Rule-based models are applicable to model the behavior of complex and nonlinear systems. Due to limited experience and randomness involving constructing information granules, an insufficient credible rules division could reduce the model's accuracy. This paper proposes a new rule-based modeling approach, which utilizes density-based spatial clustering of applications with noise (DBSCAN)-based information granules to construct the rules. First, bear in mind the advantages of density-based clustering, DBSCAN is proposed to generate data structures. Based on these data structures, two rule-based models are constructed: 1) models using DBSCAN clusters to construct granules and rules directly and 2) models generating subgranules in each DBSCAN cluster for rule formation. Experiments involving these two models are completed, and obtained results are compared with those generated with a traditional model involving fuzzy C -means-based granules. Numerical results show that the rule-based model, which builds rules from subgranules of DBSCAN structures, performs the best in analyzing system behaviors.
The acquired industrial data often contain missing outputs because of the irregularities of complicated industrial environment, which make the outputs of the training dataset incomplete. In this paper, a semi-supervised sparse Bayesian regression model is proposed for dealing with the incomplete outputs problem by employing a variational inference technique. Within the settings of specific hierarchical priors over the missing outputs, in this paper, we derive the posterior probability distribution over the uncertain variables including the missing outputs. Given that the posterior distribution is not analytically tractable, a hybrid learning procedure is designed for combining the variational inference with a gradient-based method to obtain optimal approximate posteriors. To verify the performance of the proposed method, a number of comparative experiments are conducted and analyzed by using the datasets (including artificial and real world ones) coming with different proportions of missing outputs. Compared to the existing semi-supervised regression approaches, we demonstrated the effectiveness of the proposed method.
In data analytics, system modeling, and decision-making, the aspects of interpretability and explainability are of paramount relevance; one can refer here to explainable Artificial Intelligence (XAI). The increasing complexity of systems one has to cope with, distributed nature of data with an ultimate concern about privacy and security of data and models are other challenges present in system modeling. With the proliferation of mobile devices, distributed data, and security and privacy restrictions, federated learning becomes a feasible development alternative. We advocate that there are two factors that immensely contribute to the realization of the above important requirements, namely, (i) a suitable level of abstraction along with its hierarchical aspects in describing the problem and (ii) a logic fabric of the resultant constructs. It is demonstrated that their conceptualization and the following realization can be conveniently carried out with the use of information granules (for example, fuzzy sets, sets, rough sets, and alike). Information granules are building blocks forming the interpretable environment capturing the essence of data and revealing key relationships existing there. Their emergence is supported by a systematic and focused analysis of data. At the same time, their initialization is specified by stakeholders or/and the owners and users of data. We present a comprehensive discussion of a design of information granules and their description by engaging an innovative mechanism of federated unsupervised learning in which information granules are constructed and refined with the use of collaborate schemes of clustering. For illustrative reasons, the study will be focused on the timely issues of interpretability and federated learning in the context of functional rule-based models with the rules in the form “if <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$x$</tex> is <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$A$</tex> then <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$y=f(x)$</tex> ” with the condition parts described by information granules. The interpretability mechanisms are aimed at a systematic elevation of interpretability of the conditions and conclusions of the rules. It is shown that augmenting interpretability of conditions is achieved by (i) decomposing a multivariable information granule into its one-dimensional components, (ii) delivering their symbolic characterization, and (iii) carrying out a process of linguistic approximation.
The objective of this paper is to present developed methodology for Granular Cognitive Map reconstruction. Granular Cognitive Maps model complex imprecise systems. With a proper adjustment of granularity parameters, a Granular Cognitive Map can represent given system with good balance between generality and specificity of the description. The authors present a methodology for Granular Cognitive Map reconstruction. The proposed approach takes advantage of granular information representation model. The objective of optimization is to readjust granularity parameters in order to increase coverage of targets by map responses. In this way we take full advantage of the granular information representation model and produce better, more accurate map, which maintains exactly the same balance between generality and specificity. Proposed methodology reconstructs Granular Cognitive Map without loosing its specificity. Presented approach is applied in a series of experiments that allow evaluating quality of reconstructed maps.
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.
Abstract The paper deals with a problem of decomposition of a binary fuzzy relation defined in the Cartesian product of a finite space. We propose an algorithm which produces the decomposition or indicates that the given relation is non-decomposable, within a finite sequence of steps. INDEX TERMS: Fuzzy relationdecomposition of relationfuzzy relation equation. Additional informationNotes on contributorsANTONIO DI NOLA Under the auspices of C.N.R. (G.N.S.A.G.A.), Italy. SALVATORE SESSA Under the auspices of C.N.R. (G.N.S.A.G.A.), Italy.
