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Uncertainty theory 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.
Axiom 1. (Normality Axiom) [math]\displaystyle{ \mathcal{M}\{\Gamma\}=1\text{ for the universal set }\Gamma }[/math].
Axiom 2. (Self-Duality Axiom) [math]\displaystyle{ \mathcal{M}\{\Lambda\}+\mathcal{M}\{\Lambda^c\}=1\text{ for any event }\Lambda }[/math].
Axiom 3. (Countable Subadditivity Axiom) For every countable sequence of events [math]\displaystyle{ \Lambda_1,\Lambda_2,\ldots }[/math], we have
Axiom 4. (Product Measure Axiom) Let [math]\displaystyle{ (\Gamma_k,\mathcal{L}_k,\mathcal{M}_k) }[/math] be uncertainty spaces for [math]\displaystyle{ k=1,2,\ldots,n }[/math]. Then the product uncertain measure [math]\displaystyle{ \mathcal{M} }[/math] 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.
An uncertain variable is a measurable function ξ from an uncertainty space [math]\displaystyle{ (\Gamma,L,M) }[/math] to the set of real numbers, i.e., for any Borel set B of real numbers, the set [math]\displaystyle{ \{\xi\in B\}=\{\gamma \in \Gamma\mid \xi(\gamma)\in B\} }[/math] is an event.
Uncertainty distribution is inducted to describe uncertain variables.
Definition: The uncertainty distribution [math]\displaystyle{ \Phi(x):R \rightarrow [0,1] }[/math] of an uncertain variable ξ is defined by [math]\displaystyle{ \Phi(x)=M\{\xi\leq x\} }[/math].
Theorem (Peng and Iwamura, Sufficient and Necessary Condition for Uncertainty Distribution): A function [math]\displaystyle{ \Phi(x):R \rightarrow [0,1] }[/math] is an uncertain distribution if and only if it is an increasing function except [math]\displaystyle{ \Phi (x) \equiv 0 }[/math] and [math]\displaystyle{ \Phi (x)\equiv 1 }[/math].
Definition: The uncertain variables [math]\displaystyle{ \xi_1,\xi_2,\ldots,\xi_m }[/math] are said to be independent if
for any Borel sets [math]\displaystyle{ B_1,B_2,\ldots,B_m }[/math] of real numbers.
Theorem 1: The uncertain variables [math]\displaystyle{ \xi_1,\xi_2,\ldots,\xi_m }[/math] are independent if
for any Borel sets [math]\displaystyle{ B_1,B_2,\ldots,B_m }[/math] of real numbers.
Theorem 2: Let [math]\displaystyle{ \xi_1,\xi_2,\ldots,\xi_m }[/math] be independent uncertain variables, and [math]\displaystyle{ f_1,f_2,\ldots,f_m }[/math] measurable functions. Then [math]\displaystyle{ f_1(\xi_1),f_2(\xi_2),\ldots,f_m(\xi_m) }[/math] are independent uncertain variables.
Theorem 3: Let [math]\displaystyle{ \Phi_i }[/math] be uncertainty distributions of independent uncertain variables [math]\displaystyle{ \xi_i,\quad i=1,2,\ldots,m }[/math] respectively, and [math]\displaystyle{ \Phi }[/math] the joint uncertainty distribution of uncertain vector [math]\displaystyle{ (\xi_1,\xi_2,\ldots,\xi_m) }[/math]. If [math]\displaystyle{ \xi_1,\xi_2,\ldots,\xi_m }[/math] are independent, then we have
for any real numbers [math]\displaystyle{ x_1, x_2, \ldots, x_m }[/math].
Theorem: Let [math]\displaystyle{ \xi_1,\xi_2,\ldots,\xi_m }[/math] be independent uncertain variables, and [math]\displaystyle{ f: R^n \rightarrow R }[/math] a measurable function. Then [math]\displaystyle{ \xi=f(\xi_1,\xi_2,\ldots,\xi_m) }[/math] is an uncertain variable such that
where [math]\displaystyle{ B, B_1, B_2, \ldots, B_m }[/math] are Borel sets, and [math]\displaystyle{ f( B_1, B_2, \ldots, B_m)\subset B }[/math] means [math]\displaystyle{ f(x_1, x_2, \ldots, x_m) \in B }[/math] for any[math]\displaystyle{ x_1 \in B_1, x_2 \in B_2, \ldots,x_m \in B_m }[/math].
Definition: Let [math]\displaystyle{ \xi }[/math] be an uncertain variable. Then the expected value of [math]\displaystyle{ \xi }[/math] is defined by
provided that at least one of the two integrals is finite.
Theorem 1: Let [math]\displaystyle{ \xi }[/math] be an uncertain variable with uncertainty distribution [math]\displaystyle{ \Phi }[/math]. If the expected value exists, then
Theorem 2: Let [math]\displaystyle{ \xi }[/math] be an uncertain variable with regular uncertainty distribution [math]\displaystyle{ \Phi }[/math]. If the expected value exists, then
Theorem 3: Let [math]\displaystyle{ \xi }[/math] and [math]\displaystyle{ \eta }[/math] be independent uncertain variables with finite expected values. Then for any real numbers [math]\displaystyle{ a }[/math] and [math]\displaystyle{ b }[/math], we have
Definition: Let [math]\displaystyle{ \xi }[/math] be an uncertain variable with finite expected value [math]\displaystyle{ e }[/math]. Then the variance of [math]\displaystyle{ \xi }[/math] is defined by
Theorem: If [math]\displaystyle{ \xi }[/math] be an uncertain variable with finite expected value, [math]\displaystyle{ a }[/math] and [math]\displaystyle{ b }[/math] are real numbers, then
Definition: Let [math]\displaystyle{ \xi }[/math] be an uncertain variable, and [math]\displaystyle{ \alpha\in(0,1] }[/math]. Then
is called the α-optimistic value to [math]\displaystyle{ \xi }[/math], and
is called the α-pessimistic value to [math]\displaystyle{ \xi }[/math].
Theorem 1: Let [math]\displaystyle{ \xi }[/math] be an uncertain variable with regular uncertainty distribution [math]\displaystyle{ \Phi }[/math]. Then its α-optimistic value and α-pessimistic value are
Theorem 2: Let [math]\displaystyle{ \xi }[/math] be an uncertain variable, and [math]\displaystyle{ \alpha\in(0,1] }[/math]. Then we have
Theorem 3: Suppose that [math]\displaystyle{ \xi }[/math] and [math]\displaystyle{ \eta }[/math] are independent uncertain variables, and [math]\displaystyle{ \alpha\in(0,1] }[/math]. Then we have
[math]\displaystyle{ (\xi + \eta)_{sup}(\alpha)=\xi_{sup}(\alpha)+\eta_{sup}{\alpha} }[/math],
[math]\displaystyle{ (\xi + \eta)_{inf}(\alpha)=\xi_{inf}(\alpha)+\eta_{inf}{\alpha} }[/math],
[math]\displaystyle{ (\xi \vee \eta)_{sup}(\alpha)=\xi_{sup}(\alpha)\vee\eta_{sup}{\alpha} }[/math],
[math]\displaystyle{ (\xi \vee \eta)_{inf}(\alpha)=\xi_{inf}(\alpha)\vee\eta_{inf}{\alpha} }[/math],
[math]\displaystyle{ (\xi \wedge \eta)_{sup}(\alpha)=\xi_{sup}(\alpha)\wedge\eta_{sup}{\alpha} }[/math],
[math]\displaystyle{ (\xi \wedge \eta)_{inf}(\alpha)=\xi_{inf}(\alpha)\wedge\eta_{inf}{\alpha} }[/math].
Definition: Let [math]\displaystyle{ \xi }[/math] be an uncertain variable with uncertainty distribution [math]\displaystyle{ \Phi }[/math]. Then its entropy is defined by
where [math]\displaystyle{ S(x) = -t \ln(t) - (1-t) \ln(1-t) }[/math].
Theorem 1(Dai and Chen): Let [math]\displaystyle{ \xi }[/math] be an uncertain variable with regular uncertainty distribution [math]\displaystyle{ \Phi }[/math]. Then
Theorem 2: Let [math]\displaystyle{ \xi }[/math] and [math]\displaystyle{ \eta }[/math] be independent uncertain variables. Then for any real numbers [math]\displaystyle{ a }[/math] and [math]\displaystyle{ b }[/math], we have
Theorem 3: Let [math]\displaystyle{ \xi }[/math] be an uncertain variable whose uncertainty distribution is arbitrary but the expected value [math]\displaystyle{ e }[/math] and variance [math]\displaystyle{ \sigma^2 }[/math]. Then
Theorem 1(Liu, Markov Inequality): Let [math]\displaystyle{ \xi }[/math] be an uncertain variable. Then for any given numbers [math]\displaystyle{ t \gt 0 }[/math] and [math]\displaystyle{ p \gt 0 }[/math], we have
Theorem 2 (Liu, Chebyshev Inequality) Let [math]\displaystyle{ \xi }[/math] be an uncertain variable whose variance [math]\displaystyle{ V[\xi] }[/math] exists. Then for any given number [math]\displaystyle{ t \gt 0 }[/math], we have
Theorem 3 (Liu, Holder's Inequality) Let [math]\displaystyle{ p }[/math] and [math]\displaystyle{ q }[/math] be positive numbers with [math]\displaystyle{ 1/p + 1/q = 1 }[/math], and let [math]\displaystyle{ \xi }[/math] and [math]\displaystyle{ \eta }[/math] be independent uncertain variables with [math]\displaystyle{ E[|\xi|^p]\lt \infty }[/math] and [math]\displaystyle{ E[|\eta|^q] \lt \infty }[/math]. Then we have
Theorem 4:(Liu [127], Minkowski Inequality) Let [math]\displaystyle{ p }[/math] be a real number with [math]\displaystyle{ p\leq 1 }[/math], and let [math]\displaystyle{ \xi }[/math] and [math]\displaystyle{ \eta }[/math] be independent uncertain variables with [math]\displaystyle{ E[|\xi|^p] \lt \infty }[/math] and [math]\displaystyle{ E[|\eta|^q] \lt \infty }[/math]. Then we have
Definition 1: Suppose that [math]\displaystyle{ \xi,\xi_1,\xi_2,\ldots }[/math] are uncertain variables defined on the uncertainty space [math]\displaystyle{ (\Gamma,L,M) }[/math]. The sequence [math]\displaystyle{ \{\xi_i\} }[/math] is said to be convergent a.s. to [math]\displaystyle{ \xi }[/math] if there exists an event [math]\displaystyle{ \Lambda }[/math] with [math]\displaystyle{ M\{\Lambda\} = 1 }[/math] such that
for every [math]\displaystyle{ \gamma\in\Lambda }[/math]. In that case we write [math]\displaystyle{ \xi_i\to \xi }[/math],a.s.
Definition 2: Suppose that [math]\displaystyle{ \xi,\xi_1,\xi_2,\ldots }[/math] are uncertain variables. We say that the sequence [math]\displaystyle{ \{\xi_i\} }[/math] converges in measure to [math]\displaystyle{ \xi }[/math] if
for every [math]\displaystyle{ \varepsilon\gt 0 }[/math].
Definition 3: Suppose that [math]\displaystyle{ \xi,\xi_1,\xi_2,\ldots }[/math] are uncertain variables with finite expected values. We say that the sequence [math]\displaystyle{ \{\xi_i\} }[/math] converges in mean to [math]\displaystyle{ \xi }[/math] if
Definition 4: Suppose that [math]\displaystyle{ \Phi,\phi_1,\Phi_2,\ldots }[/math] are uncertainty distributions of uncertain variables [math]\displaystyle{ \xi,\xi_1,\xi_2,\ldots }[/math], respectively. We say that the sequence [math]\displaystyle{ \{\xi_i\} }[/math] converges in distribution to [math]\displaystyle{ \xi }[/math] if [math]\displaystyle{ \Phi_i\rightarrow\Phi }[/math] at any continuity point of [math]\displaystyle{ \Phi }[/math].
Theorem 1: Convergence in Mean [math]\displaystyle{ \Rightarrow }[/math] Convergence in Measure [math]\displaystyle{ \Rightarrow }[/math] Convergence in Distribution. However, Convergence in Mean [math]\displaystyle{ \nLeftrightarrow }[/math] Convergence Almost Surely [math]\displaystyle{ \nLeftrightarrow }[/math] Convergence in Distribution.
Definition 1: Let [math]\displaystyle{ (\Gamma,L,M) }[/math] be an uncertainty space, and [math]\displaystyle{ A,B\in L }[/math]. Then the conditional uncertain measure of A given B is defined by
Theorem 1: Let [math]\displaystyle{ (\Gamma,L,M) }[/math] be an uncertainty space, and B an event with [math]\displaystyle{ M\{B\} \gt 0 }[/math]. Then M{·|B} defined by Definition 1 is an uncertain measure, and [math]\displaystyle{ (\Gamma,L,M\{\mbox{·}|B\}) }[/math]is an uncertainty space.
Definition 2: Let [math]\displaystyle{ \xi }[/math] be an uncertain variable on [math]\displaystyle{ (\Gamma,L,M) }[/math]. A conditional uncertain variable of [math]\displaystyle{ \xi }[/math] given B is a measurable function [math]\displaystyle{ \xi|_B }[/math] from the conditional uncertainty space [math]\displaystyle{ (\Gamma,L,M\{\mbox{·}|_B\}) }[/math] to the set of real numbers such that
Definition 3: The conditional uncertainty distribution [math]\displaystyle{ \Phi\rightarrow[0, 1] }[/math] of an uncertain variable [math]\displaystyle{ \xi }[/math] given B is defined by
provided that [math]\displaystyle{ M\{B\}\gt 0 }[/math].
Theorem 2: Let [math]\displaystyle{ \xi }[/math] be an uncertain variable with regular uncertainty distribution [math]\displaystyle{ \Phi(x) }[/math], and [math]\displaystyle{ t }[/math] a real number with [math]\displaystyle{ \Phi(t) \lt 1 }[/math]. Then the conditional uncertainty distribution of [math]\displaystyle{ \xi }[/math] given [math]\displaystyle{ \xi\gt t }[/math] is
Theorem 3: Let [math]\displaystyle{ \xi }[/math] be an uncertain variable with regular uncertainty distribution [math]\displaystyle{ \Phi(x) }[/math], and [math]\displaystyle{ t }[/math] a real number with [math]\displaystyle{ \Phi(t)\gt 0 }[/math]. Then the conditional uncertainty distribution of [math]\displaystyle{ \xi }[/math] given [math]\displaystyle{ \xi\leq t }[/math] is
Definition 4: Let [math]\displaystyle{ \xi }[/math] be an uncertain variable. Then the conditional expected value of [math]\displaystyle{ \xi }[/math] given B is defined by
provided that at least one of the two integrals is finite.
Original source: https://en.wikipedia.org/wiki/Uncertainty theory.
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