PARTITIONAL CLUSTERINGDeniz STN

CONTENTWHAT IS CLUSTERING ?

WHAT IS PARTITIONAL CLUSTERING ?

THE USED ALGORITHMS IN PARTITIONAL CLUSTERING

What is Clustering ?A process of clustering is classification of the objects which are similar among them, and organizing of data into groups.

The techniques for Clustering are among the unsupervised methods.

What is Partitional Clustering ?The Partitional Clustering Algorithms separate the similar objects to the Clusters.

The Partitional Clustering Algorithms are succesful to determine center based Cluster.

The Partitional Clustering Algorithms divide n objects to k cluster by using k parameter.

The techniques of the Partitional Clustering start with a randomly chosen clustering and then optimize the clustering according to some accuracy measurement.

The Used Algorithms in Partitional Clustering

K-MEANS ALGORITHM

K-MEDOIDS ALGORITHM

FUZZY C-MEANS ALGORITHM

K-MEANS ALGORITHMK-MEANS algorithm is introduced as one of the simplest unsupervised learning algorithms that resolve the clustering problems by J.B. MacQueen in 1967 (MacQueen, 1967).

K-MEANS algorithm allows that one of the data belong to only a cluster.

Therefore, this algorithm is a definite clustering algorithm.

Given the N-sample of the clusters in the N-dimensional space.

K-MEANS ALGORITHMThis space is separated, {C1,C2,,Ck} the K clusters. The vector mean (Mk) of the Ck cluster is given (Kantardzic, 2003) :where the value of Xk is i.sample belong to Ck. The square-error formula for the Ck is given :

K-MEANS ALGORITHMThe square-error formula for the Ck is called the changing in cluster.The square-error for all the clusters is the sum of the changing in clusters.The aim of the square-error method is to find the K clusters that minimize the value of the Ek2 according to the value of the given K

K-MEANS ALGORITHMEXAMPLE

GzlemlerDeiken1Deiken2Kme yeliiX132C1X223C2X378C1

K-MEANS ALGORITHMEXAMPLE

K-MEANS ALGORITHMEXAMPLE

Gzlemlerd(M1)d(M2)Kme yeliiX12,821,41C2X23,600C2X33,607,07C1

K-MEANS ALGORITHMEXAMPLE

GzlemlerDeiken1Deiken2Kme yeliiX132C2X223C2X378C1

K-MEANS ALGORITHMEXAMPLE

K-MEANS ALGORITHMEXAMPLE-1C2

C1

Gzlemlerd(M1)d(M2)Kme yeliiX17,210,7C2X27,070,7C2X307,10C1

K-MEANS ALGORITHMEXAMPLE-2

DatasetThe Number of AttributesThe Number of FeaturesThe Number of ClassSynthetic120024

K-MEANS ALGORITHMEXAMPLE-2K=2

K-MEANS ALGORITHMEXAMPLE-2K=3

K-MEANS ALGORITHMEXAMPLE-2K=4

K-MEDOIDS ALGORITHMThe aim of the K-MEDOIDS algorithm is to find the K representative objects (Kaufman and Rousseeuw, 1987).Each cluster in K-MEDOIDS algorithm is represented by the object in cluster.K-MEANS algorithm determine the clusters by the mean process. However, K-MEDOIDS algorithm find the cluster by using mid-point.

K-MEDOIDS ALGORITHMEXAMPLE-1

K-MEDOIDS ALGORITHMEXAMPLE-1Select the Randomly K-Medoids

K-MEDOIDS ALGORITHMEXAMPLE-1Allocate to Each Point to Closest Medoid

K-MEDOIDS ALGORITHMEXAMPLE-1Allocate to Each Point to Closest Medoid

K-MEDOIDS ALGORITHMEXAMPLE-1Allocate to Each Point to Closest Medoid

K-MEDOIDS ALGORITHMEXAMPLE-1Determine New Medoid for Each Cluster

K-MEDOIDS ALGORITHMEXAMPLE-1Determine New Medoid for Each Cluster

K-MEDOIDS ALGORITHMEXAMPLE-1Allocate to Each Point to Closest Medoid

K-MEDOIDS ALGORITHMEXAMPLE-1Stop Process

K-MEDOIDS ALGORITHMEXAMPLE-2

DatasetThe Number of AttributesThe Number of FeaturesThe Number of ClassSynthetic200023

K-MEDOIDS ALGORITHMEXAMPLE-2K=2

K-MEDOIDS ALGORITHMEXAMPLE-2K=3

FUZZY C-MEANS ALGORITHMFuzzy C-MEANS algorithm is the best known and widely used a method.Fuzzy C-MEANS algorithm is introduced by DUNN in 1973 and improved by BEZDEK in 1981 [Hppner vd, 2000].Fuzzy C-MEANS lets that objects are belonging to two and more cluster.The total value of the membership of a data for all the classes is equal to one.However, the value of the memebership of the cluster that contain this object is high than other clusters.This Algorithm is used the least squares method [Hppner vd, 2000].

FUZZY C-MEANS ALGORITHMThe algorithm start by using randomly membership matrix (U) and then the center vector calculate [Hppner vd, 2000].

FUZZY C-MEANS ALGORITHMAccording to the calculated center vector, the membership matrix (u) is computed by using the given as: The new membership matrix (unew) is compared with the old membership matrix (uold) and the the process continues until the difference is smaller than the value of the

FUZZY C-MEANS ALGORITHMEXAMPLE

DatasetThe Number of AttributesThe Number of FeaturesThe Number of ClassSynthetic200023

FUZZY C-MEANS ALGORITHMEXAMPLEC=3m=5=1e-6

ResultsK-MEDOIDS is the best algorithm according to K-MEANS and FUZZY C-MEANS.However, K-MEDOIDS algorithm is suitable for small datasets.K-MEANS algorithm is the best appropriate in terms of time.In FUZZY C-MEANS algorithm, a object can belong to one or more cluster.However, a object can belong to only a cluster in the other two algorithms.

References[MacQueen, 1967] J.B., MacQueen, Some Methods for Classification and Analysis of Multivariate Observations, Proc. Symp. Math. Statist.and Probability (5th), 281-297,(1967).[Kantardzic, 2003] M., Kantardzic, Data Mining: Concepts, Methods and Algorithms, Wiley, (2003).[Kaufman and Rousseeuw, 1987] L., Kaufman, P. J., Rousseeuw, Clustering by Means of Medoids, Statistical Data Analysis Based on The L1Norm and Related Methods, edited by Y. Dodge, North-Holland, 405416, (1987). [Kaufman and Rousseeuw, 1990] L., Kaufman, P. J., Rousseeuw, Finding Groups in Data: An Introduction to Cluster Analysis, John Wiley and Sons., (1990).[Hppner vd, 2000] F., Hppner, F., Klawonn, R., Kruse, T., Runkler, Fuzzy Cluster Analysis, John Wiley&Sons, Chichester, (2000).[Ik and amurcu, 2007] M., Ik, A.Y., amurcu, K-MEANS, K-MEDOIDS ve Bulank C-MEANS Algoritmalarnn Uygulamal olarak Performanslarnn Tespiti, stanbul Ticaret niversitesi Fen Bilimleri Dergisi, Say :11, 31-45, (2007).