Weakly Monotonic Averaging Functions

Wilkin, Tim; Beliakov, Gleb

doi:10.1002/int.21692

Cited by 119 publications

(39 citation statements)

References 37 publications

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“…Recently, several generalizations of aggregation functions have been introduced, considering some weaker forms of monotonicity, in particular, weakly monotone aggregation functions and preaggregation functions . Then one can consider set‐based weakly increasing aggregation functions (ie, such set‐based functions which, for each arity

n

, form an

n

‐ary weakly increasing aggregation function), or set‐based preaggregation functions, compare Example .…”

Section: Resultsmentioning

confidence: 99%

“…Indeed, for example,

F_{f} (0.2, 0.3) = 0.06 < F_{f} (0.4, 0.4) = 0.4 > F_{f} (0.5, 0.6) = 0.3 .

Observe that

F_{f} (0, \dots, 0) goodbreakinfix= 0

and

F_{f} (1, \dots, 1) goodbreakinfix= 1

, that is,

F_{f}

fulfils the boundary conditions of aggregation functions. More, for each

boldx goodbreakinfix\in {[0, 1]}^{n}

and each

boldc goodbreakinfix= (c, \dots, c) goodbreakinfix\in {[0, 1]}^{n} goodbreakinfix, n goodbreakinfix\in double-struckN

, we have

F_{f} (boldx + boldc) goodbreakinfix\geq F (x)

, that is,

F_{f}

is weakly increasing, and it can be concluded that

F_{f}

is a proper preaggregation function …”

Section: Set‐based Extended Functionsmentioning

confidence: 99%

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Set‐based extended aggregation functions

et al. 2019

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Inspired by the Zadeh approach to fuzzy connectives in fuzzy set theory and by some applications, we introduce and study set‐based extended functions, and in particular, set‐based extended aggregation functions. These functions reflect neither reordering nor repetition of input values, and, linking different arities, they introduce serious constraints for extended functions. A complete characterization of set‐based extended (aggregation) functions is given, and some constructions of such functions are also proposed, including several examples.

show abstract

n

, form an

n

‐ary weakly increasing aggregation function), or set‐based preaggregation functions, compare Example .…”

Section: Resultsmentioning

confidence: 99%

“…Indeed, for example,

F_{f} (0.2, 0.3) = 0.06 < F_{f} (0.4, 0.4) = 0.4 > F_{f} (0.5, 0.6) = 0.3 .

Observe that

F_{f} (0, \dots, 0) goodbreakinfix= 0

and

F_{f} (1, \dots, 1) goodbreakinfix= 1

, that is,

F_{f}

fulfils the boundary conditions of aggregation functions. More, for each

boldx goodbreakinfix\in {[0, 1]}^{n}

and each

boldc goodbreakinfix= (c, \dots, c) goodbreakinfix\in {[0, 1]}^{n} goodbreakinfix, n goodbreakinfix\in double-struckN

, we have

F_{f} (boldx + boldc) goodbreakinfix\geq F (x)

, that is,

F_{f}

is weakly increasing, and it can be concluded that

F_{f}

is a proper preaggregation function …”

Section: Set‐based Extended Functionsmentioning

confidence: 99%

Set‐based extended aggregation functions

et al. 2019

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show abstract

“…One may note that the averaging function EGBM is not monotone with respect to the parameters p and q . However, it is monotone in weaker sense . That is illustrated in the following theorem: Theorem For a fix

x \in {[0, 1]}^{n}

, the averaging function

{EGBM}_{p, q}

is weakly monotone with respect to the parameters p and q , that is

{italicEGBM}_{p + a, q + a} false(x false) \geq {italicEGBM}_{p, q} false(x false) for any a \in R .

…”

Section: Extended Geometric Bonferroni Meanmentioning

confidence: 99%

Aggregation of Heterogeneously Related Information with Extended Geometric Bonferroni Mean and Its Application in Group Decision Making

Chan

Guha

Niu

et al. 2017

Int. J. Intell. Syst.

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Capturing specific interrelationship among input arguments has great importance in the process of aggregation as they may change the aggregation result significantly, which can lead viable changes in the overall decision outcome. In this study, we attempt to aggregate a set of inputs with certain heterogeneous interrelationship pattern among them. To do this, we introduce a new aggregation operator, which we call the extended geometric Bonferroni mean. We investigate its properties and develop an algorithm to learn its associated parameters based on decision maker's perceived view toward the aggregation process. Moreover, to learn such heterogeneous relationship among the inputs from the data set, we provide a learning algorithm. Examples are given to illustrate the realization of algorithm and to show certain advantages over the existing aggregation operators.

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“…This is the case, for instance, of some subsethood measures (see [22]); • Many functions used for comparison of data are also directionally increasing. In particular, this is the case of those based on component-wise comparison by means of the Euclidean distance |x − y|, as for restricted equivalence functions [23]; • Weakly increasing functions ( [8]) are a particular case of directionally increasing functions, with r = (1, . .…”

Section: Directional Monotonicitymentioning

confidence: 99%

Preaggregation Functions: Construction and an Application

Lucca

Sanz

Dimuro

et al. 2016

IEEE Trans. Fuzzy Syst.

158

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In this work we introduce the notion of preaggregation function. Such a function satisfies the same boundary conditions as an aggregation function, but, instead of requiring monotonicity, only monotonicity along some fixed direction (directional monotonicity) is required. We present some examples of such functions. We propose three different methods to build pre-aggregation functions. We experimentally show that in fuzzy rule-based classification systems, when we use one of these methods, namely, the one based on the use of the Choquet integral replacing the product by other aggregation functions, if we consider the minimum or the Hamacher product t-norms for such construction, we improve the results obtained when applying the fuzzy reasoning methods obtained using two classical averaging operators like the maximum and the Choquet integral.

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Weakly Monotonic Averaging Functions

Cited by 119 publications

References 37 publications

Set‐based extended aggregation functions

Set‐based extended aggregation functions

Aggregation of Heterogeneously Related Information with Extended Geometric Bonferroni Mean and Its Application in Group Decision Making

Preaggregation Functions: Construction and an Application

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