131 research outputs found
A comprehensive study of implicator-conjunctor based and noise-tolerant fuzzy rough sets: definitions, properties and robustness analysis
© 2014 Elsevier B.V. Both rough and fuzzy set theories offer interesting tools for dealing with imperfect data: while the former allows us to work with uncertain and incomplete information, the latter provides a formal setting for vague concepts. The two theories are highly compatible, and since the late 1980s many researchers have studied their hybridization. In this paper, we critically evaluate most relevant fuzzy rough set models proposed in the literature. To this end, we establish a formally correct and unified mathematical framework for them. Both implicator-conjunctor-based definitions and noise-tolerant models are studied. We evaluate these models on two different fronts: firstly, we discuss which properties of the original rough set model can be maintained and secondly, we examine how robust they are against both class and attribute noise. By highlighting the benefits and drawbacks of the different fuzzy rough set models, this study appears a necessary first step to propose and develop new models in future research.Lynn D’eer has been supported by the Ghent University Special Research Fund, Chris Cornelis was partially supported by the Spanish Ministry of Science and Technology under the project TIN2011-28488 and the Andalusian Research Plans P11-TIC-7765 and P10-TIC-6858, and by project PYR-2014-8 of the Genil Program of CEI BioTic GRANADA and Lluis Godo has been partially supported by the Spanish MINECO project EdeTRI TIN2012-39348-C02-01Peer Reviewe
Normality of spaces of operators and quasi-lattices
We give an overview of normality and conormality properties of pre-ordered
Banach spaces. For pre-ordered Banach spaces and with closed cones we
investigate normality of in terms of normality and conormality of the
underlying spaces and .
Furthermore, we define a class of ordered Banach spaces called quasi-lattices
which strictly contains the Banach lattices, and we prove that every strictly
convex reflexive ordered Banach space with a closed proper generating cone is a
quasi-lattice. These spaces provide a large class of examples and that
are not Banach lattices, but for which is normal. In particular, we
show that a Hilbert space endowed with a Lorentz cone is a
quasi-lattice (that is not a Banach lattice if ), and
satisfies an identity analogous to the elementary Banach lattice identity
which holds for all elements of a Banach lattice. This is
used to show that spaces of operators between such ordered Hilbert spaces are
always absolutely monotone and that the operator norm is positively attained,
as is also always the case for spaces of operators between Banach lattices.Comment: Minor typos fixed. Exact solution now provided in Example 5.10. To
appear in Positivit
Distributivity of strong implications over conjunctive and disjunctive uninorms
summary:This paper deals with implications defined from disjunctive uninorms by the expression where is a strong negation. The main goal is to solve the functional equation derived from the distributivity condition of these implications over conjunctive and disjunctive uninorms. Special cases are considered when the conjunctive and disjunctive uninorm are a -norm or a -conorm respectively. The obtained results show a lot of new solutions generalyzing those obtained in previous works when the implications are derived from -conorms
On Lipschitz properties of generated aggregation functions
This article discusses Lipschitz properties of generated aggregation functions. Such generated functions include triangular norms and conorms, quasi-arithmetic means, uninorms, nullnorms and continuous generated functions with a neutral element. The Lipschitz property guarantees stability of aggregation operations with respect to input inaccuracies, and is important for applications. We provide verifiable sufficient conditions to determine when a generated aggregation function holds the k-Lipschitz property, and calculate the Lipschitz constants of power means. We also establish sufficient conditions which guarantee that a generated aggregation function is not Lipschitz. We found the only 1-Lipschitz generated function with a neutral element e ∈]0, 1[.<br /
Absorbent tuples of aggregation operators
We generalize the notion of an absorbent element of aggregation operators. Our construction involves tuples of values that decide the result of aggregation. Absorbent tuples are useful to model situations in which certain decision makers may decide the outcome irrespective of the opinion of the others. We examine the most important classes of aggregation operators in respect to their absorbent tuples, and also construct new aggregation operators with predefined sets of absorbent tuples.<br /
On extending generalized Bonferroni means to Atanassov orthopairs in decision making contexts
Extensions of aggregation functions to Atanassov orthopairs (often referred to as intuitionistic fuzzy sets or AIFS) usually involve replacing the standard arithmetic operations with those defined for the membership and non-membership orthopairs. One problem with such constructions is that the usual choice of operations has led to formulas which do not generalize the aggregation of ordinary fuzzy sets (where the membership and non-membership values add to 1). Previous extensions of the weighted arithmetic mean and ordered weighted averaging operator also have the absorbent element 〈1,0〉, which becomes particularly problematic in the case of the Bonferroni mean, whose generalizations are useful for modeling mandatory requirements. As well as considering the consistency and interpretability of the operations used for their construction, we hold that it is also important for aggregation functions over higher order fuzzy sets to exhibit analogous behavior to their standard definitions. After highlighting the main drawbacks of existing Bonferroni means defined for Atanassov orthopairs and interval data, we present two alternative methods for extending the generalized Bonferroni mean. Both lead to functions with properties more consistent with the original Bonferroni mean, and which coincide in the case of ordinary fuzzy values.<br /
- …