Fuzzy Relational Calculus Research Articles

Optimizing Sensor-Controlled Systems with Minimal Intervention: A Fuzzy Relational Calculus Approach

This article describes an approach for optimizing sensor-controlled systems through minimal intervention, utilizing fuzzy linear systems of equations (FLSEs). Starting with a generalized model of the system behavior, we incorporate an array of control units, environmental sensors, and an expert knowledge base. The described problems of detecting the level of intervention needed to change the system state to another is handled with the help of developed methods for solving the inverse problem faced by FLSEs. By achieving minimal intervention, we ensure that the system adjustments are effective, economically optimal, and non-intrusive. A MATLAB-based implementation is presented.

Student Performance Prediction

Abstract: Academic integration is now defined by how students are able to access, and effectively interact with, and are supported by engineering faculty and staff; Academic integration, which is a very important construct that contributes to a student’s persistence and ultimate graduation. But, several factors, either numeric or non-numeric can have varying degrees of influence on academic performance of students. Rankings of aforementioned factors, hopefully, will assist the management in order to initiate corrective action with a view to students' academic performance improvement. In an academic environment, teachers are frequently in close contact with students and therefore their belief about the degree of influence of factors on students’ academic performance could be of vital significance. The case study presented in this paper uses fuzzy relational calculus and the Dempster Shafer theory of evidence to examine and suggest a feasible and realistic ranking approach of only chosen factors!

On the Term Set's Semantics for Pairwise Comparisons in Fuzzy Linguistic Preference Models.

The main objective of this paper is the definition of a membership function assignment procedure based on inherent features of linguistic terms to determine their semantics when they are used for preference modelling. For this purpose, we consider what linguists say about concepts such as language complementarity, the influence of context, or the effects of the use of hedges (modifiers) on adverbs meaning. As a result, specificity, entropy and position in the universe of discourse of the functions assigned to each linguistic term are mainly determined by the intrinsic meaning of the hedges concerned. We uphold that the meaning of weakening hedges is linguistically non-inclusive because their semantics are subordinated to the proximity to the indifference meaning, whereas reinforcement hedges are linguistically inclusive. Consequently, the membership function assignment rules are different: fuzzy relational calculus and the horizon shifting model derived from the Alternative Set Theory are used to handle weakening and reinforcement hedges, respectively. The proposed elicitation method provides for the term set semantics, non-uniform distributions of non-symmetrical triangular fuzzy numbers, depending on the number of terms used and the character of the hedges involved. (This article belongs to the section "Information Theory, Probability and Statistics").

Designing of higher order information granules through clustering heterogeneous granular data

This study is devoted to the generalization of information granules by forming higher order, namely, order-2 information granules. Information granules are semantically meaningful entities, which play a central role in knowledge representation and system modeling in the framework of Granular Computing. The encountered information granules could exhibit significant heterogeneity because of the diversified formal formalisms. To facilitate an effective generalization of heterogeneous granular data when using clustering algorithms, an efficient scheme has been proposed to form a unified representation of various types of granular data by using Possibility–necessity measures. Once the clustering process has been completed in the possibility–necessity feature space, the higher order information granules come as results of decoding by involving the possibility–necessity metrics and fuzzy relational calculus. The extent to which the higher order information granules are supported by the granular data present at a lower level of hierarchy is quantified in terms of the membership degrees obtained in the clustering process. Experimental studies concerning a series of publicly available datasets coming from UCI and KEEL machine learning repositories are carried out in this study.

Lfl: An R package for linguistic fuzzy logic

This paper presents an R package that supports the use of fuzzy relational calculus and linguistic fuzzy logic in data processing applications. The lfl package enables computing compositions of fuzzy relations enhanced with distinct extensions, such as excluding features, unavoidable features, or generalized quantifiers. Furthermore, it provides tools for transformation of data into fuzzy sets representing linguistic expressions, mining of linguistic fuzzy association rules, and performing an inference on fuzzy rule bases using the Perception-based logical deduction (PbLD). The package also enables to use the Fuzzy rule-based ensemble, a tool for time series forecasting based on an ensemble of forecasts from several individual methods implemented in R. To the best of the authors' knowledge, there is no other open source software that would provide free tools covering the above-described fragments of the fuzzy modeling area. Therefore, we find highly desirable to allow the community to get familiar with the tools as well as their implementation.

Fuzzy Relational Matrix Factorization and Its Granular Characterization in Data Description

This article is concerned with a problem of relational factorization which engages fuzzy relational calculus. It forms an interesting alternative to the method of nonnegative matrix factorization that has been commonly discussed and found in numerous applications. The relational factorization takes original <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> -dimensional data located in the unit hypercube and factorizes it into data of lower dimensionality and some fuzzy relations. Owing to the logic nature of processing delivered by relational calculus, the dimensionality reduction exhibits transparency as the reduction mechanism itself is described in terms of logic expressions. Two types of factorizations mechanisms are investigated by using <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s–t</i> and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t–s</i> composition operators where <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</i> and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</i> are triangular norms and conorms, respectively. A two-level process of factorization is designed. A gradient-based learning scheme is developed. The quantification of the performance of the factorization process is realized by bringing a concept of information granularity: The obtained fuzzy relations are formed based on granular constructs and the quality of the produced factorization is assessed in terms of the coverage and specificity of the obtained granular results. A collection of experiments is included to present the performance of factorization and its parametric analysis. In addition, the proposed algorithm comes with sound interpretability in terms of both the structure of the model and an intuitive meaning of the fuzzy relations being the result of factorization.

The concept of unavoidable features in fuzzy relational compositions

Fuzzy relational calculus, and especially, the compositions of fuzzy relations, had a great impact on many theoretical as well as practical areas of fuzzy modeling and it attracted numerous researchers. Recently, the compositions had been extended in distinct directions including the incorporation of excluding features or the direct use of generalized quantifiers instead of the standard ones. This article provides an investigation of an extension that is mathematically similarly constructed to the concept of excluding features although from the semantic point of view it is opposite. In particular, we introduce the concept of unavoidable features. Formally, the concept is again constructed as conjunctive fusion of an existing fuzzy relational composition with another composition, namely the Bandler-Kohout superproduct, employing a newly defined fuzzy relation of unavoidable features for particular classes. We provide an investigation of mathematical properties of the new concept and we demonstrate the appropriateness of the proposed concept on the classification task. This is provided with help of a real biological classification problems running on a data-sets of dragonflies and amphibians. A significant improvement of the results is emphasizing how strong influence may be provided with simple ideas and concepts fused appropriately together to a more complicated yet still explainable concept.

F-FOD: Fuzzy First Order Dominance Analysis and Populations Ranking Over Ordinal Multi-Indicator Systems

In this paper, we develop a new statistical procedure for the comparison of frequency distributions on systems of ordinal indicators, based on a multidimensional fuzzy extension of the first order dominance (FOD) criterion. The procedure, named fuzzy-first order dominance (F-FOD), employs concepts and tools from partially ordered set theory and from fuzzy relational calculus and is designed to overcome the main limitations of previously developed algorithms for FOD analysis. In particular, F-FOD produces full pairwise comparison matrices, allows for partial orderings and rankings of the statistical units to be derived from the input data, is computationally sufficiently light to be applied in most cases of practical interest and is freely available in the R package PARSEC. To illustrate its effectiveness, we also show F-FOD in action on two real datasets concerning health in Denmark and child well-being in the Democratic Republic of Congo.

Television Rating Control in the Multichannel&#x0D; Environment Using Trend Fuzzy Knowledge Bases&#x0D; and Monitoring Results †

The purpose of this study is to control the ratio of programs of different genres whenforming the broadcast grid in order to increase and maintain the rating of a channel. In themultichannel environment, television rating controls consist of selecting content, the ratings ofwhich are completely restored after advertising. The hybrid approach to rule set refinement basedon fuzzy relational calculus simplifies the process of expert recommendation systems construction.By analogy with the problem of the inverted pendulum control, the managerial actions aim to retainthe balance between the fuzzy demand and supply. The increase or decrease trends of the demandand supply are described by primary fuzzy relations. The rule-based solutions of fuzzy relationalequations connect significance measures of the primary fuzzy terms. Program set refinement bysolving fuzzy relational equations allows avoiding procedures of content-based selective filtering.The solution set generation corresponds to the granulation of television time, where each solutionrepresents the time slot and the granulated rating of the content. In automated media planning,generation of the weekly TV program in the form of the granular solution provides the decrease oftime needed for the programming of the channel broadcast grid.

Determinization of fuzzy automata by factorizations of fuzzy states and right invariant fuzzy quasi-orders

In this paper we provide improvements of determinization methods, based on factorization of fuzzy states, for fuzzy finite automata that accept fuzzy languages of infinite range. The improvements are based on the usage of the fuzzy relational calculus, namely, on the usage of the right invariant fuzzy quasi-orders. Our algorithms perform better in the sense that they produce smaller automata, while require the same computation time. In addition, they can produce finite deterministic automata in cases when previous algorithms result in infinite deterministic automata. We show that the weak representable-cycles property is necessary and sufficient condition for determinization of a fuzzy automaton via a maximal factorization of fuzzy states. This condition is more general than the representable-cycles property previously determined as the necessary and sufficient condition for determinization of a fuzzy automaton via a maximal factorization of fuzzy states.

Class dependency of fuzzy relational database using relational calculus and conditional probability

In this paper, we propose a design of fuzzy relational database to deal with a conditional probability relation using fuzzy relational calculus. In the previous, there are several researches about equivalence class in fuzzy database using similarity or approximate relation. It is an interesting topic to investigate the fuzzy dependency using equivalence classes. Our goal is to introduce a formulation of a fuzzy relational database model using the relational calculus on the category of fuzzy relations. We also introduce general formulas of the relational calculus for the notion of database operations such as ’projection’, ’selection’, ’injection’ and ’natural join’. Using the fuzzy relational calculus and conditional probabilities, we introduce notions of equivalence class, redundant, and dependency in the theory fuzzy relational database.

Granular Encoders and Decoders: A Study in Processing Information Granules

Information granules are generic building blocks supporting the processing realized in granular computing and facilitating communication with the environment. In this paper, we are concerned with a fundamental problem of encoding–decoding of information granules. The essence of the problem is outlined as follows: given a finite collection of granular data X 1, X 2,…, XN (sets, fuzzy sets, etc.), construct an optimal codebook composed of information granules A 1 , A2 , …, Ac , where typically c N, so that any Xk represented in terms of A i 's and then decoded (reconstructed) with the help of this codebook leads to the lowest decoding error. A fundamental result is established, which states that in the proposed encoders and decoders, when encoding–decoding error is present, the information granule coming as a result of decoding is of a higher type than the original information granules (say, if Xk is information granule of type-1, then its decoded version becomes information granule of type-2). It would be beneficial to note that as the encoding–decoding process is not lossless (in general, with an exception of a few special cases), the lossy nature of the method is emphasized by the emergence of information granules of higher type (in comparison with the original data being processed). For instance, when realizing encoding–decoding of numeric data (viz., information granules of type-0), the losses occur and they are quantified in terms of intervals, fuzzy sets, probabilities, rough sets, etc., where, in fact, the result becomes an information granule of type-1. In light of the nature of the constructed result when Xk is an interval or a fuzzy set, an optimized performance index engages a distance between the bounds of the interval-valued membership function. We develop decoding and encoding mechanisms by engaging the theory of possibility and fuzzy relational calculus and show that the decoded information granule is either a granular interval or interval-valued fuzzy set. The optimization mechanism is realized with the aid of the particle swarm optimization (PSO). A series of experiments are reported with intent to illustrate the details of the encoding–decoding mechanisms and show that the PSO algorithm can efficiently optimize the granular codebook.

On the TL-transitivity of fuzzy similarity measures

In fuzzy neural networks, it is usual to perform rule matching by employing fuzzy similarity measures. A fuzzy similarity measure is reserved for a binary fuzzy relation. In fuzzy relational calculus, the notion of T-transitivity is indispensable because it is strongly related to the concept of a metric. In this study, we consider whether the two types of fuzzy similarity measures given by Li et al. (2014) [24] satisfy TL-transitivity. We also investigate the TL-transitivity of fuzzy equivalencies.

Fuzzy Logic-Based Formalisms for Gynecology Disease Diagnosis

AbstractThe very basis of the present article is the fact that the medical knowledge consisting of clinical presentation, diagnosis, and treatment of a disease is with imprecision and uncertainty. The overall approach in gynecological disease diagnosis could be divided into three distinct stages, and this was confirmed by seven experienced gynecologists. Stage 1 refers to an initial screening process in order to arrive at a single disease diagnosis for the patients, which is based only on the subjective information provided by patients to the physician. In stage 2, the patient who has not received a single diagnostic label in stage 1 is further investigated for a single disease diagnosis using past history criteria. If stage 2 fails to arrive at a single disease diagnosis for a patient, then physical examination and various tests like imaging tests, blood tests, etc., are conducted, and the test results are processed in stage 3. In stage 1, we have revisited fuzzy relational calculus and mathematically evaluated the perceptions of the domain experts (gynecologists) with respect to 31 gynecological diseases. The paper also presents the research findings with a case study focused on stage 2 using a type 1 fuzzy inference system. Out of 226 patients, 50 are correctly diagnosed for a single disease and 147 for multiple diseases in stage 1. The paper concludes that fuzzy relational calculus is an effective method as an “initial screening” process to arrive at a single disease diagnosis. We have identified 29 out of 226 patients satisfying past history criteria to achieve a single disease diagnosis by stage 2. Investigations for stage 3 are in progress.

Can Fuzzy Relational Calculus Bring Complex Issues in Selection of Examiners into Focus?

AbstractThe examinee and the examiner play pivotal roles in the educational grading system. Students’ academic performance evaluation by multiple experts involves epistemic uncertainty, which can be modeled using a fuzzy set theory. How many evaluators/experts are almost similar in their perceptual subjective evaluation of the students answer paper? In other words, how many experts are reliable for a particular evaluation task with a defined possibility level? In this paper, the focus is onobject’s features(students’ marks) as a basis in the subjective evaluation process to identify thedegree of similarityamong the domain experts. The case study reveals that 11 out of 20 evaluators are similar in their decision making of students’ academic performance with possibility (α-level cut, 0.98). The inter-rater reliability (κ-coefficient) among the selected 11 teachers is 0.41, which signifies a fair/moderate agreement in the evaluation process. This paper proposes an approach that is useful for the selection of experts having similar perceptions in judgment. This paper demonstrates a case study showing how it is useful to educational policy makers in the selection of examiners.

Fuzzy similarity measures as a basis for the selection of examiners

Examinee and examiner plays a pivotal role in educational grading system. Perception of experts in evaluating student's answer script is of vital significance. Multiple experts evaluating students' academic performance involves epistemic uncertainty which can be modelled using fuzzy set theory. How many evaluators/experts are almost similar in their perceptual subjective evaluation of the students answer paper? In other words, how many experts are reliable for a particular evaluation task with defined possibility level? The paper addresses the issue of the selection of similar examiners using fuzzy relational calculus. The case study reveals that all twenty evaluators agree to one another considering the similarity measures on fuzzy sets. The computed Fleiss Kappa coefficient is 0.74 which signifies excellent agreement between all the twenty experts.

Pattern Recognition Based on Fuzzy Set and Genetic Algorithm

In this paper, we consider a soft computing approach to pattern classification. Our basic tools for soft computing are fuzzy relational calculus (FRC) and genetic algorithm (GA). We introduce a new interpretation of multidimensional fuzzy implication (MFI) to represent our knowledge about the training data set. We also consider the notion of a fuzzy pattern vector to handle the fuzzy information granules of the quantized pattern space and to represent a population of training patterns in the quantized pattern space. The construction of the pattern classifier is essentially based on the estimate of a fuzzy relation Ri between the antecedent clause and consequent clause of each one-dimensional fuzzy implication. For the estimation of Ri we use floating point representation of GA. Thus a set of fuzzy relations is formed from the new interpretation of MFI. This set of fuzzy relations is termed as the core of the pattern classifier. Once the classifier is constructed the non-fuzzy features of a test pattern can be classified. The performance of the proposed scheme is tested on synthetic data. Subsequently, we use the proposed scheme for the vowel classification problem of an Indian language. Finally, a benchmark of performance is established by considering multiplayer perception (MLP), support vector machine (SVM) and the present method. The Abalone, Hosse Colic and Pima Indians data sets, obtained from UCL database repository are used for the said benchmark study. This new tool for pattern classification is very effective for classification of patterns under vegue and imprecise environment. It can provide multiple classification under overlapped classes.

Knowledge Extraction in Fuzzy Relational Systems Based on Genetic and Neural Approach

In this paper, a problem of MIMO object identification expressed mathematically in terms of fuzzy relational equations is considered. We use the multivariable relational structure based on the modular fuzzy relational equations with the multilevel composition law. The identification problem consists of extraction of an unknown relational matrix and also of parameters of membership functions included in the fuzzy knowledge base, which can be translated as a set of fuzzy IF-THEN rules. In fuzzy relational calculus this type of the problem relates to inverse problem and requires resolution for the composite fuzzy relational equations. The search for solution amounts to solving an optimization problem using the hybrid genetic and neural approach. The genetic algorithm uses all the available experimental information for the optimization, i.e., operates off-line. The essence of the approach is in constructing and training a special neuro-fuzzy network, which allows on-line correction of the extracted relations if the new experimental data is obtained. The resulting solution is linguistically interpreted as a set of possible rules bases. The approach proposed is illustrated by the computer experiment and the example from medical diagnosis.

Evidence Theory and Fuzzy Relational Calculus in Estimation of Health Effects Due to Air Pollution

Abstract.With an overall objective of establishing association between air pollutants and incidence of respiratory diseases, the environmental professionals and medical practitioners have made significant contribution, using statistical mechanics in modelling epidemiological data, population characteristics, and pollution parameters. Broadly speaking, the studies have shown that the increase in vehicular traffic has been one of the causes of respiratory diseases. However, the WHO Centre for Environment and Health, Europe in its 2005 document states: “There is little evidence for a causal relationship between asthma prevalence/incidence and air pollution in general, though the evidence is suggestive of a causal association between the prevalence/incidence of asthma symptoms and living in close proximity to traffic”. Decision making process in a real world is invariably based on perceptions which are expressed in words or may be in numeric terms and not in probability terms. In the paper, we made an attempt to model the perceptions of experienced pulmonologists in arriving at their combined degree of belief/plausibility/ignorance for all the possible combinations of identified respiratory diseases, using evidence theory and fuzzy relational calculus without collecting sizeable parametric data accumulated over a period of years. Tightening pollution norms by the regulatory authorities is an overall objective of the global efforts on greenhouse gases (GHS) reduction in general, and air pollution mitigation in particular. The concept of solar battery operated electric vehicles (SBOEV) for road transport is advocated, initially for two/three wheelers, and extending it to four wheelers, especially in the developing countries.

Fast Fuzzy Inference in Octave

Fuzzy relations are simple mathematical structures that enable a very general representation of fuzzy knowledge, and fuzzy relational calculus offers a powerful machinery for approximate reasoning. However, one of the most relevant limitations of approximate reasoning is the efficiency bottleneck. In this paper, we present two implementations for fast fuzzy inference through relational composition, with the twofold objective of being general and efficient. The two implementations are capable of working on full and sparse representations respectively. Further, a wrapper procedure is capable of automatically selecting the best implementation on the basis of the input features. We implemented the code in GNU Octave because it is a high-level language targeted to numerical computations. Experimental results show the impressive performance gain when the proposed implementation is used.