Choquet integral

The French mathematician Gustave Choquet is the creator of the Choquet integral named after him . In contrast to the Lebesgue integral , which defines the integration on arbitrary measurement spaces , no measures are used for the Choquet integral, only capacities . Choquet integrals are needed e.g. B. decision theory , the cooperative game theory , the utility theory , the data processing (for construction of aggregation functions ).

idea

Let be the basic set , a nonnegative real-valued function and a measure . The Lebesgue integral is well defined. Then the Lebesgue integral has the following representation as a Riemann integral : ${\displaystyle U}$ ${\displaystyle f|U\to [0,\infty )}$ ${\displaystyle \lambda }$ ${\displaystyle \textstyle \int fd\lambda }$

${\displaystyle \int fd\lambda =\int \limits _{0}^{\infty }\lambda (\{y|f(y)\geq x\})dx}$.

If you replace the measure by a capacitance in this representation , you already have the definition of the Choquet integral for nonnegative functions. ${\displaystyle \lambda }$ ${\displaystyle \mu }$

definition

Now let be a real-valued function, a set of subsets of and a capacity. The function is measurable, i.e. H. ${\displaystyle f|U\to \mathbb {R} }$ ${\displaystyle {\mathcal {F}}(U)}$ ${\displaystyle U}$ ${\displaystyle \mu }$ ${\displaystyle f}$

${\displaystyle \forall x\in \mathbb {R} :\quad \{y|f(y)\geq x\}\in {\mathcal {F}}(U)}$.

Then this is Choquet integral of regard. ${\displaystyle f}$ ${\displaystyle \mu }$ Follows by Riemann integrals defined:

${\displaystyle \int fd\mu :=\int \limits _{0}^{\infty }\mu (\{y|f(y)\geq x\})dx+\int \limits _{-\infty }^{0}[\mu (\{y|f(y)\geq x\})-\mu (U)]dx}$.

For positives this reduces to ${\displaystyle f}$

${\displaystyle \int fd\mu =\int \limits _{0}^{\infty }\mu (\{y|f(y)\geq x\})dx}$.

properties

See e.g. B. For applies (monotony). For is (positive homogeneity). ${\displaystyle f\leq g}$ ${\displaystyle \int fd\mu \leq \int gd\mu \quad }$ ${\displaystyle \alpha \geq 0}$ ${\displaystyle \int \alpha fd\mu =\alpha \int fd\mu \quad }$

I. general the Choquet integral is not additive , i.e. H.

${\displaystyle \int (f+g)d\mu \neq \int fd\mu +\int gd\mu }$

If 2 is monotonic then the Choquet integral is superadditive; H. ${\displaystyle \mu }$

${\displaystyle \int (f+g)d\mu \geq \int fd\mu +\int gd\mu }$.

If 2 is alternating , then the Choquet integral is subadditive; H. in the last inequality applies . ${\displaystyle \mu }$ ${\displaystyle \leq }$

Discrete Choquet integral

See e.g. B. Let and be a nonnegative function with the values . Denote the function values ​​ordered according to size, i. H. ${\displaystyle U=\{1,\dots ,n\}}$ ${\displaystyle f|U\to [0,\infty )}$ ${\displaystyle f(i)=y_{i},i=1,\dots ,n}$ ${\displaystyle y_{(1)},\dots ,y_{(n)}}$

${\displaystyle 0\leq y_{(1)}\leq y_{(2)}\leq \dots \leq y_{(n)}}$.

Since in the discrete case the defining Riemann integral degenerates into a sum, it results

${\displaystyle \int fd\mu =\sum _{i=1}^{n}(y_{(i)}-y_{(i-1)})\mu (\{j|f(j)\geq y_{(i)}\})=\sum _{i=1}^{n}y_{(i)}[\mu (A_{(i)})-\mu (A_{(i+1)})]}$;

${\displaystyle y_{(0)}=0;\quad A_{(i)}=\{(i),(i+1),\dots ,(n)\};\quad A_{(1)}=U;\quad A_{(n+1)}=\emptyset }$.

application

Discrete Choquet integrals are a flexible means of aggregating interacting criteria , see. In this case there are a lot of criteria with the characteristics . These criteria should be summarized (aggregated) into a (global) criterion by means of suitable averaging . The discrete Choquet integral forms such a mean: ${\displaystyle U=\{1,\dots ,n\}}$ ${\displaystyle n}$ ${\displaystyle f(i)=y_{i}}$ ${\displaystyle y}$

${\displaystyle y=\sum _{i=1}^{n}w_{(i)}y_{(i)}\quad {\text{mit den Gewichten}}\quad w_{(i)}=\mu (A_{(i)})-\mu (A_{(i+1)})}$

By super-additive (subadditive) can synergies ( redundancy effects are considered among the criteria). ${\displaystyle \mu }$

Web links

• M. Grabisch: An introduction to the Choquet integral

Individual evidence

1. ↑ Choquet, G. (1953). Theory of capacities . Ann.Inst.Fourier, Grenoble, 131-295 doi: 10.5802 / aif.53
2. ↑ Denneberg, D. (1994): Non-additive Measure and Integral . Kluwer, Dordrecht
3. ↑ Grabisch, M., Murofushi, T. and M. Sugeno (eds) (2000). Fuzzy Measures and Integrals - Theory and Applications . Physica publishing house
4. ↑ ^{a b} Grabisch, M., Marichal, J.-L., Mesiar, R. and E. Pap (2009): Aggregation Functions . Cambridge University Press