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A theoretical study on object-oriented and property-oriented multi-scale formal concept analysis

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Abstract

In traditional formal concept analysis, the attributes in the formal context are considered fixed. However, in the real world data set, attributes always have different levels of granularity, correspondingly, the derived concept lattice may reveal different information and patterns. Therefore, the capability to change the level of granularity of an attribute in formal concept analysis to capture relevant patterns in data is a natural requirement. In this paper, a theoretical study has been undertaken in multi-scale formal contexts, where attributes with different levels of granularity possess different attribute values. Two types of formal concepts, i.e., object-oriented and property-oriented multi-scale concepts, are introduced and studied in detail. The collection of object-oriented concept lattices and property-oriented concept lattices can be obtained at different granularity levels of attributes. It has been shown that the set of extents in the derived concept lattices increases when we choose to use a finer level of granularity. Moreover, a corresponding bidirectional approach to concept construction(i.e., from coarser to finer and from finer to coarser, respectively) is exhibited, and some characterization theorems have been obtained.

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Funding

This work is supported by the National Nature Science Foundation of China (nos. 61976244, 61472471 and 11531009), Innovation Talent Promotion Plan of Shaanxi Province for Young Sci-Tech New Star (2017KJXX-60) and Scientific Research Program of Shaanxi Provincial Education Department (no. 18JK0625).

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Authors and Affiliations

  1. College of Science, Xi’an Shiyou University, Xi’an, 710065, China

    Yanhong She, Xiaoli He, Ting Qian & Qinqin Wang

  2. College of Computer Science, Xi’an Shiyou University, Xi’an, 710065, China

    Wanglin Zeng

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  1. Yanhong She
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  2. Xiaoli He
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Correspondence to Yanhong She.

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Appendix: A detailed nomenclature

Appendix: A detailed nomenclature

\(K=(G,M,I)\): A formal context

\(\uparrow :\) Intent derivation operator

\(\downarrow :\) Extent derivation operator

\(T_{y}\): Granularity tree for attribute y

\(C_{y}\): A cut in the granularity tree for y

\((G,M_{C},I_{C}):\) The data table induced by cuts in the formal context (G, M, I)

\(\Box :\) Necessity operator

\(\diamond :\) Sufficiency operator

\(\mathbf{B }^{o}(G,M_{C},I_{c})\): The collection of object-oriented concepts induced from \((G,M_{C},I_{c})\)

\(\hbox {EXT}(\mathbf{B }^{o}(X,Y_{C},I_{c}))\): The collection of extents in \(\mathbf{B }^{o}(X,Y_{C},I_{c})\)

\(\mathbf{B }^{p}(G,M_{C},I_{c})\): The collection of property-oriented concepts induced from \((G,M_{C},I_{c})\)

\(\hbox {EXT}(\mathbf{B }^{p}(X,Y_{C},I_{c}))\): The collection of extents in \(\mathbf{B }^{p}(X,Y_{C},I_{c})\)

\(F_{C_{2}}(y)\): The father node of y in \(M_{C_{2}}\)

\(F_{C_{2}}(Y)={\{F_{C_{2}}(y)\mid y\in Y\}}\)

\(S_{C_{1}}(m)\): The set of son nodes (or successors) of m in \(M_{C_{1}}\)

\(S_{C_{1}}(M)={\{S_{C_{1}}(m)\mid m\in M\}}\)

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She, Y., He, X., Qian, T. et al. A theoretical study on object-oriented and property-oriented multi-scale formal concept analysis. Int. J. Mach. Learn. & Cyber. 10, 3263–3271 (2019). https://doi.org/10.1007/s13042-019-01015-3

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