DETECTION AND CHARACTERIZATION OF CLUSTER SUBSTRUCTURE .1. LINEAR STRUCTURE - FUZZY C-LINES
DETECTION AND CHARACTERIZATION OF CLUSTER SUBSTRUCTURE .1. LINEAR STRUCTURE - FUZZY C-LINES
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DOI：
10.1137/0140029
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发表时间：
1981-01-01
影响因子：
1.9
通讯作者：
WATSON, J
中科院分区：
文献类型：
--
作者：
BEZDEK, JC;CORAY, C;WATSON, J
在第一部分中，对模糊均值（或模糊ISODATA）聚类算法进行了推广。广义总加权平方正交误差目标函数最小的必要条件导致了一种Picard迭代方案，该方案在数据中同时产生(i)cfuzzy聚类；（ii）特征空间中一组在定义良好的意义上最适合数据的原型直线；（iii）一组原型质心（在直线上），表征每个线性模糊聚类的“核心”。利用广义聚类内模糊散点矩阵的主成分实现理论优化。给出了无限族中每个算法的收敛定理。用实际和人工数据集（本质上具有“线性”子结构）的五个数值例子来说明算法。在第二部分中，Fuzzyc-Means和Fuzzyc-Lines算法被证明是一种更一般的模糊算法的特殊情况，即Fuzzyc-Varieties族，它生成（对于原型）任意维的线性变量，直到比数据空间的维数小1维。凸组合的品种算法也将研究在第二部分[SIAM J.应用。数学。， 40(1981)，第358-372页。
In Part I, a generalization of the Fuzzyc-Means (or Fuzzy ISODATA) clustering algorithms is developed. Necessary conditions for minimization of a generalized total weighted squared orthogonal error objective function lead to a Picard iteration scheme which generates simultaneously (i)cfuzzy clusters in the data; (ii) a set ofcprototypical straight lines in feature space which best fit the data in a well-defined sense; (iii) a set ofcprototpyical centers of mass (on theclines) which characterize the “core” of each linear fuzzy cluster. Theoretical optimization is achieved using principal components of generalized within cluster fuzzy scatter matrices. A convergence theorem for each algorithm in the infinite family is given. The algorithms are exemplified by five numerical examples using both real and artificial data sets having essentially “linear” substructure. In Part II, the Fuzzyc-Means and Fuzzyc-Lines algorithms are shown to be special cases of a more general class of fuzzy algorithms, the Fuzzyc-Varieties family, which generate (for prototypes)clinear varieties of arbitrary dimension up to dimension one less than the dimension of the data space. Convex combinations of thec-Varieties algorithms will also be studied in Part II [SIAM J. Appl. Math., 40 (1981), pp. 358–372].