Federated learning (FL) facilitates collaborative training among multiple clients while preserving data privacy by eliminating raw data transmission. However, the inherent data heterogeneity among participants induces bias during collaborative learning, significantly degrading the performance of local models. Existing FL solutions face critical challenges in achieving efficient knowledge transmission, particularly with respect to insufficient information extraction or excessive communication costs, which result in slow convergence and inferior performance. To address these limitations, we propose a novel FL framework in a synergy of multi-level prototype-based contrastive learning (CL) and soft label generation, named FedMPS. The proposed method first constructs multi-level prototypes from different layers of the model to capture semantic information in high-level features and detailed information in low-level features. These prototypes are then utilized through CL to enhance intra-class discriminability and intra-class consistency in the feature space. In addition, a prototype-guided soft label generation module is introduced to model latent interclass relationships in the output space. Instead of exchanging model parameters, FedMPS transmits only prototypes and soft labels, effectively reducing global knowledge shift and communication costs. Extensive experimental studies on six publicly available datasets validate the effectiveness of the proposed method when compared to the current state-of-the-art FL approaches. The code is available at github.com/wenxinyang1026/FedMPS.

Due to the arbitrary orientation of objects in aerial images, rotation equivariance is a critical property for aerial object detectors. However, recent studies on rotation-equivariant aerial object detection remain scarce. Most detectors rely on data augmentation to enable models to learn approximately rotation-equivariant features. A few detectors have constructed rotation-equivariant networks, but due to the breaking of strict rotation equivariance by typical downsampling processes, these networks only achieve approximately rotation-equivariant backbones. Whether strict rotation equivariance is necessary for aerial image object detection remains an open question. In this paper, we implement a strictly rotation-equivariant backbone and neck network with a more advanced network structure and compare it with approximately rotation-equivariant networks to quantitatively measure the impact of rotation equivariance on the performance of aerial image detectors. Additionally, leveraging the inherently grouped nature of rotation-equivariant features, we propose a multi-branch head network that reduces the parameter count while improving detection accuracy. Based on the aforementioned improvements, this study proposes the Multi-branch head rotation-equivariant single-stage Detector (MessDet), which achieves state-of-the-art performance on the challenging aerial image datasets DOTA-v1.0, DOTA-v1.5 and DIOR-R with an exceptionally low parameter count.

In this article, we propose a novel methodology for determining accurate positions of characteristic points encountered in the analysis of bioimpedance signals. The proposed approach fully utilizes two fundamental modeling pursuits based on fuzzy rule-based models and neural networks. We take advantages of the unique capabilities of fuzzy rule-based models to characterize the nonlinear relationship between the acquired bioimpedance signals and their temporal coordinates. The fuzzy modeling approach is used to approximate the process that generates the bioimpedance signals through a collection of rules (if–then statements). In the sequel, the parameters of the models are used as the inputs of neural network models to determine the position of characteristic points. We further augment the numeric neural network to its granular counterpart to accommodate the uncertainty in the available experimental evidence by allocating a certain level of information granularity across the parameter space. The resulting granular outputs (intervals) become reflective of the quality and level of confidence associated with the prediction results. The quality of the prediction results is quantified in terms of the coverage and specificity criteria. The performance index is also enhanced to deal with the situation when the positions provided by experts are also information granules (intervals). The performance of the proposed approach is justified through a collection of experiments carried out on the collected real-world bioimpedance signals. Experimental results show that the proposed approach achieved higher accuracy in determining the position of characteristic points in comparison with other existing methods.

In this study, we establish a new design methodology of granular models realized by augmenting the existing numeric models through analyzing and modeling their associated prediction error. Several novel approaches to the construction of granular architectures through augmenting existing numeric models by incorporating modeling errors are proposed in order to improve and quantify the numeric models' prediction abilities. The resulting construct arises as a granular model that produces granular outcomes generated as a result of the aggregation of the outputs produced by the numeric model (or its granular counterpart) and the corresponding error terms. Three different architectural developments are formulated and analyzed. In comparison with the numeric models, which strive to achieve the highest accuracy, granular models are developed in a way such that they produce comprehensive prediction outcomes realized as information granules. In virtue of the granular nature of results, the coverage and specificity of the constructed information granules express the quality of the results of prediction in a more descriptive and comprehensive manner. The performance of the granular constructs is evaluated using the criteria of coverage and specificity, which are pertinent to granular outputs produced by the granular models.

Pattern recognition plays an important role in the process of knowledge discovery. The construction of easily describable and interpretable classification rules is of vital importance in pattern recognition. In this study, we propose a development of fuzzy rule-based classifier for multiclass classification problems and elaborate on a privacy-preserving realization of the proposed methodology in the presence of decentralized datasets. Fuzzy rule-based models provide an effective and efficient alternative for characterizing the complex relationship between the input variables and target classes. An overall design process of the proposed classifier consists of two main phases: (a) formation of information granules (clusters) to reveal the underlying structure of the training data, and (b) construction of local classification rules whose outputs reflect the probability distribution of the input data over all the classes. The constructed information granules form a backbone of the architecture of the classifier while the optimization of the parameters of local rules is carried out through using a gradient descent method with the guidance of the cross-entropy loss function. Furthermore, a federated gradient-based optimization mechanism is utilized to construct fuzzy classifier in a privacy-preserving approach. The originalities of the proposed methodology are twofold: first, a design of fuzzy classifier through the synergy of cluster-centric architecture and the cross-entropy loss function is presented. Second, we augment the proposed fuzzy classifier based on the concept of federated learning such that it can learn from distributed data without sacrificing data security and confidentiality. Experiments are carried out on a two-dimensional synthetic dataset and a number of real-world datasets. Experimental results show the excellent classification capability of the proposed classifier realized in the centralized way and in the federated learning environment.

This study focuses on the construction of granular Gaussian process models completed at different levels of granularity and the emergence of higher-type granular outputs through aggregating the individual prediction results. Each Gaussian process model is instantiated utilizing granular data (or information granules) to enhance algorithmic efficiency and can be tailored to specific levels of precision (granularity). The overall design methodology emphasizes human centricity in system modeling by focusing on both the interpretability and accuracy of the resulting models. First, clustering algorithms are applied to construct information granules that provide a comprehensive overview of the experimental evidence. As the number of information granules grows, the existing knowledge imbedded within data could be perceived and described at increased levels of details. Information granules are built in an augmented feature space constructed by concatenating the input and output variables. Next, Gaussian process models are constructed on a basis of the information granules formed at different levels of abstraction. Subsequently, the confidence intervals are transformed to intervals and the reconciliation of the predictions produced by individual models, which offer different perspectives on the system, leads to the emergence of more abstract entities (such as type-2 intervals/fuzzy sets, etc.) rather than plain numbers. The efficacy of the comprehensive model is measured by the coverage and specificity criteria of the granular outputs. Experimental studies conducted on a synthetic dataset and a number of real-world datasets validated the effectiveness and adaptability of the proposed methodology.

Rule–based fuzzy models have modular architectures and come with well-developed design methodologies such that they can build accurate models with good interpretabilities in system modeling. However, a large amount of private data needs to be used for statistical analysis and forecasting in rule–based fuzzy models. The purpose of this study is to build an intelligent model with high accuracy and versatility under the premise of data privacy and model security. To mitigate the risk of malicious attacks on privacy data during the analysis process, we have employed highly regarded differential privacy techniques to devise a novel rule-based fuzzy modeling approach. We propose a function approximation mechanism to reconstruct the objective function and add a perturbation mechanism to the objective function in Takagi-Sugeno rules model. Taking into account the delicate balance between data privacy and utility, we have innovatively introduced a Takagi-Sugeno rule-based model based on differential privacy. This model is applicable to both linear and nonlinear systems, offering protection to sensitive data privacy and model security within the system. We investigate the relationship between the interpretability of the model and the degree of privacy protection. By constructing a reasonable rule base, we achieve higher accuracy than other system modeling methods based on differential privacy. This paper compares the influence of the number of rules on differential privacy, and considers the algorithm performance under various noise distributions. It is shown that the Takagi-Sugeno rules model based on differential privacy has a strong ability to predict and analyze data.

This article presents a design and realization of fuzzy rule-based regression models based on standard decision trees. A two-phase design of rule-based model is offered in this study to provide a good alternative to cope with high dimensional data. We first build a standard decision tree on the basis of variables in order to discover homogeneous subsets of the data. Subsequently, a collection of fuzzy rules is induced by the decision tree with the aim of reflecting the underlying phenomenon. The calculation of membership degrees and the refinement of fuzzy rules on the basis of data located in each partition exhibit a substantial level of originality and innovation. The introduction of fuzziness into decision rules helps to characterize and quantify the continuous change of output values near the boundary areas. The constructed fuzzy rules could efficiently handle the ambiguity and vagueness in the experimental evidence and offer an accurate characterization of the nonlinearities of the input–output relationships. The developed fuzzy models could achieve much higher prediction accuracy in comparison with traditional decision trees of the same size and fuzzy rule-based models with the same number of rules. Another advantage of the proposed methodology comes with the evident readability of the formed fuzzy rules. A series of experiments is reported to demonstrate the superiority of the proposed architecture of fuzzy rule-based models over traditional fuzzy rule-based models and decision trees.

This study is devoted to the design of gradient boosted fuzzy rule-based models for regression problems. Fuzzy rule-based models are built on the basis of information granules formed in the input and output spaces whose structure involves a family of conditional ‘if-then’ statements. The architecture of fuzzy rule-based models contributes to the realization of a sound tradeoff between modeling accuracy and interpretability and computing overhead. Gradient boosting paradigm has emerged as a powerful learning method realized through sequentially fitting additive base learners to current residuals in the steepest descent way. However, surprisingly, studies on the design and analysis of gradient boosted fuzzy rule-based models are still lacking. In this study, fuzzy rule-based model is regarded as a base learner. Different loss functions and their influence on the performance of the final models are explored. We also thoroughly investigate an impact of the initial quality of the rule-based model (implied by the number of rules) on the process of gradient boosting. The performance of the proposed approach is illustrated by a series of experimental studies concerning synthetic and publicly available datasets.

In this study, we elaborate on a realization of fuzzy clustering and the construction of fuzzy rule-based models on the basis of vertically partitioned datasets in a privacy-preserving federated learning approach. The main focus of the overall design process is to construct a family of information granules (clusters) and the corresponding fuzzy rules in the presence of a collection of vertically partitioned datasets without compromising data privacy. These datasets are composed of the same data but are described by different features, and due to security considerations, data cannot be shared. The vertical federated fuzzy clustering can be realized as an iterative optimization process composed of successive cycles: 1) computation (update) of the prototypes and partition matrices performed on the basis of local datasets and 2) an integration of the local sources of knowledge carried out on a central coordinator-server. The update of the partition matrices can be completed using a distance-based or gradient-based approach. The communication of findings between local clients and the coordinator-server is realized through exchanging partition matrices, which are more general than numeric data and can avoid leakage of data privacy. Fuzzy models are optimized in a similar manner through exchanging the gradients of the performance index computed with respect to the parameters between the clients and the global coordinator. The proposed mechanism exhibits significant originality since the realization of fuzzy modeling in a vertical federated learning environment has not been studied. Experimental studies show that the proposed federated clustering and fuzzy model design could effectively reveal the structure of the entire dataset and achieve high performance compared with the results obtained in a centralized manner.

Designing effective and efficient classifiers is a challenging task given the facts that data may exhibit different geometric structures and complex intrarelationships may exist within data. As a fundamental component of granular computing, information granules play a key role in human cognition. Therefore, it is of great interest to develop classifiers based on information granules such that highly interpretable human-centric models with higher accuracy can be constructed. In this study, we elaborate on a novel design methodology of granular classifiers in which information granules play a fundamental role. First, information granules are formed on the basis of labeled patterns following the principle of justifiable granularity. The diversity of samples embraced by each information granule is quantified and controlled in terms of the entropy criterion. This design implies that the information granules constructed in this way form sound homogeneous descriptors characterizing the structure and the diversity of available experimental data. Next, granular classifiers are built in the presence of formed information granules. The classification result for any input instance is determined by summing the contents of the related information granules weighted by membership degrees. The experiments concerning both synthetic data and publicly available datasets demonstrate that the proposed models exhibit better prediction abilities than some commonly encountered classifiers (namely, linear regression, support vector machine, Naïve Bayes, decision tree, and neural networks) and come with enhanced interpretability.

Understanding the rationale behind the predictions produced by machine learning models is a necessary prerequisite for human to build confidence and trust for the intelligent systems. To tackle the problem of interpretability faced by black-box models, a fuzzy local surrogate model is proposed in this study to articulate the rationale for predictions to enhance the interpretability of the results of machine learning models. Fuzzy rule-based model comes with high interpretability since it is composed of a collection of readable rules, and thus is suitable for prediction interpretation. The general scheme of fuzzy local surrogate model is composed of the following phases: i) select data points around the instance of interest, for which we wish to explain the prediction result produced by the predictive model; ii) generate predictions for these newly selected data and weight the selected data based on the distance from the instance of interest; and iii) a fuzzy rule-based model composed a collection of interpretable is constructed to approximate the weighted data and offer meaningful interpretation to the prediction result of the given instance. The proposed fuzzy model for explaining predictions is model-agnostic and could provide high estimation accuracy. The proposed methodology offers a significant original contribution to the interpretation of machine learning models. Experimental studies demonstrate the usefulness of the proposed fuzzy local surrogate model in providing local explanations.

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.

Information granulation and degranulation play a fundamental role in granular computing (GrC). Given a collection of information granules (referred to as reference information granules), the essence of the granulation process (encoding) is to represent each data (either numeric or granular) in terms of these reference information granules. The degranulation process (decoding) that realizes the reconstruction of original data is associated with a certain level of reconstruction error. An important issue is how to reduce the reconstruction error such that the data could be reconstructed more accurately. In this study, the granulation process is realized by involving fuzzy clustering. A novel neural network is leveraged in the consecutive degranulation process, which could help significantly reduce the reconstruction error. We show that the proposed degranulation architecture exhibits improved capabilities in reconstructing original data in comparison with other methods. A series of experiments with the use of synthetic data and publicly available datasets coming from the machine-learning repository demonstrates the superiority of the proposed method over some existing alternatives.

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.

In this article, we are concerned with the formation of type-2 information granules in a two-stage approach. We present a comprehensive algorithmic framework which gives rise to information granules of a higher type (type-2, to be specific) such that the key structure of the local granular data, their topologies, and their diversities become fully reflected and quantified. In contrast to traditional collaborative clustering where local structures (information granules) are obtained by running algorithms on the local datasets and communicating findings across sites, we propose a way of characterizing granular data (formed) by forming a suite of higher type information granules to reveal an overall structure of a collection of locally available datasets. Information granules built at the lower level on a basis of local sources of data are weighted by the number of data they represent while the information granules formed at the higher level of hierarchy are more abstract and general, thus facilitating a formation of a hierarchical description of data realized at different levels of detail. The construction of information granules is completed by resorting to fuzzy clustering algorithms (more specifically, the well-known Fuzzy C-Means). In the formation of information granules, we follow the fundamental principle of granular computing, viz., the principle of justifiable granularity. Experimental studies concerning selected publicly available machine-learning datasets are reported.

In this article, we elaborate on a design and realization of fuzzy rule-based model in the horizontal federated learning framework. Traditional machine learning in distributed environment often involves sharing sensitive information with other sites or transferring data to a central server on which a global model is trained. These situations increase the communication overhead and pose serious threats to the privacy of sensitive data. Federated learning opens up the possibility for collaboratively training a global model on a basis of distributed on-site data without sacrificing data privacy. While fuzzy rule-based models have been used in system modeling due to their substantial modeling abilities and good interpretability, the implementation of fuzzy rule-based models in a distributed environment without compromising data privacy still requires careful consideration. This article proposes a two-step federated learning approach to train a global model on a basis of private data located across different sites without their centralization. The first step concerns the determination of the structure of the data through federated collaborative clustering. Subsequently, a shared global model is trained jointly by all the participating clients. An advantage of the proposed method is that it achieves high accuracy without violating data privacy. A series of experimental studies are conducted to gain a detailed insight into the realization steps and demonstrate the effectiveness of the proposed method.

While granular computing has experienced rapid growth in the past decades and some milestones have been reached, a comprehensive study of the representation capabilities delivered by numeric prototypes and granular prototypes produced by different techniques still calls for comprehensive research and a comparative analysis. Well-constructed information granules are reflective of the nature of the numeric evidence and serve as backbones of granular classifiers and granular models. The objective of this study is to review a number of clustering paradigms aimed at the construction of information granules, discuss the development of granular prototypes, and conduct a comprehensive evaluation of quality of numeric prototypes and their corresponding augmentations coming in the form of granular prototypes. We have been witnessing many studies devoted to the construction of information granules, but a comparative analysis of the quality of information granules constructed on a basis of prototypes produced by different clustering algorithms is still lacking. In this regard, the review of the clustering algorithms supporting the formation of information granules and the comprehensive comparative study of their usefulness in classification and modeling tasks offered in this study make sense. This will promote the usage of information granules in various future works, especially classification problem and 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.

In this article, we elaborate on a design methodology and the detailed realization of hierarchically structured granular models by engaging the fundamental principles and concepts of granular computing. The existing models are elevated to a more abstract (general) level by allocating a certain level of information granularity throughout the parameter space. In a concise way, the essence of the overall architecture of the proposed modeling mechanism could be generalized as follows: Numeric model (granular model of type-0)→granular model of type-1→granular model of type-2→…→granular model of higher type. The results of the granular models come in the form of type-0, type-1, or higher type information granules, which are decided by the overall level of hierarchy of the corresponding granular model. The specificity of granular outputs becomes a more comprehensive and sound quantification of the prediction accuracy and precision of the model and the quality of the specific prediction outputs. The proposed method facilitates effective communication with humans, who could get actively involved in the modeling process and determine the suitable level of abstraction depending upon the requirements of the problem. The determination of a suitable level of information granularity is realized with the guidance of the principle of justifiable granularity. A number of experimental studies concerning publicly available datasets are presented to illustrate the development methodology and show the effectiveness of the approach to form hierarchically structured solutions (reflecting different levels of abstraction) to the problem.

In this article, we propose a design and evaluation framework of granular neural networks realized in the presence of information granules. Neural networks realized in this manner are able to process both nonnumerical data, such as information granules as well as numerical data. Information granules are meaningful and semantically sound entities formed by organizing existing knowledge and available experimental data. The directional nature of mapping between the input and output data needs to be considered when building information granules. The development of neural networks advocated in this article is realized as a two-phase process. First, a collection of information granules is formed through granulation of numeric data in the input and output spaces. Second, neural networks are constructed on the basis of information granules rather than original (numeric) data. The proposed method leads to the construction of neural networks in a completely new way. In comparison with traditional (numeric) neural networks, the networks developed in the presence of granular data require shorter learning time. They also produce the results (outputs) that are information granules rather than numeric entities. The quality of granular outputs generated by our neural networks is evaluated in terms of the coverage and specificity criteria that are pertinent to the characterization of the information granules.

Fuzzy clustering has emerged as one of the fundamental conceptual and algorithmic frameworks supporting the development of information granules. Generic fuzzy clustering such as fuzzy C-means (FCM) has been utilized in a broad range of applications. However, the constructs resulting from fuzzy clustering, namely a partition matrix and prototypes, are numeric and as such are not capable of fully capturing the essence of the overall data. In this study, we propose an alternative augmented way of building information granules by generating hypercube-like information granules. A collection of hypercubes is referred to as a family of ε-information granules. This family is constructed around numeric prototypes generated through a modified version of the FCM algorithm whose running time is linear with respect to the number of clusters. By admitting a certain level of information granularity (ε), a collection of hypercubes is formed around the prototypes. The quality of information granules realized in this way is assessed by involving them in the granulation-degranulation process as well as determining a value of the coverage criterion. The level of information granularity and the number of the granular prototypes in the family of ε-information granules form an important design asset directly impacting the obtained coverage level of the data. The computational facet of the approach is stressed. It has been demonstrated that the granular enhancements of the description of data come with a very limited computing overhead. Experimental studies involve synthetic data as well as data coming from the UCI Machine Learning repository. The granular reconstruction capabilities delivered by the family of ε-information granules are discussed.

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-à-vis experimental (numeric) data. Experimental results are reported for both synthetic datasets and publicly available data sets coming from the UCI machine learning repository.

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.

Granular computing (GrC) has emerged as a unified conceptual and processing framework. Information granules are fundamental constructs that permeate concepts and models of GrC. This paper is concerned with a design of a collection of meaningful, easily interpretable ellipsoidal information granules with the use of the principle of justifiable granularity by taking into consideration reconstruction abilities of the designed information granules. The principle of justifiable granularity supports designing of information granules based on numeric or granular evidence, and aims to achieve a compromise between justifiability and specificity of the information granules to be constructed. A two-stage development strategy behind the construction of justifiable information granules is considered. First, a collection of numeric prototypes is determined with the use of fuzzy clustering. Second, the lengths of the semi-axes of ellipsoidal information granules to be formed around such prototypes are optimized. Two optimization criteria are introduced and studied. Experimental studies involving synthetic data set and data sets coming from the machine learning repository are reported.

Information granules are generic building blocks supporting the processing realized in granular computing and facilitating communication with the environment. In this paper, we are concerned with a fundamental problem of encoding-decoding of information granules. The essence of the problem is outlined as follows: given a finite collection of granular data X1, X2,...,XN (sets, fuzzy sets, etc.), construct an optimal codebook composed of information granules A 1, A2, ..., Ac, where typically c <;<; N, so that any Xk represented in terms of A i's and then decoded (reconstructed) with the help of this codebook leads to the lowest decoding error. A fundamental result is established, which states that in the proposed encoders and decoders, when encoding-decoding error is present, the information granule coming as a result of decoding is of a higher type than the original information granules (say, if Xk is information granule of type-1, then its decoded version becomes information granule of type-2). It would be beneficial to note that as the encoding-decoding process is not lossless (in general, with an exception of a few special cases), the lossy nature of the method is emphasized by the emergence of information granules of higher type (in comparison with the original data being processed). For instance, when realizing encoding-decoding of numeric data (viz., information granules of type-0), the losses occur and they are quantified in terms of intervals, fuzzy sets, probabilities, rough sets, etc., where, in fact, the result becomes an information granule of type-1. In light of the nature of the constructed result when Xkis an interval or a fuzzy set, an optimized performance index engages a distance between the bounds of the interval-valued membership function. We develop decoding and encoding mechanisms by engaging the theory of possibility and fuzzy relational calculus and show that the decoded information granule is either a granular interval or interval-valued fuzzy set. The optimization mechanism is realized with the aid of the particle swarm optimization (PSO). A series of experiments are reported with intent to illustrate the details of the encoding-decoding mechanisms and show that the PSO algorithm can efficiently optimize the granular codebook.

Formed as generic building blocks being reflective of domain knowledge and experimental numeric evidence, information granules play a pivotal role in processing realized in Granular Computing and facilitating communication with the environment. In this study, we are concerned with a fundamental problem of constructing a collection of meaningful, easily interpretable spherical information granules with the use of the principle of justifiable granularity. The design process is formulated as an optimization problem. First, a series of numeric prototypes are determined around which information granules are constructed. Second, the values of radii of these information granules are optimized aiming at maximizing a certain performance index. Two alternatives of determining centers of information granules are compared, i.e., randomly selected numeric prototypes and prototypes generated with the aid of clustering. Two optimization criteria are also introduced and studied. Experimental studies involving synthetic data as well as data coming from the UCI Machine Learning repository are reported.

This poster paper addresses the fitness calculation problem fornested if-else constructs. A new term "optimism level" is incorporated into the fitness function to assess the branch distance of nested branches for a given test data.