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:
- (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 .
Other properties
A capacity is called superadditive if
- ,
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
- .
It is the complement to . If is superadditive, then is subadditive and vice versa. Be and . A capacitance is called k-monotonic if
- ,
it is called completely monotonic if it is k-monotonic for each . A capacitance is called k-alternating if
- ,
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).
Examples
Hit and miss probabilities for random sets
Let be a random (compact) set and a fixed compact set. Be
- .
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 .
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.
-Fuzzy dimensions from Sugeno
They were introduced by Sugeno in 1974. A capacity is called a -uzzy measure if the following applies to with :
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.
- decomposable dimensions
They were introduced by Siegfried Weber in 1984. Be a -conorm . A capacity is called decomposable if:
For example, a Possibilität is -dekomposabel respect. And -Fuzzy measure is dekomposabel respect.
- .
Bibliography
- Grabisch, M .: Set Functions, Games and Capacities in Decision Theory , Springer 2016
Individual evidence
- ↑ Choquet, G. (1953). Theory of capacities . Ann.Inst. Fourier, Grenoble, pp. 131-295, doi: 10.5802 / aif.53 .
- ↑ ^{a } ^{b} Sugeno, M. (1974). Theory of Fuzzy Integrals and its Application . PhD thesis, Tokyo Institute of Technology
- ^ Matheron, G. (1975) Random Sets and Integral Geometry . J. Wiley & Sons, New York.
- ^ Molchanov, I. (2005) The Theory of Random Sets . Springer, New York
- ^ Shafer, G. (1976): A Mathematical Theory of Evidence , Princeton, University Press
- ↑ Dubois, D. and Prade, H. (1988): Possibility Theory: An Approach to Computerized Processing of Uncertainty , New York: Plenum Press
- ↑ 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 .
- ↑ Weber, S. (1984), Decomposable Measures and Integrals for Archimedean -conorms , Journal of Mathematical Analysis and Application 101, 114-138 (full text)