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模糊理論集群法
Fuzzy Clustering
此方法利用模糊(fuzzy)理論來對資料分群,此類型的方法又稱為軟分群(soft clustering),每一個個體可能屬於一個或多個群體。在模糊理論集群法中,依據資料的距離測量方式差異而分群結果會有所差別,最被廣泛使用的是 Fuzzy C-Means (FCM)方法,由Bezdek於1981所提出,其距離測量方式採歐基里德(Euclidean)平方。

分群模式-方法簡介

本方法使用之R相關套件與參考文獻:
相關套件:stats、base、graphics、cluster
參考文獻(依套件名稱排序):
  1. R Core Team (2013). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. URL:http://www.R-project.org/.
  2. Maechler, M., Rousseeuw, P., Struyf, A., Hubert, M., Hornik, K.(2013). cluster: Cluster Analysis Basics and Extensions. R package version 1.14.4.
範例F-6:

鳶尾花(iris)資料,最早由英國統計學家費雪(R. A. Fisher, 1890 – 1962)用於多變量分析(multivariate analysis)中的判別分析(discriminant analysis),故常稱為費雪鳶尾花資料。此資料是由美國植物學家安德生(E. S. Anderson, 1897 – 1969)所收集,故也稱為安德生鳶尾花資料。此資料記錄了鳶尾花三個亞種及其特徵,三亞種分別為山鳶尾(setosa)、變色鳶尾(versicolor)及維吉尼亞鳶尾(virginica),花的特徵則包含花萼(sepal)與花瓣(petal)的長度與寬度。

表:鳶尾花資料
變數名稱 花萼長度 花萼寬度 花瓣長度 花瓣寬度 品種
1 5.1 3.5 1.4 0.2 setosa
2 4.9 3.0 1.4 0.2 setosa
3 4.7 3.2 1.3 0.2 setosa
150 5.9 3.0 5.1 1.8 virginica

Q1:資料中記錄有花萼與花瓣的長度與寬度以及花的亞種,植物學家想了解,若記錄的資料中僅有花的特徵(花萼與花瓣的長度與寬度),能否以此資料分辨出花的亞種數目?
統計方法:問題中想利用花萼與花瓣的長度與寬度來了解該花種的亞種數目,而亞種個數為未知,故適合使用具有分群能力的分析方法。具有分群能力的方法歸類於集群分析(clustering analysis)中,以下列出適用於此鳶尾花資料的方法,使用者可比較在相同的目的下,不同分析方法所獲得結果的差異性。

分群模式 集群分析 K組平均集群法(K-Means Clustering)
模糊理論集群法(Fuzzy Clustering)[包含C-Means方法]
階層分群法(Hierarchical Clustering)
自組織映射圖(Kohonen SOM)
模糊理論集群法 - 分析結果
  • 分析方法:模糊理論集群法
  • 資料名稱:範例F-6
  • 變數名稱:Sepal.Length, Sepal.Width, Petal.Length, Petal.Width
  • 距離矩陣測量方式:歐氏距離
  • 計算時間:0.305秒

  • 樣本敘述統計量I
    變數名稱
    Variable
    樣本數
    Count
    平均數
    Mean
    中位數
    Median
    最小值
    Minimum
    最大值
    Maximum
    標準差
    Std. dev.
    Sepal.Length1505.84335.84.37.90.8281
    Sepal.Width1503.0573324.40.4359
    Petal.Length1503.7584.3516.91.7653
    Petal.Width1501.19931.30.12.50.7622
    I:樣本敘述統計量皆不包含遺失值

  • 分群的成員係數(members coefficient)表I:(下載完整CSV檔)
    成員
    123
    191.423.64.97
    285.955.858.2
    387.015.467.53
    484.266.569.18
    590.454.035.53
    1465.6870.3723.95
    1476.6251.4641.93
    1484.7370.5624.71
    1497.1764.4228.4
    1506.7347.8745.4
    I:表中分數以百分比計算

  • 模糊係數(fuzzy coefficient):
    Dunn's係數 標準化係數
    0.5679 0.3519

  • 分群後集群成員(members):(下載完整CSV檔)
    觀察值編號集群
    11
    21
    31
    41
    51
    1462
    1472
    1482
    1492
    1502

  • 輪廓(silhouette)圖形資訊:(下載完整CSV檔)
    樣本編號集群
    cluster
    鄰居
    neighbor
    輪廓寬度
    silhouette width
    8130.8499
    1130.8491
    50130.8481
    18130.8471
    40130.8462
    84320.1601
    128320.148
    102320.1386
    143320.1386
    87320.0656

  • 平均輪廓(silhouette)寬度:
    集群
    cluster
    個數
    numbers
    平均輪廓寬度
    Average of silhouette width
    1500.7928
    2450.3823
    3550.4267
    Total1500.5354

  • 分群輪廓圖:

  • 二維分群圖:

  • 樣本相異(dissimilarities)矩陣:I
    123456789101112131415161718192021222324252627282930
    100.53850.50990.64810.14140.61640.51960.17320.9220.4690.37420.37420.59160.9950.88321.10450.54770.10.74160.33170.43590.30.64810.4690.59160.54770.31620.14140.14140.5385
    20.538500.30.33170.60831.09090.50990.42430.50990.17320.8660.45830.14140.67821.36011.62791.05360.54771.17470.83670.70710.76160.7810.55680.64810.22360.50.59160.50.3464
    30.50990.300.24490.50991.08630.26460.41230.43590.31620.88320.37420.26460.51.36381.58751.010.51961.23690.7550.83070.70.50990.64810.64030.4690.50990.61640.54770.3
    40.64810.33170.244900.64811.16620.33170.50.30.316210.37420.26460.51961.52971.71461.16620.65571.32290.8660.87750.80620.70710.64810.53850.42430.54770.72110.67820.1732
    50.14140.60830.50990.648100.61640.45830.22360.9220.52920.42430.34640.64030.97470.91651.08630.54770.17320.79370.26460.53850.26460.56570.52920.57450.63250.34640.24490.28280.5385
    60.61641.09091.08631.16620.616400.9950.71.45951.010.34640.81241.16191.57160.67820.61640.40.59160.33170.38730.53850.41231.12250.67820.83071.010.64810.52920.64811.0149
    70.51960.50990.26460.33170.45830.99500.42430.54770.47960.8660.30.48990.61641.36011.49330.95390.50991.20830.64810.86020.60.45830.62450.54770.60830.45830.62450.60830.3162
    80.17320.42430.41230.50.22360.70.424300.78740.33170.50.22360.4690.90551.0441.23690.70.20.83670.42430.44720.37420.67080.38730.44720.41230.22360.22360.22360.3742
    90.9220.50990.43590.30.9221.45950.54770.787400.55681.28450.67080.42430.34641.79161.99751.43180.92741.61251.14891.15761.08630.83070.9110.81240.64030.83071.0050.94340.469
    100.4690.17320.31620.31620.52921.010.47960.33170.556800.78740.34640.17320.7281.31151.55561.010.51.10.7550.62450.70.77460.52920.51960.20.44720.50990.44720.2646
    110.37420.8660.883210.42430.34640.8660.51.28450.787400.67820.93271.36750.58310.78740.34640.38730.38730.33170.36060.36060.94870.61640.7810.81240.54770.28280.37420.866
    120.37420.45830.37420.37420.34640.81240.30.22360.67080.34640.678200.45830.81851.23291.36380.86020.38730.9950.51960.60830.47960.66330.44720.30.44720.28280.42430.44720.2236
    130.59160.14140.26460.26460.64031.16190.48990.4690.42430.17320.93270.458300.58311.43181.69411.12690.61641.2570.88320.78740.82460.7550.65570.64810.30.57450.65570.57450.3162
    140.9950.67820.50.51960.97471.57160.61640.90550.34640.7281.36750.81850.583101.80832.04211.46631.011.73211.21661.31911.17470.68561.1181.02960.8660.9951.10911.03440.6782
    150.88321.36011.36381.52970.91650.67821.36011.0441.79161.31150.58311.23291.43181.808300.54770.4690.88880.55680.79370.87750.84261.28061.14891.36011.34161.09540.83670.87181.4177
    161.10451.62791.58751.71461.08630.61641.49331.23691.99751.55560.78741.36381.69412.04210.547700.61641.09090.64030.85441.08170.9221.46291.27281.41771.58111.22471.04881.14021.578
    170.54771.05361.011.16620.54770.40.95390.71.43181.010.34640.86021.12691.46630.4690.616400.51960.51960.38730.67080.41230.92740.78741.0051.04880.70710.52920.58311.0536
    180.10.54770.51960.65570.17320.59160.50990.20.92740.50.38730.38730.61641.010.88881.09090.519600.73480.31620.44720.24490.65570.41230.60.55680.26460.17320.17320.5477
    190.74161.17471.23691.32290.79370.33171.20830.83671.61251.10.38730.9951.2571.73210.55680.64030.51960.734800.63250.50990.64811.32290.80621.011.07240.81850.62450.71411.1747
    200.33170.83670.7550.8660.26460.38730.64810.42431.14890.7550.33170.51960.88321.21660.79370.85440.38730.31620.632500.54770.14140.74160.57450.64810.81850.43590.33170.43590.7348
    210.43590.70710.83070.87750.53850.53850.86020.44721.15760.62450.36060.60830.78741.31910.87751.08170.67080.44720.50990.547700.50991.08170.43590.63250.57450.45830.30.36060.7348
    220.30.76160.70.80620.26460.41230.60.37421.08630.70.36060.47960.82461.17470.84260.9220.41230.24490.64810.14140.509900.74160.45830.61640.74160.33170.30.38730.6782
    230.64810.7810.50990.70710.56571.12250.45830.67080.83070.77460.94870.66330.7550.68561.28061.46290.92740.65571.32290.74161.08170.741600.95920.94340.93810.77460.78740.74830.728
    240.4690.55680.64810.64810.52920.67820.62450.38730.9110.52920.61640.44720.65571.1181.14891.27280.78740.41230.80620.57450.43590.45830.959200.47960.44720.20.42430.44720.5196
    250.59160.64810.64030.53850.57450.83070.54770.44720.81240.51960.7810.30.64811.02961.36011.41771.0050.61.010.64810.63250.61640.94340.479600.53850.41230.57450.64030.3742
    260.54770.22360.4690.42430.63251.010.60830.41230.64030.20.81240.44720.30.8661.34161.58111.04880.55681.07240.81850.57450.74160.93810.44720.538500.44720.54770.48990.3606
    270.31620.50.50990.54770.34640.64810.45830.22360.83070.44720.54770.28280.57450.9951.09541.22470.70710.26460.81850.43590.45830.33170.77460.20.41230.447200.31620.34640.4123
    280.14140.59160.61640.72110.24490.52920.62450.22361.0050.50990.28280.42430.65571.10910.83671.04880.52920.17320.62450.33170.30.30.78740.42430.57450.54770.316200.14140.5916
    290.14140.50.54770.67820.28280.64810.60830.22360.94340.44720.37420.44720.57451.03440.87181.14020.58310.17320.71410.43590.36060.38730.74830.44720.64030.48990.34640.141400.5745
    300.53850.34640.30.17320.53851.01490.31620.37420.4690.26460.8660.22360.31620.67821.41771.5781.05360.54771.17470.73480.73480.67820.7280.51960.37420.36060.41230.59160.57450
    I:因資料筆數較多,網頁僅顯示部分矩陣。

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