In clustering we often face the situation that only a subset of the available attributes is relevant for forming clusters, even though this may not be known beforehand. In such cases it is desirable to have a clustering algorithm that automatically weights attributes or even selects a proper subset. In this paper I study such an approach for fuzzy clustering, which is based on the idea to trans...

Fuzzy logic and fuzzy set theory enable us to use experts’ uncertain (“fuzzy”) knowledge in decision making. The corresponding methods usually generalize known methods of decision making which are based on the crisp (non-fuzzy) knowledge about the environment. These crisp methods enable us to come up with a crisp decision: e.g., whether we should build a plant or not. When we generalize these m...

In a modular neural network, the conflicting information supplied by the information sources, i.e., the outputs of the subnetworks, can be combined by applying the concept of fuzzy integral. To compute the fuzzy ints gal, it is essential to know the importance of each subset of the information sources in a quantified form. In practice, it is very difficult to determine the worth of the informat...

Let d be a metric on the set FP(X) of fuzzy subsets of a finite set X. A midpoint of two fuzzy subsets μ, ν ∈ FP(X) is any fuzzy subset ξ ∈ FP(X) such that d(ξ, μ) = d(ξ, ν) = 1 2d(μ, ν). These midpoints can be used to represent “middle ways” or “compromises” between two situations described by the fuzzy subsets μ and ν. Now, the imprecise knowledge of a nucleic acid or protein sequence of leng...

Rough set theory has been extensively discussed in machine learning and pattern recognition. It provides us another important theoretical tool for feature selection. In this paper, we construct a novel rough set model for feature subset selection. First, we define the fuzzy decision of a sample by using the concept of fuzzy neighborhood. A parameterized fuzzy relation is introduced to character...

The purpose of this study is to define the concepts of Tsallis entropy and conditional Tsallis entropy of fuzzy partitions and to obtain some results concerning this kind entropy. We show that the Tsallis entropy of fuzzy partitions has the subadditivity and concavity properties. We study this information measure under the refinement and zero mode subset relations. We check the chain rules for ...

The concept of fuzzy set, which was published by Zadeh in his classic paper [24] of 1965, was applied by many researchers to generalize some of the basic concepts of algebra. The fuzzy algebraic structures play a central role in mathematics with wide applications in many other branches such as theoretical physics, computer sciences, control engineering, information sciences, coding theory, topo...

A systematic approach for fine-tuning fuzzy controllers has been developed and evaluated for an aeration control system implemented in a WWTP. The challenge with the application of fuzzy controllers to WWTPs is simply that they contain many parameters, which need to be adjusted for different WWTP applications. To this end, a methodology based on model simulations is used that employs three stat...

We characterize a fuzzy lattice through a fuzzy partial order relation, define ideals and filters of fuzzy lattices, characterize an ideal of fuzzy lattice using its level set and its support and show that a subset of a fuzzy lattice is an ideal if and only if its support is an ideal. Similarly, we show the same for its level set. Lastly, we define some kinds of ideals and filters of fuzzy latt...

It is well known that almost all random subset sum instances with density less than 0.6463... can be solved with an l2-norm SVP oracle by Lagarias and Odlyzko. Later, Coster et al. improved the bound to 0.9408... by using a different lattice. In this paper, we generalize this classical result to lp-norm. More precisely, we show that for p ∈ Z, an lp-norm SVP oracle can be used to solve almost a...