Inner product reduction in formal context

doi:10.11772/j.issn.1001-9081.2022030328

Journal of Computer Applications ›› 2023, Vol. 43 ›› Issue (4): 1079-1085.DOI: 10.11772/j.issn.1001-9081.2022030328

• Data science and technology • Previous Articles

Inner product reduction in formal context

Qing WANG(), Xiuwei GAO, Yehai XIE, Guilong LIU

School of Information Science，Beijing Language and Culture University，Beijing 100083，China

Received:2022-03-21 Revised:2022-07-05 Accepted:2022-07-11 Online:2023-01-11 Published:2023-04-10
Contact: Qing WANG
About author:GAO Xiuwei， born in 1991， Ph. D. candidate. Her research interests include rough set， formal concept analysis.
XIE Yehai， born in 1984， Ph. D. candidate. His research interests include rough set， formal concept analysis.
LIU Guilong， born in 1962， Ph. D.， professor. His research interests include formal concept analysis， rough set， computational intelligence， data mining.
Supported by:
National Natural Science Foundation of China(61972052)

形式背景的内积约简

王清(), 高秀巍, 谢业海, 刘贵龙

北京语言大学信息科学学院，北京 100083

通讯作者: 王清
作者简介:高秀巍（1991—），女，辽宁鞍山人，博士研究生，主要研究方向：粗糙集、形式概念分析；
谢业海（1984—），男，海南琼中人，博士研究生，主要研究方向：粗糙集、形式概念分析；
刘贵龙（1962—），男，江西广昌人，教授，博士，主要研究方向：形式概念分析、粗糙集、计算智能、数据挖掘。
基金资助:
国家自然科学基金资助项目(61972052)

Abstract

Abstract:

Formal concept analysis is an important tool for knowledge representation and mining， and formal context is one of the basic concepts in formal concept analysis. A new attribute reduction — inner product reduction was proposed to solve the problem of whether the object set in the formal context has the same attribute in a given attribute set， and also to solve the problem of how to eliminate irrelevant attributes in the calculation. Firstly， the concept of inner product was given in formal context. Then， the reduction theory and method in relation system were used to define the inner product reduction， and the inner product reduction algorithm based on discernibility matrix was proposed to obtain all the reduction results in the formal context， and the reduction core was obtained through the intersection operation based on the results. In addition， when attributes increased， an incremental inner product reduction algorithm was designed. Finally， the application of inner product reduction was explored in infectious disease network. In the simulated case， 6 attributes were reduced to 2 attributes. Simulation outcomes demonstrate that the inner product reduction method is feasible， interpretable， and successful in achieving the knowledge reduction goal.

Key words: inner product reduction, formal context, discernibility matrix, incremental inner product reduction, infectious disease network

摘要：

形式概念分析是知识表示与挖掘的重要工具，而形式背景是形式概念分析中的基本概念之一。针对形式背景中对象集在给定属性集内是否拥有相同属性的问题，同时也为解决如何在计算中消除无关属性的问题，提出了新的属性约简类型，即内积约简。首先在形式背景中给出了内积概念；然后利用关系系统的约简理论和方法定义内积约简，并提出了基于差别矩阵的内积约简算法，以得到形式背景下的全部约简，在此基础上，通过交运算可以得到约简核；当属性增加时，设计了增量内积约简算法。最后结合传染病网络探索内积约简的应用，在模拟案例中属性数量从6个约简到2个。模拟结果表明，内积约简具有可行性和可解释性，达到了知识约简目的。

关键词: 内积约简, 形式背景, 差别矩阵, 增量内积约简, 传染病网络

CLC Number:

TP18

Qing WANG, Xiuwei GAO, Yehai XIE, Guilong LIU. Inner product reduction in formal context[J]. Journal of Computer Applications, 2023, 43(4): 1079-1085.

王清, 高秀巍, 谢业海, 刘贵龙. 形式背景的内积约简[J]. 《计算机应用》唯一官方网站, 2023, 43(4): 1079-1085.

Figures/Tables 9

Tab. 1 Formal context 1 （U，A，I）

$U$	$a 1$	$a 2$	$a 3$	$a 4$
$x 1$	1	0	1	1
$x 2$	0	1	1	0
$x 3$	1	0	1	0
$x 4$	0	0	0	1

Tab. 1 Formal context 1 （U，A，I）

$U$	$a 1$	$a 2$	$a 3$	$a 4$
$x 1$	1	0	1	1
$x 2$	0	1	1	0
$x 3$	1	0	1	0
$x 4$	0	0	0	1

Tab. 2 Formal context 2 （U，A，I）

$U$	$a 1$	$a 2$	$a 3$	$a 4$	$a 5$	$a 6$
$x 1$	1	0	1	1	1	0
$x 2$	0	1	1	0	1	1
$x 3$	1	0	1	0	0	1
$x 4$	0	0	0	1	0	0

Tab. 2 Formal context 2 （U，A，I）

$U$	$a 1$	$a 2$	$a 3$	$a 4$	$a 5$	$a 6$
$x 1$	1	0	1	1	1	0
$x 2$	0	1	1	0	1	1
$x 3$	1	0	1	0	0	1
$x 4$	0	0	0	1	0	0

Tab. 3 Discernibility matrix

$U$	$x 1$	$x 2$	$x 3$	$x 4$
$x 1$	$M 1, 3, 4, 5$	$M 3, 5$	$M 1, 3$	$M 4$
$x 2$		$M 2, 3, 5, 6$	$M 3, 6$	$∅$
$x 3$			$M 1, 3, 6$	$∅$
$x 4$				$M 4$

Tab. 3 Discernibility matrix

$U$	$x 1$	$x 2$	$x 3$	$x 4$
$x 1$	$M 1, 3, 4, 5$	$M 3, 5$	$M 1, 3$	$M 4$
$x 2$		$M 2, 3, 5, 6$	$M 3, 6$	$∅$
$x 3$			$M 1, 3, 6$	$∅$
$x 4$				$M 4$

Tab. 4 Formal context 3 （U，A1，I1）

$U$	$a 1$	$a 2$	$a 3$	$a 4$	$a 5$	$a 6$	$a 1 +$	$a 2 +$
$x 1$	1	0	1	1	1	0	1	0
$x 2$	0	1	1	0	1	1	0	1
$x 3$	1	0	1	0	0	1	0	0
$x 4$	0	0	0	1	0	0	0	1

Tab. 4 Formal context 3 （U，A1，I1）

$U$	$a 1$	$a 2$	$a 3$	$a 4$	$a 5$	$a 6$	$a 1 +$	$a 2 +$
$x 1$	1	0	1	1	1	0	1	0
$x 2$	0	1	1	0	1	1	0	1
$x 3$	1	0	1	0	0	1	0	0
$x 4$	0	0	0	1	0	0	0	1

Tab. 5 New discernibility matrix

$U$	$x 1$	$x 2$	$x 3$	$x 4$
$x 1$	$M 1, 3, 4, 5, M 1 +$	$M 3, 5$	$M 1, 3$	$M 4$
$x 2$		$M 2, 3, 5, 6, M 2 +$	$M 3, 6$	$M 2 +$
$x 3$			$M 1, 3, 6$	$∅$
$x 4$				$M 4, M 2 +$

Tab. 5 New discernibility matrix

$U$	$x 1$	$x 2$	$x 3$	$x 4$
$x 1$	$M 1, 3, 4, 5, M 1 +$	$M 3, 5$	$M 1, 3$	$M 4$
$x 2$		$M 2, 3, 5, 6, M 2 +$	$M 3, 6$	$M 2 +$
$x 3$			$M 1, 3, 6$	$∅$
$x 4$				$M 4, M 2 +$

Tab. 6 Formal context 4 （U，A，I）

$U$	$a 1$	$a 2$	$a 3$	$a 4$	$a 5$	$a 6$
$x 1$	1	0	1	1	1	0
$x 2$	0	1	1	0	1	1
$x 3$	1	0	1	0	0	1
$x 4$	0	0	0	1	0	0

Tab. 6 Formal context 4 （U，A，I）

$U$	$a 1$	$a 2$	$a 3$	$a 4$	$a 5$	$a 6$
$x 1$	1	0	1	1	1	0
$x 2$	0	1	1	0	1	1
$x 3$	1	0	1	0	0	1
$x 4$	0	0	0	1	0	0

Fig. 1 Infectious disease network diagram

Fig. 2 Reduction result 1

Fig. 3 Reduction result 2

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Inner product reduction in formal context

形式背景的内积约简

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