Definition of Algoritma Fuzzy C-means

Fuzzy c-means merupakan pengelompokan suatu data yang menggunakan salah satu jenis soft clustering.Fuzzy C-Means adalah salah satu metode fuzzy clustering untuk mengelompokkan dataset menjadi K cluster.

Jumlah cluster (K) harus lebih besar dari 1. Fuzzy C-Means adalah pengembangan dari metode k-means dengan beberapa modifikasi.

Stages of Fuzzy C means

berikut ini adalah tahapan algotitma fuzzy C-Means

1.Menentukan banyak cluster (k) yang akan dibuat.

2.Menentukan nilai proporsi setiap data poin secara random untuk masuk dalam suatu cluster.

3.Menghitung nilai centroid. Dalam menghitung nilai centroid, menggunakan formula berikut : \[C_j = \frac{\sum u_{ij}^mx}{\sum u_{ij}^m}\]

4.Menghtung ulang nilai proporsi setiap data poin untuk masuk pada setiap cluster. menggunakan formula berikut : \[u_{ij}^m = \frac{1}{\sum (\frac{|x_i - c_j|}{|x_i - c_k|})^\frac{2}{m-1}}\]

Eksperimen Algoritma Fuzzy C-means

Library

library (ppclust)

library (factoextra)

## Loading required package: ggplot2

## Welcome! Want to learn more? See two factoextra-related books at https://goo.gl/ve3WBa

library (dplyr)

## 
## Attaching package: 'dplyr'

## The following objects are masked from 'package:stats':
## 
##     filter, lag

## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union

library (cluster)

library(fclust)

Import Data

Sebelum melakukan clustering,Import terlebih dahulu data yang kita butuhkan. Pada kali ini kita menggunakan dataset Penguin.

library(readxl)
data <- read_excel("D:/dataset/Data Penguin.xlsx")

data

data_penguin = data[-1]

data_penguin

library (psych)

## 
## Attaching package: 'psych'

## The following objects are masked from 'package:ggplot2':
## 
##     %+%, alpha

## The following object is masked from 'package:ppclust':
## 
##     pca

Fuzzy C-Means

library(ppclust)
res.fcm <-fcm (data_penguin, centers = 5)

Fuzzy Membership Matrix

as.data.frame(res.fcm$u)[1:10,]

Initial and final cluster prototypes matrices

res.fcm$v0

##           Culmen.Length Culmen.Depth Flipper.Length Body.Mass
## Cluster 1          48.5         15.0            219      4850
## Cluster 2          38.1         17.0            181      3175
## Cluster 3          55.9         17.0            228      5600
## Cluster 4          42.7         18.3            196      4075
## Cluster 5          46.8         16.5            189      3650

res.fcm$v

##           Culmen.Length Culmen.Depth Flipper.Length Body.Mass
## Cluster 1      45.91966     15.21606       213.0069  4847.306
## Cluster 2      39.59229     17.54392       187.8470  3231.889
## Cluster 3      50.11423     15.80701       222.3897  5606.411
## Cluster 4      44.01479     17.74764       200.8494  4263.106
## Cluster 5      42.54310     18.34748       191.2722  3722.744

summary(res.fcm)

## Summary for 'res.fcm'
## 
## Number of data objects:  342 
## 
## Number of clusters:  5 
## 
## Crisp clustering vector:
##   [1] 5 5 2 2 5 5 1 2 4 2 5 2 5 4 5 2 4 2 4 2 5 5 5 5 5 5 2 2 5 2 5 2 5 2 4 5 5
##  [38] 2 1 2 5 2 4 2 1 2 2 2 4 5 4 2 4 2 5 5 5 2 5 2 4 5 4 2 5 2 4 2 4 5 5 5 4 5
##  [75] 4 5 5 5 4 2 1 5 4 2 5 5 5 5 5 5 4 2 4 2 4 5 4 2 4 5 1 2 4 2 5 5 5 2 1 5 1
## [112] 2 4 5 4 2 5 2 2 2 5 2 5 2 4 2 4 2 4 2 5 5 4 2 5 2 5 2 4 2 2 2 5 2 5 4 2 2
## [149] 5 5 4 5 5 5 5 5 5 2 5 4 5 5 5 5 4 5 4 2 5 2 4 5 2 2 5 2 4 2 5 5 4 2 4 2 4
## [186] 5 5 5 1 2 4 5 5 5 5 5 4 2 4 2 5 2 5 5 4 2 2 2 4 5 5 5 5 5 4 2 5 4 5 4 3 4
## [223] 3 3 4 1 1 4 1 1 3 1 3 4 3 4 3 1 3 3 1 4 1 1 1 4 3 1 3 3 1 1 3 1 3 1 3 4 3
## [260] 5 3 4 1 3 1 4 3 1 3 1 3 4 1 1 1 4 1 4 3 4 3 4 3 1 3 1 3 1 3 1 1 1 1 1 3 1
## [297] 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 1 1 1 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3
## [334] 4 3 1 3 1 1 3 1 3
## 
## Initial cluster prototypes:
##           Culmen.Length Culmen.Depth Flipper.Length Body.Mass
## Cluster 1          48.5         15.0            219      4850
## Cluster 2          38.1         17.0            181      3175
## Cluster 3          55.9         17.0            228      5600
## Cluster 4          42.7         18.3            196      4075
## Cluster 5          46.8         16.5            189      3650
## 
## Final cluster prototypes:
##           Culmen.Length Culmen.Depth Flipper.Length Body.Mass
## Cluster 1      45.91966     15.21606       213.0069  4847.306
## Cluster 2      39.59229     17.54392       187.8470  3231.889
## Cluster 3      50.11423     15.80701       222.3897  5606.411
## Cluster 4      44.01479     17.74764       200.8494  4263.106
## Cluster 5      42.54310     18.34748       191.2722  3722.744
## 
## Distance between the final cluster prototypes
##           Cluster 1 Cluster 2 Cluster 3 Cluster 4
## Cluster 2 2610251.1                              
## Cluster 3  576347.3 5639665.3                    
## Cluster 4  341446.8 1063598.8 1804973.2          
## Cluster 5 1265133.8  240959.7 3549236.5  292086.3
## 
## Difference between the initial and final cluster prototypes
##           Culmen.Length Culmen.Depth Flipper.Length  Body.Mass
## Cluster 1     -2.580337    0.2160584      -5.993145  -2.694207
## Cluster 2      1.492292    0.5439221       6.846952  56.888612
## Cluster 3     -5.785767   -1.1929917      -5.610251   6.411371
## Cluster 4      1.314789   -0.5523628       4.849409 188.106417
## Cluster 5     -4.256901    1.8474850       2.272209  72.743568
## 
## Root Mean Squared Deviations (RMSD): 93.9904 
## Mean Absolute Deviation (MAD): 297.7592 
## 
## Membership degrees matrix (top and bottom 5 rows): 
##     Cluster 1   Cluster 2   Cluster 3   Cluster 4   Cluster 5
## 1 0.000708671 0.003181015 0.000247691 0.003238805 0.992623818
## 2 0.005192422 0.017658436 0.001745818 0.026544496 0.948858828
## 3 0.000148537 0.997718099 0.000068250 0.000369270 0.001695844
## 4 0.013960078 0.572636354 0.005861659 0.041229723 0.366312185
## 5 0.003526914 0.028931337 0.001321089 0.013450693 0.952769967
## ...
##      Cluster 1   Cluster 2   Cluster 3   Cluster 4   Cluster 5
## 338 0.96798199 0.002039450 0.012592211 0.013342114 0.004044234
## 339 0.99992528 0.000004904 0.000022447 0.000037267 0.000010103
## 340 0.02425893 0.003117231 0.958874264 0.008940266 0.004809307
## 341 0.50420497 0.016189767 0.379426944 0.071443495 0.028734824
## 342 0.11675218 0.007586244 0.835395400 0.027590316 0.012675862
## 
## Descriptive statistics for the membership degrees by clusters
##           Size       Min        Q1      Mean    Median        Q3       Max
## Cluster 1   64 0.5037126 0.7183871 0.8197986 0.8692388 0.9456157 0.9999388
## Cluster 2   71 0.4756544 0.7521029 0.8345414 0.8879665 0.9673164 0.9977756
## Cluster 3   49 0.5001229 0.8210079 0.8629821 0.9108374 0.9803829 0.9997904
## Cluster 4   64 0.4600043 0.6692306 0.8075202 0.8607078 0.9671469 0.9987311
## Cluster 5   94 0.5023993 0.7251313 0.8254332 0.8627523 0.9758493 0.9998970
## 
## Dunn's Fuzziness Coefficients:
## dunn_coeff normalized 
##  0.7451125  0.6813907 
## 
## Within cluster sum of squares by cluster:
##       1       2       3       4       5 
## 2135028 2498762 2807058 1626772 1906951 
## (between_SS / total_SS =  95.08%) 
## 
## Available components: 
##  [1] "u"          "v"          "v0"         "d"          "x"         
##  [6] "cluster"    "csize"      "sumsqrs"    "k"          "m"         
## [11] "iter"       "best.start" "func.val"   "comp.time"  "inpargs"   
## [16] "algorithm"  "call"

Run FCM with Multiple Starts

res.fcm <- fcm(data_penguin, centers=4, nstart=10)

res.fcm$func.val

##  [1] 11739109 11739109 11739109 11739109 11739109 11739109 11739109 11739109
##  [9] 11739109 11739109

res.fcm$iter

##  [1] 130 127 132 132 129 132 134 135  90 131

res.fcm$best.start

## [1] 1

summary(res.fcm)

## Summary for 'res.fcm'
## 
## Number of data objects:  342 
## 
## Number of clusters:  4 
## 
## Crisp clustering vector:
##   [1] 1 1 4 4 1 1 3 4 1 4 1 4 1 3 1 4 3 4 1 4 4 1 1 1 1 4 4 4 1 4 1 4 1 4 1 1 4
##  [38] 4 3 4 1 4 3 4 3 4 4 4 1 4 3 4 1 4 1 4 1 4 1 4 3 4 1 4 1 4 1 4 3 4 1 4 1 1
##  [75] 1 1 1 4 1 4 3 1 1 4 4 1 4 1 4 4 3 4 3 4 3 1 3 4 1 1 3 4 1 4 4 1 1 4 3 1 3
## [112] 4 1 1 1 4 1 4 4 4 4 4 1 4 1 4 3 4 1 4 4 4 3 4 1 4 1 4 1 4 4 4 1 4 1 1 4 4
## [149] 1 1 1 4 1 1 4 1 1 4 1 1 1 1 1 1 1 4 1 4 1 4 3 4 4 4 1 4 1 4 1 1 3 4 3 4 1
## [186] 4 1 1 3 4 3 1 1 4 4 1 3 4 3 4 1 4 1 4 1 4 4 4 1 1 4 1 1 1 1 4 1 1 1 3 2 3
## [223] 2 2 3 3 2 3 2 3 2 3 2 1 2 1 2 3 2 2 3 3 3 3 3 1 2 3 2 2 3 3 2 2 2 3 2 3 2
## [260] 1 2 3 3 2 3 1 2 3 2 3 2 3 3 3 3 3 3 3 2 1 2 3 2 3 2 3 2 3 2 3 3 3 2 3 2 3
## [297] 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 3 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2
## [334] 3 2 3 2 3 3 2 2 2
## 
## Initial cluster prototypes:
##           Culmen.Length Culmen.Depth Flipper.Length Body.Mass
## Cluster 1          41.3         21.1            195      4400
## Cluster 2          48.7         15.7            208      5350
## Cluster 3          37.0         16.9            185      3000
## Cluster 4          52.7         19.8            197      3725
## 
## Final cluster prototypes:
##           Culmen.Length Culmen.Depth Flipper.Length Body.Mass
## Cluster 1      43.27426     18.45995       194.5110  3895.212
## Cluster 2      49.82006     15.74988       221.8621  5557.892
## Cluster 3      45.29771     15.78941       209.1899  4688.798
## Cluster 4      40.01420     17.70322       188.5384  3307.929
## 
## Distance between the final cluster prototypes
##           Cluster 1 Cluster 2 Cluster 3
## Cluster 2 2765305.4                    
## Cluster 3  630005.6  755506.4          
## Cluster 4  344947.4 5063544.0 1907255.7
## 
## Difference between the initial and final cluster prototypes
##           Culmen.Length Culmen.Depth Flipper.Length Body.Mass
## Cluster 1      1.974261  -2.64004585     -0.4890197 -504.7884
## Cluster 2      1.120056   0.04988389     13.8620970  207.8923
## Cluster 3      8.297707  -1.11058594     24.1898972 1688.7977
## Cluster 4    -12.685795  -2.09678437     -8.4616403 -417.0707
## 
## Root Mean Squared Deviations (RMSD): 911.7446 
## Mean Absolute Deviation (MAD): 2895.527 
## 
## Membership degrees matrix (top and bottom 5 rows): 
##     Cluster 1   Cluster 2   Cluster 3  Cluster 4
## 1 0.877555766 0.005712083 0.021174800 0.09555735
## 2 0.950263601 0.002813389 0.011002442 0.03592057
## 3 0.008077252 0.000631219 0.001624163 0.98966736
## 4 0.091073867 0.004062859 0.011763285 0.89309999
## 5 0.630347092 0.010416243 0.035133544 0.32410312
## ...
##      Cluster 1  Cluster 2  Cluster 3   Cluster 4
## 338 0.04336857 0.11484274 0.82419871 0.017589978
## 339 0.02614376 0.04757857 0.91625347 0.010024193
## 340 0.01021753 0.95267286 0.03121532 0.005894287
## 341 0.04700803 0.62434521 0.30629103 0.022355727
## 342 0.01035685 0.93791483 0.04636996 0.005358357
## 
## Descriptive statistics for the membership degrees by clusters
##           Size       Min        Q1      Mean    Median        Q3       Max
## Cluster 1  102 0.4799810 0.7487436 0.8423650 0.8882931 0.9517204 0.9998587
## Cluster 2   55 0.5162548 0.8184940 0.8831996 0.9379148 0.9866536 0.9998800
## Cluster 3   88 0.4597556 0.6853438 0.8094714 0.8620170 0.9775407 0.9995412
## Cluster 4   97 0.4820959 0.7864087 0.8511891 0.8930451 0.9694579 0.9997757
## 
## Dunn's Fuzziness Coefficients:
## dunn_coeff normalized 
##  0.7673983  0.6898644 
## 
## Within cluster sum of squares by cluster:
##       1       2       3       4 
## 3416139 3707131 4904659 4361364 
## (between_SS / total_SS =  92.73%) 
## 
## Available components: 
##  [1] "u"          "v"          "v0"         "d"          "x"         
##  [6] "cluster"    "csize"      "sumsqrs"    "k"          "m"         
## [11] "iter"       "best.start" "func.val"   "comp.time"  "inpargs"   
## [16] "algorithm"  "call"

Pairwise Scatter Plots

plotcluster(res.fcm, cp=3, trans=TRUE)

gap index

library(clusterSim)

## Loading required package: MASS

## 
## Attaching package: 'MASS'

## The following object is masked from 'package:dplyr':
## 
##     select

## Registered S3 method overwritten by 'e1071':
##   method       from  
##   print.fclust fclust

Davies-Bouldin’s index

cl2 <- pam(data_penguin[,1:4], 5)
print(index.DB(data_penguin[,1:4], cl2$clustering, centrotypes="centroids"))

## $DB
## [1] 0.6054911
## 
## $r
## [1] 0.6707952 0.6707952 0.5626484 0.5626484 0.5605683
## 
## $R
##           [,1]      [,2]      [,3]      [,4]      [,5]
## [1,]       Inf 0.6707952 0.2912618 0.5544725 0.2169387
## [2,] 0.6707952       Inf 0.2353416 0.3367806 0.1953817
## [3,] 0.2912618 0.2353416       Inf 0.5626484 0.5605683
## [4,] 0.5544725 0.3367806 0.5626484       Inf 0.3027583
## [5,] 0.2169387 0.1953817 0.5605683 0.3027583       Inf
## 
## $d
##           1         2         3         4         5
## 1    0.0000  496.7568 1047.8005  500.6397 1797.3615
## 2  496.7568    0.0000 1544.5279  997.3800 2294.0980
## 3 1047.8005 1544.5279    0.0000  547.1632  749.5729
## 4  500.6397  997.3800  547.1632    0.0000 1296.7225
## 5 1797.3615 2294.0980  749.5729 1296.7225    0.0000
## 
## $S
## [1] 137.4574 195.7647 167.7269 140.1336 252.4599
## 
## $centers
##          [,1]     [,2]     [,3]     [,4]
## [1,] 42.61000 18.39889 191.9556 3766.111
## [2,] 39.79375 17.67375 188.4750 3269.375
## [3,] 45.79516 15.25968 212.0323 4813.710
## [4,] 44.34035 17.80877 200.9474 4266.667
## [5,] 49.73962 15.74906 221.8679 5563.208