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Fuzzy Clustering Techniques: Fuzzy C-Means andFuzzy Min-Max Clustering Neural Networks

Benjamin ames Bush!!"# $%& Term 'a(er) Fall *+%*

|1| INTRODUCTION,ata clustering is a data (rocessing strategy which aims to organize a collection odata (oints .here/y sim(ly called (oints0 into grou(s1 Traditionally) the data set is(artitioned so that each (oint /elongs to one and only one cluster1 2owe3er) unlessthe data is 3ery highly clustered) it is oten the case that some (oints do notcom(letely /elong to any one cluster1 4ith the arri3al o Fuzzy clustering) these(oints could /e assigned a set o mem/ershi( degrees associated with eachcluster instead o arti5cially (igeonholing it as /elonging to only one1 The 3olumeo literature a3aila/le on uzzy clustering is immense6 a general re3iew o the

literature is outside the sco(e o this term (a(er1 This (a(er discusses only *a((roaches to uzzy clustering: the u/iquitous Fuzzy C-Means Clustering algorithmand the less well known /ut interesting Fuzzy Min-Max Clustering Neural Network1These a((roaches are discussed in sections * and 7) res(ecti3ely1 "n (art 8 " will/rie9y discuss se3eral a((lications which use the uzzy clustering techniquesco3ered here1

|2| THE FUZZY C-MEANS (FCM) CLUSTERINGALGORITHMFuzzy C-Means) also known as Fuzzy -Means and Fuzzy "!;,

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The (rocess is initialized /y (icking diAerent ?centroids@ at random rom thes(ace in which the (oints are em/edded1 From here) the -Means (rocess can /edi3ided into two (hases:

34+# 1! F"r C,u#$r#. #ach centroid is associated with a diAerent cluster1 To

orm these clusters) each (oint in the data set is e3aluated in turn1 4hene3aluated) a (oint is assigned to the cluster corres(onding to the closest centroid134+# 2! M"5 C'$r"i*#.#ach o the centroid is now mo3ed to the (ositiono/tained /y taking the mean o each o the (oints in the cluster associated withthe centroid1

These two (hases are re(eated in turn until con3ergence is reached .i1e1 until the3alue o the cost unction sto(s decreasing signi5cantly01 "t should /e noted thatthere is no guarantee that the cost unction will /e minimized1 The outcomede(ends on initial conditions1 < 9ow chart is (ro3ided /elow to aid the reader>sunderstanding o the algorithm1

Figur 2! A 6"7 &4+r$ #u+rii'g $4 K-M+'# C,u#$ri'g A,g"ri$4.

=nderstanding o the -means clustering algorithm can /e urther enhanced /y3iewing a series o animated "F images (roduced /y

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Figur 8! K %r+# %r" +' +'i+$i"' "' -+'# &,u#$ri'g A'*r A. S4++,i'9 34D

|2.2| FUZZY C-MEANS CLUSTERING (FCM)FCM is a generalization o -Means1 4hile -Means assigns each (oint to one andonly one cluster) FCM allows clusters to /e uzzy sets) so that each (oint /elongs toall clusters to 3arying degrees) with the ollowing restriction: The sum o allmem/ershi( degrees or any gi3en data (oint is equal to %1 The cost unction usedin FCM .shown in 5gure 70 is 3ery similar to the one used /y -Means) /ut there aresome key diAerences: The inner sum contains a term or each data (oint in the set1

#ach o these terms is weighed /y a mem/ershi( degree raised to the (ower o auzziness ex(onent1

Figur :! C"#$ %u'&$i"' %"r FCM. Figur +*+;$* %r" /10. A''"$+$i"'# . C";+r 7i$4 Figur1.

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Eike -means) FCM is initialized /y choosing a 5xed num/er o centroids at random1

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Figur >! T4 +i' 7i'*"7 "% fcmdemo+%$r ru''i'g i$ "' *+$+ #$ 2 7i$4 C ? 8 +'* ? 2.

Figur @! Mr#4i; %u'&$i"' ;,"$# +%$r ru''i'g %&*" 7i$4 %ui'## ;"''$ ? 1.