cmeans: Fuzzy C-Means Clustering
In e1071: Misc Functions of the Department of Statistics, Probability Theory Group (Formerly: E1071), TU Wien
Description
Usage
Arguments
Details
Value
Author(s)
References
Examples
Description
The fuzzy version of the known kmeans clustering algorithm as
well as an on-line variant (Unsupervised Fuzzy Competitive learning).
Usage
1
2
3
Arguments
x
The data matrix where columns correspond to 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
If "cmeans", then we have the c-means fuzzy
clustering method, if "ufcl" we have the on-line update.
Abbreviations are also accepted.
m
A number greater than 1 giving the degree of fuzzification.
rate.par
A number between 0 and 1 giving the parameter of the
learning rate for the on-line variant. The default corresponds to
0.3.
weights
a numeric vector with non-negative case weights.
Recycled to the number of observations in x if necessary.
control
a list of control parameters. See Details.
Details
The data given by x is clustered by generalized versions of the
fuzzy c-means algorithm, which use either a fixed-point or an
on-line heuristic for minimizing the objective function
∑_i ∑_j w_i u_{ij}^m d_{ij},
where w_i is the weight of observation i, u_{ij} is
the membership of observation i in cluster j, and
d_{ij} is the distance (dissimilarity) between observation
i and center j. The dissimilarities used are the sums of
squares ("euclidean") or absolute values ("manhattan")
of the element-wise differences.
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, or when the algorithm is unable to reduce
the current value val of the objective function by
reltol * (abs(val) * reltol) at a step. The relative
convergence tolerance reltol can be specified as the
reltol component of the list of control parameters, and
defaults to sqrt(.Machine$double.eps).
If verbose is TRUE, each iteration displays its number
and the value of the objective function.
If method is "cmeans", then we have the c-means
fuzzy clustering method, see for example Bezdek (1981). If
"ufcl", we have the On-line Update (Unsupervised Fuzzy
Competitive Learning) method due to Chung and Lee (1992), see also Pal
et al (1996). This method works by performing an update directly
after each input signal (i.e., for each single observation).
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.
Value
An object of class "fclust" which is a list with components:
centers
the final cluster centers.
size
the number of data points in each cluster of the closest
hard clustering.
cluster
a vector of integers containing the indices of the
clusters where the data points are assigned to for the closest hard
clustering, as obtained by assigning points to the (first) class with
maximal membership.
iter
the number of iterations performed.
membership
a matrix with the membership values of the data points
to the clusters.
withinerror
the value of the objective function.
call
the call used to create the object.
Author(s)
Evgenia Dimitriadou and Kurt Hornik
References
J. C. Bezdek (1981).
Pattern recognition with fuzzy objective function algorithms.
New York: Plenum.
Fu Lai Chung and Tong Lee (1992).
Fuzzy competitive learning.
Neural Networks, 7(3), 539–551.
Nikhil R. Pal, James C. Bezdek, and Richard J. Hathaway (1996).
Sequential competitive learning and the fuzzy c-means clustering
algorithms.
Neural Networks, 9(5), 787–796.
Examples
1
2
3
4
5
6
7
8
9
10
11
12
# a 2-dimensional example
x<-rbind(matrix(rnorm(100,sd=0.3),ncol=2),
matrix(rnorm(100,mean=1,sd=0.3),ncol=2))
cl<-cmeans(x,2,20,verbose=TRUE,method="cmeans",m=2)
print(cl)
# a 3-dimensional example
x<-rbind(matrix(rnorm(150,sd=0.3),ncol=3),
matrix(rnorm(150,mean=1,sd=0.3),ncol=3),
matrix(rnorm(150,mean=2,sd=0.3),ncol=3))
cl<-cmeans(x,6,20,verbose=TRUE,method="cmeans")
print(cl)
Example output
Iteration: 1, Error: 0.1646046972
Iteration: 2, Error: 0.1543607452
Iteration: 3, Error: 0.1542090708
Iteration: 4, Error: 0.1542074436
Iteration: 5, Error: 0.1542074251
Iteration: 6 converged, Error: 0.1542074248
Fuzzy c-means clustering with 2 clusters
Cluster centers:
[,1] [,2]
1 -0.1263853 -0.08036624
2 0.9699374 1.13365887
Memberships:
1 2
[1,] 0.8041721115 0.195827889
[2,] 0.9624985570 0.037501443
[3,] 0.9422326865 0.057767313
[4,] 0.9358055047 0.064194495
[5,] 0.9494495198 0.050550480
[6,] 0.8087931231 0.191206877
[7,] 0.9833660711 0.016633929
[8,] 0.9819309179 0.018069082
[9,] 0.9894646400 0.010535360
[10,] 0.9715390275 0.028460973
[11,] 0.9311442900 0.068855710
[12,] 0.9375432876 0.062456712
[13,] 0.9130738829 0.086926117
[14,] 0.9192294617 0.080770538
[15,] 0.9724676099 0.027532390
[16,] 0.9317887302 0.068211270
[17,] 0.9820251726 0.017974827
[18,] 0.9853295839 0.014670416
[19,] 0.8442704429 0.155729557
[20,] 0.9441364738 0.055863526
[21,] 0.9153389923 0.084661008
[22,] 0.9826214149 0.017378585
[23,] 0.9892078061 0.010792194
[24,] 0.8759832492 0.124016751
[25,] 0.9540134279 0.045986572
[26,] 0.9347523673 0.065247633
[27,] 0.9973578267 0.002642173
[28,] 0.8199341875 0.180065812
[29,] 0.9360137796 0.063986220
[30,] 0.9361926486 0.063807351
[31,] 0.9446261582 0.055373842
[32,] 0.9912608295 0.008739171
[33,] 0.6940101181 0.305989882
[34,] 0.6148928515 0.385107149
[35,] 0.7882044852 0.211795515
[36,] 0.9814336729 0.018566327
[37,] 0.9323308133 0.067669187
[38,] 0.9723538916 0.027646108
[39,] 0.5611358982 0.438864102
[40,] 0.9506471475 0.049352853
[41,] 0.9420410152 0.057958985
[42,] 0.9475289679 0.052471032
[43,] 0.9380632237 0.061936776
[44,] 0.9928575856 0.007142414
[45,] 0.9790199311 0.020980069
[46,] 0.9217313557 0.078268644
[47,] 0.8883284811 0.111671519
[48,] 0.9816556987 0.018344301
[49,] 0.9549250997 0.045074900
[50,] 0.8871454670 0.112854533
[51,] 0.2114530375 0.788546962
[52,] 0.0862448751 0.913755125
[53,] 0.0510784637 0.948921536
[54,] 0.0004877015 0.999512298
[55,] 0.0414776249 0.958522375
[56,] 0.0259554941 0.974044506
[57,] 0.0014311945 0.998568806
[58,] 0.0516033727 0.948396627
[59,] 0.0688317563 0.931168244
[60,] 0.0352494627 0.964750537
[61,] 0.0129179845 0.987082016
[62,] 0.1849344045 0.815065595
[63,] 0.0100294106 0.989970589
[64,] 0.0003615129 0.999638487
[65,] 0.0271209451 0.972879055
[66,] 0.0169744698 0.983025530
[67,] 0.0623126968 0.937687303
[68,] 0.0183115960 0.981688404
[69,] 0.0212901927 0.978709807
[70,] 0.1699873547 0.830012645
[71,] 0.0762438670 0.923756133
[72,] 0.0274513011 0.972548699
[73,] 0.0346728814 0.965327119
[74,] 0.0165709526 0.983429047
[75,] 0.1757232598 0.824276740
[76,] 0.0137118788 0.986288121
[77,] 0.1526938167 0.847306183
[78,] 0.0094204114 0.990579589
[79,] 0.0679097455 0.932090254
[80,] 0.0586380314 0.941361969
[81,] 0.1199307203 0.880069280
[82,] 0.0321827248 0.967817275
[83,] 0.0456242573 0.954375743
[84,] 0.0047418541 0.995258146
[85,] 0.0470258756 0.952974124
[86,] 0.0436314118 0.956368588
[87,] 0.0091670303 0.990832970
[88,] 0.0529815880 0.947018412
[89,] 0.0454349758 0.954565024
[90,] 0.0292213963 0.970778604
[91,] 0.0298330734 0.970166927
[92,] 0.0353835883 0.964616412
[93,] 0.0278109131 0.972189087
[94,] 0.0810262427 0.918973757
[95,] 0.0348255780 0.965174422
[96,] 0.0444705016 0.955529498
[97,] 0.1344526231 0.865547377
[98,] 0.0021772383 0.997822762
[99,] 0.0000134157 0.999986584
[100,] 0.0145777392 0.985422261
Closest hard clustering:
[1] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
[38] 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
[75] 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
Available components:
[1] "centers" "size" "cluster" "membership" "iter"
[6] "withinerror" "call"
Iteration: 1, Error: 0.1298456810
Iteration: 2, Error: 0.1229062610
Iteration: 3, Error: 0.1212776711
Iteration: 4, Error: 0.1206833559
Iteration: 5, Error: 0.1204243043
Iteration: 6, Error: 0.1202854412
Iteration: 7, Error: 0.1201900100
Iteration: 8, Error: 0.1201088417
Iteration: 9, Error: 0.1200290587
Iteration: 10, Error: 0.1199452623
Iteration: 11, Error: 0.1198574450
Iteration: 12, Error: 0.1197696703
Iteration: 13, Error: 0.1196878506
Iteration: 14, Error: 0.1196170832
Iteration: 15, Error: 0.1195598756
Iteration: 16, Error: 0.1195160069
Iteration: 17, Error: 0.1194835469
Iteration: 18, Error: 0.1194600296
Iteration: 19, Error: 0.1194431820
Iteration: 20, Error: 0.1194311862
Fuzzy c-means clustering with 6 clusters
Cluster centers:
[,1] [,2] [,3]
1 2.02713891 2.06187217 1.60006313
2 1.85480130 1.87632883 2.17935541
3 2.22873211 1.95218615 2.13934215
4 0.01608874 -0.04380011 -0.04180612
5 0.89331264 0.86158238 0.73616487
6 1.02580340 1.10541890 1.07396566
Memberships:
1 2 3 4 5 6
[1,] 0.031182129 0.029894678 0.026495493 0.7332621295 0.103531727 0.075633843
[2,] 0.006293163 0.005924999 0.005132503 0.9267356727 0.034798201 0.021115462
[3,] 0.043602468 0.044905678 0.038483940 0.5466457597 0.190425146 0.135937009
[4,] 0.005105510 0.004863401 0.004261589 0.9474284213 0.023209348 0.015131730
[5,] 0.006103718 0.005809667 0.005036668 0.9285204463 0.033923423 0.020606077
[6,] 0.017003813 0.016006522 0.014390295 0.8440141699 0.064673464 0.043911736
[7,] 0.017438711 0.015493820 0.013659739 0.7791045373 0.113410722 0.060892471
[8,] 0.009643208 0.009098969 0.008049831 0.9079532465 0.038988411 0.026266335
[9,] 0.019550633 0.018918679 0.016756501 0.8041357613 0.084502979 0.056135446
[10,] 0.008592215 0.008217756 0.007259424 0.9145936237 0.036898237 0.024438745
[11,] 0.010661819 0.010161943 0.009022874 0.8973319686 0.043582275 0.029239120
[12,] 0.001968249 0.001818769 0.001604215 0.9791992418 0.009492673 0.005916853
[13,] 0.026868012 0.023393047 0.021126600 0.7166762073 0.133365085 0.078571048
[14,] 0.009109111 0.008750079 0.007705480 0.9087687621 0.039475392 0.026191175
[15,] 0.023057138 0.021472378 0.019119586 0.7355531097 0.126344937 0.074452851
[16,] 0.002633931 0.002495302 0.002193376 0.9723208433 0.012427799 0.007928748
[17,] 0.017499578 0.015893572 0.013951170 0.7607045624 0.126489801 0.065461317
[18,] 0.006475889 0.006130995 0.005419997 0.9313758000 0.031038764 0.019558555
[19,] 0.004021132 0.003698344 0.003293002 0.9572521600 0.019695333 0.012040028
[20,] 0.021167789 0.018747250 0.016454449 0.7207328912 0.146194038 0.076703582
[21,] 0.015191664 0.014569793 0.012755628 0.8249640223 0.082278444 0.050240450
[22,] 0.015278086 0.014108936 0.012372400 0.8464962886 0.067589863 0.044154426
[23,] 0.023409303 0.021243179 0.018460081 0.7389067068 0.122648950 0.075331780
[24,] 0.018354571 0.017716551 0.015746596 0.8285614700 0.070790536 0.048830275
[25,] 0.024683662 0.023130683 0.020729744 0.7395746142 0.118277428 0.073603869
[26,] 0.012717975 0.012298906 0.010731588 0.8732647680 0.054257537 0.036729226
[27,] 0.007258978 0.006708534 0.005834637 0.9077472281 0.046453562 0.025997061
[28,] 0.002594901 0.002440015 0.002133353 0.9720402621 0.012760208 0.008031261
[29,] 0.018890251 0.017666928 0.015280866 0.7292689892 0.143614355 0.075278611
[30,] 0.012900199 0.011729916 0.010331469 0.8658674451 0.060899950 0.038271021
[31,] 0.036466119 0.031517850 0.028223149 0.6386216233 0.163231814 0.101939443
[32,] 0.013800337 0.012723795 0.011434194 0.8592258262 0.063277270 0.039538578
[33,] 0.024344525 0.021375481 0.018844778 0.6782372377 0.169436053 0.087761925
[34,] 0.031996321 0.030473882 0.026652136 0.5827042994 0.209588630 0.118584732
[35,] 0.029479272 0.026669067 0.023793585 0.7355012190 0.109599015 0.074957842
[36,] 0.018746110 0.016965205 0.015329572 0.8280458258 0.072794177 0.048119110
[37,] 0.014597778 0.013593325 0.011946801 0.8578010470 0.061215787 0.040845263
[38,] 0.028993857 0.026464929 0.023837538 0.6799064993 0.151392293 0.089404883
[39,] 0.019117446 0.016924331 0.014915445 0.7699441256 0.114953979 0.064144674
[40,] 0.043404573 0.039525829 0.034318994 0.5822910602 0.178034902 0.122424641
[41,] 0.018387567 0.018017805 0.015745729 0.8208723040 0.075066183 0.051910412
[42,] 0.016013662 0.014533084 0.012831873 0.7933206646 0.106384945 0.056915772
[43,] 0.007152092 0.006691343 0.005924854 0.9208292207 0.036990924 0.022411566
[44,] 0.018980023 0.017737290 0.015171784 0.7434440337 0.131246443 0.073420427
[45,] 0.016660708 0.015102723 0.013237562 0.7712993488 0.121146931 0.062552727
[46,] 0.009072265 0.008438969 0.007326118 0.8971035266 0.048420388 0.029638733
[47,] 0.016143132 0.014837984 0.013174850 0.8480377614 0.064598381 0.043207892
[48,] 0.001461545 0.001360121 0.001205122 0.9846420968 0.006976088 0.004355027
[49,] 0.003889862 0.003598323 0.003166631 0.9539975364 0.022397091 0.012950557
[50,] 0.025546153 0.022666226 0.020104830 0.7425877245 0.115955351 0.073139716
[51,] 0.046264481 0.050020174 0.038122189 0.0723822558 0.390034039 0.403176861
[52,] 0.022398404 0.017378263 0.014055030 0.0175561146 0.437933695 0.490678494
[53,] 0.040790104 0.030126180 0.024672268 0.0193174308 0.274042849 0.611051169
[54,] 0.097090861 0.061930822 0.053121560 0.0456466324 0.338853080 0.403357045
[55,] 0.013621711 0.011874554 0.009281567 0.0083909430 0.154631275 0.802199951
[56,] 0.017984481 0.015808344 0.012618935 0.0315904960 0.674298401 0.247699344
[57,] 0.007982512 0.007389885 0.005570039 0.0049027593 0.077041243 0.897113561
[58,] 0.014716581 0.012567412 0.009718921 0.0074140747 0.112520288 0.843062723
[59,] 0.066330058 0.072011936 0.049381434 0.0395576960 0.231980534 0.540738342
[60,] 0.016148092 0.015325914 0.011932409 0.0231989244 0.566017593 0.367377068
[61,] 0.094702673 0.066800197 0.053067531 0.0436193056 0.292140958 0.449669335
[62,] 0.043883182 0.046166458 0.036418800 0.0899405331 0.440064743 0.343526283
[63,] 0.074616152 0.077718776 0.053477323 0.0417847577 0.234382055 0.518020935
[64,] 0.031491957 0.030752342 0.021609401 0.0154976154 0.142536975 0.758111709
[65,] 0.025892384 0.023118021 0.017329297 0.0107491390 0.122030374 0.800880784
[66,] 0.025239407 0.024829620 0.019145749 0.0402951378 0.497912538 0.392577547
[67,] 0.039712733 0.029560393 0.025523199 0.0498948665 0.586087027 0.269221782
[68,] 0.010469482 0.008953223 0.007382476 0.0238647208 0.833637009 0.115693089
[69,] 0.006065325 0.005311596 0.004297732 0.0113591210 0.879046751 0.093919474
[70,] 0.077607075 0.072613887 0.062556569 0.0637461983 0.357279910 0.366196361
[71,] 0.032702798 0.033256389 0.024121076 0.0270331962 0.250930364 0.631956177
[72,] 0.088320777 0.060600439 0.051776052 0.0312208199 0.282433843 0.485648069
[73,] 0.132549617 0.080261986 0.064696239 0.0300207410 0.218810550 0.473660867
[74,] 0.095507479 0.066312074 0.051209043 0.0316726056 0.237560329 0.517738470
[75,] 0.036015320 0.035384359 0.028546497 0.0479844869 0.473256481 0.378812857
[76,] 0.007424191 0.006646909 0.004880437 0.0037358645 0.048664587 0.928648012
[77,] 0.016145154 0.013076890 0.011167018 0.0288478359 0.783289205 0.147473896
[78,] 0.164385878 0.114526695 0.080918552 0.0269426317 0.165999574 0.447226669
[79,] 0.048046933 0.043204244 0.033150523 0.0560528071 0.391474699 0.428070793
[80,] 0.037925512 0.028470164 0.024819000 0.0475649521 0.602440899 0.258779473
[81,] 0.005792613 0.005076313 0.003949968 0.0042043150 0.107842108 0.873134683
[82,] 0.008604478 0.007148191 0.005605474 0.0051527659 0.108818405 0.864670686
[83,] 0.014037073 0.011221505 0.009497201 0.0211651253 0.794840427 0.149238669
[84,] 0.030858002 0.028310640 0.019992563 0.0123610285 0.117437362 0.791040404
[85,] 0.099485054 0.077760680 0.070865305 0.0579117000 0.341426021 0.352551241
[86,] 0.025695831 0.022154454 0.016747888 0.0186691063 0.250587469 0.666145251
[87,] 0.016448974 0.015151108 0.011063205 0.0098768195 0.121020049 0.826439846
[88,] 0.032185612 0.030158552 0.025423209 0.1915921704 0.521733727 0.198906730
[89,] 0.151710685 0.092716833 0.072724749 0.0267479691 0.187686144 0.468413620
[90,] 0.020181706 0.018164711 0.013149027 0.0100196650 0.111300509 0.827184383
[91,] 0.038405705 0.030084256 0.022509666 0.0147804250 0.156489575 0.737730372
[92,] 0.019793522 0.017765762 0.014209369 0.0174236781 0.400109736 0.530697934
[93,] 0.037840481 0.031417104 0.027069307 0.0423520599 0.551231545 0.310089503
[94,] 0.028563079 0.024413929 0.019354550 0.0391115942 0.535048107 0.353508741
[95,] 0.041877741 0.038463595 0.031014845 0.0291958238 0.333959546 0.525488450
[96,] 0.016342248 0.013236416 0.010861752 0.0142389920 0.547523239 0.397797354
[97,] 0.035345437 0.032474963 0.027517445 0.0686327355 0.569677445 0.266351975
[98,] 0.015665510 0.015056775 0.011473896 0.0133240788 0.233714114 0.710765627
[99,] 0.004448651 0.003928881 0.003041295 0.0034666777 0.098064450 0.887050045
[100,] 0.016116657 0.015419471 0.011847780 0.0151503431 0.298191925 0.643273824
[101,] 0.011390208 0.031100197 0.955233303 0.0002895831 0.000794716 0.001191993
[102,] 0.101005309 0.517271049 0.338644478 0.0058379108 0.014971840 0.022269413
[103,] 0.537157317 0.134003561 0.152533581 0.0182200140 0.063919110 0.094166417
[104,] 0.052551293 0.079712038 0.859162408 0.0010488391 0.002994473 0.004530948
[105,] 0.772785916 0.076987015 0.124881941 0.0030583547 0.008943314 0.013343459
[106,] 0.064438785 0.824437771 0.093072961 0.0018840446 0.006016784 0.010149655
[107,] 0.191239837 0.227533527 0.528513330 0.0076855065 0.018750337 0.026277462
[108,] 0.088972659 0.665849646 0.211855031 0.0042052456 0.011445408 0.017672010
[109,] 0.687488273 0.116033595 0.140547595 0.0068438530 0.019821758 0.029264926
[110,] 0.388123884 0.344894932 0.231131561 0.0037046915 0.012094379 0.020050553
[111,] 0.377065992 0.185370621 0.369398209 0.0079728535 0.024143024 0.036049301
[112,] 0.093040707 0.295689574 0.570649531 0.0057146175 0.014305032 0.020600539
[113,] 0.840054746 0.055771664 0.085799030 0.0021694603 0.006463303 0.009741797
[114,] 0.030326654 0.058565984 0.903794644 0.0009509014 0.002573018 0.003788798
[115,] 0.093801211 0.451891363 0.405203728 0.0066627377 0.017134386 0.025306575
[116,] 0.059618474 0.813125846 0.100700933 0.0029746585 0.008904841 0.014675247
[117,] 0.086835359 0.085555088 0.059224372 0.0223276299 0.160343862 0.585713689
[118,] 0.160627654 0.357077235 0.422861857 0.0085459899 0.020896919 0.029990346
[119,] 0.087621452 0.114171490 0.776713389 0.0028688073 0.007623383 0.011001479
[120,] 0.430101179 0.344888670 0.183804997 0.0043916969 0.013912458 0.022900999
[121,] 0.106490706 0.338872767 0.498837647 0.0078748245 0.019657413 0.028266642
[122,] 0.156810105 0.468170616 0.228891120 0.0197367689 0.050127715 0.076263674
[123,] 0.274972235 0.218162221 0.451941244 0.0063251623 0.019236015 0.029363123
[124,] 0.286068836 0.187813888 0.432238735 0.0135411159 0.034184255 0.046153169
[125,] 0.266304581 0.148908869 0.550297133 0.0042775826 0.012196905 0.018014929
[126,] 0.057602234 0.253785235 0.677191249 0.0013569706 0.003943460 0.006120851
[127,] 0.059954603 0.807677178 0.114266598 0.0019211436 0.006068918 0.010111560
[128,] 0.354755446 0.151574932 0.445309559 0.0063574430 0.017299562 0.024703058
[129,] 0.080631055 0.240388710 0.666645562 0.0014544418 0.004257755 0.006622478
[130,] 0.018413308 0.941551666 0.035077104 0.0005429127 0.001668422 0.002746587
[131,] 0.303469936 0.318632210 0.285883828 0.0125488419 0.032286194 0.047178990
[132,] 0.122789071 0.617211259 0.226059062 0.0035595781 0.011414350 0.018966680
[133,] 0.103812642 0.629297321 0.193707112 0.0080710044 0.024530448 0.040581474
[134,] 0.291390446 0.489524548 0.171465860 0.0047167318 0.015768891 0.027133524
[135,] 0.021089637 0.035384688 0.939366319 0.0005421933 0.001462666 0.002154497
[136,] 0.204182703 0.488294428 0.284340586 0.0025863257 0.007901570 0.012694386
[137,] 0.080933651 0.636295311 0.237090108 0.0055337962 0.015635702 0.024511431
[138,] 0.165862228 0.632985321 0.146749478 0.0055073664 0.017918986 0.030976620
[139,] 0.196280700 0.286170100 0.440995583 0.0117092568 0.027207365 0.037636996
[140,] 0.937566144 0.027773908 0.026312882 0.0008602356 0.002846972 0.004639859
[141,] 0.137257276 0.147575403 0.688903073 0.0035824940 0.009304407 0.013377348
[142,] 0.043607930 0.782426346 0.161654670 0.0015038453 0.004225908 0.006581300
[143,] 0.063169190 0.297391699 0.621846069 0.0021295718 0.006092289 0.009371181
[144,] 0.101205599 0.355274019 0.493763094 0.0070428126 0.017491707 0.025222769
[145,] 0.813353793 0.072735522 0.068523021 0.0044769875 0.015557716 0.025352961
[146,] 0.616823895 0.118588621 0.146667272 0.0134167754 0.042802929 0.061700508
[147,] 0.126145069 0.255696994 0.572723184 0.0056608105 0.015950209 0.023823733
[148,] 0.121478723 0.265004378 0.584640086 0.0038963158 0.010127414 0.014853083
[149,] 0.284671071 0.327601458 0.160945416 0.0190844246 0.071171191 0.136526439
[150,] 0.047253642 0.701891894 0.233753137 0.0020622925 0.005872042 0.009166992
Closest hard clustering:
[1] 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
[38] 4 4 4 4 4 4 4 4 4 4 4 4 4 6 6 6 6 6 5 6 6 6 5 6 5 6 6 6 5 5 5 5 6 6 6 6 6
[75] 5 6 5 6 6 5 6 6 5 6 6 6 6 5 6 6 6 6 5 5 6 5 5 6 6 6 3 2 1 3 1 2 3 2 1 1 1
[112] 3 1 3 2 2 6 3 3 1 3 2 3 3 3 3 2 3 3 2 2 2 2 2 3 2 2 2 3 1 3 2 3 3 1 1 3 3
[149] 2 2
Available components:
[1] "centers" "size" "cluster" "membership" "iter"
[6] "withinerror" "call"
e1071 documentation built on Oct. 23, 2020, 5:13 p.m.
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Description Usage Arguments Details Value Author(s) References Examples
Description
The fuzzy version of the known kmeans clustering algorithm as well as an on-line variant (Unsupervised Fuzzy Competitive learning).
Usage
1 2 3 |
Arguments
x |
The data matrix where columns correspond to variables and rows to observations. |
centers |
Number of clusters or initial values for cluster centers. |
iter.max |
Maximum number of iterations. |
verbose |
If |
dist |
Must be one of the following: If |
method |
If |
m |
A number greater than 1 giving the degree of fuzzification. |
rate.par |
A number between 0 and 1 giving the parameter of the learning rate for the on-line variant. The default corresponds to 0.3. |
weights |
a numeric vector with non-negative case weights.
Recycled to the number of observations in |
control |
a list of control parameters. See Details. |
Details
The data given by x is clustered by generalized versions of the
fuzzy c-means algorithm, which use either a fixed-point or an
on-line heuristic for minimizing the objective function
∑_i ∑_j w_i u_{ij}^m d_{ij},
where w_i is the weight of observation i, u_{ij} is
the membership of observation i in cluster j, and
d_{ij} is the distance (dissimilarity) between observation
i and center j. The dissimilarities used are the sums of
squares ("euclidean") or absolute values ("manhattan")
of the element-wise differences.
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, or when the algorithm is unable to reduce
the current value val of the objective function by
reltol * (abs(val) * reltol) at a step. The relative
convergence tolerance reltol can be specified as the
reltol component of the list of control parameters, and
defaults to sqrt(.Machine$double.eps).
If verbose is TRUE, each iteration displays its number
and the value of the objective function.
If method is "cmeans", then we have the c-means
fuzzy clustering method, see for example Bezdek (1981). If
"ufcl", we have the On-line Update (Unsupervised Fuzzy
Competitive Learning) method due to Chung and Lee (1992), see also Pal
et al (1996). This method works by performing an update directly
after each input signal (i.e., for each single observation).
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.
Value
An object of class "fclust" which is a list with components:
centers |
the final cluster centers. |
size |
the number of data points in each cluster of the closest hard clustering. |
cluster |
a vector of integers containing the indices of the clusters where the data points are assigned to for the closest hard clustering, as obtained by assigning points to the (first) class with maximal membership. |
iter |
the number of iterations performed. |
membership |
a matrix with the membership values of the data points to the clusters. |
withinerror |
the value of the objective function. |
call |
the call used to create the object. |
Author(s)
Evgenia Dimitriadou and Kurt Hornik
References
J. C. Bezdek (1981). Pattern recognition with fuzzy objective function algorithms. New York: Plenum.
Fu Lai Chung and Tong Lee (1992). Fuzzy competitive learning. Neural Networks, 7(3), 539–551.
Nikhil R. Pal, James C. Bezdek, and Richard J. Hathaway (1996). Sequential competitive learning and the fuzzy c-means clustering algorithms. Neural Networks, 9(5), 787–796.
Examples
1 2 3 4 5 6 7 8 9 10 11 12 | # a 2-dimensional example
x<-rbind(matrix(rnorm(100,sd=0.3),ncol=2),
matrix(rnorm(100,mean=1,sd=0.3),ncol=2))
cl<-cmeans(x,2,20,verbose=TRUE,method="cmeans",m=2)
print(cl)
# a 3-dimensional example
x<-rbind(matrix(rnorm(150,sd=0.3),ncol=3),
matrix(rnorm(150,mean=1,sd=0.3),ncol=3),
matrix(rnorm(150,mean=2,sd=0.3),ncol=3))
cl<-cmeans(x,6,20,verbose=TRUE,method="cmeans")
print(cl)
|
Example output
Iteration: 1, Error: 0.1646046972
Iteration: 2, Error: 0.1543607452
Iteration: 3, Error: 0.1542090708
Iteration: 4, Error: 0.1542074436
Iteration: 5, Error: 0.1542074251
Iteration: 6 converged, Error: 0.1542074248
Fuzzy c-means clustering with 2 clusters
Cluster centers:
[,1] [,2]
1 -0.1263853 -0.08036624
2 0.9699374 1.13365887
Memberships:
1 2
[1,] 0.8041721115 0.195827889
[2,] 0.9624985570 0.037501443
[3,] 0.9422326865 0.057767313
[4,] 0.9358055047 0.064194495
[5,] 0.9494495198 0.050550480
[6,] 0.8087931231 0.191206877
[7,] 0.9833660711 0.016633929
[8,] 0.9819309179 0.018069082
[9,] 0.9894646400 0.010535360
[10,] 0.9715390275 0.028460973
[11,] 0.9311442900 0.068855710
[12,] 0.9375432876 0.062456712
[13,] 0.9130738829 0.086926117
[14,] 0.9192294617 0.080770538
[15,] 0.9724676099 0.027532390
[16,] 0.9317887302 0.068211270
[17,] 0.9820251726 0.017974827
[18,] 0.9853295839 0.014670416
[19,] 0.8442704429 0.155729557
[20,] 0.9441364738 0.055863526
[21,] 0.9153389923 0.084661008
[22,] 0.9826214149 0.017378585
[23,] 0.9892078061 0.010792194
[24,] 0.8759832492 0.124016751
[25,] 0.9540134279 0.045986572
[26,] 0.9347523673 0.065247633
[27,] 0.9973578267 0.002642173
[28,] 0.8199341875 0.180065812
[29,] 0.9360137796 0.063986220
[30,] 0.9361926486 0.063807351
[31,] 0.9446261582 0.055373842
[32,] 0.9912608295 0.008739171
[33,] 0.6940101181 0.305989882
[34,] 0.6148928515 0.385107149
[35,] 0.7882044852 0.211795515
[36,] 0.9814336729 0.018566327
[37,] 0.9323308133 0.067669187
[38,] 0.9723538916 0.027646108
[39,] 0.5611358982 0.438864102
[40,] 0.9506471475 0.049352853
[41,] 0.9420410152 0.057958985
[42,] 0.9475289679 0.052471032
[43,] 0.9380632237 0.061936776
[44,] 0.9928575856 0.007142414
[45,] 0.9790199311 0.020980069
[46,] 0.9217313557 0.078268644
[47,] 0.8883284811 0.111671519
[48,] 0.9816556987 0.018344301
[49,] 0.9549250997 0.045074900
[50,] 0.8871454670 0.112854533
[51,] 0.2114530375 0.788546962
[52,] 0.0862448751 0.913755125
[53,] 0.0510784637 0.948921536
[54,] 0.0004877015 0.999512298
[55,] 0.0414776249 0.958522375
[56,] 0.0259554941 0.974044506
[57,] 0.0014311945 0.998568806
[58,] 0.0516033727 0.948396627
[59,] 0.0688317563 0.931168244
[60,] 0.0352494627 0.964750537
[61,] 0.0129179845 0.987082016
[62,] 0.1849344045 0.815065595
[63,] 0.0100294106 0.989970589
[64,] 0.0003615129 0.999638487
[65,] 0.0271209451 0.972879055
[66,] 0.0169744698 0.983025530
[67,] 0.0623126968 0.937687303
[68,] 0.0183115960 0.981688404
[69,] 0.0212901927 0.978709807
[70,] 0.1699873547 0.830012645
[71,] 0.0762438670 0.923756133
[72,] 0.0274513011 0.972548699
[73,] 0.0346728814 0.965327119
[74,] 0.0165709526 0.983429047
[75,] 0.1757232598 0.824276740
[76,] 0.0137118788 0.986288121
[77,] 0.1526938167 0.847306183
[78,] 0.0094204114 0.990579589
[79,] 0.0679097455 0.932090254
[80,] 0.0586380314 0.941361969
[81,] 0.1199307203 0.880069280
[82,] 0.0321827248 0.967817275
[83,] 0.0456242573 0.954375743
[84,] 0.0047418541 0.995258146
[85,] 0.0470258756 0.952974124
[86,] 0.0436314118 0.956368588
[87,] 0.0091670303 0.990832970
[88,] 0.0529815880 0.947018412
[89,] 0.0454349758 0.954565024
[90,] 0.0292213963 0.970778604
[91,] 0.0298330734 0.970166927
[92,] 0.0353835883 0.964616412
[93,] 0.0278109131 0.972189087
[94,] 0.0810262427 0.918973757
[95,] 0.0348255780 0.965174422
[96,] 0.0444705016 0.955529498
[97,] 0.1344526231 0.865547377
[98,] 0.0021772383 0.997822762
[99,] 0.0000134157 0.999986584
[100,] 0.0145777392 0.985422261
Closest hard clustering:
[1] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
[38] 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
[75] 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
Available components:
[1] "centers" "size" "cluster" "membership" "iter"
[6] "withinerror" "call"
Iteration: 1, Error: 0.1298456810
Iteration: 2, Error: 0.1229062610
Iteration: 3, Error: 0.1212776711
Iteration: 4, Error: 0.1206833559
Iteration: 5, Error: 0.1204243043
Iteration: 6, Error: 0.1202854412
Iteration: 7, Error: 0.1201900100
Iteration: 8, Error: 0.1201088417
Iteration: 9, Error: 0.1200290587
Iteration: 10, Error: 0.1199452623
Iteration: 11, Error: 0.1198574450
Iteration: 12, Error: 0.1197696703
Iteration: 13, Error: 0.1196878506
Iteration: 14, Error: 0.1196170832
Iteration: 15, Error: 0.1195598756
Iteration: 16, Error: 0.1195160069
Iteration: 17, Error: 0.1194835469
Iteration: 18, Error: 0.1194600296
Iteration: 19, Error: 0.1194431820
Iteration: 20, Error: 0.1194311862
Fuzzy c-means clustering with 6 clusters
Cluster centers:
[,1] [,2] [,3]
1 2.02713891 2.06187217 1.60006313
2 1.85480130 1.87632883 2.17935541
3 2.22873211 1.95218615 2.13934215
4 0.01608874 -0.04380011 -0.04180612
5 0.89331264 0.86158238 0.73616487
6 1.02580340 1.10541890 1.07396566
Memberships:
1 2 3 4 5 6
[1,] 0.031182129 0.029894678 0.026495493 0.7332621295 0.103531727 0.075633843
[2,] 0.006293163 0.005924999 0.005132503 0.9267356727 0.034798201 0.021115462
[3,] 0.043602468 0.044905678 0.038483940 0.5466457597 0.190425146 0.135937009
[4,] 0.005105510 0.004863401 0.004261589 0.9474284213 0.023209348 0.015131730
[5,] 0.006103718 0.005809667 0.005036668 0.9285204463 0.033923423 0.020606077
[6,] 0.017003813 0.016006522 0.014390295 0.8440141699 0.064673464 0.043911736
[7,] 0.017438711 0.015493820 0.013659739 0.7791045373 0.113410722 0.060892471
[8,] 0.009643208 0.009098969 0.008049831 0.9079532465 0.038988411 0.026266335
[9,] 0.019550633 0.018918679 0.016756501 0.8041357613 0.084502979 0.056135446
[10,] 0.008592215 0.008217756 0.007259424 0.9145936237 0.036898237 0.024438745
[11,] 0.010661819 0.010161943 0.009022874 0.8973319686 0.043582275 0.029239120
[12,] 0.001968249 0.001818769 0.001604215 0.9791992418 0.009492673 0.005916853
[13,] 0.026868012 0.023393047 0.021126600 0.7166762073 0.133365085 0.078571048
[14,] 0.009109111 0.008750079 0.007705480 0.9087687621 0.039475392 0.026191175
[15,] 0.023057138 0.021472378 0.019119586 0.7355531097 0.126344937 0.074452851
[16,] 0.002633931 0.002495302 0.002193376 0.9723208433 0.012427799 0.007928748
[17,] 0.017499578 0.015893572 0.013951170 0.7607045624 0.126489801 0.065461317
[18,] 0.006475889 0.006130995 0.005419997 0.9313758000 0.031038764 0.019558555
[19,] 0.004021132 0.003698344 0.003293002 0.9572521600 0.019695333 0.012040028
[20,] 0.021167789 0.018747250 0.016454449 0.7207328912 0.146194038 0.076703582
[21,] 0.015191664 0.014569793 0.012755628 0.8249640223 0.082278444 0.050240450
[22,] 0.015278086 0.014108936 0.012372400 0.8464962886 0.067589863 0.044154426
[23,] 0.023409303 0.021243179 0.018460081 0.7389067068 0.122648950 0.075331780
[24,] 0.018354571 0.017716551 0.015746596 0.8285614700 0.070790536 0.048830275
[25,] 0.024683662 0.023130683 0.020729744 0.7395746142 0.118277428 0.073603869
[26,] 0.012717975 0.012298906 0.010731588 0.8732647680 0.054257537 0.036729226
[27,] 0.007258978 0.006708534 0.005834637 0.9077472281 0.046453562 0.025997061
[28,] 0.002594901 0.002440015 0.002133353 0.9720402621 0.012760208 0.008031261
[29,] 0.018890251 0.017666928 0.015280866 0.7292689892 0.143614355 0.075278611
[30,] 0.012900199 0.011729916 0.010331469 0.8658674451 0.060899950 0.038271021
[31,] 0.036466119 0.031517850 0.028223149 0.6386216233 0.163231814 0.101939443
[32,] 0.013800337 0.012723795 0.011434194 0.8592258262 0.063277270 0.039538578
[33,] 0.024344525 0.021375481 0.018844778 0.6782372377 0.169436053 0.087761925
[34,] 0.031996321 0.030473882 0.026652136 0.5827042994 0.209588630 0.118584732
[35,] 0.029479272 0.026669067 0.023793585 0.7355012190 0.109599015 0.074957842
[36,] 0.018746110 0.016965205 0.015329572 0.8280458258 0.072794177 0.048119110
[37,] 0.014597778 0.013593325 0.011946801 0.8578010470 0.061215787 0.040845263
[38,] 0.028993857 0.026464929 0.023837538 0.6799064993 0.151392293 0.089404883
[39,] 0.019117446 0.016924331 0.014915445 0.7699441256 0.114953979 0.064144674
[40,] 0.043404573 0.039525829 0.034318994 0.5822910602 0.178034902 0.122424641
[41,] 0.018387567 0.018017805 0.015745729 0.8208723040 0.075066183 0.051910412
[42,] 0.016013662 0.014533084 0.012831873 0.7933206646 0.106384945 0.056915772
[43,] 0.007152092 0.006691343 0.005924854 0.9208292207 0.036990924 0.022411566
[44,] 0.018980023 0.017737290 0.015171784 0.7434440337 0.131246443 0.073420427
[45,] 0.016660708 0.015102723 0.013237562 0.7712993488 0.121146931 0.062552727
[46,] 0.009072265 0.008438969 0.007326118 0.8971035266 0.048420388 0.029638733
[47,] 0.016143132 0.014837984 0.013174850 0.8480377614 0.064598381 0.043207892
[48,] 0.001461545 0.001360121 0.001205122 0.9846420968 0.006976088 0.004355027
[49,] 0.003889862 0.003598323 0.003166631 0.9539975364 0.022397091 0.012950557
[50,] 0.025546153 0.022666226 0.020104830 0.7425877245 0.115955351 0.073139716
[51,] 0.046264481 0.050020174 0.038122189 0.0723822558 0.390034039 0.403176861
[52,] 0.022398404 0.017378263 0.014055030 0.0175561146 0.437933695 0.490678494
[53,] 0.040790104 0.030126180 0.024672268 0.0193174308 0.274042849 0.611051169
[54,] 0.097090861 0.061930822 0.053121560 0.0456466324 0.338853080 0.403357045
[55,] 0.013621711 0.011874554 0.009281567 0.0083909430 0.154631275 0.802199951
[56,] 0.017984481 0.015808344 0.012618935 0.0315904960 0.674298401 0.247699344
[57,] 0.007982512 0.007389885 0.005570039 0.0049027593 0.077041243 0.897113561
[58,] 0.014716581 0.012567412 0.009718921 0.0074140747 0.112520288 0.843062723
[59,] 0.066330058 0.072011936 0.049381434 0.0395576960 0.231980534 0.540738342
[60,] 0.016148092 0.015325914 0.011932409 0.0231989244 0.566017593 0.367377068
[61,] 0.094702673 0.066800197 0.053067531 0.0436193056 0.292140958 0.449669335
[62,] 0.043883182 0.046166458 0.036418800 0.0899405331 0.440064743 0.343526283
[63,] 0.074616152 0.077718776 0.053477323 0.0417847577 0.234382055 0.518020935
[64,] 0.031491957 0.030752342 0.021609401 0.0154976154 0.142536975 0.758111709
[65,] 0.025892384 0.023118021 0.017329297 0.0107491390 0.122030374 0.800880784
[66,] 0.025239407 0.024829620 0.019145749 0.0402951378 0.497912538 0.392577547
[67,] 0.039712733 0.029560393 0.025523199 0.0498948665 0.586087027 0.269221782
[68,] 0.010469482 0.008953223 0.007382476 0.0238647208 0.833637009 0.115693089
[69,] 0.006065325 0.005311596 0.004297732 0.0113591210 0.879046751 0.093919474
[70,] 0.077607075 0.072613887 0.062556569 0.0637461983 0.357279910 0.366196361
[71,] 0.032702798 0.033256389 0.024121076 0.0270331962 0.250930364 0.631956177
[72,] 0.088320777 0.060600439 0.051776052 0.0312208199 0.282433843 0.485648069
[73,] 0.132549617 0.080261986 0.064696239 0.0300207410 0.218810550 0.473660867
[74,] 0.095507479 0.066312074 0.051209043 0.0316726056 0.237560329 0.517738470
[75,] 0.036015320 0.035384359 0.028546497 0.0479844869 0.473256481 0.378812857
[76,] 0.007424191 0.006646909 0.004880437 0.0037358645 0.048664587 0.928648012
[77,] 0.016145154 0.013076890 0.011167018 0.0288478359 0.783289205 0.147473896
[78,] 0.164385878 0.114526695 0.080918552 0.0269426317 0.165999574 0.447226669
[79,] 0.048046933 0.043204244 0.033150523 0.0560528071 0.391474699 0.428070793
[80,] 0.037925512 0.028470164 0.024819000 0.0475649521 0.602440899 0.258779473
[81,] 0.005792613 0.005076313 0.003949968 0.0042043150 0.107842108 0.873134683
[82,] 0.008604478 0.007148191 0.005605474 0.0051527659 0.108818405 0.864670686
[83,] 0.014037073 0.011221505 0.009497201 0.0211651253 0.794840427 0.149238669
[84,] 0.030858002 0.028310640 0.019992563 0.0123610285 0.117437362 0.791040404
[85,] 0.099485054 0.077760680 0.070865305 0.0579117000 0.341426021 0.352551241
[86,] 0.025695831 0.022154454 0.016747888 0.0186691063 0.250587469 0.666145251
[87,] 0.016448974 0.015151108 0.011063205 0.0098768195 0.121020049 0.826439846
[88,] 0.032185612 0.030158552 0.025423209 0.1915921704 0.521733727 0.198906730
[89,] 0.151710685 0.092716833 0.072724749 0.0267479691 0.187686144 0.468413620
[90,] 0.020181706 0.018164711 0.013149027 0.0100196650 0.111300509 0.827184383
[91,] 0.038405705 0.030084256 0.022509666 0.0147804250 0.156489575 0.737730372
[92,] 0.019793522 0.017765762 0.014209369 0.0174236781 0.400109736 0.530697934
[93,] 0.037840481 0.031417104 0.027069307 0.0423520599 0.551231545 0.310089503
[94,] 0.028563079 0.024413929 0.019354550 0.0391115942 0.535048107 0.353508741
[95,] 0.041877741 0.038463595 0.031014845 0.0291958238 0.333959546 0.525488450
[96,] 0.016342248 0.013236416 0.010861752 0.0142389920 0.547523239 0.397797354
[97,] 0.035345437 0.032474963 0.027517445 0.0686327355 0.569677445 0.266351975
[98,] 0.015665510 0.015056775 0.011473896 0.0133240788 0.233714114 0.710765627
[99,] 0.004448651 0.003928881 0.003041295 0.0034666777 0.098064450 0.887050045
[100,] 0.016116657 0.015419471 0.011847780 0.0151503431 0.298191925 0.643273824
[101,] 0.011390208 0.031100197 0.955233303 0.0002895831 0.000794716 0.001191993
[102,] 0.101005309 0.517271049 0.338644478 0.0058379108 0.014971840 0.022269413
[103,] 0.537157317 0.134003561 0.152533581 0.0182200140 0.063919110 0.094166417
[104,] 0.052551293 0.079712038 0.859162408 0.0010488391 0.002994473 0.004530948
[105,] 0.772785916 0.076987015 0.124881941 0.0030583547 0.008943314 0.013343459
[106,] 0.064438785 0.824437771 0.093072961 0.0018840446 0.006016784 0.010149655
[107,] 0.191239837 0.227533527 0.528513330 0.0076855065 0.018750337 0.026277462
[108,] 0.088972659 0.665849646 0.211855031 0.0042052456 0.011445408 0.017672010
[109,] 0.687488273 0.116033595 0.140547595 0.0068438530 0.019821758 0.029264926
[110,] 0.388123884 0.344894932 0.231131561 0.0037046915 0.012094379 0.020050553
[111,] 0.377065992 0.185370621 0.369398209 0.0079728535 0.024143024 0.036049301
[112,] 0.093040707 0.295689574 0.570649531 0.0057146175 0.014305032 0.020600539
[113,] 0.840054746 0.055771664 0.085799030 0.0021694603 0.006463303 0.009741797
[114,] 0.030326654 0.058565984 0.903794644 0.0009509014 0.002573018 0.003788798
[115,] 0.093801211 0.451891363 0.405203728 0.0066627377 0.017134386 0.025306575
[116,] 0.059618474 0.813125846 0.100700933 0.0029746585 0.008904841 0.014675247
[117,] 0.086835359 0.085555088 0.059224372 0.0223276299 0.160343862 0.585713689
[118,] 0.160627654 0.357077235 0.422861857 0.0085459899 0.020896919 0.029990346
[119,] 0.087621452 0.114171490 0.776713389 0.0028688073 0.007623383 0.011001479
[120,] 0.430101179 0.344888670 0.183804997 0.0043916969 0.013912458 0.022900999
[121,] 0.106490706 0.338872767 0.498837647 0.0078748245 0.019657413 0.028266642
[122,] 0.156810105 0.468170616 0.228891120 0.0197367689 0.050127715 0.076263674
[123,] 0.274972235 0.218162221 0.451941244 0.0063251623 0.019236015 0.029363123
[124,] 0.286068836 0.187813888 0.432238735 0.0135411159 0.034184255 0.046153169
[125,] 0.266304581 0.148908869 0.550297133 0.0042775826 0.012196905 0.018014929
[126,] 0.057602234 0.253785235 0.677191249 0.0013569706 0.003943460 0.006120851
[127,] 0.059954603 0.807677178 0.114266598 0.0019211436 0.006068918 0.010111560
[128,] 0.354755446 0.151574932 0.445309559 0.0063574430 0.017299562 0.024703058
[129,] 0.080631055 0.240388710 0.666645562 0.0014544418 0.004257755 0.006622478
[130,] 0.018413308 0.941551666 0.035077104 0.0005429127 0.001668422 0.002746587
[131,] 0.303469936 0.318632210 0.285883828 0.0125488419 0.032286194 0.047178990
[132,] 0.122789071 0.617211259 0.226059062 0.0035595781 0.011414350 0.018966680
[133,] 0.103812642 0.629297321 0.193707112 0.0080710044 0.024530448 0.040581474
[134,] 0.291390446 0.489524548 0.171465860 0.0047167318 0.015768891 0.027133524
[135,] 0.021089637 0.035384688 0.939366319 0.0005421933 0.001462666 0.002154497
[136,] 0.204182703 0.488294428 0.284340586 0.0025863257 0.007901570 0.012694386
[137,] 0.080933651 0.636295311 0.237090108 0.0055337962 0.015635702 0.024511431
[138,] 0.165862228 0.632985321 0.146749478 0.0055073664 0.017918986 0.030976620
[139,] 0.196280700 0.286170100 0.440995583 0.0117092568 0.027207365 0.037636996
[140,] 0.937566144 0.027773908 0.026312882 0.0008602356 0.002846972 0.004639859
[141,] 0.137257276 0.147575403 0.688903073 0.0035824940 0.009304407 0.013377348
[142,] 0.043607930 0.782426346 0.161654670 0.0015038453 0.004225908 0.006581300
[143,] 0.063169190 0.297391699 0.621846069 0.0021295718 0.006092289 0.009371181
[144,] 0.101205599 0.355274019 0.493763094 0.0070428126 0.017491707 0.025222769
[145,] 0.813353793 0.072735522 0.068523021 0.0044769875 0.015557716 0.025352961
[146,] 0.616823895 0.118588621 0.146667272 0.0134167754 0.042802929 0.061700508
[147,] 0.126145069 0.255696994 0.572723184 0.0056608105 0.015950209 0.023823733
[148,] 0.121478723 0.265004378 0.584640086 0.0038963158 0.010127414 0.014853083
[149,] 0.284671071 0.327601458 0.160945416 0.0190844246 0.071171191 0.136526439
[150,] 0.047253642 0.701891894 0.233753137 0.0020622925 0.005872042 0.009166992
Closest hard clustering:
[1] 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
[38] 4 4 4 4 4 4 4 4 4 4 4 4 4 6 6 6 6 6 5 6 6 6 5 6 5 6 6 6 5 5 5 5 6 6 6 6 6
[75] 5 6 5 6 6 5 6 6 5 6 6 6 6 5 6 6 6 6 5 5 6 5 5 6 6 6 3 2 1 3 1 2 3 2 1 1 1
[112] 3 1 3 2 2 6 3 3 1 3 2 3 3 3 3 2 3 3 2 2 2 2 2 3 2 2 2 3 1 3 2 3 3 1 1 3 3
[149] 2 2
Available components:
[1] "centers" "size" "cluster" "membership" "iter"
[6] "withinerror" "call"
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