We explore switching techniques between q-fractional fuzzy sets (qFr sets) and various other classes of fuzzy sets to establish connections and provide a comprehensive framework. In particular, we examine the relationships between qFr sets and interval-valued fuzzy sets (IVFS), type 2 fuzzy sets(T2FS), intuitionistic fuzzy sets(IFS), Pythagorean fuzzy sets(PFS), q-rung orthopair fuzzy sets (q-ROFS), and linear diophantine fuzzy sets(LDFS). By examining these interconnections, we aim to understand better qFr sets and their applications in a wide range of fuzzy systems. It is possible to convert qFr sets into other fuzzy set models using the derived switching techniques, facilitating the utilization of existing methods and algorithms. The versatility of qFr sets, combined with the bridging techniques presented, holds promise for addressing complex problems in decision-making, pattern recognition, and other applications where uncertainty and imprecision play significant roles. Through case studies and practical applications, we illustrate the effectiveness and usefulness of the proposed switching techniques, showcasing their potential impact on real-world scenarios.
The study introduces a concept of mappings realized in presence of information granules and offers a design framework supporting the formation of such mappings. Information granules are conceptually meaningful entities formed on a basis of a large number of experimental input–output numeric data available for the construction of the model. We develop a conceptually and algorithmically sound way of forming information granules. Considering the directional nature of the mapping to be formed, this directionality aspect needs to be taken into account when developing information granules. The property of directionality implies that while the information granules in the input space could be constructed with a great deal of flexibility, the information granules formed in the output space have to inherently relate to those built in the input space. The input space is granulated by running a clustering algorithm; for illustrative purposes, the focus here is on fuzzy clustering realized with the aid of the fuzzy C-means algorithm. The information granules in the output space are constructed with the aid of the principle of justifiable granularity (being one of the underlying fundamental conceptual pursuits of Granular Computing). The construct exhibits two important features. First, the constructed information granules are formed in the presence of information granules already constructed in the input space (and this realization is reflective of the direction of the mapping from the input to the output space). Second, the principle of justifiable granularity does not confine the realization of information granules to a single formalism such as fuzzy sets but helps form the granules expressed any required formalism of information granulation. The quality of the granular mapping (viz. the mapping realized for the information granules formed in the input and output spaces) is expressed in terms of the coverage criterion (articulating how well the experimental data are “covered” by information granules produced by the granular mapping for any input experimental data). Some parametric studies are reported by quantifying the performance of the granular mapping (expressed in terms of the coverage and specificity criteria) versus the values of a certain parameters utilized in the construction of output information granules through the principle of justifiable granularity. The plots of coverage–specificity dependency help determine a knee point and reach a sound compromise between these two conflicting requirements imposed on the quality of the granular mapping. Furthermore, quantified is the quality of the mapping with regard to the number of information granules (implying a certain granularity of the mapping). A series of experiments is reported as well.
The learning abilities and high transparency are the two important and highly desirable features of any model of software quality. The transparency and user-centricity of quantitative models of software engineering are of paramount relevancy as they help us gain a better and more comprehensive insight into the revealed relationships characteristic to software quality and software processes. In this study, we are concerned with logic-driven architectures of logic models based on fuzzy multiplexers (fMUXs). Those constructs exhibit a clear and modular topology whose interpretation gives rise to a collection of straightforward logic expressions. The design of the logic models is based on the genetic optimization and genetic algorithms, in particular. Through the prudent usage of this optimization framework, we address the issues of structural and parametric optimization of the logic models. Experimental studies exploit software data that relates software metrics (measures) to the number of modifications made to software modules.
In this paper, we develop a comprehensive conceptual and algorithmic framework to cope with a problem of clustering homogeneous information granules. While there have been several approaches to coping with granular (viz. non-numeric) data, the origin of granular data themselves considered there is somewhat unclear and, as a consequence, the results of clustering start lacking some full-fledged interpretation. In this paper, we offer a holistic view at clustering information granules and an evaluation of the results of clustering. We start with a process of forming information granules with the use of the principle of justifiable granularity (PJG). With this regard, we discuss a number of parameters used in this development of information granules as well as quantify the quality of the granules produced in this manner. In the sequel, Fuzzy C -Means is applied to cluster the derived information granules, which are represented in a parametric manner and associated with weights resulting from the usage of the PJG. The quality of clustering results is evaluated through the use of the reconstruction criterion (quantifying the concept of information granulation and degranulation). A suite of experiments using synthetic and publicly available datasets is reported to quantify the performance of the proposed approach and highlight its key features.
In this paper, we elaborate on a new design approach to the development and analysis of granular input spaces and ensuing granular modeling. Given a numeric model (no matter what specific design methodology has been used to construct it and what architecture has been adopted), we form a granular input space through allocating a certain level of information granularity across the input variables. The formation of granular input space helps us gain a better insight into the ranking of input variables with respect to their precision (the variables with a lower level of information granularity need to be specified in a precise way when estimating the inputs). As a consequence, for granular inputs, the outputs of the granular model are also information granules (say, intervals, fuzzy sets, rough sets, etc.). It is shown that the process of forming granular input space can be sought as an optimization of allocation of information granularity across the input variables so that the specificity of the corresponding granular outputs of the granular model becomes the highest while coverage of data becomes maximized. The construction of granular input space dwells upon two fundamental principles of granular computing-the principle of justifiable granularity and the optimal allocation of information granularity. The quality of the granular input space is quantified in terms of the two conflicting criteria, that is, the specificity of the results produced by the granular model and the coverage of experimental data delivered by this model. In the ensuing optimization problem, one maximizes a product of specificity and coverage. Differential evolution is engaged in this optimization task. The experimental studies involve both synthetic dataset and data coming from the machine learning repository.
The identification of fuzzy rule-based systems is considered. By their nature, these fuzzy models are geared toward capturing relationships between information granules — fuzzy sets. The level of granularity of fuzzy sets helps establish a required level of detail that is of interest in the given modelling environment. The form of the information granules themselves (in particular their distribution and type of membership functions) becomes an important design feature of the fuzzy model, contributing to its structural as well as parametric optimisation. This, in turn, calls for a comprehensive and efficient framework of information (data) granulation, and the one introduced in the study involves a hard C-means (HCM) clustering method and genetic algorithms (GAs). HCM produces an initial collection of information granules (clusters) that are afterwards refined in a parametric way with the aid of a genetic algorithm. The rules of the fuzzy model assume the form `if x1 is A and x2 is B and · · · and xn is W then y=phis(x1 , x2 ,…, xn , param) and come in two forms: a simplified one that involves conclusions that are fixed numeric values (that is, phis is a constant function), and a linear one where the conclusion part (phis) is viewed as a linear function of inputs. The parameters of the rules are optimised through a standard method of linear regression (least square error method). An aggregate objective function with weighting factor used in this study helps maintain a balance between the performance of the model for training and testing data. The proposed identification framework is illustrated with the use of two representative numerical examples.
Fuzzy cognitive maps (FCMs) are a convenient tool for modeling of dynamic systems by means of concepts connected by cause-effect relationships. The FCM models can be developed either manually (by the experts) or using an automated learning method (from data). Some of the methods from the latter group, including recently proposed Nonlinear Hebbian Learning (NHL) algorithm, use Hebbian law and a set of conditions imposed on output concepts. In this paper, we propose a novel approach named data-driven NHL (DD-NHL) that extends NHL method by using historical data of the input concepts to provide improved quality of the learned FCMs. DD-NHL is tested on both synthetic and real-life data, and the experiments show that if historical data are available, then the proposed method produces better FCM models when compared with those formed by the generic NHL method.
The study is devoted to linguistic data mining, an endeavor that exploits the concepts, constructs, and mechanisms of fuzzy set theory. The roles of information granules, information granulation, and the techniques therein are discussed in detail. Particular attention is given to the manner in which these information granules are represented as fuzzy sets and manipulated according to the main mechanisms of fuzzy sets. We introduce unsupervised learning (clustering) where optimization is supported by the linguistic granules of context, thereby giving rise to so-called context-sensitive fuzzy clustering. The combination of neuro, evolutionary, and granular computing in the context of data mining is explored. Detailed numerical experiments using well-known datasets are also included and analyzed.
