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Information Structures and Uncertainty measures in a Hybrid Information System: Gaussian Kernel Method

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Abstract

A hybrid information system is an information system where there exist varieties of data, and its information structures reflect the internal features of this kind of information system. This paper researches information structures and uncertainty measurement in an HIS based on Gaussian kernel. According to the viewpoint, an HIS is seen as a multi-source information system, and the distance under each attribute is proposed which is combined into a hybrid distance. Then, the fuzzy \(\tau _{\text {cos}}\)-equivalence relation by using Gaussian kernel that is based on this hybrid distance is obtained. Next, information structures are described via set vectors, and dependence between information structures is studied. Moreover, uncertainty measures in an HIS are investigated based on its information structures. Eventually, the optimal selection of information structures in an HIS based on \(\delta \)-information granulation or \(\delta \)-rough entropy is given. These studies may be useful for exploring the essence of granular computing.

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Abbreviations

U :

A finite universe \((U=\{u_1,u_2,\ldots ,u_n\})\)

I :

The unit interval [0, 1]

F :

A fuzzy set (\(F:U\rightarrow I\))

R :

A fuzzy relation on U

\(I^U\) :

The family of all fuzzy sets in U

\(I^{U\times U}\) :

The set of all fuzzy relations on U

(U, A):

An information system

\((U, \overrightarrow{\mathbf{A }})\) :

A multi-source information system

HIS:

A hybrid information system

HD(u, v):

The hybrid distance

\(d_{\overrightarrow{\scriptstyle{\mathbf{P}}}}(u,v)\) :

The hybrid distance between two objects u and v in \((U,\overrightarrow{\mathbf{P}})\)

\(G(u_,v)\) :

Gaussian kernel

\(\delta \) :

A given threshold (\(\delta \in (0,1]\))

\(M(R_{\overrightarrow{\scriptstyle{\mathbf{P}}}}^{G}(\delta ))\) :

Gaussian kernel matrix of \((U,\overrightarrow{\mathbf{P }})\) with respect to \(\delta \)

\(R_{\overrightarrow{\scriptstyle{\mathbf{P }}}}^{G}(\delta )\) :

A fuzzy \(\tau _{\text {cos}}\)-equivalence relation on U

\(\mathbf {S}^\delta (U,\overrightarrow{\mathbf{A }})\) :

The information structure base of \((U,\overrightarrow{\mathbf{A }})\) with respect to \(\delta \)

\(D(S^\delta (\overrightarrow{\mathbf{Q }})/S^\delta (\overrightarrow{\scriptstyle{\mathbf{P }}}))\) :

Inclusion degree

\(M^\delta \) :

\(\delta \)-information granulation function in \((U,\overrightarrow{\mathbf{A }})\)

\(G^\delta (\overrightarrow{\mathbf{P }})\) :

\(\delta \)-information granulation of \((U,\overrightarrow{\mathbf{P }})\)

\(H^\delta (\overrightarrow{\mathbf{P }})\) :

\(\delta \)-information entropy of \((U,\overrightarrow{\mathbf{P }})\)

\((E_\text {r})^\delta (\overrightarrow{\mathbf{P }})\) :

\(\delta \)-rough entropy of \((U,\overrightarrow{\mathbf{P }})\)

\(E^\delta (\overrightarrow{\mathbf{P }})\) :

\(\delta \)-information amount of \((U,\overrightarrow{\mathbf{P }})\)

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Acknowledgements

The authors would like to thank the editors and the anonymous reviewers for their valuable comments and suggestions which have helped immensely in improving the quality of this paper. This work is supported by National Natural Science Foundation of China (11971420), Natural Science Foundation of Guangxi (2018GXNSFDA294003, 2018GXNSFAA294134) and High Level Innovation Team Program from Guangxi Higher Education Institutions ([2019] 52).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Hunan University of Technology and Business, Changsha, 410205, Hunan, People’s Republic of China

    Jiasheng Zeng

  2. Key Laboratory of Complex System Optimization and Big Data Processing in Department of Guangxi Education, Yulin Normal University, Yulin, 537000, Guangxi, People’s Republic of China

    Zhaowen Li & Pei Wang

  3. School of Information Science and Technology, Southwest Jiaotong University, Chengdu, 611756, Sichuan, People’s Republic of China

    Pengfei Zhang

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  1. Jiasheng Zeng
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Correspondence to Zhaowen Li.

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Zeng, J., Li, Z., Zhang, P. et al. Information Structures and Uncertainty measures in a Hybrid Information System: Gaussian Kernel Method. Int. J. Fuzzy Syst. 22, 212–231 (2020). https://doi.org/10.1007/s40815-019-00779-8

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