A Subspace Clustering Algorithm of Categorical Data Using Multiple Attribute Weights
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摘要: 采用多属性频率权重以及多目标簇集质量聚类准则，提出一种分类数据子空间聚类算法.该算法利用粗糙集理论中的等价类，定义了一种多属性权重计算方法，有效地提高了属性的聚类区分能力；在多目标簇集质量函数的基础上，采用层次凝聚策略，迭代合并子簇，有效地度量了各类尺度的聚类簇；利用区间离散度，解决了使用阈值删除噪音点所带来的参数问题；利用属性对簇的依附程度，确定了聚类簇的属性相关子空间，提高了聚类簇的可理解性.最后，采用人工合成、UCI和恒星光谱数据集，实验验证了该聚类算法的可行性和有效性.Abstract: In this paper, we propose a subspace clustering algorithm using frequencies of multiple attributes of categorical data. An attribute-value weight is calculated on the basis of equivalence class in rough set theory by using frequencies of multiple attributes. Attribute-value weights offer ample opportunities to improve classiffication ability of clustering. The well-known parameter problem, which is caused by using a threshold to delete noise points, is solved by the virtue of interval dispersion degrees. By adopting the hierarchical clustering method to iteratively merge sub-clusters, we effectively measure various scale clusters on the basis of a multi-objective clusters quality function. An attribute subspace is determined with the relevance degree of dimension to a cluster, so as to improve the cluster's interpretability. Finally, we validate the feasibility and effectiveness of our algorithm through extensive experiments using synthetic data as well as UCI and stellar spectral data sets.1) 本文责任编委 曾志刚
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图 7 各算法在四种数据集上的(ARI)对比
Fig. 7 Contrast experiment on four data sets (ARI) for all algorithms
表 1 符号表示及含义
Table 1 Symbol and notation
符号表示 符号含义 U 数据集 $x_k$ 第 $k$ 个数据点 $a_j$ 第 $j$ 维属性 $NOS$ 由噪音点组成的集合 $SA_s$ 簇 $C_s$ 的相关属性子空间 A 属性集 $x_{ki}$ 数据 $x_k$ 在第 $i$ 维属性上的属性取值 $C_s$ 第 $s$ 个类簇 $SD_s$ 簇 $C_s$ 的数据集合 $V_{a_i}$ 属性 $a_i$ 的值域 S 一个分类信息系统 ${[x_k]}_{a_i}$ 包含数据点 $x_k$ 的 $a_i$ 等价类 $n$ 数据点的总数 $d$ 属性空间的总维数 SC 子簇集 
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表 2 数据集
Table 2 Dataset
类别 数据集 数据量 维数 簇数 平衡数据 实验目的 合成数据 Type 1 10 000 50 6 是 抗噪性对比实验 Type 2 10 000 200 6 是 维度可扩展性对比实验 Type 3 60 000 50 6 是 数据量可扩展性对比实验 Type 4 10 000 50 6 否 非平衡数据集聚类效果对比实验 UCI Voting 435 16 2 否 属性值域元素少对聚类效果的影响 Splice 3 190 60 3 否 高维UCI数据集的效果对比实验 Mushroom 8 124 22 2 是 大数据量聚类效果对比实验 Zoo 101 17 7 否 多簇数据集性能实验; 属性子空间实验 真实数据 光谱数据 8 315 208 7 否 真实数据聚类效果对比实验
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表 3 SAC在光谱数据上性能指标变化
Table 3 Noise immunity on the spectral data for SAC
加入噪音数 去除噪音数 ARI $(\%)$ Purity $(\%)$ Jaccard $(\%)$ RI $(\%)$ 1 000 993 2.56 $\downarrow$ 0.13 $\downarrow$ 2.28 $\downarrow$ 2.08 $\downarrow$ 2 000 1 984 2.94 $\downarrow$ 0.57 $\downarrow$ 2.72 $\downarrow$ 2.21 $\downarrow$ 3 000 2 971 3.25 $\downarrow$ 0.92 $\downarrow$ 3.01 $\downarrow$ 2.52 $\downarrow$
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表 4 UCI数据集上的算法性能
Table 4 Algorithm performance on UCI
算法 UCI ARI $(\%)$ Purity $(\%)$ Jaccard $(\%)$ RI $(\%)$ DHCC Voting 53.67 86.67 63.17 76.84 Splice 41.75 46.08 38.96 75.11 Mushroom 29.90 89.50 38.22 63.52 Zoo 73.79 73.26 68.05 89.21 AT-DC Voting 38.5 80.92 54.23 69.28 Splice 19.78 40.22 35.31 61.28 Mushroom 45.20 93.21 48.81 72.63 Zoo 92.00 92.08 88.55 97.03 PROCAD Voting 58.49 88.28 66.39 79.11 Splice 16.57 63.29 34.31 59.30 Mushroom 61.10 89.11 68.22 80.56 Zoo 10.91 42.57 23.79 28.81 SAC Voting 84.88 96.09 86.80 92.47 Splice 86.26 94.76 84.23 93.54 Mushroom 46.21 84.25 59.62 73.13 Zoo 85.37 92.08 79.53 94.95
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表 5 Voting集上的实验结果
Table 5 Experimental results on Voting
算法 簇的总数 簇编号 democrat republican DHCC 2 0 49 159 1 218 9 AT-DC 2 0 186 1 1 81 166 PROCAD 2 0 226 10 1 41 158 SAC 2 0 2 153 1 265 15
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表 6 Splice集上的实验结果
Table 6 Experimental results on Splice
算法 簇的总数 簇编号 EI IE Neither DHCC 6 0 15 10 419 1 668 19 28 2 40 3 393 3 11 0 369 4 28 728 34 5 5 8 412 AT-DC 3 0 55 78 513 1 151 644 1 058 2 561 46 84 PROCAD 4 0 304 0 0 1 462 747 687 2 1 20 909 3 0 1 59 SAC 7 0 765 82 37 1 2 0 1 2 0 682 45 3 0 1 1 566 4 0 2 0 5 0 1 4 6 0 0 2
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表 7 Zoo集上的实验结果
Table 7 Experimental results on Zoo
算法 簇的总数 簇编号 Mammal Bird Reptile Fish Amphibian Insect Invertebrate DHCC 3 0 41 0 0 0 0 0 0 1 0 0 5 13 4 0 8 2 0 20 0 0 0 8 2 AT-DC 7 0 41 0 2 0 1 0 0 1 0 0 0 0 0 0 7 2 0 0 0 0 3 0 0 3 0 0 0 0 0 8 2 4 0 20 0 0 0 0 0 5 0 0 3 13 0 0 0 6 0 0 0 0 0 0 1 PROCAD 2 0 41 18 5 12 4 7 10 1 0 2 0 1 0 1 0 SAC 9 0 37 0 0 0 0 0 0 1 4 0 0 13 0 0 0 2 0 20 0 0 3 0 0 3 0 0 0 0 0 0 6 4 0 0 0 0 0 8 1 5 0 0 2 0 3 0 0 6 0 0 3 0 1 0 0 7 0 0 0 0 0 0 2 8 0 0 0 0 0 0 1
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表 8 Mushroom集上的实验结果
Table 8 Experimental results on Mushroom
算法 簇的总数 簇编号 poisonous edible DHCC 10 0 24 32 1 1 728 0 2 1 296 0 3 0 16 4 736 2 880 5 72 808 6 0 216 7 44 0 8 16 64 9 0 192 AT-DC 8 0 360 3 536 1 0 288 2 1 734 0 3 456 192 4 1 296 0 5 64 0 6 0 129 7 6 0 PROCAD 2 0 3 050 20 1 866 4 188 SAC 3 0 1 258 4 185 1 2 615 23 2 43 0
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表 9 Zoo数据集中的相关子空间
Table 9 Relevant subspace on Zoo data set
簇号 簇内数据分布 子空间上的属性及其取值 $C_1$ 100%Mammal hair = 1; eggs = 0; milk = 1; fins = 0; legs = 4 $C_2$ 76.5%Fish; 23.5%Mammal fins = 1; legs = 0; aquatic = 1 $C_3$ 100%Bird feathers = 1; airborne = 1; legs = 2 $C_4$ 100%Invertebrate backbone = 0; legs = 0 $C_5$ 88.8%Insect; 11.2%Invertebrate backbone = 0; legs = 6 $C_6$ 60%Amphibian; 40%Reptile hair = 0; predator = 1; toothed = 1 $C_7$ 75%Reptile; 25%Amphibian breathes = 1; legs = 4; tail = 1 $C_8$ 100%Invertebrate legs = 8 $C_9$ 100%Invertebrate legs = 5
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表 10 各种离散化方法下的SAC聚类性能指标对比
Table 10 Clustering performance of different discretization methods
ARI $(\%)$ Purity $(\%)$ Jaccard $(\%)$ RI $(\%)$ 7等距离散 45.18 67.55 36.86 85.20 11等距离散 46.65 84.02 36.12 87.38 15等距离散 41.75 91.34 31.06 87.13 18等距离散 36.68 91.61 26.61 86.46 22等距离散 31.40 92.71 22.22 85.78
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表 11 各算法在光谱数据上的对比实验
Table 11 Algorithm performance on spectral data
ARI $(\%)$ Purity $(\%)$ Jaccard $(\%)$ RI $(\%)$ SAC 46.65 84.02 36.12 87.38 PROCAD 43.41 80.18 32.41 82.72 DHCC 40.79 78.31 31.23 79.67 AT-DC 41.38 79.84 31.86 80.21 EWKM 33.47 70.08 30.11 71.64 CLICKS 40.65 71.66 30.83 75.79
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