup}(\alpha)=\xi_{sup}(\alpha)+\eta_{sup}{\alpha}</math>,
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| <math>(\xi + \eta)_{inf}(\alpha)=\xi_{inf}(\alpha)+\eta_{inf}{\alpha}</math>,
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| <math>(\xi \vee \eta)_{sup}(\alpha)=\xi_{sup}(\alpha)\vee\eta_{sup}{\alpha}</math>,
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| <math>(\xi \vee \eta)_{inf}(\alpha)=\xi_{inf}(\alpha)\vee\eta_{inf}{\alpha}</math>,
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| <math>(\xi \wedge \eta)_{sup}(\alpha)=\xi_{sup}(\alpha)\wedge\eta_{sup}{\alpha}</math>,
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| <math>(\xi \wedge \eta)_{inf}(\alpha)=\xi_{inf}(\alpha)\wedge\eta_{inf}{\alpha}</math>.
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| | |
| ==Entropy==
| |
| '''Definition''': Let <math>\xi</math> be an uncertain variable with uncertainty distribution <math>\Phi</math>. Then its entropy is defined by
| |
| ::<math>H[\xi]=\int_{-\infty}^{+\infty}S(\Phi(x))dx</math>
| |
| where <math>S(x)=-t\mbox{ln}(t)-(1-t)\mbox{ln}(1-t)</math>.
| |
| | |
| '''Theorem 1'''(''Dai and Chen''): Let <math>\xi</math> be an uncertain variable with regular uncertainty distribution <math>\Phi</math>. Then
| |
| ::<math>H[\xi]=\int_0^1\Phi^{-1}(\alpha)\mbox{ln}\frac{\alpha}{1-\alpha}d\alpha</math>.
| |
| | |
| '''Theorem 2''': Let <math>\xi</math> and <math>\eta</math> be independent uncertain variables. Then for any real numbers <math>a</math> and <math>b</math>, we have
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| ::<math>H[a\xi+b\eta]=|a|E[\xi]+|b|E[\eta]</math>.
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| | |
| '''Theorem 3''': Let <math>\xi</math> be an uncertain variable whose uncertainty distribution is arbitrary but the expected value <math>e</math> and variance <math>\sigma^2</math>. Then
| |
| ::<math>H[\xi]\leq\frac{\pi\sigma}{\sqrt{3}}</math>.
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| | |
| ==Inequalities==
| |
| '''Theorem 1'''(''Liu'', Markov Inequality): Let <math>\xi</math> be an uncertain variable. Then for any given numbers <math>t > 0</math> and <math>p > 0</math>, we have
| |
| ::<math>M\{|\xi|\geq t\}\leq \frac{E[|\xi|^p]}{t^p}</math>.
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| | |
| '''Theorem 2''' (''Liu'', Chebyshev Inequality) Let <math>\xi</math> be an uncertain variable whose variance <math>V[\xi]</math> exists. Then for any given number<math> t > 0</math>, we have
| |
| ::<math>M\{|\xi-E[\xi]|\geq t\}\leq \frac{V[\xi]}{t^2}</math>.
| |
| | |
| '''Theorem 3''' (''Liu'', Holder’s Inequality) Let <math>p</math> and <math>q</math> be positive numbers with <math>1/p + 1/q = 1</math>, and let <math>\xi</math> and <math>\eta</math> be independent uncertain variables with <math>E[|\xi|^p]< \infty</math> and <math>E[|\eta|^q] < \infty</math>. Then we have
| |
| ::<math>E[|\xi\eta|]\leq \sqrt[p]{E[|\xi|^p]} \sqrt[p]{E[\eta|^p]}</math>.
| |
| | |
| '''Theorem 4''':(Liu [127], Minkowski Inequality) Let <math>p</math> be a real number with <math>p\leq 1</math>, and let <math>\xi</math> and <math>\eta</math> be independent uncertain variables with <math>E[|\xi|^p]< \infty</math> and <math>E[|\eta|^q] < \infty</math>. Then we have
| |
| ::<math>\sqrt[p]{E[|\xi+\eta|^p]}\leq \sqrt[p]{E[|\xi|^p]}+\sqrt[p]{E[\eta|^p]}</math>.
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| ==Convergence concept==
| |
| '''Definition 1''': Suppose that <math>\xi,\xi_1,\xi_2,\ldots</math> are uncertain variables defined on the uncertainty space <math>(\Gamma,L,M)</math>. The sequence <math>\{\xi_i\}</math> is said to be convergent a.s. to <math>\xi</math> if there exists an event <math>\Lambda</math> with <math>M\{\Lambda\} = 1</math> such that
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| ::<math>\mbox{lim}_{i\rightarrow\infty}|\xi_i(\gamma)-\xi(\gamma)|=0</math>
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| for every <math>\gamma\in\Lambda</math>. In that case we write <math>\xi_i\rightarrow \xi</math>,a.s.
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| '''Definition 2''': Suppose that <math>\xi,\xi_1,\xi_2,\ldots</math> are uncertain variables. We say that the sequence <math>\{\xi_i\}</math> converges in measure to <math>\xi</math> if
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| ::<math>\mbox{lim}_{i\rightarrow\infty}M\{|\xi_i-\xi|\leq \varepsilon \}=0</math>
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| for every <math>\varepsilon>0</math>.
| |
| | |
| '''Definition 3''': Suppose that <math>\xi,\xi_1,\xi_2,\ldots</math> are uncertain variables with finite expected values. We say that the sequence <math>\{\xi_i\}</math> converges in mean to <math>\xi</math> if
| |
| ::<math>\mbox{lim}_{i\rightarrow\infty}E[|\xi_i-\xi|]=0</math>.
| |
| | |
| '''Definition 4''': Suppose that <math>\Phi,\phi_1,\Phi_2,\ldots</math> are uncertainty distributions of uncertain variables <math>\xi,\xi_1,\xi_2,\ldots</math>, respectively. We say that the sequence <math>\{\xi_i\}</math> converges in distribution to <math>\xi</math> if <math>\Phi_i\rightarrow\Phi</math> at any continuity point of <math>\Phi</math>.
| |
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| '''Theorem 1''': Convergence in Mean <math>\Rightarrow</math> Convergence in Measure <math>\Rightarrow</math> Convergence in Distribution.
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| However, Convergence in Mean <math>\nLeftrightarrow</math> Convergence Almost Surely <math>\nLeftrightarrow</math> Convergence in Distribution.
| |
| | |
| ==Conditional uncertainty==
| |
| '''Definition 1''': Let <math>(\Gamma,L,M)</math> be an uncertainty space, and <math>A,B\in L</math>. Then the conditional uncertain measure of A given B is defined by
| |
| | |
| ::<math>\mathcal{M}\{A\vert B\}=\begin{cases} \displaystyle\frac{\mathcal{M}\{A\cap B\} }{\mathcal{M}\{B\} }, &\displaystyle\text{if }\frac{\mathcal{M}\{A\cap B\} }{\mathcal{M}\{B\} }<0.5 \\ \displaystyle 1 - \frac{\mathcal{M}\{A^c\cap B\} }{\mathcal{M}\{B\} }, &\displaystyle\text{if } \frac{\mathcal{M}\{A^c\cap B\} }{\mathcal{M}\{B\} }<0.5 \\ 0.5, & \text{otherwise} \end{cases}</math>
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| ::<math>\text{provided that } \mathcal{M}\{B\}>0</math>
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| | |
| '''Theorem 1''': Let <math>(\Gamma,L,M)</math> be an uncertainty space, and B an event with <math>M\{B\} > 0</math>. Then M{·|B} defined by Definition 1 is an uncertain measure, and <math>(\Gamma,L,M\{\mbox{·}|B\})</math>is an uncertainty space.
| |
| | |
| '''Definition 2''': Let <math>\xi</math> be an uncertain variable on <math>(\Gamma,L,M)</math>. A conditional uncertain variable of <math>\xi</math> given B is a measurable function <math>\xi|_B</math> from the conditional uncertainty space <math>(\Gamma,L,M\{\mbox{·}|_B\})</math> to the set of real numbers such that
| |
| ::<math>\xi|_B(\gamma)=\xi(\gamma),\forall \gamma \in \Gamma</math>.
| |
| | |
| '''Definition 3''': The conditional uncertainty distribution <math>\Phi\rightarrow[0, 1]</math> of an uncertain variable <math>\xi</math> given B is defined by
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| ::<math>\Phi(x|B)=M\{\xi\leq x|B\}</math>
| |
| provided that <math>M\{B\}>0</math>.
| |
| | |
| '''Theorem 2''': Let <math>\xi</math> be an uncertain variable with regular uncertainty distribution <math>\Phi(x)</math>, and <math>t</math> a real number with <math>\Phi(t) < 1</math>. Then the conditional uncertainty distribution of <math>\xi</math> given <math>\xi> t</math> is
| |
| ::<math>\Phi(x\vert(t,+\infty))=\begin{cases} 0, & \text{if }\Phi(x)\le\Phi(t)\\ \displaystyle\frac{\Phi(x)}{1-\Phi(t)}\and 0.5, & \text{if }\Phi(t)<\Phi(x)\le(1+\Phi(t))/2 \\ \displaystyle\frac{\Phi(x)-\Phi(t)}{1-\Phi(t)}, & \text{if }(1+\Phi(t))/2\le\Phi(x) \end{cases}</math>
| |
| | |
| '''Theorem 3''': Let <math>\xi</math> be an uncertain variable with regular uncertainty distribution <math>\Phi(x)</math>, and <math>t</math> a real number with <math>\Phi(t)>0</math>. Then the conditional uncertainty distribution of <math>\xi</math> given <math>\xi\leq t</math> is
| |
| ::<math>\Phi(x\vert(-\infty,t])=\begin{cases} \displaystyle\frac{\Phi(x)}{\Phi(t)}, & \text{if }\Phi(x)\le\Phi(t)/2 \\ \displaystyle\frac{\Phi(x)+\Phi(t)-1}{\Phi(t)}\or 0.5, & \text{if }\Phi(t)/2\le\Phi(x)<\Phi(t) \\ 1, & \text{if }\Phi(t)\le\Phi(x) \end{cases}</math>
| |
| | |
| '''Definition 4''': Let <math>\xi</math> be an uncertain variable. Then the conditional expected value of <math>\xi</math> given B is defined by
| |
| ::<math>E[\xi|B]=\int_0^{+\infty}M\{\xi\geq r|B\}dr-\int_{-\infty}^0M\{\xi\leq r|B\}dr</math>
| |
| provided that at least one of the two integrals is finite.
| |
| | |
| ==References==
| |
| {{reflist}}
| |
| | |
| * Xin Gao, Some Properties of Continuous Uncertain Measure, ''[[International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems]]'', Vol.17, No.3, 419-426, 2009.
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| * Cuilian You, Some Convergence Theorems of Uncertain Sequences, ''Mathematical and Computer Modelling'', Vol.49, Nos.3-4, 482-487, 2009.
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| * 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.
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| * 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.
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| * 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.
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| * 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.
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| * 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.
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| * 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.
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| * 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.
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| * 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.
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| * 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.
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| * 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.
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| {{DEFAULTSORT:Uncertainty Theory}}
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| [[Category:Probability theory]]
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| [[Category:Fuzzy logic]]
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