Discrete Sugeno Integral Research Articles

A new characterization of congruences and the discrete Sugeno integral on LB-valued general fuzzy automata

The present study aimed at exploring the idempotent and compatible aggregation functions on a complete infinitely distributive lattice B of propositions about the LB-valued general fuzzy automata (for simplicity, LB-valued GFA), where the compatibility is related to the congruences on B. Further, we have shown that all aggregation functions on B can be obtained as usual composition of lattice operations ∨,∧ and certain unary and binary aggregation functions. Moreover, our results yield a new characterization of discrete Sugeno integrals on B.

Remarks on Sugeno Integrals on Bounded Lattices

A discrete Sugeno integral on a bounded distributive lattice L is defined as an idempotent weighted lattice polynomial. Another possibility for axiomatization of Sugeno integrals is to consider compatible aggregation functions, uniquely extending the L-valued fuzzy measures. This paper aims to study the mentioned unique extension property concerning the possible extension of a Sugeno integral to non-distributive lattices. We show that this property is equivalent to the distributivity of the underlying bounded lattice. As a byproduct, an alternative proof of Iseki’s result, stating that a lattice having prime ideal separation property for every pair of distinct elements is distributive, is provided.

A New Characterization of Congruence and the Discrete Sugeno Integral on Lb-Valued Gfa

A New Characterization of Congruence and the Discrete Sugeno Integral on Lb-Valued Gfa

DC optimization for constructing discrete Sugeno integrals and learning nonadditive measures

Defined solely by means of order-theoretic operations meet (min) and join (max), weighted lattice polynomial functions are particularly useful for modelling data on an ordinal scale. A special case, the discrete Sugeno integral, defined with respect to a nonadditive measure (a capacity), enables accounting for the interdependencies between input variables. However, until recently the problem of identifying the fuzzy measure values with respect to various objectives and requirements has not received a great deal of attention. By expressing the learning problem as the difference of convex functions, we are able to apply DC (difference of convex) optimization methods. Here we formulate one of the global optimization steps as a local linear programming problem and investigate the improvement under different conditions. Abbreviation: DC: difference of convex; LP: linear programming

Fitting Sugeno integral for learning fuzzy measures using PAVA isotone regression

The discrete Sugeno integral is an aggregation function particularly suited to aggregation of ordinal inputs. It accounts for inputs interactions, such as redundancy and complementarity, and its learning from empirical data is a challenging optimisation problem. The methods of ordinal regression involve an expensive objective function, whose complexity is quadratic in the number of data. We formulate ordinal regression using a much less expensive objective computed in linear time by the pool-adjacent-violators algorithm. We investigate the learning problem numerically and show the superiority of the new algorithm.

A note on some algebraic properties of discrete Sugeno integrals

Based on the link between Sugeno integrals and fuzzy measures, we discuss several algebraic properties of discrete Sugeno integrals. We recall that the composition of Sugeno integrals is again a Sugeno integral, and that each Sugeno integral can be obtained as a composition of binary Sugeno integrals. In particular, we discuss the associativity, dominance, commuting and bisymmetry of Sugeno integrals.

From quantitative to qualitative orness for lattice OWA operators

This paper deals with OWA (ordered weighted average) operators defined on an arbitrary finite lattice endowed with a t-norm and a t-conorm. A qualitative orness measure for any OWA operator is suggested, based on its proximity to the OR operator that yields the maximum of the given data. In the particular case of a finite distributive lattice, considering the t-norm given by the meet and the t-conorm given by the join, this qualitative measure agrees with the value that some discrete Sugeno integral takes on the vector consisting of all the members of the lattice. Some applications of the qualitative orness of OWA operators to decision-making problems are shown. In addition, OWA operators defined on a finite product lattice are also applied in image processing. We analyze the effect of several OWA operators with respect to their orness.

H-Index and Other Sugeno Integrals: Some Defects and Their Compensation

The famous Hirsch index has been introduced just ca. ten years ago. Despite that, it is already widely used in many decision-making tasks, like in evaluation of individual scientists, research grant allocation, or even production planning. It is known that the $h$ -index is related to the discrete Sugeno integral and the Ky Fan metric introduced in the 1940s. The aim of this paper is to propose a few modifications of this index as well as other fuzzy integrals—also on bounded chains—that lead to better discrimination of some types of data that are to be aggregated. All of the suggested compensation methods try to retain the simplicity of the original measure.

Congruences and the discrete Sugeno integrals on bounded distributive lattices

We study compatible aggregation functions on a general bounded distributive lattice L, where the compatibility is related to the congruences on L. As a by-product, a new proof of an earlier result of G. Grätzer is obtained. Moreover, our results yield a new characterization of discrete Sugeno integrals on bounded distributive lattices.

A new characterization of the discrete Sugeno integral

We introduce a new property of the discrete Sugeno integrals which can be seen as their characterization, too. This property, compatibility with respect to congruences on [0, 1], stresses the importance of the Sugeno integrals in multicriteria decision support as well.

Multiattribute preference models with reference points

In the context of multiple attribute decision making, preference models making use of reference points in an ordinal way have recently been introduced in the literature. This text proposes an axiomatic analysis of such models, with a particular emphasis on the case in which there is only one reference point. Our analysis uses a general conjoint measurement model resting on the study of traces induced on attributes by the preference relation and using conditions guaranteeing that these traces are complete. Models using reference points are shown to be a particular case of this general model. The number of reference points is linked to the number of equivalence classes distinguished by the traces. When there is only one reference point, the induced traces are quite rough, distinguishing at most two distinct equivalence classes. We study the relation between the model using a single reference point and other preference models proposed in the literature, most notably models based on concordance and models based on a discrete Sugeno integral.

OWA operators defined on complete lattices

In this paper the concept of an ordered weighted average (OWA) operator is extended to any complete lattice endowed with a t-norm and a t-conorm. In the case of a complete distributive lattice it is shown to agree with a particular case of the discrete Sugeno integral. As an application, we show several ways of aggregating closed intervals by using OWA operators. In a complementary way, the notion of generalized Atanassov's operators is weakened in order to be extended to intervals contained in any lattice. This new approach allows us to build a kind of binary aggregation functions for complete lattices, including OWA operators.

On the axiomatization of some classes of discrete universal integrals

Following the ideas of the axiomatic characterization of the Choquet integral due to [D. Schmeidler, Integral representation without additivity, Proc. Amer. Math. Soc. 97 (1986) 255–261] and of the Sugeno integral given in [J.-L. Marichal, An axiomatic approach of the discrete Sugeno integral as a tool to aggregate interacting criteria in a qualitative framework, IEEE Trans. Fuzzy Syst. 9 (2001) 164–172], we provide a general axiomatization of some classes of discrete universal integrals, including the case of discrete copula-based universal integrals (as usual, the product copula corresponds just to the Choquet integral, and the minimum to the Sugeno integral).

Characterizations of discrete Sugeno integrals as polynomial functions over distributive lattices

We give several characterizations of discrete Sugeno integrals over bounded distributive lattices, as particular cases of lattice polynomial functions, that is, functions which can be represented in the language of bounded lattices using variables and constants. We also consider the subclass of term functions as well as the classes of symmetric polynomial functions and weighted infimum and supremum functions, and present their characterizations, accordingly. Moreover, we discuss normal form representations of these functions.

Weighted lattice polynomials

We define the concept of weighted lattice polynomial functions as lattice polynomial functions constructed from both variables and parameters. We provide equivalent forms of these functions in an arbitrary bounded distributive lattice. We also show that these functions include the class of discrete Sugeno integrals and that they are characterized by a median-based decomposition formula.

The Use of the Discrete Sugeno Integral in Decision-Making A Survery

The Use of the Discrete Sugeno Integral in Decision-Making A Survery

THE USE OF THE DISCRETE SUGENO INTEGRAL IN DECISION-MAKING: A SURVERY

An overview of the use of the discrete Sugeno integral as either an aggregation tool or a preference functional is presented in the qualitative framework of two decision paradigms: multi-criteria decision-making and decision-making under uncertainty. The parallelism between the representation theorems in both settings is stressed, even if a basic requirement like the idempotency of the aggregation scheme should be explicitely stated in multi-criteria decision-making, while its counterpart is implicit in decision under uncertainty by equating the utility of a constant act with the utility of its consequence. Important particular cases of Sugeno integrals such as prioritized minimum and maximum operators, their ordered versions, and Boolean max-min functions are studied.

An axiomatic approach of the discrete Sugeno integral as a tool to aggregate interacting criteria in a qualitative framework

We present a model allowing us to aggregate decision criteria when the available information is of a qualitative nature. The use of the Sugeno integral as an aggregation function is justified by an axiomatic approach. It is also shown that the mutual preferential independence of criteria reduces the Sugeno integral to a dictatorial aggregation.