Fuzzy C-Shell Clustering — cshell • e1071

The c-shell clustering algorithm, the shell prototype-based version (ring prototypes) of the fuzzy kmeans clustering method.

Usage

cshell(x, centers, iter.max=100, verbose=FALSE, dist="euclidean",
       method="cshell", m=2, radius = NULL)

Arguments

x: The data matrix, were columns correspond to the variables and rows to observations.
centers: Number of clusters or initial values for cluster centers
iter.max: Maximum number of iterations
verbose: If TRUE, make some output during learning
dist: Must be one of the following: If "euclidean", the mean square error, if "manhattan", the mean absolute error is computed. Abbreviations are also accepted.
method: Currently, only the "cshell" method; the c-shell fuzzy clustering method
m: The degree of fuzzification. It is defined for values greater than 1
radius: The radius of resulting clusters

Details

The data given by x is clustered by the fuzzy c-shell algorithm.

If centers is a matrix, its rows are taken as the initial cluster centers. If centers is an integer, centers rows of x are randomly chosen as initial values.

The algorithm stops when the maximum number of iterations (given by iter.max) is reached.

If verbose is TRUE, it displays for each iteration the number the value of the objective function.

If dist is "euclidean", the distance between the cluster center and the data points is the Euclidean distance (ordinary kmeans algorithm). If "manhattan", the distance between the cluster center and the data points is the sum of the absolute values of the distances of the coordinates.

If method is "cshell", then we have the c-shell fuzzy clustering method.

The parameters m defines the degree of fuzzification. It is defined for real values greater than 1 and the bigger it is the more fuzzy the membership values of the clustered data points are.

The parameter radius is by default set to 0.2 for every cluster.

Value

cshell returns an object of class "cshell".

centers: The final cluster centers.
size: The number of data points in each cluster.
cluster: Vector containing the indices of the clusters where the data points are assigned to. The maximum membership value of a point is considered for partitioning it to a cluster.
iter: The number of iterations performed.
membership: a matrix with the membership values of the data points to the clusters.
withinerror: Returns the sum of square distances within the clusters.
call: Returns a call in which all of the arguments are specified by their names.

Author

Evgenia Dimitriadou

References

Rajesh N. Dave. Fuzzy Shell-Clustering and Applications to Circle Detection in Digital Images. Int. J. of General Systems, Vol. 16, pp. 343-355, 1996.

Examples

## a 2-dimensional example
x <- rbind(matrix(rnorm(50, sd = 0.3), ncol = 2),
           matrix(rnorm(50, mean = 1, sd=0.3), ncol = 2))
cl <- cshell(x, 2, 20, verbose = TRUE, method = "cshell", m = 2)
#> Iteration:   1    Error:    0.0317656192
#> Iteration:   2    Epsi2:   18.0300922366
#> Iteration:   2    Error:    0.0284355344
#> Iteration:   3    Epsi2:   14.5909299138
#> Iteration:   3    Error:    0.0246995601
#> Iteration:   4    Epsi2:    8.7592561949
#> Iteration:   4    Error:    0.0223561159
#> Iteration:   5    Epsi2:    6.2345946961
#> Iteration:   5    Error:    0.0194313739
#> Iteration:   6    Epsi2:    2.7899574363
#> Iteration:   6    Error:    0.0181781516
#> Iteration:   7    Epsi2:    1.4094098460
#> Iteration:   7    Error:    0.0178983110
#> Iteration:   8    Epsi2:    0.8611228287
#> Iteration:   8    Error:    0.0178152877
#> Iteration:   9    Epsi2:    0.5602410942
#> Iteration:   9    Error:    0.0177740024
#> Iteration:  10    Epsi2:    0.2662051168
#> Iteration:  10    Error:    0.0177588922
#> Iteration:  11    converged, Error:    0.1642056212
#> Iteration:  11    Error:    0.0177476041
print(cl)
#> Fuzzy c-means clustering with 2 clusters
#> 
#> Cluster centers:
#>        [,1]      [,2]
#> 1 0.3369830 0.2915182
#> 2 0.7677189 0.5842502
#> 
#> Memberships:
#>                  1            2
#>  [1,] 0.0016701803 0.9983298197
#>  [2,] 0.0222895667 0.9777104333
#>  [3,] 0.3166518915 0.6833481085
#>  [4,] 0.0014978196 0.9985021804
#>  [5,] 0.0457113907 0.9542886093
#>  [6,] 0.0047635492 0.9952364508
#>  [7,] 0.0015842603 0.9984157397
#>  [8,] 0.9410556387 0.0589443613
#>  [9,] 0.0352848813 0.9647151187
#> [10,] 0.2017209903 0.7982790097
#> [11,] 0.0083336245 0.9916663755
#> [12,] 0.9221261901 0.0778738099
#> [13,] 0.1007964286 0.8992035714
#> [14,] 0.2443427985 0.7556572015
#> [15,] 0.9803239258 0.0196760742
#> [16,] 0.9909381427 0.0090618573
#> [17,] 0.9487001405 0.0512998595
#> [18,] 0.4289579508 0.5710420492
#> [19,] 0.4959662024 0.5040337976
#> [20,] 0.1408096741 0.8591903259
#> [21,] 0.0834521110 0.9165478890
#> [22,] 0.0615471157 0.9384528843
#> [23,] 0.0017123893 0.9982876107
#> [24,] 0.9562406912 0.0437593088
#> [25,] 0.0142216528 0.9857783472
#> [26,] 0.5524127294 0.4475872706
#> [27,] 0.0480705399 0.9519294601
#> [28,] 0.0242270284 0.9757729716
#> [29,] 0.9570376167 0.0429623833
#> [30,] 0.9209252221 0.0790747779
#> [31,] 0.9610408777 0.0389591223
#> [32,] 0.9734715747 0.0265284253
#> [33,] 0.3714824375 0.6285175625
#> [34,] 0.0001690426 0.9998309574
#> [35,] 0.0524782103 0.9475217897
#> [36,] 0.9708420973 0.0291579027
#> [37,] 0.9995023861 0.0004976139
#> [38,] 0.0311154433 0.9688845567
#> [39,] 0.8121020057 0.1878979943
#> [40,] 0.8834971596 0.1165028404
#> [41,] 0.9968247213 0.0031752787
#> [42,] 0.9920258288 0.0079741712
#> [43,] 0.9822035817 0.0177964183
#> [44,] 0.0923990411 0.9076009589
#> [45,] 0.4607847491 0.5392152509
#> [46,] 0.2480535209 0.7519464791
#> [47,] 0.0604587600 0.9395412400
#> [48,] 0.9896420904 0.0103579096
#> [49,] 0.7395691023 0.2604308977
#> [50,] 0.6735220365 0.3264779635
#> 
#> Closest hard clustering:
#>  [1] 2 2 2 2 2 2 2 1 2 2 2 1 2 2 1 1 1 2 2 2 2 2 2 1 2 1 2 2 1 1 1 1 2 2 2 1 1 2
#> [39] 1 1 1 1 1 2 2 2 2 1 1 1
#> 
#> Available components:
#> [1] "centers"     "radius"      "size"        "cluster"     "iter"       
#> [6] "membership"  "withinerror" "call"