Study on the Approximate Reasoning Models of Decision Implication in Formal Decision Context
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摘要: 决策蕴涵分析是形式概念分析研究的重要方面, 基于形式背景获取决策蕴涵、概念规则等知识是数据分析、机器学习的重要研究内容之一. 首先, 利用属性逻辑语义对决策蕴涵的特性进行刻画. 其次, 在经典二值逻辑框架下分析决策蕴涵、概念规则的基于全蕴涵三I推理思想及分离规则(Modus ponens, MP)和逆分离规则(Modus tonens, MT)的近似推理模式的特征, 证明决策蕴涵的MP、MT近似推理结论是决策蕴涵, 概念规则的MP、MT近似推理结论是概念规则等结论. 引进属性逻辑公式的伪距离, 在属性逻辑伪距离空间中分析推理对象范围参数变化对决策蕴涵MP、MT近似推理结论的影响. 最后, 提出若干通过MP、MT近似推理生成决策蕴涵、概念规则及拟决策蕴涵的模式和方法, 数值实验验证了所提方法的有效性.Abstract: Decision implication analysis is an important aspect of formal concept analysis, and it is one of the contents of data analysis and machine learning to obtain the knowledge of decision implication and conceptual rule from formal decision context. Firstly, attribute logic semantics is used to analyze the characteristics of decision implication. Secondly, in the classical binary logic framework, the characteristics of modus ponens (MP) and modus tonens (MT) approximate reasoning models based on the all implication three-I inference idea is analyzed. It is proved that the MP and MT approximate reasoning conclusions of decision implications are decisions implications, the MP and MT approximate reasoning conclusions of concept rules are concept rules. The pseudo-metric of attribute logic formula is introduced, and the approximate reasoning of decision implication and its properties are analyzed in attribute logical pseudo-metric space in machine learning environment. Finally, some models and methods of generating decision implication, concept rules and quasi-decision implication through MP and MT approximate reasoning based on existing decision implication, concept rules and quasi-decision implication are proposed, numerical experiments show that the proposed methods are effective.
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表 1 形式背景$K=(G,M,I)$
Table 1 A formal context $K=(G,M,I)$
$G$ $a_1$ $a_2$ $a_3$ $a_4$ $a_5$ $u_1$ 1 1 0 0 1 $u_2$ 0 0 1 1 0 $u_3$ 1 0 1 0 0 $u_4$ 0 1 0 1 0 $u_5$ 0 1 0 1 1 $u_6$ 1 0 1 0 1 
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表 2 决策形式背景$K=(G,C,D,I,J)$
Table 2 A formal decision context $K=(G,C,D,I,J)$
$G$ $a_1$ $a_2$ $a_3$ $a_4$ $d_1$ $d_2$ $d_3$ $u_1$ 1 1 1 1 1 0 1 $u_2$ 0 0 1 0 0 1 0 $u_3$ 1 0 0 0 0 1 1 $u_4$ 0 1 1 0 1 0 0
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表 3 决策形式背景$K=(G,C,D,I,J)$
Table 3 A formal decision context $K=(G,C,D,I,J)$
$G$ $a_1$ $a_2$ $a_3$ $a_4$ $a_5$ $a_6$ $d_1$ $d_2$ $d_3$ $u_1$ 1 1 0 0 1 1 1 0 1 $u_2$ 0 0 1 1 1 1 1 1 0 $u_3$ 1 0 1 0 0 1 0 0 1 $u_4$ 0 1 0 1 0 1 1 0 0 $u_5$ 0 1 0 1 1 1 1 1 0 $u_6$ 1 0 1 0 1 1 0 1 1 $u_7$ 0 1 1 1 0 0 1 1 1
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表 4 生成的拟决策蕴涵个数
Table 4 The number of generated quasi-decision implications
数据组别 生成的拟决策蕴涵个数 数据组1 62 数据组2 58 数据组3 74 数据组4 71
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表 5 生成的拟决策蕴涵后件与后件合取公式的伪距离
Table 5 The metric between the consequent of generated quasi-decision implications and the consequent conjunctive
数据组别 最大伪距离 最小伪距离 数据组1 0.195 0.147 数据组2 0.189 0.160 数据组3 0.197 0.152 数据组4 0.194 0.161
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表 6 测试数据变化生成的拟决策蕴涵表
Table 6 A table of generated quasi-decision implication as test data changes
数据组别 拟决策蕴涵个数 最小伪距离 数据组1 3 0.0082 数据组2 5 0.0157 数据组3 8 0.0256 数据组4 12 0.0324 数据组5 16 0.0418 数据组6 21 0.0527 数据组7 25 0.0619 数据组8 29 0.0718 数据组9 34 0.0821 数据组10 39 0.0913
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表 7 后件集对结论的支持度和获取拟决策蕴涵时间成本对比
Table 7 Comparison of the support degree of consequent set to the conclusion and time consumption of obtaining quasi-decision implications
数据组别 $L$的后件集对结论支持度$\tau_{\Delta}$ 方式1)总时间成本(s) 方式2)总时间成本 (s) 增量获取所用时间成本(s) 数据组1 0.933 0.204 0.223 0.103 数据组2 0.923 0.283 0.304 0.147 数据组3 0.914 0.361 0.389 0.198 数据组4 0.903 0.450 0.484 0.258 数据组5 0.893 0.539 0.258 0.326
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