# 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: ${\ displaystyle U}$ ${\ displaystyle {\ mathcal {P}} (U)}$ ${\ displaystyle \ mu | {\ mathcal {P}} (U) \ to [0, \ infty)}$ ${\ displaystyle \ mu}$ ${\ displaystyle \ mu (\ emptyset) = 0}$ ${\ displaystyle \ mu (A) \ leq \ mu (B), \ quad {\ text {if}} \ quad A \ subseteq B}$ (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)${\ displaystyle \ bot}$ ${\ displaystyle t}$ ${\ displaystyle \ bot}$ 