Optimal Allocation Of Information Granularity Research Articles

Consensus Model Driven by Interpretable Rules in Large-Scale Group Decision Making With Optimal Allocation of Information Granularity

In group decision making (GDM), consensus level is regarded as a critical criterion to measure the effectiveness and availability of the final group decision solution. Consensus model is aimed at conducting the decision group to reach agreement through the process of group negotiation, advice feedback, and opinion modification, which is time-consuming and rests with the willingness and behavior of individual decision makers. Thus, to guide the shift in the opinions of decision makers within a limited time, it is essential to design an effective, interpretable, and fair consensus mechanism in GDM, which is particularly vital when a mass of decision makers (e.g., more than 30) are involved in the decision process, viz., we encounter a large-scale GDM (LSGDM). With the involvement of information granulation, this study presents a rule-based consensus model in LSGDM by optimally allocating the level of information granularity to each decision maker. The opinions of decision makers in LSGDM are divided into different clusters by engaging the fuzzy <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$C$ </tex-math></inline-formula> -means method. Inspired by a generic fuzzy rule-based model, the radius of the individual preference granule (PG) is calculated by a weighted linear combination of the granularity levels allocated to the clusters. Then, a consensus model with the optimal allocation of information granularity (CMOIG) is built to determine the granularity level for each cluster by minimizing the sum of radii of individual PG. An interactive consensus reaching process is proposed with the proposed CMOIG and fuzzy modification rules. The CMOIG and fuzzy modification rules simultaneously guarantees high efficiency and interpretability, and the generation method of PGs leads to high fairness due to low discrepancy among the decision group. Finally, numerical and comparative experiments are conducted in detail to verify the validity and superiority of the presented models in terms of the efficiency, interpretability, and fairness.

Design and Development of Granular Fuzzy Rule-Based Models for Knowledge Transfer

As an effective way for knowledge representation and processing, fuzzy rule-based models have been extensively studied and widely used in practice. In many circumstances, a very limited amount of data or insufficient computational resources make the construction of accurate models a genuine challenge. In this study, a granular augmentation of fuzzy rule-based models is proposed with intent to realize knowledge transfer in system modeling. This research mainly focuses on how to effectively exploit the existing fuzzy model, which has been constructed on extensive previously acquired experimental evidence and could be regarded as source of knowledge, in a new environment where only very limited experimental evidence is available. Rather than constructing a new model from scratch, knowledge conveyed by the existing model could be retained and reused in the target domain. The originality and innovation of this study lies in the adaption of the existing model to the new environment through optimal allocation of information granularity to produce granular fuzzy models, which are more abstract and general than the original numeric constructs. The granular fuzzy models yield results in a granular form whose quality is evaluated using the coverage and specificity criteria.

Building a consensus for the best-worst method in group decision-making with an optimal allocation of information granularity

In a multiple-criteria group decision-making process, it may be unrealistic to assume that all experts are specialized in all aspects of the entire problem and can reach full agreement. To obtain a suitable result with more flexibility and robustness in group decision-making, this study concerns the introduction of information granules to the best-worst method model, regarded as an essential design asset to reach a consensus in a group decision-making scenario. More specifically, each pairwise comparison is performed through information granules instead of single numbers, by using an optimal allocation of information granularity. Further, we develop an optimization model that considers both consistency and consensus in one problem. The particle swarm optimization algorithm is used as an optimization method to find the optimal particle of the granular decision model. Furthermore, a new convergent iterative algorithm is developed to obtain the desired decision matrix. Next, an experimental study is presented to demonstrate the effectiveness, flexibility, and essence of the proposed model. Finally, a case study of autohome.com is conducted to select suitable automobiles for a company based on online reviews. The solving procedures are demonstrated, to further illustrate the feasibility of the proposed method.

A Differential Evolution-Based Consistency Improvement Method in AHP With an Optimal Allocation of Information Granularity.

In the analytic hierarchy process (AHP), the reciprocal matrix is generated based on the pairwise comparisons completed among all the alternatives or attributes under consideration. To ensure reliability and validity of the decision solution, a certain modification of entries of the matrix is usually needed to improve the consistency of the reciprocal matrix. This study aims to present a consistency improvement method by admitting some level of information granularity in the evaluation process. This gives rise to a granular rather than numeric matrix of pairwise comparisons. First, with a given average level of information granularity, we present an optimal granularity model that is characterized by maximal consistency. One can maximize the consistency degree by invoking a process of allocation of information granularity across the corresponding modifications of the reciprocal matrix. Based on the optimal granularity model, an interactive consistency improvement process is presented with the involvement of the decision maker. Then, an adaptive differential evolution algorithm is applied to optimize entries of the modified reciprocal matrix. Detailed experiments along with a thorough comparative analysis are completed to demonstrate the effectiveness of the proposed method.

Granular Fuzzy Modeling Guided Through the Synergy of Granulating Output Space and Clustering Input Subspaces

As an augmentation of classic fuzzy models, granular fuzzy models (GFMs) have been applied to many fields being in rapport with experimental data, models, and users. However, most of the existing methods used to construct GFMs are based on the principle of optimal allocation of information granularity, which requires that a numeric model be provided in advance. In this paper, a straightforward and convincing modeling method is proposed to directly construct GFM on a basis of experimental data. The method first granulates the output space to form some interval information granules with distinct semantics and then uses them to partition the entire input space into a series of input subspaces. Subsequently, an initial GFM is emerged by using "If-Then" rules to relate with those interval information granules positioned in the output space and structures expressed in prototypes that are produced by clustering individual input subspaces. Further, the initial GFM is also refined by continuously migrating prototypes in individual input subspaces. The experimental studies using the synthetic dataset and several real-world datasets are reported. They offer a useful insight into the feasibility and effectiveness of the proposed modeling method and reveal the impact of parameters on the performance of the ensuing GFMs. An application example is also presented to exhibit the advantages of the resulting GFM.

A Development of Granular Input Space in System Modeling.

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.

A Granular Approach to Interval Output Estimation for Rule-Based Fuzzy Models.

Rule-based fuzzy models play a dominant role in fuzzy modeling and come with extensive applications in the system modeling area. Due to the presence of system modeling error, it is impossible to construct a model that fits exactly the experimental evidence and, at the same time, exhibits high generalization capabilities. To alleviate these problems, in this study, we elaborate on a realization of granular outputs for rule-based fuzzy models with the aim of effectively quantifying the associated modeling errors. Through analyzing the characteristics of modeling errors, an error model is constructed to characterize deviations among the estimated outputs and the expected ones. The resulting granular model comes into play as an aggregation of the regression model and the error model. Information granularity plays a central role in the construction of granular outputs (intervals). The quality of the produced interval estimates is quantified in terms of the coverage and specificity criteria. The optimal allocation of information granularity is determined through a combined index involving these two criteria pertinent to the evaluation of interval outputs. A series of experimental studies is provided to demonstrate the effectiveness of the proposed approach and show its superiority over the traditional statistical-based method.

Granular Models and Granular Outliers

In this study, we propose a new design methodology of granular fuzzy models, introduce its further generalization in the form of granular fuzzy models of higher type, and discuss detection and characterization of outliers expressed with regard to the constructed information granules. In recent years, various models that describe the system from different perspectives have been built to resolve the growing challenges brought on by real-world systems. These models usually aim to achieve the highest accuracy at the cost of model interpretability. To improve the interpretability of models, a concept of granular models has been developed in the setting of granular computing. We focus on the formation of a general granular model at the higher level of hierarchy by taking advantage of existing models developed at the lower (numeric) level. Here, information granularity is regarded as an important design asset whose optimal allocation across the parameters of the original model gives rise to granular models. Next, through an allocation of information granularity to the existing type-1 granular model, we create an interesting and useful augmentation of the granular fuzzy model by forming a granular fuzzy model of type-2. Higher type granular models are also realized through the optimal allocation of information granularity. We examine the problem of outlier detection in granular models where outliers are expressed with regard to the constructed information granules. Experimental results demonstrate that granular fuzzy models provide significant improvement to the model's interpretability, and the proposed outlier detection method based on granular models of higher type is effective.

A Design of Granular Takagi–Sugeno Fuzzy Model Through the Synergy of Fuzzy Subspace Clustering and Optimal Allocation of Information Granularity

Fuzzy models have been commonly used in system modeling and model-based control. Among various fuzzy models, Takagi–Sugeno (TS) fuzzy models form one of the intensively studied and applied categories of models. In this study, we are concerned with a development of a granular TS fuzzy model realized on a basis of numerical evidence and completed through a combination of fuzzy subspace clustering and the principle of optimal allocation of information granularity. The TS fuzzy models are built with the use of the fuzzy subspace clustering algorithm. Information granularity is regarded as a crucial design asset whose optimal allocation gives rise to granular fuzzy models and makes the constructed models to become better in rapport with experimental data. In comparison with fuzzy models, granular fuzzy models produce results (outputs) that are information granules rather than numeric entities being encountered in fuzzy models. In contrast with the commonly used optimization criteria, which emphasize the highest accuracy encountered at the numeric level, the performance of the granular TS fuzzy model is quantified in terms of the coverage and specificity criteria where such criteria are of interest in the evaluation of quality of information granules vis-a-vis experimental (numeric) data. Experimental results are reported for both synthetic datasets and publicly available data sets coming from the UCI machine learning repository.

From fuzzy rule-based models to their granular generalizations

In recent years, granular fuzzy models have become an intensively studied category of fuzzy models. Granular fuzzy models help elevate the existing models to the higher level of abstraction subsequently making the resulting constructs in a better rapport with real-world systems. In contrast to numeric models, granular models produce results in the form of information granules (such as intervals, fuzzy sets, rough sets and alike). A number of studies have been focused on efficient designing granular fuzzy models where the information granularity has been formalized in the form of intervals. In this study, we propose a novel conceptual and algorithmic approach by elevating existing fuzzy models to granular fuzzy models in the form of fuzzy sets. Concentrating on Takagi-Sugeno fuzzy rule-based models (being commonly used in the literature), we establish a way of an optimal allocation of information granularity across the parameters of the original model and making them fuzzy sets (fuzzy numbers). As a result, the outputs of the granular fuzzy model become fuzzy sets. In the process of allocation of information granularity, we resort ourselves to some population-based meta-heuristic optimization techniques such as particle swarm optimization and differential evolution. To evaluate the performance of granular fuzzy models, we engage the principle of justifiable information granularity leading to the assessment and optimization of the coverage and specificity of the resulting fuzzy set. Besides, we involve the defuzzification (decoding) process to evaluate the impact of granular parameters on the quality of the numeric manifestation of the model. Comprehensive experimental studies involving synthetic and publicly available data sets are reported to demonstrate the performance of the granular fuzzy models.

The development of granular rule-based systems: a study in structural model compression

In this study, we develop a comprehensive design process of granular fuzzy rule-based systems. These constructs arise as a result of a structural compression of fuzzy rule-based systems in which a subset of originally existing rules is retained. Because of the reduced subset of the originally existing rules, the remaining rules are made more abstract (general) by expressing their conditions in the form of granular fuzzy sets (such as interval-valued fuzzy sets, rough fuzzy sets, probabilistic fuzzy sets, etc.), hence the name of granular fuzzy rule-based systems emerging during the compression of the rule bases. The design of these systems dwells upon an important mechanism of allocation of information granularity using which the granular fuzzy rules are formed. The underlying optimization consists of two phases: structural (being of combinatorial character in which a subset of rules is selected) and parametric (when the conditions of the selected rules are made granular through an optimal allocation of information granularity). We implement the cooperative particle swarm optimization to solve optimization problem. A number of experimental studies are reported; those include fuzzy rule-based systems.

Optimal allocation of information granularity in system modeling through the maximization of information specificity: A development of granular input space

In this study, we introduce a concept of a granular input space in system modeling, in particular in fuzzy rule-based modeling. The underlying problem can be succinctly formulated in the following way: given is a numeric model, develop an efficient way of forming granular input variables so that the corresponding granular outputs of the model achieve the highest level of specificity. The rationale behind the formulation of the problem is offered along with several illustrative examples. In conjunction with the underlying idea, developed is an algorithmic framework supporting an optimization of the specificity of the model exposed to granular inputs (data). It is dwelled upon one of the principles of Granular Computing, namely an optimal allocation of information granularity. For illustrative purposes, the study is focused on information granules formalized in terms of intervals (however the proposed approach becomes equally relevant for other formalism of information granules). Some comparative analysis with the existing idea of global sensitivity analysis is also carried out by contrasting the essential differences among the two approaches and analyzing the results of computational experiments.

The granular extension of Sugeno-type fuzzy models based on optimal allocation of information granularity and its application to forecasting of time series

The Sugeno-type fuzzy models are used frequently in system modeling. The idea of information granulation inherently arises in the design process of Sugeno-type fuzzy model, whereas information granulation is closely related with the developed information granules. In this paper, the design method of Sugeno-type granular model is proposed on a basis of an optimal allocation of information granularity. The overall design process initiates with a well-established Sugeno-type numeric fuzzy model (the original Sugeno-type model). Through assigning soundly information granularity to the related parameters of the antecedents and the conclusions of fuzzy rules of the original Sugeno-type model (i.e. granulate these parameters in the way of optimal allocation of information granularity becomes realized), the original Sugeno-type model is extended to its granular counterpart (granular model). Several protocols of optimal allocation of information granularity are also discussed. The obtained granular model is applied to forecast three real-world time series. The experimental results show that the method of designing Sugeno-type granular model offers some advantages yielding models of good prediction capabilities. Furthermore, those also show merits of the Sugeno-type granular model: (1) the output of the model is an information granule (interval granule) rather than the specific numeric entity, which facilitates further interpretation; (2) the model can provide much more flexibility than the original Sugeno-type model; (3) the constructing approach of the model is of general nature as it could be applied to various fuzzy models and realized by invoking different formalisms of information granules.

From Fuzzy Models to Granular Fuzzy Models

AbstractIn this study, we offer a general view at the area of fuzzy modeling and elaborate on a new direction of system modeling by introducing a concept of granular models. Those models constitute a generalization of existing fuzzy models and, in contrast to existing models, generate results in the form of information granules (such as intervals, fuzzy sets, rough sets and others). We present a rationale and some key motivating arguments behind the emergence of granular models and discuss their underlying design process. Central to the development of granular models are granular spaces, namely a granular space of parameters of the models and a granular input space. The development of the granular model is completed through an optimal allocation of information granularity, which optimizes criteria of coverage and specificity of granular information. The emergence of granular models of type-2 and type-n, in general, is discussed along with an elaboration on their formation. It is shown that achieving a sou...

Granular fuzzy rule-based architectures: Pursuing analysis and design in the framework of granular computing

In this study, we propose a new concept of granular rule-based models whose rules assume a format if G(Ai) then G(fi) w hereG(.)s are granular generalizations of the numeric conditions and conclusions of the rules. Those generalizations can be expressed e.g., in terms of interval-valued, type-2 or probabilistic fuzzy sets. We discuss several classes of fuzzy models depending upon available information granules and offer a motivation present behind their emergence. The design of these granular architectures exploits the essentials of Granular Computing such as a principle of justifiable granularity and an optimal allocation of information granularity. Detailed investigations of the performance indexes (objective functions) along with the related optimization schemes are covered as well.

Corrigendum to “A characterization of electrocardiogram signals through optimal allocation of information granularity” [Artif. Intell. Med. 54 (2012) 125–134

Corrigendum to “A characterization of electrocardiogram signals through optimal allocation of information granularity” [Artif. Intell. Med. 54 (2012) 125–134

Concepts and Design Aspects of Granular Models of Type-1 and Type-2

In this study, we pursue a new direction for system modeling by introducing the concept of granular models, which produce results in the form of information granules (such as intervals, fuzzy sets, and rough sets). We present a rationale and several key motivating arguments behind the use of granular models and discuss their underlying design processes. The development of the granular model includes optimal allocation of information granularity through optimizing the criteria of coverage and specificity. The emergence and construction of granular models of type-2 and type-n (in general) is discussed. It is shown that achieving a suitable coverage-specificity tradeoff (compromise) is essential for developing granular models.

From logic descriptors to granular logic descriptors: a study in allocation of information granularity

Local sources of knowledge structured in the form of logic descriptors—constructs of fuzzy logic, are arranged together (structured) in the form of a global model coming as a high-level granular logic descriptor. The inherent granularity of the global descriptor of this nature arises as a manifestation of the diversity of the locally available descriptors. The granular descriptor can be expressed with the aid of any of the formal models of information granules including sets, fuzzy sets, rough sets, probabilistic granules and others. The architectural essence of the granular descriptor, which supports a quantification of the variability among the sources of knowledge, is realized through an optimal allocation of information granularity. Information granularity is treated as an important design asset and its allocation throughout the parameters of the logic descriptors helps quantify the diversity of individual sources of knowledge. Various protocols of allocation of information granularity along with an overall quantification of their effectiveness are discussed along with their numeric characterization.

Knowledge transfer in system modeling and its realization through an optimal allocation of information granularity

In this study, we introduce and discuss a concept of knowledge transfer in system modeling. In a nutshell, knowledge transfer is about forming ways on how a source of knowledge (namely, an existing model) can be used in presence of new, very limited experimental evidence. In virtue of the nature of the problem at hand (a situation encountered quite commonly, e.g. in project cost estimation), new data could be very limited and this scarcity of data makes it insufficient to construct a new model. At the same time, the new data originate from a similar (but not the same) phenomenon (process) for which the original model has been constructed so the existing model, even though it could applied, has to be treated with a certain level of reservation. Such situations can be encountered, e.g. in software engineering where in spite existing similarities, each project, process, or product exhibits its own unique characteristics. Taking this into consideration, the existing model is generalized (abstracted) by forming its granular counterpart – granular model where its parameters are regarded as information granules rather than numeric entities, viz. their non-numeric (granular) version is formed based on the values of the numeric parameters present in the original model. The results produced by the granular model are also granular and in this manner they become reflective of the differences existing between the current phenomenon and the process for which the previous model has been formed.In the study on knowledge transfer and reusability, information granularity is viewed as an important design asset and as such it is subject to optimization. We formulate an optimal information granularity allocation problem: assuming a certain level of granularity, distribute it optimally among the parameters of the model (making them granular) so that a certain data coverage criterion is maximized. While the underlying concept is general and applicable to a variety of models, in this study, we discuss its use to fuzzy neural networks with intent to clearly visualize the advantages of the approach and emphasize various ways of forming granular versions of the weights (parameters) of the connections of the network. Several granularity allocation protocols (ranging from a uniform distribution of granularity, symmetric and asymmetric schemes of allocation) are discussed and the effectiveness of each of them is quantified. The use of Particle Swarm Optimization (PSO) as the underlying optimization tool to realize optimal granularity allocation is discussed.

An Optimization of Allocation of Information Granularity in the Interpretation of Data Structures: Toward Granular Fuzzy Clustering

Clustering forms one of the most visible conceptual and algorithmic framework of developing information granules. In spite of the algorithm being used, the representation of information granules-clusters is predominantly numeric (coming in the form of prototypes, partition matrices, dendrograms, etc.). In this paper, we consider a concept of granular prototypes that generalizes the numeric representation of the clusters and, in this way, helps capture more details about the data structure. By invoking the granulation-degranulation scheme, we design granular prototypes being reflective of the structure of data to a higher extent than the representation that is provided by their numeric counterparts (prototypes). The design is formulated as an optimization problem, which is guided by the coverage criterion, meaning that we maximize the number of data for which their granular realization includes the original data. The granularity of the prototypes themselves is treated as an important design asset; hence, its allocation to the individual prototypes is optimized so that the coverage criterion becomes maximized. With this regard, several schemes of optimal allocation of information granularity are investigated, where interval-valued prototypes are formed around the already produced numeric representatives. Experimental studies are provided in which the design of granular prototypes of interval format is discussed and characterized.