A Novel Fuzzy Clustering Algorithm Based on Similarity of Attribute Space
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摘要: 模糊C均值(FCM)聚类算法及其相关改进算法基于最大模糊隶属度原则确定聚类结果，没有充分利用迭代后的模糊隶属度矩阵和簇类中心的样本属性特征信息，影响聚类准确度。针对这个问题，该文提出一种新的改进思路：改进FCM算法输出定类原则。给出二元属性拓扑子空间中属性相似度的定义，最终提出一种基于属性空间相似性的改进FCM算法(FCM-SAS)：首先，选择FCM算法聚类后模糊隶属度低于聚类置信度的样本作为存疑样本；然后，计算存疑样本与聚类后聚类中心的属性相似度；最后，基于最大属性相似度原则更新存疑样本的簇类标签。通过UCI数据集实验，证明算法不仅有效，还较一些基于最大模糊隶属度原则定类的改进算法具有更优的聚类评价指标。Abstract: With the attribute feature information of the fuzzy membership matrix and cluster centers after the iteration not fully utilized, the results of Fuzzy C-Means (FCM) Clustering and related modified algorithms are determined based on the principle of maximum fuzzy membership, causing bad influence on the clustering accuracy. To solve this problem, the improvement ideas are proposed: to improve classification principle of FCM. The formula definition of attribute similarity in binary topological subspaces is given. Then, the improved FCM algorithm based on the Similarity of Attribute Space (FCM-SAS) is proposed: First, samples with fuzzy membership degree lower than the clustering reliability are selected as suspicious samples. Next, the attribute similarity between the suspicious samples and the cluster centers after clustering are calculated. Finally, cluster labels of suspicious samples based on the principle of maximum attribute similarity are updated. The validity and superiority of the proposed algorithm is verified by the UCI sample set experiments and comparisons with other modified algorithms based on the principle of maximum fuzzy membership.
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表 1 FCM-SAS算法具体步骤
输入： 样本集${\text{X}}$、样本数$n$，聚类个数$c$、加权指数$m$、迭代阈
值$\varepsilon $、最大迭代次数T、聚类存疑率$\xi $、属性占比率$\kappa $；输出： 样本标签集${{\text{X}}_l}'$； 1 按表1的传统FCM算法步骤得到迭代后模糊隶属度矩阵${\text{U}}$、簇类中心${\text{V}}$和样本标签集${{\text{X}}_l}$，令$j = 0$； 2 计算所有样本点${\text{x}}$的模糊隶属度最大值，按递增顺序排序并组成数组，选出此数组中第$\left[ {\xi \times n} \right]$个元素的数值作为聚类置信度$\eta $； 3 令$j = j + 1$，判断第$j$个样本的模糊隶属度$\max \left( {\left\{ {{u_{ij}}\left| {i = 1, 2, ·\!·\!· , c} \right.} \right\}} \right)$是否不大于聚类置信度$\eta $，若是则此样本为存疑样本，转步骤4；否则转步骤8； 4 按式(8)计算第$j$个样本与各个簇类中心在2元属性拓扑子空间中的拓扑相似度集合$\gamma \left( {{{\text{x}}_j}, {{\text{v}}_i}} \right)$，将所有集合中的元素取绝对值后按递增的顺序排序并组成数组，计算此数组中第$\left[ {n \times {\gamma _{dis}} \times \kappa } \right]$个元素与数值1之间差的绝对值作为邻域半径$\delta $； 5 以$\delta $为邻域半径，按式(10)计算第$j$个样本与各个簇类中心的属性相似度$\psi \left( {{{\text{x}}_j}, {{\text{v}}_i}} \right)$； 6 若最大属性相似度$\max \left( {\psi \left( {{{\text{x}}_j}, {{\text{v}}_i}} \right)} \right)$只有一个，则选出最大属性相似度时的簇类中心所在的类别作为此样本更新后的标签$x_{lj}'$，转步骤8；否则，转步骤7； 7 若最大属性相似度不止一个，则选择这些簇类中最大拓扑相似度集合之和$\widehat S = \max \left( {\Sigma \gamma \left( {{{\text{x}}_j}, {{\text{v}}_i}} \right)} \right)$时的簇类中心所在的类别作为$x_{lj}'$； 8 判断$j < n$，若是则转步骤3，否则输出更新后的样本标签集${{\text{X}}_l}'$。 
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表 2 算法输入参数设置
参数 数值 加权指数$m$ 2 迭代阈值$\varepsilon $ ${10^{ - 3}}$ 最大迭代次数$T\;$ $100$ 聚类存疑率$\xi $ $0.3$ 属性占比率$\kappa $ $0.5$
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表 3 UCI数据集的统计描述
数据集 样本数 维数 簇类 各类占比 Iris 150 4 3 50:50:50 Wine 178 13 3 59:71:48 Seeds 210 7 3 70:70:70 Breast 683 9 2 444:239 Glass 214 9 6 70:17:76:13:9:29
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表 4 UCI数据集聚类结果评价指标对比(1)
FCM RL-FCM RCA WFCM FRCM FCM-SAS(标准化样本) FCM-SAS(未标准化样本) Iris AR 0.893 0.907 0.967 0.957 0.960 0.987 0.953 RI 0.880 0.892 0.958 0.934 0.952 0.983 0.942 NMI 0.750 – – 0.831 0.873 0.949 0.8498 Seeds AR 0.895 0.895 0.903 0.895 0.895 0.919 0.900 RI 0.874 0.874 0.884 0.873 0.876 0.899 0.877 NMI 0.695 – – 0.677 0.697 0.717 0.671 Breast AR 0.937 0.953 0.655 0.938 0.947 0.965 0.946 RI 0.876 0.910 0.548 0.884 0.911 0.932 0.897 NMI 0.730 – – 0.736 0.755 0.782 0.688
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表 5 UCI数据集聚类结果评价指标对比(2)
FCM PSO-IFCM GA-IFCM ABC-IFCM KFCM WGFCM FCM-SAS(标准化样本) FCM-SAS(未标准化样本) Iris AR 0.893 0.807 0.849 0.787 0.895 0.973 0.987 0.953 Wine AR 0.949 0.655 0.652 0.642 0.942 0.966 0.955 0.781 Glass AR 0.421 0.419 0.393 0.467 0.460 0.733 0.533 0.472
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表 6 FCM-SAS算法聚类过程统计数据
样本集 1次定类错误样本数 1次定类正确的存疑样本数 1次定类错误的存疑样本数 存疑样本数 2次定类正确样本数 2次定类错误样本数 Iris 16 29 16 45 43 2 Seeds 21 44 19 63 48 15 Breast 43 173 27 200 191 9 Wine 9 45 8 53 47 6 Glass 126 21 42 63 45 18
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