Elsevier

Applied Soft Computing

Volume 115, January 2022, 108062
Applied Soft Computing

Granular models as networks of associations of information granules: A development scheme via augmented principle of justifiable granularity

https://doi.org/10.1016/j.asoc.2021.108062Get rights and content

Highlights

  • Granular fuzzy models based on an augmented principle of justifiable granularity.

  • Information in the output variables to eliminate overlaps among information granules.

  • A granular net to predict granular outputs through an elaborate inference scheme.

Abstract

This study proposes an approach to the construction of granular models directly based on information granules expressed both in input and output spaces. Associating these information granules, the constructed granular models come in the framework of three layers networks: input granules, an inference scheme and output granules. The proposed approach consists of two stages. First, an augmented principle of justifiable granularity is proposed and applied to construct information granules in an input space. This principle constructs information granules not only through establishing a sound balance between two criteria, i.e., coverage and specificity, but also by optimizing those information granules on the basis of their homogeneity assessed with respect to data localized in output space. At the second stage, we propose an inference scheme by analyzing a location of an input datum in relation with the already formed information granules in an input space. The computed relation can be quantified as membership grades, thus yielding aggregation results involving information granules in an output space. The performance of the proposed granular model is supported by the mechanisms of granular computing and the principle of justifiable granularity. Experimental studies concerning synthetic and publicly available data are performed and some comparative analysis involving rule-based models is given.

Introduction

The increasing importance of data and experimental evidence in control engineering [1], image processing [2], [3], [4], network science [5], [6], etc, has become apparent. It is highly desired to capture the essence of data, build their essential descriptors, and reveal key relationships among input and output variables at a suitable level of abstraction.

As an interesting paradigm of information processing, one of the key issues in granular computing is to construct granules based on similarity, indistinguishability, and functionality [7]. The basic ingredients of granular computing are information granules, which could be further refined into abstracts of higher or smaller. The refinement of information granules makes them flexible and adaptive to suit some problems at hand. Thus, granular computing can be regarded as a powerful platform for supporting an analysis and design of human-centric intelligent models. This feature attracts researchers to transform data into information granules for further analysis. In human exploration, analysis and decision-making activities, information granules [8] serve as a bridge for iterations between human beings and their surroundings. When complex phenomena appear, information granules emerge as a useful tool that convert data and existing knowledge into user-oriented information at a higher level of abstraction. This abstracted information could be helpful in the ensuing process of describing phenomena and making informed decisions. Information granules play a key role in data representation and processing.

The concept of information granules can be formalized in different ways, such as fuzzy sets [9], rough sets [10], shadowed sets [11], and probabilistic sets [12] among others. For example, generalized theory of uncertainty by Zadeh [13] treats information as a generalized constraint and operate on information described in natural language; Type-2 information granules is a generalization of Type-1 granules [14]: the extra degrees of freedom make Type-2 granules more flexible but higher computing complexity and the underlying estimation process. Extended fuzzy logic [15] is a kind of logic that can deal with incomplete or imprecise information. Z-numbers [16] relate to the issue of reliability of information. In this work, Type-1 information granules are constructed, which can abstract information and reflect upon existing data structure. Deciding how to represent or describe a phenomenon with a suitable hierarchy and abstracted level is a key issue to be solved. Also, as a fundamental construct of granular computing [17], [18], information granules [19], [20] are abstract entities capturing essential features of numeric data in an efficient and concise manner. Hence, well-designed information granules are required in the first place. To confront this issue, we propose an augmented principle of justifiable granularity (APJG) as an improved version of the principle of justifiable granularity (PJG). With APJG, there are two additional factors that influence the construction of information granules in the input space: (1) information contained in output variables manages overlaps among information granules in the input space; and (2) the proportion is controllable for coverage and specificity, which are the two basic parameters that impact the construction of information granules.

The increasing need for in human-centric information processing motivates us to carefully design information granules. These information granules could abstract and compress information at an appropriate level to limit memory, time and computation cost. These elaborated information granules, as basic entities, guarantee the quality of granular models under the frame of granular computing. These granular models produce results in a granular form, which is flexible to fit user-oriented and problem-driven interest, such as forecasting stock prices [21].

Bearing this in mind, the objective of this study is to develop a granular model that is built directly on the basis of information granules formed previously and produces granular outputs. Those constructed granules are associated together in a network structure. Then, an elaborate inference scheme is developed as a bridge linking input and output spaces. Being different from numeric models, the final constructed model predicts granular outputs. To the best of our knowledge, there exist only a few studies [21], [22] to build information granules based on output variable clustering, which motivates us to proposes a granular modeling approach. The associations of information granules and the linkage between input and output space make the whole framework of granular models be like networks.

The objective and motivation mentioned above come with evident aspects of originality. First, additional source of knowledge contained in the output variables acts as the augmentation of generic PJG. Second, overlaps among constructed information granules could be eliminated. Third, a computationally efficient inference scheme is developed that associates information granules both in input and output spaces.

Clearly, the main contribution of this work is to propose a general principle named APJG that helps to build efficient net-like granular models. Its key contributions include (1) APJG takes advantage of information contained in the output variables when constructing granules in input space; (2) Weight coefficient of each parameter is controllable, which makes the formed granules compact and balanced in terms of coverage and specificity and (3) an elaborate inference scheme processes information granules as a network at a higher granular model. This work also presents a new method that allows us to develop a granular model directly from experimental data without any requirement of predefined numeric models, namely the proposed granular model (PGM). The model formed produces results at the granular level, which is more useful and meaningful in some domains. For example, in stock market, a predictor interval could contain more information than a simple numeric point.

Section 2 reviews the state of the art relevant to this study. An augmented principle of justifiable granularity is presented in Section 3, where the incorporation of modifications to keep information granules disjoint is highlighted. The formulation of PGM is then presented in Section 4. The inference and performance evaluation schemes are provided and quantified in Section 5. In Section 6, experimental studies are reported along with a thorough comparative analysis. Finally, conclusions and some open issues for future studies are covered in Section 7.

Section snippets

Related work

Granular models, established on a basis of granular computing, are about a paradigm of representing, constructing, and processing information granules [9]. Compared with numerical data, information granules are more general entities and as such they play a pivotal role in knowledge representation and processing [23]. Compared with some deep learning approaches, such as CNN [24], Recurrent neural network [25], deep generative approach [26], granular model has an advantage of interpretability.

Design of focal information granules: An augmented principle of justifiable granularity

The granular model proposed here is built directly on information granules with the aid of APJG. The quality of focal information granules implies the performance of granular model. Built on PJG [44], [45], information granules are constructed by achieving a sound balance between coverage and specificity. The augmentation of PJG is to include an additional criterion of heterogeneity to be applied to the related output data and the weights for both coverage and specificity. In the realization of

Granular neural model as a collection of associations among information granules

A granular neural model is composed of three layers: (1) Input layer. It is formed by associating information granules in an input space; (2) Hidden layer. It is the inference scheme that links input and output spaces; and (3) Output layer. It gives the prediction results via information granules. Thus, the framework of the granular model is similar as a neural network. For the sake of convenience, it is named granular neural model. In this section, two critical issues on PGM are detailed. The

Inference scheme and performance index

In this section, an inference scheme is proposed to complete the remaining part of PGM. A generic performance index scheme is presented to evaluate the performance of PGM. This inference influences the accuracy of our model and should be carefully designed.

Experimental studies

We report on results of the completed experimental studies: (1) the complete construction of PGM is accomplished through synthetic data; (2) Comparative experiments with other rule-based models are conducted on publicly available datasets. As to the FCM algorithm, the fuzzification exponent m is set to 2.0. All the datasets are split into training and testing sets in the 80–20 split. In all considerations, we assume that experimental input–output data have been normalized in the range [0,1].

Conclusions

This study has focused on the development of granular models that associate information granules as networks. To make these information granules more meaningful and compact for our granular neural models, information contained in the output variable is involved by using APJG. Furthermore, an additional process is invoked to keep these information granules disjointed. With the accomplishment of collection information granules in the input space, a net in the input space is formed through linking

CRediT authorship contribution statement

TaiLong Jing: Software, Resource, Writing – original draft. Cong Wang: Formal analysis, Data curation. Witold Pedrycz: Conceptualization, Methodology, Supervision. ZhiWu Li: Writing – review & editing. Giancarlo Succi: Validation. MengChu Zhou: Methodology, Supervision.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

TaiLong Jing received the B.S. degree in automation and the M.S. degree in control theory and control engineering from Xidian University, Xi’an China, in 2012 and 2017, respectively. He is currently pursuing the Ph.D. degree in control theory and control engineering at Xidian University, Xi’an, China. He was a visiting Ph.D. student in the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB, Canada. His research interests include granular computing, data

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    TaiLong Jing received the B.S. degree in automation and the M.S. degree in control theory and control engineering from Xidian University, Xi’an China, in 2012 and 2017, respectively. He is currently pursuing the Ph.D. degree in control theory and control engineering at Xidian University, Xi’an, China. He was a visiting Ph.D. student in the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB, Canada. His research interests include granular computing, data mining, control engineering, and reinforcement learning.

    Cong Wang received the B.S. degree in automation and the M.S. degree in mathematics from Hohai University, Nanjing, China, in 2014 and 2017, respectively. He received the Ph.D. degree in mechatronic engineering from Xidian University, Xi’an, China in 2021. He is now an assistant professor in School of Artificial Intelligence, OPtics and ElectroNics (iOPEN), Northwestern Polytechnical University, Xi’an, China. He was a Visiting Ph.D. Student at the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB, Canada, and the Department of Electrical and Computer Engineering, National University of Singapore, Singapore. He was also a Research Assistant at the School of Computer Science and Engineering, Nanyang Technological University, Singapore. His current research interests include wavelet analysis and its applications, granular computing, as well as image processing.

    Witold Pedrycz received the MS.c. degree in computer science and technology, the Ph.D. degree in computer engineering, and the D.Sci. degree in systems science from the Silesian University of Technology, Gliwice, Poland, in 1977, 1980, and 1984, respectively. He is a Professor and the Canada Research Chair in Computational Intelligence with the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB, Canada. He is also with the Systems Research Institute of the Polish Academy of Sciences, Warsaw, Poland. He is a foreign member of the Polish Academy of Sciences. He has authored 15 research monographs covering various aspects of computational intelligence, data mining, and software engineering. His current research interests include computational intelligence, fuzzy modeling, and granular computing, knowledge discovery and data mining, fuzzy control, pattern recognition, knowledge-based neural networks, relational computing, and software engineering. He has published numerous papers in the above areas. Dr. Pedrycz was a recipient of the IEEE Canada Computer Engineering Medal, the Cajastur Prize for Soft Computing from the European Centre for Soft Computing, the Killam Prize, and the Fuzzy Pioneer Award from the IEEE Computational Intelligence Society. He is intensively involved in editorial activities. He is an Editor-in-Chief of Information Sciences, an Editor-in-Chief of WIREs Data Mining and Knowledge Discovery (Wiley) and the International Journal of Granular Computing (Springer). He currently serves as a member of a number of editorial boards of other international journals. He is a Fellow of IEEE and the Royal Society of Canada.

    ZhiWu Li received the B.S. degree in mechanical engineering, the M.S. degree in automatic control, and the Ph.D. degree in manufacturing engineering from Xidian University, Xi’an, China, in 1989, 1992, and 1995, respectively. He joined Xidian University in 1992. He was a Visiting Professor with the University of Toronto, Toronto, ON, Canada; the Technion—Israel Institute of Technology, Haifa, Israel; the Martin-Luther University of Halle–Wittenburg, Halle, Germany; the Conservatoire National des Arts et Métiers, Paris, France; and Meliksah Universitesi, Kayseri, Turkey. He is currently with the Institute of Systems Engineering, Macau University of Science and Technology, Macau, China. His current research interests include Petri net theory and applications, supervisory control of discrete-event systems, workflow modeling and analysis, system reconfiguration, game theory, and data and process mining. Dr. Li was a recipient of the Alexander von Humboldt Research Grant and the Alexander von Humboldt Foundation, Germany. He is listed in Marquis Who’s Who in the World (27th ed., 2010). He serves as a Frequent Reviewer for more than 70 international journals, including Automatica and a number of the IEEE Transactions as well as many international conferences. He is the Founding Chair of the Xi’an Chapter of the IEEE Systems, Man, and Cybernetics Society. He is a member of Discrete-Event Systems Technical Committee of the IEEE Systems, Man, and Cybernetics Society, and was on the IFAC Technical Committee on Discrete-Event and Hybrid Systems, from 2011 to 2014. He is a Fellow of IEEE.

    Giancarlo Succi is currently Professor and Dean of Faculty of Computer Science and Software Engineering. He is also the of Head of Laboratory for Improving the Production of Software at Innopolis University, Russia. He holds a Laurea Degree in Electrical Engineering from the University of Genova, an M.Sc. in Computer Science from the State University of New York at Buffalo, and a PhD in Computer and Electrical Engineering again from the University of Genoa, Italy (December 1993). He has passed the habilitation certification as professional engineering both in Italy and in Canada and he has consulted for several organizations worldwide. He has been a Professor at the Free University of Bolzano-Bozen, Italy, University of Alberta, Edmonton, Canada, University of Calgary, Canada holding also various administrative positions. Giancarlo Succi has taught a variety of academic and industrial courses throughout his career in Software Engineering, Programming Languages, and Mobile, Distributed, and Centralized Operating Systems. He has organized various international conferences and other scientific and educational events. His research interests are in empirical software engineering, open source, mobile and energy aware systems, software reuse, and software product lines. He is the author of 5 and editor of 12 books and over than 370 publications.

    MengChu Zhou received the B.S. degree in control engineering from Nanjing University of Science and Technology, Nanjing, China in 1983, M.S. degree in automatic control from Beijing Institute of Technology, Beijing, China in 1986, and Ph.D. degree in computer and systems engineering from Rensselaer Polytechnic Institute, Troy, NY, USA in 1990. He joined New Jersey Institute of Technology (NJIT), Newark, NJ, in 1990, and is now a Distinguished Professor of Electrical and Computer Engineering. His research interests are in Petri nets, intelligent automation, Internet of Things, big data, web services, and intelligent transportation. He has over 900 publications including 12 books, 600+ journal papers (500+ in IEEE Transactions, 29 patents and 29 book-chapters. He is the founding Editor of IEEE Press Book Series on Systems Science and Engineering, Editor-in-Chief of IEEE/CAA Journal of Automatica Sinica, and Associate Editor of IEEE Internet of Things Journal, IEEE Transactions on Intelligent Transportation Systems, IEEE Transactions on Systems, Man, and Cybernetics: Systems, and Frontiers of Information Technology & Electronic Engineering. He served as Associate Editor of IEEE Transactions on Robotics and Automation, IEEE Transactions on Automation Science and Engineering, and IEEE Transactions on Industrial Informatics, and Editor of IEEE Transactions on Automation Science and Engineering. He served as a Guest-Editor for many journals including IEEE Internet of Things Journal, IEEE Transactions on Industrial Electronics, and IEEE Transactions on Semiconductor Manufacturing. He is founding Chair/Co-chair of Technical Committee on AI-based Smart Manufacturing Systems of IEEE Systems, Man, and Cybernetics Society, Technical Committee on Semiconductor Manufacturing Automation and Technical Committee on Digital Manufacturing and Human-Centered Automation of IEEE Robotics and Automation Society. He was General Chair of IEEE Conf. on Automation Science and Engineering, Washington D.C., August 23-26, 2008, General Co-Chair of 2003 IEEE International Conference on System, Man and Cybernetics (SMC), Washington DC, October 5-8, 2003 and 2019 IEEE International Conference on SMC, Bari, Italy, Oct. 6-9, 2019, Founding General Co-Chair of 2004 IEEE Int. Conf. on Networking, Sensing and Control, Taipei, March 21-23, 2004, and General Chair of 2006 IEEE Int. Conf. on Networking, Sensing and Control, Ft. Lauderdale, Florida, U.S.A. April 23-25, 2006. He was Program Chair of 2010 IEEE International Conference on Mechatronics and Automation, August 4-7, 2010, Xi’an, China, 1998 and 2001 IEEE International Conference on SMC and 1997 IEEE International Conference on Emerging Technologies and Factory Automation. Dr. Zhou has led or participated in over 50 research and education projects with total budget over $12M, funded by National Science Foundation, Department of Defense, NIST, New Jersey Science and Technology Commission, and industry. He is a recipient of Excellence in Research Prize and Medal from NJIT, Humboldt Research Award for US Senior Scientists from Alexander von Humboldt Foundation, and Franklin V. Taylor Memorial Award and the Norbert Wiener Award from IEEE SMC Society, Computer-Integrated Manufacturing UNIVERSITY-LEAD Award from Society of Manufacturing Engineers, and Edison Patent Award from the Research & Development Council of New Jersey. He has been among most highly cited scholars since 2012 and ranked top one in the field of engineering worldwide in 2012 by Web of Science. He is a life member of Chinese Association for Science and Technology-USA and served as its President in 1999. He is a Fellow of IEEE, International Federation of Automatic Control (IFAC), American Association for the Advancement of Science (AAAS) and Chinese Association of Automation (CAA).

    This work was supported in part by the National Key R&D Project of China under Grant No. 2018YFB1700104, in part by the National Natural Science Foundation of China under Grants Nos. 61873342, 62076189, in part by the China National Postdoctoral Program for Innovative Talents under Grant No. BX2021249, in part by the fellowship of China Postdoctoral Science Foundation under Grant No. 2021M702678, in part by the Recruitment Program of Global Experts, Canada Research Chair, in part by the Natural Sciences and Engineering Research Council of Canada, in part by the Science and Technology Development Fund, MSAR, under Grant No. 0012/2019/A1, in part by the National Natural Science Foundation of China, under Grant No. 62076182 (W. Pedrycz), in part by the FDCT (Fundo para o Desenvolvimento das Ciencias e da Tecnologia) under Grant No. 0047/2021/A1, and in part by the Ministry of Science and Higher Education of the Russian Federation as part of World-class Research Center program: Advanced Digital Technologies under Contract No. 075-15-2020-903.

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