Granular Fuzzy Model Research Articles

Granular fuzzy rule-based model construction under the collaboration of multiple organizations

In the real world, phenomena are often observed and recorded by multiple organizations which results in multiple sources of data. When dealing with such data, the centralized modeling approach aims to achieve collaborative modeling by fusing multiple sources of data into a single data set, which may pose challenges to data privacy. Unlike centralized modeling, the distributed modeling approach can effectively solve the privacy issue. However, modeling approaches based on this idea still suffer from either low prediction accuracy or high communication costs. In this study, we propose a collaborative modeling strategy for multi-source data based on fuzzy rule-based models (FRBMs) to balance the needs of both model prediction accuracy and efficiency. First, we adopt the concept and algorithm of collaborative fuzzy clustering (CFC) to improve prediction accuracy and reduce communication costs by improving the CFC algorithm. Then, we construct a granular FRBM for multi-source data based on the principle of justifiable granularity (PJG) by integrating local models into a more robust and perfect global model. Finally, we improve the performance evaluation index of the existing granular FRBM and propose two model optimization schemes to further improve the performance of the model. We conduct experiments on both synthetic and publicly available data sets to demonstrate the effectiveness of the proposed method.

A Design and Its Application of Multi-Granular Fuzzy Model with Hierarchical Tree Structures

This paper is concerned with the design of a context-based fuzzy C-means (CFCM)-based multi-granular fuzzy model (MGFM) with hierarchical tree structures. For this purpose, we propose three types of hierarchical tree structures (incremental, aggregated, and cascaded types) in the design of MGFM. In general, the conventional fuzzy inference system (FIS) has problems, such as time consumption and an exponential increase in the number of if–then rules when processing large-scale multivariate data. Meanwhile, the existing granular fuzzy model (GFM) reduces the number of rules that increase exponentially. However, the GFM not only has overlapping rules as the cluster centers become closer but also has problems that are difficult to interpret due to many input variables. To solve these problems, the CFCM-based MGFM can be designed as a smaller tree of interconnected GFMs. Here, the inputs of the high-level GFMs are taken from the output to the low-level GFMs. The hierarchical tree structure is more computationally efficient and easier to understand than a single GFM. Furthermore, since the output of the CFCM-based MGFM is a triangular fuzzy number, it is evaluated based on a performance measurement method suitable for the GFM. The prediction performance is analyzed from the automobile fuel consumption and Boston housing database to present the validity of the proposed approach. The experimental results demonstrate that the proposed CFCM-based MGFM based on the hierarchical tree structure creates a small number of meaningful rules and solves prediction-related problems by making them explainable.

Fuzzy Rule-Based Models: A Design with Prototype Relocation and Granular Generalization

Fuzzy rule-based models and the extension of classical fuzzy models have been widely used in many domains. From a holistic perspective, regardless of the design methods and rules adopted in a fuzzy model, the determination of fuzzy sets is a pivotal issue. In the proposed methods, instead of traditional data clustering with no directional tendency, we introduce an optimization algorithm that can adjust the position of the prototypes of zero- and first-order fuzzy models to learn internal structure information from the data in the process of parameter identification. Furthermore, to build a granular fuzzy model, the prototypes are then scaled to more robust intervals by generating information granularity with specific semantics such that they split the whole output space. Particle swarm optimization algorithm is applied to adjust both the locations of the prototypes and the allocation of information granularity to improve the performance of the data-driven models. Experimental studies on synthetic and real-world datasets are provided to demonstrate the effectiveness of these methods.

From granulation-degranulation mechanisms to fuzzy rule-based models: Augmentation of granular-based models with a double fuzzy clustering

In this study, we develop a novel clustering with double fuzzy factors to enhance the performance of the granulation-degranulation mechanism, with which a fuzzy rule-based model is designed and demonstrated to be an enhanced one. The essence of the developed scheme is to optimize the construction of the information granules so as to eventually improve the performance of the fuzzy rule-based models. In the design process, a prototype matrix is defined to express the Fuzzy C-Means based granulation-degranulation mechanism in a clear manner. We assume that the dataset degranulated from the formed information granules is equal to the original numerical dataset. Then, a clustering method with double fuzzy factors is derived. We also present a detailed mathematical proof for the proposed approach. Subsequently, on the basis of the enhanced version of the granulation-degranulation mechanism, we design a granular fuzzy model. The whole design is mainly focused on an efficient application of the fuzzy clustering to build information granules used in fuzzy rule-based models. Comprehensive experimental studies demonstrate the performance of the proposed scheme.

Granular Aggregation of Fuzzy Rule-Based Models in Distributed Data Environment

Quite often, complex systems or phenomena are observed from various points of view yielding the particular subsets of data usually being composed of locally available attributes. Such datasets give rise to individual models. As is reflective of the local behavior of the system (global data), each model can produce different, albeit similar results. A critical issue is to aggregate the results coming from the individual models. In virtue of the diversity of the produced results, the aggregation process has to be reflective of this variety. Equally important is a way of quantifying the diversity of the individual results. In this article, we provide an efficient and original way of aggregation of the results by engaging a principle of justifiable granularity and in this manner leading to interval-valued results summarizing the results produced by a collection of models. We develop an overall design process and discuss the associated optimization mechanism leading to a granular fuzzy model of a global nature. The detailed scheme of the principle of justifiable granularity is discussed along with the related performance indexes; in particular, two modes of design of information granules are investigated. The quality of the granular model is quantified with the aid of the criteria of coverage and specificity.

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.

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.

Granular description of data in a non-stationary environment

When developing models of non-stationary data, it becomes imperative to endow them with some meaningful update mechanisms, viz to provide with sufficient capabilities to accommodate changing characteristics of the environment. These mechanisms make the models of data evolvable. In the study, we introduce and discuss a class of evolvable models of data with the main objective to describe and interpret the incoming data. We advocate that information granularity and ensuing information granules are central to the characterization and interpretation of the dynamics and variability of numeric data. The relevance of information granules describing the data is evaluated in their abilities to construct the associated Takagi–Sugeno rule-based models. It is shown how the condition part of the rules formed by information granules changes when exposed to data of varying characteristics. Along with the structural facet of evolvability discussed is its parametric manifestation present in terms of the changes (updates) of the parameters of the local models standing in the conclusion part of the rules. The continuity of the evolving information granules (being crucial to their interpretability) is assured by running the clustering scheme initialized on the basis of the previously formed clusters (conditions of the rules) rather than starting FCM from some random configuration. We introduce some graph-oriented visualization means to provide a concise insight into the dynamics of information granules. As an interesting alternative, we introduce a granular fuzzy model where the added granularity of the parameters of a stationary fuzzy model is considered as a way to compensate for the non-stationary of the described system. A series of experiments is reported on with intent to demonstrate the performance of the model, analyze mechanisms of evolution of information granules, and deliver some useful comparative analysis.

Fuzzy rule-based models with interactive rules and their granular generalization

Processing realized in fuzzy rule-based models is associated with matching individual rules with the existing input data and aggregation of the levels of matching and conclusions of the rules. Rules are processed as individual entities. In this study, we introduce an augmentation of fuzzy models by facilitating interaction among the rules leading to more flexible type membership functions of fuzzy sets forming conditions of the rules (thus resulting in substantially advanced topology of the partition of the input space). Different ways of realizing interaction among the rules are studied. In the sequel, we develop a granular fuzzy model implied by the rule-based model showing how granular parameters of the original rule-based model enhance its quality expressed in terms of coverage of experimental data. The two evaluation criteria of the constructed granular model, namely coverage and specificity are studied. Experimental results are reported for a series of publicly available 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.

Granular fuzzy models: a study in knowledge management in fuzzy modeling

In system modeling, knowledge management comes vividly into the picture when dealing with a collection of individual models. These models being considered as sources of knowledge, are engaged in some collective pursuits of a collaborative development to establish modeling outcomes of global character. The result comes in the form of a so-called granular fuzzy model, which directly reflects upon and quantifies the diversity of the available sources of knowledge (local models) involved in knowledge management. In this study, several detailed algorithmic schemes are presented along with related computational aspects associated with Granular Computing. It is also shown how the construction of information granules completed through the use of the principle of justifiable granularity becomes advantageous in the realization of granular fuzzy models and a quantification of the quality (specificity) of the results of modeling. We focus on the design of granular fuzzy models considering that the locally available models are those fuzzy rule-based. It is shown that the model quantified in terms of two conflicting criteria, that is (a) a coverage criterion expressing to which extent the resulting information granules “cover” include data and (b) specificity criterion articulating how detailed (specific) the obtained information granules are. The overall quality of the granular model is also assessed by determining an area under curve (AUC) where the curve is formed in the coverage-specificity coordinates. Numeric results are discussed with intent of displaying the most essential features of the proposed methodology and algorithmic developments.