Granular Fuzzy Models Research Articles

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.

Online structural clustering based on DBSCAN extension with granular descriptors

In online structural clustering, general density-based clustering algorithms have problems of low scalability and high computation cost, especially in big data analysis, this paper proposed a DBSCAN extension algorithm with consideration of granule computing to handle these problems. This algorithm mainly makes use of advantages of DBSCAN and granular descriptors to realize effective and efficient structural online clustering. Frist, to extract structural clusters effectively, DBSCAN is considered as the basic clustering algorithm in this research. Second, since DBSCAN’s results are not numerical for online testing, this paper proposes to apply granule computing (GrC) to construct information granules describing arbitrarily-shaped clusters from DBSCAN. Third, to realize an efficient online structural clustering, especially in big data analysis, a series of granular fuzzy models are built with consideration of structural information, then a rule-based model is formed for guiding online clustering of new testing data. Through the proposed method, the online clustering ability of DBSCAN is developed with reduced computation cost, meanwhile the structural clustering ability is also retained in online testing. Experiments on synthetic data, publicly available data and real-world data are discussed, online testing accuracy and computation time are evaluated to validate the feasibility and effectiveness of the proposed method.

Granular Fuzzy Rule-Based Modeling With Incomplete Data Representation.

Incomplete data are frequently encountered and bring difficulties when it comes to further processing. The concepts of granular computing (GrC) help deliver a higher level of abstraction to address this problem. Most of the existing data imputation and related modeling methods are of numeric nature and require prior numeric models to be provided. The underlying objective of this study is to introduce a novel and straightforward approach that uses information granules as a vehicle to effectively represent missing data and build granular fuzzy models directly from resulting hybrid granular and numeric data. The evaluation and optimization of this method are guided by the principle of justifiable granularity engaging the coverage and specificity criteria and carried out with the help of particle swarm optimization. We provide a collection of experimental studies using a synthetic dataset and several publicly available real-world datasets to demonstrate the feasibility and analyze the main features of this 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.

Landslide Risk Assessment Using Granular Fuzzy Rule-Based Modeling: A Case Study on Earthquake-Triggered Landslides

Landslides are one type of destructive and recurring natural calamities in the mountainous regions. The landslide occurrences often lead to immense damage to local infrastructure and loss of land, human lives and livestock. Data-driven risk assessment of landslide risk plays a crucial role in preventing the incoming landslide occurrences and recurrences. In this research, we developed a human-centric framework using information-granules to perform risk assessment of a group of landslides. First, the density-based spatial clustering of applications with noise (DBSCAN) has been selected as the backbone unsupervised learning method to subclusters for landslide risk indication. The clustering outcomes are visualized via t-distributed stochastic neighbor embedding (t-SNE) in the 2-D embedding space. Second, the prototype points within the subclusters produced by DBSCAN are computed for granular construction. Third, interval-based information-granules are constructed and measured via coverage, specificity and area under the coverage-specificity curve (AUC). Last, with the optimal information-granules constructed, two risk measures namely Value-at-Risk (VaR) and Conditional-Value-at-Risk (CVaR) are computed to interpret the rule-based information-granules with respect to the key attributes. Comparative experiments have also been performed against other benchmarking clustering approaches. Computational results indicate that the information-granules constructed from DBSCAN subclusters offered enhanced performance in reveal meaningful information-granules and provide promising results. The proposed approach can capture the main essence of landslide pattern with higher interpretability and help to reduce the computing overhead.

Ensemble of evolving optimal granular experts, OWA aggregation, and time series prediction

This paper presents an online-learning ensemble framework for nonstationary time series prediction. Optimal granular fuzzy rule-based models with different objective functions and constraints are evolved from data streams. Evolving optimal granular systems (eOGS) consider multiobjective optimization, the specificity of information, model compactness, and variability and coverage of the data within the process of modeling data streams. Forecasts of individual base eOGS models are combined using averaging aggregation functions: ordered weighted averaging (OWA), weighted arithmetic mean, median, and linear non-inclusive centered OWA. Some aggregation functions use specific weights for the relevance of the base models and exclude extreme values and outliers. The weights of other aggregation functions are adapted over time based on a quadratic programming problem and the data within a sliding window. This paper investigates whether an online-learning ensemble can outperform individual eOGS models, and which aggregation function provides the most accurate forecasts. Real multivariate weather time series, particularly time series of daily mean temperature, air humidity, and wind speed from different weather stations, such as Paris–Orly, Frankfurt–Main, Reykjavik, and Oslo–Blindern, are used for evaluation. The results show that ensemble schemes outperform individual models. The proposed linear non-inclusive centered OWA function provided the most accurate numerical predictions.

Granular Fuzzy Modeling for Multidimensional Numeric Data: A Layered Approach Based on Hyperbox

At present, the development of most of the granular fuzzy models depends upon some well-established numeric ones. In this study, a layered approach used to directly construct granular fuzzy models based on multidimensional numeric data is presented by engaging design methodology of granular computing. The crux of the approach involves a construction of interval information granules in the output space and the corresponding hyperbox information granules in the input space. A method of constructing these information granules and the hyperbox-based granular fuzzy model formed around them is studied in detail. Two different schemes to decode the formed hyperbox-based granular fuzzy model are also presented. Furthermore, a measure of a composite quality of the formed hyperbox-based granular fuzzy model is proposed along with the concept of coverage and specificity of resulting information granules. A number of experimental studies are reported, which offer a useful insight into the effectiveness of the presented approach, as well as reveal the impact of critical parameters on the performance of the established models.

Interval Granular Fuzzy Models: Concepts and Development

In this paper, we present a concept of interval granular fuzzy models and elaborate on their detailed development scheme and performance evaluation. The crux of the underlying design is that the information granules and information granularity which play a pivotal role in human cognitive and decision-making activities are incorporated into existing fuzzy modeling methods to realize system modeling at the level of information granules. The overall development process of interval granular fuzzy model includes two stages. At the first stage, we form input-error data by computing residual error resulting from the well-established numeric model and then establish granular structures positioned in the input space by clustering input-error data and invoking the principle of justifiable information granularity for weighted data. These granular structures discovered in the input space become condition parts of the rules of the developed interval granular fuzzy model. At the second stage, the interval information granules positioned in the error space are directly induced with the aid of granular structures discovered in the input space. These interval information granules describe the range of residual error produced by the numeric model and are exploited to form the conclusion part of the corresponding rules of the developed interval granular fuzzy model. So far, the interval granular fuzzy model is completely constructed, whose rules help compensate the discrepancies of the numeric model. The output of the developed model is an interval information granule showing a range of possible residual error produced by the numeric model. The several performance indices are presented to evaluate the developed interval granular fuzzy model. A series of numeric experiments completed for synthetic data and real-world data coming from the machine learning repository provide a useful insight into the effectiveness of the presented development scheme, reveal the impact of some parameters on the performance of the developed model and demonstrate its advantages.

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.

Granular fuzzy rough sets based on fuzzy implicators and coimplicators

This paper introduces granular fuzzy rough sets in the view of fuzzy implicators and fuzzy coimplicators, and discusses the constructive and axiomatic approach to fuzzy granules based on fuzzy implicators and coimplicators. Moreover, we study the connection between fuzzy granules and fuzzy relations and discuss the relationship between existing granular fuzzy rough set models and that proposed in this paper. Considering the absolute error limit, we introduce the concept of the granular variable precision fuzzy rough sets based on fuzzy implicators and coimplicators. Then we present four propositions to ensure that the approximation operators can be efficiently calculated.

Granular Fuzzy Rule-Based Models: A Study in a Comprehensive Evaluation and Construction of Fuzzy Models

Fuzzy models are regarded as numeric constructs and as such are optimized and evaluated at the numeric level. In this study, we depart from this commonly accepted position and propose a granular evaluation of fuzzy models and present an augmentation of fuzzy models by forming information granules around numeric values of the parameters and constructions of the models. The concepts and algorithms of granular fuzzy models are discussed in the setting of Takagi–Sugeno rule-based architectures. We show how different protocols of forming and allocating information granules lead to the improvement of the granular performance of the models. Different from the standard numeric performance measure of fuzzy models coming in the form of the root mean squared error index, two performance measures are introduced that are pertinent to granular constructs, namely coverage and specificity. Furthermore, we propose a global indicator implied by these two measures, called an area under the curve, being computed for the characteristics of the granular model expressed in the coverage-specificity coordinates. A series of experimental studies is reported, which offers a comprehensive overview of the introduced performance measure criteria as well as the underlying realization of the granular fuzzy models.

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.

SYSTEM MODELING WITH FUZZY MODELS: FUNDAMENTAL DEVELOPMENTS AND PERSPECTIVES

In this study, we offer a general view at the area of fuzzy modeling and fuzzymodels, identify the visible development phases and elaborate on a new and promisingdirections of system modeling by introducing a concept of granular models. Granularmodels, especially granular fuzzy models constitute an important generalization of existingfuzzy models and, in contrast to the existing models, generate results in the form ofinformation granules (such as intervals, fuzzy sets, rough sets and others). We present arationale and deliver some key motivating arguments behind the emergence of granularmodels and discuss their underlying design process. Central to the development of granularmodels are granular spaces, namely a granular space of parameters of the models and agranular input space. The development of the granular model is completed through anoptimal allocation of information granularity, which optimizes criteria of coverage andspecificity 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 thatachieving a sound coverage-specificity tradeoff (compromise) is of paramount relevance inthe realization of the granular models.

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...

Design of rule-based models through information granulation

In this study, we explore the combination of two well defined topics in fuzzy systems research: fuzzy rule based systems, and information granulation. Rule based systems are a powerful and well-studied form of knowledge representation, due to their approximation abilities and interpretability. In recent years, these types of systems have become increasingly powerful with regards to modeling accuracy; however, many of these improvements come at the cost of model interpretability. This recent direction of research has left an unexplored avenue towards the generation of increasingly interpretable fuzzy rule based models, which we intend to explore. Information granulation is a relatively new, yet very promising area of research in human centric systems. As a form of knowledge representation, information granulation is very well suited to fuzzy rule based systems, where rules represent linguistic quantities in a, intuitively understandable format. It is notable that the combination of these two concepts has been left largely unstudied. We aim to explore this union by defining a methodology for the construction of a partially granular fuzzy rule based model. The aim of this novel model format is to provide a first step in the improvement of fuzzy model interpretability, through the use of information granulation. We are additionally interested in studying new ways of generating fuzzy rules; hence, we will also look at the use of hierarchical clustering as a potential alternative to the tried and tested Fuzzy C Means clustering algorithm. The models created using hierarchical clustering are then compared with those generated using Fuzzy C Means to evaluate the effectiveness of this algorithm. As a result of these experiments, we demonstrate that partially granular fuzzy rules are capable of providing a significant improvement to fuzzy rule interpretability, and we believe that granular fuzzy models present an exciting avenue of future research in human centric systems.

Granular fuzzy models: Analysis, design, and evaluation

The study is concerned with a design of granular fuzzy models. We exploit a concept of information granularity by developing a model coming as a network of intuitively structured collection of interval information granules described in the output space and a family of induced information granules (in the form of fuzzy sets) formed in the input space. In contrast to most fuzzy models encountered in the literature, the results produced by granular models are information granules rather than plain numeric entities. The design of the model concentrates on a construction of information granules that form a backbone of the overall construct. Interval information granules positioned in the output space are built by considering intervals of equal length, equal probability, and developing an optimized version of the intervals. The induced fuzzy information granules localized in the input space are realized by running a conditional Fuzzy C-Means (FCM). The performance of the model is assessed by considering criteria of coverage and information specificity (information granularity). Further optimization of the model is proposed along the line of an optimal re-distribution of input information granules induced by the individual interval information granules located in the output space. Experimental results involve some synthetic low-dimensional data and publicly available benchmark data sets.

Granular fuzzy modeling with evolving hyperboxes in multi-dimensional space of numerical data

Clustering has been applied to numerous areas, including signal and image processing. Many approaches have been developed over the years to efficiently construct granular models on a basis of numerical experimental data. In this study, we propose a novel approach to construct a granular model that is fundamentally designed around information granules regarded as hyperboxes. Several studies have been focused on building a set of hyperboxes around data; one of them being a Min-Max Neural Network (NN) algorithm. Here we develop two different methods to construct these information granules, nevertheless some essential similarities to previous studies can be found. In particular, hyperboxes are constructed by using some reference data, and they are endowed with some parametric flexibility to facilitate controlling their size, whereas the construction of the hyperboxes involve elimination or reduction of possible overlaps between them. In the proposed approach, given a set of input and output data pairs, we construct interval-based information granules to partition the output space (viz. the space of the output variable). On a basis of these intervals, we carry out a so-called context-based Fuzzy C-Means algorithm to construct cluster centers (prototypes) in the multivariable input space. These prototypes serve as hyperbox cores. To construct the information granules, two methods are studied: one develops a family of hyperboxes by realizing some constrictions, while the other one engages Differential Evolution (DE) to realize further optimization. To reduce overlap, two methods are tested: one being previously proposed for the min-max NN and a new one, which engages DE to optimize the overlap reduction. Experimental studies involve synthetic data and publicly available real-world data. The results are compared with the outcomes produced by the algorithm proposed by Simpson. The performance of the method is quantified and it is demonstrated that the obtained results are substantially better when dealing with multi-dimensional data.

Designing granular fuzzy models: A hierarchical approach to fuzzy modeling

In this study, we elaborate on a distributed fuzzy modeling and ensuing granular fuzzy modeling. Such modeling is realized in the presence of separate and locally available data while the ensuing fuzzy rule-based models constructed on their basis are regarded as individual sources of knowledge. In virtue of an inherent diversity of these sources (models) and in an attempt to quantify it, a global model being formed at the higher level of hierarchy is becoming more abstract than those at the lower level is referred to as a granular fuzzy model. An essential concept of this class of models is introduced and their enhanced functionality is studied. Furthermore, we show interesting linkages of these models with type-2 fuzzy models studied in the literature. We highlight a number of arguments motivating a need and justifiable relevance of higher type of information granules. A detailed discussion on fuzzy rule-based models exhibiting an interesting aspect of an incremental format of the rules (whose rules capture an incremental description of input–output relationships formed with respect to some simple reference model (say, constant or linear) is presented. The design practice of the models is elaborated on by highlighting in this context the use of augmented fuzzy clustering. The construction of a granular fuzzy model is guided by the principle of justifiable granularity using which we show how granular parameters of the models are formed. The performance of the model is quantified with respect to the two criteria, namely coverage of experimental data and specificity of granular results. Experimental studies are reported for both synthetic and publicly available data sets.

From fuzzy data analysis and fuzzy regression to granular fuzzy data analysis

This note offers some personal views on the two pioneers of fuzzy sets, late Professors Hideo Tanaka and Kiyoji Asai. The intent is to share some personal memories about these remarkable researchers and humans, highlight their long-lasting research accomplishments and stress a visible impact on the fuzzy set community.The note elaborates on new and promising research avenues initiated by fuzzy regression and identifies future developments of these models emerging within the realm of Granular Computing and giving rise to a plethora of granular fuzzy models and higher-order and higher-type granular constructs.