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Fuzzy Integrals

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A Practical Guide to Averaging Functions

Part of the book series: Studies in Fuzziness and Soft Computing ((STUDFUZZ,volume 329))

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Abstract

This chapter presents two main types of fuzzy integrals, the Choquet integral and the Sugeno integral. Fuzzy measures are introduced and their main properties and special cases are discussed. Various indices which characterize fuzzy measures are presented. The topic of fitting fuzzy measures to empirical data is treated in detail. Induced fuzzy integrals are also presented.

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Notes

  1. 1.

    A set function is a function whose domain consists of all possible subsets of \(\mathcal N\). For example, for \(n=3\), a set function is specified by \(2^3=8\) values at \(v(\emptyset )\), \(v(\{1\})\), \(v(\{2\})\), \(v(\{3\})\), \(v(\{1,2\})\), \(v(\{1,3\})\), \(v(\{2,3\})\), \(v(\{1,2,3\})\).

  2. 2.

    In general, this definition applies to any set function.

  3. 3.

    Such an array is based on a Hasse diagram of the inclusion relation defined on the set of subsets of \(\mathcal N\).

  4. 4.

    Two vectors \(\mathbf {x}, \mathbf {y} \in {\mathbb R}^n\) are called comonotone if there exists a common permutation P of \(\{1,2,\ldots ,n\}\), such that \(x_{P(1)}\le x_{P(2)} \le \cdots \le x_{P(n)}\) and \(y_{P(1)}\le y_{P(2)} \le \cdots \le y_{P(n)}\). Equivalently, this condition is frequently expressed as \((x_i-x_j)(y_i-y_j)\ge 0\) for all \(i,j \in \{1,\ldots ,n\}\).

  5. 5.

    As a consequence, this property holds for a linear convex combination of any number of fuzzy measures.

  6. 6.

    Since Choquet integral is a homogeneous aggregation function, we can calculate it directly on \([0,20]^n\) rather than scaling the inputs to \([0,1]^n\).

  7. 7.

    For a fixed \(m\ge 1\) this condition is called m-monotonicity (simple monotonicity for \(m=1\)), and if it holds for all \(m\ge 1\), it is called total monotonicity [Den00, Gra00]. For a fixed m, condition in Definition 4.35 is called m-alternating monotonicity. 2-monotone fuzzy measures are called supermodular (see Definition 4.14), also called convex, whereas 2-alternating fuzzy measures are called submodular. If a fuzzy measure is m-monotone, its dual is m-alternating and vice versa.

  8. 8.

    A set E is convex if \(\alpha x+(1-\alpha )y\in E\) for all \(x,y \in E, \alpha \in [0,1]\).

  9. 9.

    Weakly monotone refers here to this property: \(\forall \mathcal A, \mathcal B \subseteq \mathcal N\), \(\mathcal A \subseteq \mathcal B\) implies \(v_k(\mathcal A^k) \le v_k(\mathcal B^k)\).

  10. 10.

    Such interactions are well known in game theory. For example, contributions of the efforts of workers in a group can be greater or smaller than the sum of their separate contributions (if working independently).

  11. 11.

    See discussion in [GNW95], p. 318.

  12. 12.

    We remind that in the definition OWA, we used a non-increasing permutation of the components of \(\mathbf {x}\), \(\mathbf {x}_{\searrow }\), whereas in Choquet integral we use a non-decreasing permutation \(\mathbf {w}_{\nearrow }\). Then OWA is expressed as

    $$ C_v(\mathbf {x})=\sum _{i=1}^n x_{\nearrow (i)}\left( Q(\frac{n-i+1}{n})-Q(\frac{n-i}{n})\right) =\sum _{i=1}^n x_{\searrow (i)}\left( Q(\frac{i}{n})-Q(\frac{i-1}{n})\right) . $$

    We also remind that \(Q:[0,1]\rightarrow ][0,1], Q(0)=0, Q(1)=1\) is a RIM quantifier which determines values of v as \(v(\mathcal A) = Q\left( \frac{|\mathcal A|}{n}\right) \).

  13. 13.

    See footnote 4 on p. 150.

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Authors and Affiliations

  1. School of Information Technology, Deakin University, Burwood, VIC, Australia

    Gleb Beliakov

  2. Departamento de Automática y Computación, Universidad Pública de Navarra, Pamplona, Spain

    Humberto Bustince Sola

  3. Departamento de Ciencias de la Computación, Universidad de Alcalá, Alcalá de Henares, Madrid, Spain

    Tomasa Calvo Sánchez

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  1. Gleb Beliakov
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  2. Humberto Bustince Sola
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  3. Tomasa Calvo Sánchez
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Correspondence to Gleb Beliakov .

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Beliakov, G., Bustince Sola, H., Calvo Sánchez, T. (2016). Fuzzy Integrals. In: A Practical Guide to Averaging Functions. Studies in Fuzziness and Soft Computing, vol 329. Springer, Cham. https://doi.org/10.1007/978-3-319-24753-3_4

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  • DOI: https://doi.org/10.1007/978-3-319-24753-3_4

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