模糊群集分析(例如：模糊c均值法(fuzzy c-means) 和以模型基礎分群法(model-based clustering))是以隸屬程度方式呈現每個觀測值被分到每一群集的結果，比硬分群帶來更多訊息，然而其結果卻很少直接被用來分析或者可視化。較多的情況是依照擁有最大隸屬程度的該群轉換成類別標籤，如此便可以利用現有處理硬分群結果的方法進一步分析。本論文提出一個模糊群集分析結果視覺化的方法，並實作一個R語言的套件，命名為vfclust，用以執行各種模糊分群的演算法以及其可視化結果，演算法包括：模糊切片逆迴歸法(fuzzy sliced inverse regression)、修正塞門映射(modified Sammon mapping)等等。視覺化方法則結合維度縮減方法和地形等高線圖用以呈現在低維度空間中資料的模糊群集趨勢。同時我們也加入群集分析有效性的評估指標。我們的目的在於能夠以視覺化方法反映關於模糊群集結構所含包的訊息，並且提供使用者一個探索性資料分析的圖形化工具。我們將會描述這個套件的功能並且用真實與模擬的資料來演示這套件。

The fuzzy clustering algorithms such as fuzzy c-means and model-based clustering assign each object to multiple clusters according to their degrees of membership and thus give more information than those obtained with hard clustering methods. However, the resulting class membership matrix is rarely visualized and analyzed, but converted to the class labels with the highest membership value so that the existing approaches for hard clustering can be applied. This study presents the package vfclust for performing the various fuzzy clustering algorithms and plots to visualizing results in R. These include the fuzzy sliced inverse regression and the modified Sammon mapping (FUZZSAM). We combine the dimension reduction techniques and the contour map to visualize the fuzzy clustering results in the lower dimensional subspace. We also implemented some fuzzy cluster validity indices. We aim to provide a graphical tool which reflect additional information regarding the fuzzy cluster structures for the exploratory data analysis. We describe the functions of the package and demonstrate on real and artificial data sets accompanying the package.

目錄
1 緒論............................................1
2 模糊群集分析演算法...............................3
 2.1 模糊 c 均值法................................3
 2.2 模糊 c-shell 分群法..........................3
 2.3 模型基礎分群法................................4
 2.4 Gustafson-Kessel 分群法......................5
 2.5 Gath-Geva 分群法.............................5
 2.6 模糊 C 均值法之變化: fanny....................6
 2.7 引導聚集算法..................................6
3 維度縮減方法.....................................6
 3.1 主成份分析....................................6
 3.2 多元尺度法....................................7
 3.3 塞門映射法....................................7
 3.4 FUZZSAM 演算法...............................7
 3.5 切片逆迴歸法..................................8
 3.6 模糊切片逆迴歸法..............................8
4 模糊群集分析結果之視覺化..........................9
5 於 R 語言實行....................................10
 5.1 R 語言套件：vfclust...........................10
 5.2 vfclust 語法說明..............................10
 5.2.1 vfclust 輸入指令............................10
 5.2.2 vfclust 輸出物件............................12
 5.3 實際演示......................................14
  5.3.1 以 Wine 資料集為例.........................14
  5.3.2 以 Motorcycle 資料集為例...................25
 5.4 與 R 套件 fclust 比較.........................28
6 結果與討論.......................................30
圖目錄
1 vfclust 以 Wine 資料集範例圖形....................17
2 Wine 資料集以不同分群方法之視覺化圖形..............21
3 Wine 資料集以不同投影方法之視覺化圖形..............23
4 Wine 資料集各別群集之視覺化圖形....................25
5 vfclust 以 Motorcycle 資料集範例圖形..............28
6 Wine 與 Motorcycle 資料集使用 fclust 執行
  模糊分群視覺化圖形................................29
7 Wine 資料集執行 Gustafson-Kessek 分群法之
  視覺化結果........................................34
8 Wine 資料集執行 Gustafson-Kessek 分群法之
  視覺化結果........................................35
9 Wine 資料集執行 fanny 分群法之視覺化結果............37
10 Wine 資料集執行模糊 c-shell 分群法之視覺化結果.....38
11 Wine 資料集執行模型基礎分群法之視覺化結果...........40
12 Wine 資料集執行引導聚集算法分群之視覺化結果.........41
表目錄
1 vclust 之 FC.method 指令表........................12
2 vclust 之 DR.method 指令表........................12
3 Wine 資料集各個模糊分群方法之有效性指標............. 21

Abonyi J, Babuska R (2004). “FUZZSAM - Visualization of Fuzzy Clustering Results by Modified Sammon Mapping”, IEEE International Conference on Fuzzy Systems. Budapest, Hungary, 365–370.
Berthod MR, Hall LO (2003). “Visualizing Fuzzy Points in Parallel Coordinates”, IEEE Transaction on Fuzzy Systems, 11(3), 369–374.
Bezdek JC, Dunn JC (1975). “Optimal Fuzzy Partitions: A Heuristic for Estimating The Parameters in a Mixture of Normal Distributions”, IEEE Transactions on Computers,
C24(8), 835–838.
Bezdek JC, Ehrlich R, Full W (1984). “FCM: The Fuzzy C-Means Clustering Algorithm”, Computers & Geosciences, 10(2–3), 191–203
Breiman L (1996). “Bagging Predictors”, Machine learning, 24(2), 123–140.
Dave RN (1990). “Fuzzy Shell-Clustering and Applications to Circle Detection in Digital Images”, International Journal Of General System, 16(4), 343–355.
Ferrigno S, Gannoun A, Saracco J (2008). “Inverse Regression Methods based on Fuzzy Partitions”, In International Journal of Pure and Applied Mathematics, 43(1), 43–62.
Fraley C, Raftery AE (2002). “Model-Based Clustering, Discriminant Analysis, and Density Estimation”, Journal of the American Statistical Association, 97(458), 611–631.
Gao C, Zhang C (2010). “A Novel Method for Clustering Results”, Communications in Statistics - Simulation and Computation, 39(5), 1049–1056.
Gath I, Geva AB (1989). “Unsupervised Optimal Fuzzy Clustering”, IEEE Transactions on Pattern Analysis and Machine Intelligence, 11(7), 773–780.
Gustafson DE ,Kessel WC (1979). “Fuzzy Clustering with a Fuzzy Covariance Matrix”, Decision and Control including the 17th Symposium on Adaptive Processes, 761–766.
Kruskal JB (1964). “Multidimensional Scaling by Optimizing Goodness of Fit to a Nonmetric Hypothesis”, Psychometrika, 29(1), 1–27.
Li KC (1991). “Sliced Inverse Regression for Dimension Reduction”, Journal of the American Statistical Association, 86, 316–342.
Sammon JW (1969).“A Nonlinear Mapping for Data Structure Analysis”, IEEE Transactions on Computers, C18(5), 401–409.
Sharko J, Grinstein G (2009). “Visualizing Fuzzy Clusters Using RadViz”, 13th International Conference Information Visualization, 107–316.
Wold S, Esbensen K (1987), “Principal Component Analysis”, Chemometrics and Intelligent Laboratory Systems, 2(1–3), 37–52.
Zhang Y, Rueda L (2006). “Geometric Visualization of Clusters Obtained from Fuzzy Clustering Algorithms”, Pattern Recognition, 39(8), 1415–1429.