Capacity (math)

A capacity (Engl. Capacity ) is a monotone set function and starting point of many mathematical investigations z. B. in measure theory , probability theory , evidence theory . The term capacity goes back to the French mathematician Gustave Choquet , which is why one often speaks of Choquet capacities . Based on a work by Sugeno, capacities used to be called fuzzy measures , although they have nothing to do with blurring.

definition

Let be the basic set , its power set and a set function. The quantity function is called capacity if: ${\displaystyle U}$ ${\displaystyle {\mathcal {P}}(U)}$ ${\displaystyle \mu |{\mathcal {P}}(U)\to [0,\infty )}$ ${\displaystyle \mu }$

${\displaystyle \mu (\emptyset )=0}$

${\displaystyle \mu (A)\leq \mu (B),\quad {\text{wenn}}\quad A\subseteq B}$ (Monotony)

A measure is a special capacity, because the additivity of (i.e. ) gives rise to monotony. If true, then the capacity is called normalized . ${\displaystyle \mu }$ ${\displaystyle \mu (A\cup B)=\mu (A)+\mu (B),\quad A\cap B=\emptyset }$ ${\displaystyle \mu (U)=1}$

Other properties

A capacity is called superadditive if ${\displaystyle \mu }$

${\displaystyle \mu (A\cup B)\geq \mu (A)+\mu (B),\quad A\cap B=\emptyset ,\quad A,B\in {\mathcal {P}}(U)}$,

it is called subadditive if the inequality sign is reversed. Synergy effects can be modeled through superadditive capacities and redundancy effects through subadditive capacities . The too dual (also conjugated ) capacity is defined by ${\displaystyle \mu }$ ${\displaystyle {\overline {\mu }}}$

${\displaystyle {\overline {\mu }}(A)=1-\mu ({\overline {A}})}$.

It is the complement to . If is superadditive, then is subadditive and vice versa. Be and . A capacitance is called k-monotonic if ${\displaystyle {\overline {A}}}$ ${\displaystyle A\in {\mathcal {P}}(U)}$ ${\displaystyle \mu }$ ${\displaystyle {\overline {\mu }}}$ ${\displaystyle A_{1},\dotsc ,A_{k}\in {\mathcal {P}}(U)}$ ${\displaystyle k\geq 2}$ ${\displaystyle \mu }$

${\displaystyle \mu (\bigcup _{i=1}^{k}A_{i})\geq \sum _{I\subseteq \{1,\dotsc ,k\},I\neq \emptyset }(-1)^{|I|+1}\mu (\bigcap _{i\in I}A_{i})}$,

it is called completely monotonic if it is k-monotonic for each . A capacitance is called k-alternating if ${\displaystyle k\geq 2}$ ${\displaystyle \mu }$

${\displaystyle \mu (\bigcap _{i=1}^{k}A_{i})\leq \sum _{I\subseteq \{1,\dotsc ,k\},I\neq \emptyset }(-1)^{|I|+1}\mu (\bigcup _{i\in I}A_{i})}$,

it is called fully alternating if it is k-alternating for each . A k-monotonic capacity is superadditive, a k-alternating capacity is subadditive. A capacity is k-monotonic (k-alternating) if and only if the dual capacity is k-alternating (k-monotonic). ${\displaystyle k\geq 2}$ ${\displaystyle \mu }$ ${\displaystyle {\overline {\mu }}}$

Examples

Hit and miss probabilities for random sets

Let be a random (compact) set and a fixed compact set. Be ${\displaystyle X}$ ${\displaystyle A\in {\mathcal {P}}(U)}$

${\displaystyle P^{*}(A)=P(X\cap A\neq \emptyset ),\quad P_{*}(A)=P(X\subset A)=P(X\cap {\overline {A}}=\emptyset )}$.

${\displaystyle P^{*}}$is the probability that the set “hits” and is therefore called the hit probability . is the probability that the set “does not hit” and is therefore called a miss probability . It is . is a normalized completely alternating capacitance, is a normalized completely monotonic capacitance. Hit & Miss probabilities produce the distribution of the random set in a unique way . ${\displaystyle X}$ ${\displaystyle A}$ ${\displaystyle P_{*}}$ ${\displaystyle X}$ ${\displaystyle {\overline {A}}}$ ${\displaystyle P^{*}(A)>P_{*}(A)}$ ${\displaystyle P^{*}}$ ${\displaystyle P_{*}}$ ${\displaystyle X}$

Belief and plausibility

Belief and plausibility are basic concepts in Glenn Shafer's evidence theory . A delivery function is a standardized, completely monotonous capacity and a plausibility is a standardized, completely alternating capacity. The dual capacity to the delivery function is a plausibility and vice versa. The for the possibility theory basic Possibility is a special plausibility that its dual necessity a special Belieffunktion.

Lower and upper probabilities

Dempster's lower and upper probabilities are constructed similarly to the above hit & miss probabilities. Lower probabilities are therefore normalized, completely monotonic and upper probabilities are normalized, completely alternating capacities. Belief functions are special lower probabilities and plausibilities are special upper probabilities.

${\displaystyle \lambda }$-Fuzzy dimensions from Sugeno

They were introduced by Sugeno in 1974. A capacity is called a -uzzy measure if the following applies to with : ${\displaystyle \mu _{\lambda }}$ ${\displaystyle \lambda }$ ${\displaystyle A,B\in {\mathcal {P}}(U)}$ ${\displaystyle A\cap B=\emptyset }$

${\displaystyle \mu _{\lambda }(A\cup B)=\min\{\mu _{\lambda }(A)+\mu _{\lambda }(B)+\lambda \mu _{\lambda }(A)\mu _{\lambda }(B),1\},\quad \lambda >-1}$

For is a probability measure, for a belief function and for a plausibility. In a sense, the parameter measures the deviation from the probability measure. ${\displaystyle \lambda =0}$ ${\displaystyle \mu _{\lambda }}$ ${\displaystyle \lambda >0}$ ${\displaystyle -1<\lambda <0}$ ${\displaystyle \lambda }$

${\displaystyle \bot }$- decomposable dimensions

They were introduced by Siegfried Weber in 1984. Be a -conorm . A capacity is called decomposable if: ${\displaystyle \bot }$ ${\displaystyle t}$ ${\displaystyle \mu }$ ${\displaystyle \bot }$

${\displaystyle \mu (A\cup B)=\bot (\mu (A),\mu (B)),\quad A\cap B=\emptyset }$

For example, a Possibilität is -dekomposabel respect. And -Fuzzy measure is dekomposabel respect. ${\displaystyle \bot }$ ${\displaystyle \bot =\max }$ ${\displaystyle \lambda }$

${\displaystyle \bot (u,v)=\min(u+v+\lambda uv,1);\quad \lambda >-1;\quad u,v\in [0,1]}$.

Bibliography

• Grabisch, M .: Set Functions, Games and Capacities in Decision Theory , Springer 2016

Individual evidence

1. ↑ Choquet, G. (1953). Theory of capacities . Ann.Inst. Fourier, Grenoble, pp. 131-295, doi: 10.5802 / aif.53 .
2. ↑ ^{a b} Sugeno, M. (1974). Theory of Fuzzy Integrals and its Application . PhD thesis, Tokyo Institute of Technology
3. ^ Matheron, G. (1975) Random Sets and Integral Geometry . J. Wiley & Sons, New York.
4. ^ Molchanov, I. (2005) The Theory of Random Sets . Springer, New York
5. ^ Shafer, G. (1976): A Mathematical Theory of Evidence , Princeton, University Press
6. ↑ Dubois, D. and Prade, H. (1988): Possibility Theory: An Approach to Computerized Processing of Uncertainty , New York: Plenum Press
7. ↑ Dempster, AP (1967): Upper and lower probabilities induced by a multivalued mapping , Annals of Mathematical Statistics 38, pp. 325–339, doi: 10.1214 / aoms / 1177698950 .
8. ↑ Weber, S. (1984), Decomposable Measures and Integrals for Archimedean -conorms , Journal of Mathematical Analysis and Application 101, 114-138 (full text) ${\displaystyle \bot }$ ${\displaystyle t}$ ${\displaystyle \bot }$