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The construction of attribute (object)-oriented multi-granularity concept lattices

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Abstract

How to reduce the complexity of lattice construction is an important research topic in formal concept analysis. Based on granularity tree, the relationship between the extent and the intent of the attribute (object)-oriented concept before and after granularity transformation are investigated. Then, zoom algorithms for attribute (object)-oriented concept lattices are proposed. Specifically, zoom-in algorithm is applied to change the attribute granularity from coarse-granularity to fine-granularity, and zoom-out algorithm achieves changing the attribute granularity from fine-granularity to coarse-granularity. Zoom algorithms deal with the problems of fast construction of the attribute (object)-oriented multi-granularity concept lattices. By using zoom algorithms, the attribute (object)-oriented concept lattice based on different attribute granularity can be directly generated through the existing attribute (object)-oriented concept lattice. The proposed algorithms not only reduce the computational complexity of concept lattice construction, but also facilitate further data mining and knowledge discovery in formal contexts. Furthermore, the transformation algorithms among three kinds of concept lattice are proposed.

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Acknowledgements

This work was supported by grants from the National Natural Science Foundation of China (nos. 61673396, 61573321), and Shandong Province National Natural Science Foundation (nos. ZR2017MF032, ZR2015FM022), the Fundamental Research Funds for the Central Universities under Grant (no. 18CX02133A).

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Authors and Affiliations

  1. College of Computer and Communication Engineering, China University of Petroleum, Qingdao, 266580, Shandong, People’s Republic of China

    Ming-Wen Shao, Meng-Meng Lv & Ken-Wen Li

  2. Department of Mathematics, Bohai University, Jinzhou, 121000, People’s Republic of China

    Chang-Zhong Wang

Authors
  1. Ming-Wen Shao
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  2. Meng-Meng Lv
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  4. Chang-Zhong Wang
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Correspondence to Ming-Wen Shao or Meng-Meng Lv.

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Appendix: the demonstrations of the zoom-in and zoom-out algorithms

Appendix: the demonstrations of the zoom-in and zoom-out algorithms

The formal contexts are presented by Tables 3 and  4 with the object set is \(\{x_1,x_2,x_3,x_4,x_5,x_6\}\), attribute sets are \(\{a,b,c,d,e\}\) and \(\{a,b,c_1,c_2,d,e\}\) respectively. In the sequence of graphs, the green concept indicates the concept currently being processed and the red concept represents the processed concept.

Table 3 Formal context (coarse-granularity)
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Table 4 Formal context (fine-granularity)
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The first figure sequence (Figs. 4, 5, 6, 7, 8) is a zoom-in algorithm presentation for the attribute-oriented concept lattice, that is, from the attribute-oriented concept lattice corresponding to Table 3 to the attribute-oriented concept lattice corresponding to Table 4.

Fig. 4
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Attribute-oriented concept lattice

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Fig. 5
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Attribute-oriented concept lattice

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Fig. 6
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Attribute-oriented concept lattice

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Fig. 7
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Attribute-oriented concept lattice

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Fig. 8
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Attribute-oriented concept lattice

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The second figure sequence (Fig. 9, 10, 11, 12, 13, 14, 15, 16) is a zoom-out algorithm presentation for attribute-oriented concept lattice, that is, from the attribute-oriented concept lattice corresponding to Table 4 to the attribute-oriented concept lattice corresponding to Table 3.

Fig. 9
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Attribute-oriented concept lattice

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Fig. 10
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Attribute-oriented concept lattice

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Fig. 11
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Attribute-oriented concept lattice

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Fig. 12
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Attribute-oriented concept lattice

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Fig. 13
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Attribute-oriented concept lattice

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Fig. 14
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Attribute-oriented concept lattice

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Fig. 15
figure 15

Attribute-oriented concept lattice

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Fig. 16
figure 16

Attribute-oriented concept lattice

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Shao, MW., Lv, MM., Li, KW. et al. The construction of attribute (object)-oriented multi-granularity concept lattices. Int. J. Mach. Learn. & Cyber. 11, 1017–1032 (2020). https://doi.org/10.1007/s13042-019-00955-0

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