Abstract
The so-called fuzzy integrals, which are defined with respect to fuzzy measures, allow interaction effects between variables to be accounted for when averaging data. This chapter introduces these concepts in the framework of aggregating discrete sets of inputs. After providing rationale for the study of discrete fuzzy measures along with the basic definitions, we give a short overview of aggregation functions, their properties, and some prototypical examples.
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Notes
- 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 \(\mu (\emptyset )\), \(\mu (\{1\})\), \(\mu (\{2\})\), \(\mu (\{3\})\), \(\mu (\{1,2\})\), \(\mu (\{1,3\})\), \(\mu (\{2,3\})\), \(\mu (\{1,2,3\})\).
- 2.
Such an array is based on a Hasse diagram of the inclusion relation defined on the set of subsets of \(\mathcal N\).
- 3.
In general, this definition applies to any set function.
- 4.
The interval [0, 1] can be substituted with any interval \([a,b] \subset \mathbb {R}\) using a simple transformation.
- 5.
Two vectors \(\mathbf x, \mathbf y \in [0,1]^n\) are not comparable if there exists an i such that \(x_i > y_i\) and \(j \ne i\) such that \(x_j < y_j\).
- 6.
- 7.
Proof: Take any \(\mathbf {x} \in {\mathbb I}^n\), and let p, q denote the values \(p=\min (\mathbf {x}), q=\max (\mathbf {x})\). By monotonicity, \(p=f(p,p,\ldots ,p)\leqslant f(\mathbf {x})\leqslant f(q,q,\ldots ,q)=q\). Hence \(\min (\mathbf {x})\leqslant f(\mathbf {x})\leqslant \max (\mathbf {x})\). The converse: let \(\min (\mathbf {x})\leqslant f(\mathbf {x})\leqslant \max (\mathbf {x})\). By taking \(\mathbf {x}=(t,t,\ldots ,t)\), \(\min (\mathbf {x})=\max (\mathbf {x})=f(\mathbf {x})=t\), hence idempotency.
- 8.
Proof: Assume f has two neutral elements e and u. Then \(u=f(e, u) =e\), therefore \(e=u\). For n variables, assume \(e<u\). By monotonicity, \(e=f(e,u,\ldots ,u,\ldots , u) \geqslant f(e,e,\ldots ,e,u,e\ldots ,e)=u\), hence we have a contradiction. The case \(e>u\) leads to a similar contradiction.
- 9.
A frequently used term is bijection: a bijection is a function \(f:A\rightarrow B\), such that for every \(y \in B\) there is exactly one \(x \in A\), such that \(y=f(x)\), i.e., it defines a one-to-one correspondence between A and B. Because N is strictly monotone, it is a one-to-one function. Its range is [0,1], hence it is an onto mapping, and therefore a bijection.
- 10.
Automorphism is another useful term: An automorphism is a strictly increasing bijection of an intervalonto itself \([a,b]\rightarrow [a,b]\).
- 11.
A real function of n arguments is continuous if for any sequences \(\{x_{ij}\}, i=1,\ldots ,n\) such that \( \lim \limits _{j \rightarrow \infty }x_{ij}=y_i \) it holds that \(\lim \limits _{j \rightarrow \infty } f(x_{1j},\ldots x_{nj})= f(y_1,\ldots ,y_n)\). Because the domain \({\mathbb I}^n\) is a compact set, continuity is equivalent to its stronger version, uniform continuity. An aggregation function is uniformly continuous if and only if it is continuous in each argument (i.e., we can check continuity by fixing all variables but one, and checking continuity of each univariate function. However, general non-monotone functions can be continuous in each variable without being continuous).
- 12.
A distance between objects x, y from a set \(\mathcal S\) is a function defined on \(\mathcal S \times \mathcal S\), whose values are non-negative real numbers, with the properties: (1) \(d(x, y)=0\) if and only if \(x=y\), (2) \(d(x,y)=d(y,x)\) (symmetry), and (3) \(d(x,z)\leqslant d(x,y)+d(y,z)\) (triangular inequality). Such a distance is called a metric.
- 13.
A norm is a function f on a vector space with the properties: (1) \(f(\mathbf {x}) > 0\) for all nonzero \(\mathbf {x}\) and \(f(\mathbf {0})=0\) , (2) \(f(a \mathbf {x})=|a|f(\mathbf {x})\), and (3)\(f(\mathbf {x} + \mathbf {y}) \leqslant f(\mathbf {x})+ f(\mathbf {y})\).
- 14.
I.e., it is differentiable on its entire domain, except for a subset of measure zero.
- 15.
Take \(f(x_1,x_2)=\sqrt{x_1 x_2}\), which is continuous for \(x_1,x_2 \geqslant 0\), and let \(x_2=1\). \(f(t,1)=\sqrt{t}\) is continuous but not Lipschitz. To see this, let \(t=0\) and \(u>0\). Then \(|\sqrt{0}-\sqrt{u}|=\sqrt{u} > Mu=M|0-u|\), or \(u^{-\frac{1}{2}}>M\), for whatever choice of M, if we make u sufficiently small. Hence the Lipschitz condition fails.
- 16.
Note the difference to the increasing ordering used for the Choquet and Sugeno integrals.
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Beliakov, G., James, S., Wu, JZ. (2020). Introduction. In: Discrete Fuzzy Measures. Studies in Fuzziness and Soft Computing, vol 382. Springer, Cham. https://doi.org/10.1007/978-3-030-15305-2_1
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