Uncertainty theory is a branch of mathematics based on normality, monotonicity, self-duality, countable subadditivity, and product measure axioms. Mathematical measures of the likelihood of an event being true include probability theory, capacity, fuzzy logic, possibility, and credibility, as well as uncertainty.
Four axioms
Axiom 1. . Axiom 2. . Axiom 3. For every countable sequence of events Λ1, Λ2,..., we have Axiom 4. Let be uncertainty spaces for. Then the product uncertain measure is an uncertain measure on the product σ-algebra satisfying 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 A function is an uncertain distribution if and only if it is an increasing function except and.
Independence
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 law
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 Value
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
Variance
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 value
Definition: Let be an uncertain variable, and. Then is called the α-optimistic value to, and is called the α-pessimistic value to. 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
Entropy
Definition: Let be an uncertain variable with uncertainty distribution. Then its entropy is defined by where. Theorem 1: 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
Inequalities
Theorem 1: Let be an uncertain variable. Then for any given numbers and, we have Theorem 2 Let be an uncertain variable whose variance exists. Then for any given number, we have Theorem 3 Let and be positive numbers with, and let and be independent uncertain variables with and. Then we have Theorem 4: Let be a real number with, and let and be independent uncertain variables with and. Then we have
Convergence concept
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 MeanConvergence in MeasureConvergence in Distribution. However, Convergence in Mean ConvergenceAlmost Surely Convergence in Distribution.
Conditional uncertainty
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 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.
