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 支撐向量機（support vector machines）在近年來成為最熱門的一種機器學習演算法。支撐向量群聚演算法( support vector clustering )簡稱SVC便是以此為靈感而發展形成。 SVC是一種非監督式學習的演算法。它把要分群的點先映到其它高維度的空間，接著找一最小包含球( minimum enclosing ball )包住這些點，當這在高維度形成的球映回原來的空間變成數個輪廓，每個輪廓中包含個數不一的資料點，同一輪擴的資料點理所當然成為一個群。這是SVC的主要分群概念。SVC主要的缺點在於其群聚分配( cluster labeling )時間複雜度過高，因此許多的方法被提出來目的在改善這個缺點。 本篇論文結合支撐向量( support vector )的特性、 最小包含球以及k-means演算法， 把這些概念作一適當的分配利用達到降低時間複雜度的目的。 在實驗的部分我們創造一些分布在2維或3維空間的資料集以利可以馬上觀察出各個資料集的群聚分布。並把我們提出的方法跟其他現有的方法作比較，不管是時間花費度跟潔果準確度我們的方法均有較突出且令人滿意的表現。
 Clustering analysis which is categorized as unsupervised learning in machine learning means based on speci‾c features creating groups of objects in such a way that the objects grouping into the same clusters are similar and those belonging in diferent clusters are dissimilar. Support vector clustering (SVC) is an unsupervised and kernel-based clustering algorithm. SVC could naturally separate dataset with any shape into diferent clusters. SVC separates the dataset into appropriate clusters by tuning two parameters. Suppose number of data points is n, the time complexity of labeling data points becomes O(n2).It becomes the bottleneck of SVC and this is the major drawback why SVC always takes more time than other clustering algorithms.Focus this drawback, this thesis suggests a novel cluster labeling algorithm combining with the concept of minimum enclosing balls (MEBs), the property of support vector and k means to improve the e±ciency. In the later section, we will test our proposed method on synthetic datasets either in R2 and R3 space in order to visualize the clustering results, and demonstrate the e±ciency of our proposed method by comparing with other cluster labeling algorithms. Besides we also find a flaw in the original SVC mathematical programming model, another issue of this thesis is to discuss the flaw and solve it.
 1 Introduction 11.1 Clustering Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Related Clustering Algorithms . . . . . . . . . . . . . . . . . . . . . . 21.2.1 Hierarchical algorithms . . . . . . . . . . . . . . . . . . . . . . . 21.2.2 Partitional algorithms . . . . . . . . . . . . . . . . . . . . . . . . 31.3 A Novel Clustering Algorithm : Support Vector Clustering . . . . 41.4 Organization of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Related Work 62.1 Support Vector Machines for Classi‾cation . . . . . . . . . . . . . . 62.1.1 Support Vector Classi‾cation . . . . . . . . . . . . . . . . . . . . . 62.1.2 Idea of Kernel Functions . . . . . . . . . . . . . . . . . . . . . . . . 82.1.3 One-Class SVM . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Support Vector Clustering . . . . . . . . . . . . . . . . . . . . . . . .10I2.2.1 Introduction of SVC . . . . . . . . . . . . . . . . . . . . . . . . . 102.2.2 Strategies of Cluster Labeling . . . . . . . . . . . . . . . . . . . 153 SVC via Minimum Enclosing Balls 193.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . 193.2 Cluster Labeling via MEBs . . . . . . . . . . . . . . . . . . . . . . . . 204 Experiments and Results 274.1 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . .275 Conclusion and Further Work 32A SVC Type Algorithm Via MEBs Implementation for Matlab Code 34
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