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Fuzzy Integrals

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Discrete Fuzzy Measures

Part of the book series: Studies in Fuzziness and Soft Computing ((STUDFUZZ,volume 382))

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Abstract

While the Choquet and Sugeno integrals are the most widely adopted when it comes to applying fuzzy measures for aggregation, there are a number of alternative approaches to integration with respect to measures that may not be necessarily additive. This chapter gives an overview of some of the main types proposed along with their properties, semantic interpretations and generalisations. For each of the integrals presented, we outline important considerations for computation and provide examples.

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Notes

  1. 1.

    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 \(x_{P(1)}\leqslant x_{P(2)} \leqslant \cdots \leqslant x_{P(n)}\) and \(y_{P(1)}\leqslant y_{P(2)} \leqslant \cdots \leqslant y_{P(n)}\). Equivalently, this condition is frequently expressed as \((x_i-x_j)(y_i-y_j)\geqslant 0\) for all \(i,j \in \{1,\ldots ,n\}\).

  2. 2.

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

  3. 3.

    See footnote 1 on p. 94.

  4. 4.

    With respect to a measure m, an integral was sought satisfying:

    (S1) \(\int (\mathbf 1 | \mathcal A) \ \mathrm d m = m(\mathcal A),\) where \((\mathbf 1 | \mathcal A)\) represents the constant function 1 restricted to the set \(\mathcal A\), e.g. in the discrete case if \(n = 3\) and \(\mathcal A = \{2,3\}\) then \((\mathbf 1 | \mathcal A)=(0,1,1)\). In other words, integrating over the characteristic function of a set \(\mathcal A\) returns the measure of \(\mathcal A\);

    (S2) \(\int c f \ \mathrm d m = c \int f \ \mathrm d m, c \geqslant 0\), homogeneity with respect to a multiplying constant;

    (S3) \(\int \bigcup \limits _{n=1}^\infty f_n \ \mathrm d m= \bigcup \limits _{n=1}^\infty \int f_n \ \mathrm d m\), consistency in terms of integrating over sequences of functions \(f_n\) that converge to f; the notation of \(\cup \) here denotes both lowest upper bound and union.

  5. 5.

    For example, with the Lebesgue measure it holds that \(L(0.3,0) + L(0,0.5) = L(0.3,0.5)\) (vertical partition) and also that \(L(0.3,0.3) + L(0,0.2) = L(0.3,0.5)\) (horizontal partition).

    For the Shilkret integral, if the measure is maxitive, it holds that \(\max \{ Sh(0.3,0),Sh(0,0.5) \} = Sh(0.3,0.5)\) (vertical partition) and also that \(\max \{Sh(0.3,0.3), Sh(0,0.5)\} = Sh(0.3,0.5)\) (horizontal partition).

  6. 6.

    In the more general case, the Pan integral is usually defined using \(\sup \) and \(\inf \) with conditions on the operations ensuring closure and other properties. We do not require these in the discrete case.

  7. 7.

    A function UI on \([0,\infty ]\) is called a universal integral with respect to a measure m if the following axioms hold [KMP10]:

    (I1) The function is nondecreasing with respect to the measure m and with respect to the input function f;

    (I2) There exists a pseudo-multiplication \(\otimes : [0,\infty ]^2 \rightarrow [0,\infty ]\) such that \(UI_m(\mathbf c |\mathcal A) = c\cdot m(\mathcal A)\) where \(\mathbf c|\mathcal A\) denotes the constant function c restricted to the set \(\mathcal A\), e.g. in the discrete case if \(n=3\) and \(\mathcal A = \{2,3\}\) then \(\mathbf c|\mathcal A = (0,c,c)\);

    (I3) For all integral equivalent pairs \((m_1, f_1), (m_2,f_2)\) we have \(UI_{m_1}(f_1) = UI_{m_2}(f_2) \).

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Authors and Affiliations

  1. Deakin University, Burwood, VIC, Australia

    Gleb Beliakov & Simon James

  2. Ningbo University, Ningbo, Zhejiang, China

    Jian-Zhang Wu

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  1. Gleb Beliakov
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  2. Simon James
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Correspondence to Gleb Beliakov .

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Beliakov, G., James, S., Wu, JZ. (2020). Fuzzy Integrals. In: Discrete Fuzzy Measures. Studies in Fuzziness and Soft Computing, vol 382. Springer, Cham. https://doi.org/10.1007/978-3-030-15305-2_5

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