Fuzzy Lattice Research Articles

More on fuzzy lattices

In this paper, the concept of an (∈,∈∨qk)-fuzzy sublattice (ideal, filter) and (∈¯,∈¯∨q¯k)-fuzzy sublattice (ideal, filter) in a lattice is introduced, which is a generalization of the concept as given by Davvaz and Kazancı(2011) [18]. Characterizations for an (∈,∈∨qk)-fuzzy sublattice (ideal, filter) and (∈¯,∈¯∨q¯k)-fuzzy sublattice (ideal, filter) are provided. Different classes of lattices are characterized by the properties of these fuzzy sublattices (ideals, filters). Using the notions of a fuzzy sublattice (ideal, filter) with thresholds, characterization of a fuzzy sublattice (ideal, filter), an (∈,∈∨qk)-fuzzy sublattice (ideal, filter) and (∈¯,∈¯∨q¯k)-fuzzy sublattice (ideal, filter) are discussed.

Classification of power quality disturbances using non-linear dimension reduction

The proposed technique is different from others in respect that it is based on the concept of local non-linear relation and uses non-linear fuzzy functions to extract the feature-specific data. To extract any change during change in the patterns of power quality (PQ) events, non-linear Gaussian functions have been used which results in the formation of fuzzy lattices. The fuzzy lattices have been expressed in the form of Schrodinger equation to find the kinetic energy (KE) used corresponding to any change occurring in the power quality disturbances. Finally, the KE value embedded in two-dimension space has been used to distinguish PQ events. The method is applied to classify the various PQ events such as transient, sag, swell and harmonics and results are simulated using MATLAB version 7.3. The simulated results validate that the proposed algorithm can efficiently distinguish the PQ events in a single cycle and work perfectly in real time.

Fuzzy lattice classifier and its application to bearing fault diagnosis

In this work, we present a novel classification scheme named fuzzy lattice classifier (FLC) based on the lattice framework and apply it to the bearing faults diagnosis problem. Different from the fuzzy lattice reasoning (FLR) model developed in literature, there is no need to tune any parameter and to compute the inclusion measure in the training procedure in our new FLC model. It can converge rapidly in a single pass through training patterns with a few induced rules. A series of experiments are conducted on five popular benchmark datasets and three bearing datasets to evaluate and compare the presented FLC with the FLR model as well as some other widely used classification methods. Experimental results indicate that the FLC yields a satisfactory classification performance with higher computation efficiency than other classifiers. It is very desirable to utilize the FLC scheme for on-line condition monitoring of bearings and other mechanical systems.

Similarity reasoning for the semantic web based on fuzzy concept lattices: An informal approach

Similarity Reasoning in the presence of vague information is becoming fundamental in several research areas and, in particular, in the Semantic Web. Fuzzy Formal Concept Analysis (FFCA) is a generalization of Formal Concept Analysis (FCA) for modeling uncertainty information. Although FFCA has become very interesting for supporting different activities for the development of the Semantic Web, in the literature it is usually addressed at a technical level and intended for a restricted audience. This paper proposes a similarity measure for FFCA concepts. The key notions underlying the proposed approach are presented informally, in order to reach a broad audience of readers.

Definability and textures

This paper aims to give a new perspective for definability in rough set theory. First, a counterpart of definability is introduced in textural approximation spaces. Then a complete field of sets for texture spaces is defined and using textural arguments, some new results are obtained for rough sets. It is shown that definability can be also discussed in terms of a complete field of fuzzy sets on a fuzzy lattice for the various fuzzy approximation spaces. It is also given a partial affirmative answer to an open problem posed by Wei-Zhi Wu in On some mathematical structures of T-fuzzy rough set algebras in infinite universes of discourse in Fundamenta Informaticae 108 (3–4) 2011 337–369.

Fuzzy Soft Sets and Fuzzy Soft Lattices

Abstract The theory of soft sets was introduced by Molodtsov in 1999 in order to deal with uncertainties. In this paper, the notion of fuzzy soft lattice is defined and some related properties are derived, which extends the notion of a fuzzy lattice to include the algebraic structures of soft sets. Then the lattice structure of fuzzy soft lattices are discussed. At last, the concept of fuzzy soft ideal over a lattice is presented and the lattice structures of fuzzy soft ideal over a lattice are discussed.

A Method for Decomposition of Fuzzy Formal Context

Recently, fuzzy FCA becomes popular in researchers for representing the knowledge precisely. In this study we present a method for decomposition of given fuzzy formal context into several crisp formal contexts with help of distinct granules of fuzzy relation. The presented method reduces the lattice of given fuzzy formal context into several lattices for refining the interested patterns. From the obtained crisp formal contexts, we can generate the formal concepts in order to visualize them in lattice structure for describing the knowledge of given research field. Proposed method can be helpful for the researchers for reducing number of concepts, visualization of concepts for distinct fuzzy relations, and reduction of fuzzy lattice into several lattices in fuzzy context related fields

Fuzzy Lattice Reasoning for Pattern Classification Using a New Positive Valuation Function

This paper describes an enhancement of fuzzy lattice reasoning (FLR) classifier for pattern classification based on a positive valuation function. Fuzzy lattice reasoning (FLR) was described lately as a lattice data domain extension of fuzzy ARTMAP neural classifier based on a lattice inclusion measure function. In this work, we improve the performance of FLR classifier by defining a new nonlinear positive valuation function. As a consequence, the modified algorithm achieves better classification results. The effectiveness of the modified FLR is demonstrated by examples on several well-known pattern recognition benchmarks.

Fuzzy lattice based technique for classification of power quality disturbances

SUMMARYIn this paper, a new technique is proposed for classification of power quality disturbances. The proposed technique is based on the concept of local non‐linear relation to extract the feature specific data. To extract any change occurring in the patterns of power quality events, non‐linear Gaussian functions have been used that result in formation of fuzzy lattices. These fuzzy lattices are expressed in the form of Schrödinger equation to find the kinetic energy involved corresponding to any change occurring in the power quality disturbances. Finally, the kinetic energy parameter embedded in two dimension space has been used to distinguish various power quality events. The proposed technique efficiently distinguishes various real time power quality events in a single cycle. Copyright © 2011 John Wiley & Sons, Ltd.

An Interactive Method for Multi Stage Fuzzy Lattice decision making problems using Triangular fuzzy numbers

method for finding interactive multi- stage lattice fuzzy decision making problems in which all the parameters are represented by fuzzy numbers. In this interactive method a new algorithm is proposed to fuzzy lattice decision makings together with new representations of triangular fuzzy numbers. This paper will show the advantages of using proposed representations over the existing representations of fuzzy numbers and will present with great clarity the proposed method and illustrate numerical example.

Possibility-theoretic extension of derivation operators in formal concept analysis over fuzzy lattices

Formal concept analysis (FCA) associates a binary relation between a set of objects and a set of properties to a lattice of formal concepts defined through a Galois connection. This relation is called a formal context, and a formal concept is then defined by a pair made of a subset of objects and a subset of properties that are put in mutual correspondence by the connection. Several fuzzy logic approaches have been proposed for inducing fuzzy formal concepts from L-contexts based on antitone L-Galois connections. Besides, a possibility-theoretic reading of FCA which has been recently proposed allows us to consider four derivation powerset operators, namely sufficiency, possibility, necessity and dual sufficiency (rather than one in standard FCA). Classically, fuzzy FCA uses a residuated algebra for maintaining the closure property of the composition of sufficiency operators. In this paper, we enlarge this framework and provide sound minimal requirements of a fuzzy algebra w.r.t. the closure and opening properties of antitone L-Galois connections as well as the closure and opening properties of isotone L-Galois connections. We apply these results to particular compositions of the four derivation operators. We also give some noticeable properties which may be useful for building the corresponding associated lattices.

Applying the fuzzy lattice neurocomputing (FLN) classifier model to gear fault diagnosis

Gear faults are among the main causes of rotating machines breakdown in industrial applications. Intelligent condition monitoring for fault diagnosis can be helpful for detecting gear faults in an early stage so as to reduce production loss and, in addition, improve operation safety and reliability. In this work, we present an intelligent gear fault diagnosis scheme based on a novel classification model, namely the fuzzy lattice neurocomputing (FLN) classifier model. Five gear states including one healthy state and four defective states are tested in a two-stage gearbox. Statistical parameters in both the time domain and the frequency domain of vibration signals, acquired from gearbox, are used as features. We conducted experiments on a benchmark dataset as well as on a gear faults dataset to evaluate both the classification performance and the computational cost of the FLN classifier comparatively with alternative classification methods from the literature including artificial neural networks, support vector machines and decision trees. Our study has demonstrated that the FLN model yields better classification performance with smaller computational cost than the aforementioned alternative methods. The FLN classifier can further be used for condition monitoring and fault diagnosis in other mechanical systems.

Intuitionistic Fuzzy Equipotent Sublattices of Lattice Ordered Groups with Respect to SNorms

this paper, we introduce the notion of intuitionistic fuzzy equipotent lattice in a fuzzy lattice and then some basic properties are investigated. Characterization of intuitionistic fuzzy equipotent lattices are given. Using a collection of lattices, an intuitionistic fuzzy equipotent lattice is established. The notion of fuzzy equipotent lattice relation on the family of all intuitionistic fuzzy sub lattices of L are discussed upper and lower level sets of fuzzy equipotent lattices are studied.

Binary Image 2D Shape Learning and Recognition Based on Lattice-Computing (LC) Techniques

This work introduces a Type-II fuzzy lattice reasoning (FLRtypeII) scheme for learning/generalizing novel 2D shape representations. A 2D shape is represented as an element--induced from populations of three different shape descriptors--in the product lattice (F 3,?), where (F,?) denotes the lattice of Type-I intervals' numbers (INs). Learning is carried out by inducing Type-II INs, i.e. intervals in (F,?). Our proposed techniques compare well with alternative classification methods from the literature in three benchmark classification problems. Competitive advantages include an accommodation of granular data as well as a visual representation of a class. We discuss extensions to gray/color images, etc.

Constructing Domain Ontology Based on Fuzzy Set and Concept Lattice

The major content in FCA is to extract formal concepts and connections between them from data in form of formal context so as to form a lattice structure of formal concepts. Fuzzy set theory and fuzzy logic are acknowledged as an appropriate formalism for capturing imprecise and vague knowledge. The paper offers a methodology for building ontology for knowledge sharing and reusing based on fuzzy concept lattices union. This paper makes up these defects by applying formal concept analysis theory and fuzzy sets to construct concept hierarchies of ontology, and the experiments shows the CPU Time in the attribute numbers, indicating that FFCA is superior to FCA in building the ontology of semantic web.

A Lattice-Computing ensemble for reasoning based on formal fusion of disparate data types, and an industrial dispensing application

By “fusion” this work means integration of disparate types of data including (intervals of) real numbers as well as possibility/probability distributions defined over the totally-ordered lattice (R,⩽) of real numbers. Such data may stem from different sources including (multiple/multimodal) electronic sensors and/or human judgement. The aforementioned types of data are presented here as different interpretations of a single data representation, namely Intervals’ Number (IN). It is shown that the set F of INs is a partially-ordered lattice (F,⪯) originating, hierarchically, from (R,⩽). Two sound, parametric inclusion measure functions σ:FN×FN→[0,1] result in the Cartesian product lattice (FN,⪯) towards decision-making based on reasoning. In conclusion, the space (FN,⪯) emerges as a formal framework for the development of hybrid intelligent fusion systems/schemes. A fuzzy lattice reasoning (FLR) ensemble scheme, namely FLR pairwise ensemble, or FLRpe for short, is introduced here for sound decision-making based on descriptive knowledge (rules). Advantages include the sensible employment of a sparse rule base, employment of granular input data (to cope with imprecision/uncertainty/vagueness), and employment of all-order data statistics. The advantages as well as the performance of our proposed techniques are demonstrated, comparatively, by computer simulation experiments regarding an industrial dispensing application.

Fuzzy lattice and function approximation in image processing

Lattice theory plays a key role in image processing with the help of isomorphic fuzzy lattices. The fuzzy lattice L is considered and all its properties are transformed. The fuzzy measure induces entropy and it is defined by means of isotone valuation. For the minimisation process in image enhancement problem the three parameters such as intensification parameter (t), the crossover approximation fuzzy point (C(x,l)) and the fuzzyfier (f

The Construction of Fuzzy Concept Lattices Based on (θ, σ)-Fuzzy Rough Approximation Operators

Formal concept analysis and rough set analysis are two complementary approaches for analyzing data. This paper studies approaches to constructing fuzzy concept lattices based on generalized fuzzy rough approximation operators. For a residual implicator t satisfying t(a, b) = t(1 − b, 1 − a) and its dual σ, a pair of (t, σ)-fuzzy rough approximation operators is defined. We then propose three kinds of fuzzy operators, and examine some of their basic properties. Thus, three complete fuzzy concept lattices can be produced, for which the properties are analogous to those of the classical concept lattices.

Textures and Fuzzy Rough Sets

In this paper, we consider the Alexandroff topology for texture spaces. We prove that there exists a one-to-one correspondence between the Alexandroff ditopologies, and the reflexive and transitive direlations on a given texture. Using textural fuzzy direlations on a fuzzy lattice, we obtain a fuzzy rough set algebra where the inverse fuzzy relation and inverse fuzzy corelation are the upper approximation and lower approximation, respectively. In a special case, this gives us the fuzzy rough sets which are calculated with respect to the min-t norm introduced by S. Gottwald, or the fuzzy rough sets which are considered by D. Pei. (This work has been supported by the Turkish Scientific and Technological Research Council under the project TBAG 109T683.)

FL-GrCCA: A granular computing classification algorithm based on fuzzy lattices

Defining a relation between granules and computing ever-changing granules are two important issues in granular computing. In view of this, this work proposes a partial order relation and lattice computing , respectively, for dealing with the aforementioned issues. A fuzzy lattice granular computing classification algorithm, or FL-GrCCA for short, is proposed here in the framework of fuzzy lattices. Algorithm FL-GrCCA computes a fuzzy inclusion relation between granules by using an inclusion measure function based on both a nonlinear positive valuation function, namely arctan , and an isomorphic mapping between lattices. Changeable classification granules are computed with a dilation operator using, conditionally, both the fuzzy inclusion relation between two granules and the size of a dilated granule. We compare the performance of FL-GrCCA with the performance of popular classification algorithms, including support vector machines (SVMs) and the fuzzy lattice reasoning (FLR) classifier, for a number of two-class problems and multi-class problems. Our computational experiments showed that FL-GrCCA can both speed up training and achieve comparable generalization performance.