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.
Four axioms
Axiom 1. (Normality Axiom)
.
Axiom 2. (Self-Duality Axiom)
.
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.
Uncertain variables
An uncertain variable is a measurable function ξ from an uncertainty space
to the set of real numbers, i.e., for any Borel set B of real numbers, the set
is an event.
Independence
Definition: The uncertain variables
are said to be independent if
![{\displaystyle M\{\cap _{i=1}^{m}(\xi \in B_{i})\}={\mbox{min))_{1\leq i\leq m}M\{\xi _{i}\in B_{i}\))](https://wikimedia.org/api/rest_v1/media/math/render/svg/1cab5d067c6bcec99ccccb85e5ea24a5c40e7012)
for any Borel sets
of real numbers.
Theorem 1: The uncertain variables
are independent if
![{\displaystyle M\{\cup _{i=1}^{m}(\xi \in B_{i})\}={\mbox{max))_{1\leq i\leq m}M\{\xi _{i}\in B_{i}\))](https://wikimedia.org/api/rest_v1/media/math/render/svg/59c7d88c20ae02fdf82488ea68c4101f6de60b35)
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
![{\displaystyle \Phi (x_{1},x_{2},\ldots ,x_{m})={\mbox{min))_{1\leq i\leq m}\Phi _{i}(x_{i})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/45941f928a97f315459af9d727b4a974df28d864)
for any real numbers
.
Operational law
Theorem: Let
be independent uncertain variables, and
a measurable function. Then
is an uncertain variable such that
![{\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}\},&{\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}\}>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}\},&{\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}\}>0.5\\0.5,&{\text{otherwise))\end{cases))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/41ce444908bf6be908041432cbea1cee31a93985)
where
are Borel sets, and
means
for any
.
Critical value
Definition: Let
be an uncertain variable, and
. Then
![{\displaystyle \xi _{sup}(\alpha )=\sup\{r\mid M\{\xi \geq r\}\geq \alpha \))](https://wikimedia.org/api/rest_v1/media/math/render/svg/3df520ad6931bd5b613e1bd37ddbeb2859ab4343)
is called the α-optimistic value to
, and
![{\displaystyle \xi _{inf}(\alpha )=\inf\{r\mid M\{\xi \leq r\}\geq \alpha \))](https://wikimedia.org/api/rest_v1/media/math/render/svg/bd634492042b9b5e3ff45fa69c0628ddd5058a64)
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
,
,
,
,
,
.
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
![{\displaystyle \lim _{i\to \infty }|\xi _{i}(\gamma )-\xi (\gamma )|=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b39c1eb4b19a261b362e0e177cceac8ac42c8dfd)
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
![{\displaystyle \lim _{i\to \infty }M\{|\xi _{i}-\xi |\leq \varepsilon \}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9374781cb31d9687358b393ec6abb0ca149f91c0)
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 uncertainty
Definition 1: Let
be an uncertainty space, and
. Then the conditional uncertain measure of A given B is defined by
![{\displaystyle {\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))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/012b697706950db6e09f9b5d3c251cb00de5880d)
![{\displaystyle {\text{provided that )){\mathcal {M))\{B\}>0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9043cf3a8fd10bf42e49e8cd416b9961dd50b8b5)
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
![{\displaystyle \Phi (x|B)=M\{\xi \leq x|B\))](https://wikimedia.org/api/rest_v1/media/math/render/svg/5c2ae7fadb3c788f7050fbc9ae853f7677665a4e)
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
![{\displaystyle \Phi (x\vert (t,+\infty ))={\begin{cases}0,&{\text{if ))\Phi (x)\leq \Phi (t)\\\displaystyle {\frac {\Phi (x)}{1-\Phi (t)))\land 0.5,&{\text{if ))\Phi (t)<\Phi (x)\leq (1+\Phi (t))/2\\\displaystyle {\frac {\Phi (x)-\Phi (t)}{1-\Phi (t))),&{\text{if ))(1+\Phi (t))/2\leq \Phi (x)\end{cases))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de659b167ace67296c3559a2d5f35cd65dbaf7c6)
Theorem 3: Let
be an uncertain variable with regular uncertainty distribution
, and
a real number with
. Then the conditional uncertainty distribution of
given
is
![{\displaystyle \Phi (x\vert (-\infty ,t])={\begin{cases}\displaystyle {\frac {\Phi (x)}{\Phi (t))),&{\text{if ))\Phi (x)\leq \Phi (t)/2\\\displaystyle {\frac {\Phi (x)+\Phi (t)-1}{\Phi (t)))\lor 0.5,&{\text{if ))\Phi (t)/2\leq \Phi (x)<\Phi (t)\\1,&{\text{if ))\Phi (t)\leq \Phi (x)\end{cases))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ab25ac495b7fabe9920768cfa8be2c98a3d62438)
Definition 4: Let
be an uncertain variable. Then the conditional expected value of
given B is defined by
![{\displaystyle E[\xi |B]=\int _{0}^{+\infty }M\{\xi \geq r|B\}dr-\int _{-\infty }^{0}M\{\xi \leq r|B\}dr}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b465ad50dcba935be93702afe642b664f7cc5bf)
provided that at least one of the two integrals is finite.
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