K-means Clustering Algorithm with Matlab Source code

The K-means Clustering
Algorithm 1
K-means is a method of clustering observations into a speciﬁc number of disjoint clusters.
The ”K” refers to the number of clusters speciﬁed. Various distance measures exist to deter-
mine which observation is to be appended to which cluster. The algorithm aims at minimiz-
ing the measure between the centroide of the cluster and the given observation by iteratively
appending an observation to any cluster and terminate when the lowest distance measure
is achieved.
1.1 Overview Of Algorithm
1. The sample space is intially partitioned into K clusters and the observations are ran-
domly assigned to the clusters.
2. For each sample:
• Calculate the distance from the observation to the centroide of the cluster.
• IF the sample is closest to its own cluster THEN leave it ELSE select another
cluster.
3. Repeat steps 1 and 2 untill no observations are moved from one cluster to another
When step 3 terminates the clusters are stable and each sample is assigned a cluster which
results in the lowest possible distance to the centriode of the cluster.
Common distance measures include the Euclidean distance, the Euclidean squared distance
and the Manhattan or City distance.
The Euclidean measure corresponds to the shortest geometric distance between to points.
d =
N
∑
i=1
(xi −yi)2
1
(19.1)
19.2 Distance measures
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19.1

A faster way of determining the distance is by use of the squared Euclidean distance which
calculates the above distance squared, i.e.
dsq =
N
∑
i=1
(xi −yi)2
The Manhattan measure calculates a distance between points based on a grid and is illus-
Euclidean measure Manhattan measure
For applications in speech processing the squared Euclidean distance is widely used.
K-means can be used to cluster the extracted features from speech signals. The extracted
features from the signal include for instance mel frequency cepstral coefﬁcients or line spec-
trum pairs. This allows speech signals with similar spectral characteristics to be positioned
into the same position in the codebook. In this way similar narrow band signals will be
predicted likewise thereby limiting the size of the codebook.
The following ﬁgures illustrate the K-means algoritm on a 2-dimensional data set.
2
Figure 19.1:Comparision between theEuclideanandtheManhattan measure.
CHAPTER 19. THE K-MEANS CLUSTERING ALGORITHM
(19.2)
trated in Figure 19.1.
19.3 Application of K-means
19.4 Example of K-means Clustering

−20 −15 −10 −5 0 5 10 15 20
−20
−15
−10
−5
0
5
10
15
20
−20 −15 −10 −5 0 5 10 15 20
−20
−15
−10
−5
0
5
10
15
20
marked with a cross.
3
Figure 19.2:Exampleof signal datamadefrom GaussianWhiteNoise.
Figure 19.3:Thesignal dataareseperated intoseven clusters.Thecentroidsare
19.4. EXAMPLE OF K-MEANS CLUSTERING

0 0.2 0.4 0.6 0.8 1
1
2
3
4
5
6
7
Silhouette Value
Cluster
into the seven clusters. If the distance from one point to two centroids is the
same, it means the point could belong to both centroids. The result is a conﬂict
which gives a negative value in the Silhouette diagram. The positive part of
the Silhuoette diagram, shows that there is a clear seperation of the points
between the clusters.
4
Figure 19.4:TheSilhouettediagramshowshow well thedataareseperated

1 close a l l
2 clear a l l
3 clc
4
5 Limit = 2 0 ;
6
7 X = [ 1 0 ∗ randn (400 ,2) ; 1 0 ∗ randn (400 ,2) ] ;
8 plot (X ( : , 1 ) ,X ( : , 2 ) , ’k . ’ )
9 length (X ( : , 1 ) )
10 figure
11 %i =1;
12 k=1;
13 for i =1: length (X ( : , 1 ) )
14 i f ( sqrt (X( i , 1 ) ^2+X( i , 2 ) ^2) ) > Limit ;
15 X( i , 1 ) =0;
16 X( i , 2 ) =0;
17 else
18 Y( k , 1 ) =X( i , 1 ) ;
19 Y( k , 2 ) =X( i , 2 ) ;
20 k=k+1;
21 end
22 end
23 plot (Y ( : , 1 ) ,Y ( : , 2 ) , ’k . ’ )
24 figure
25
26 [ cidx , c t r s ] = kmeans (Y , 7 , ’ d i s t ’ , ’ sqEuclidean ’ , ’ rep ’ ,5 , ’ disp ’ , ’
f i n a l ’ , ’ EmptyAction ’ , ’ singleton ’ ) ;
27
28 plot (Y( cidx ==1 ,1) ,Y( cidx ==1 ,2) , ’ r . ’ , . . .
29 Y( cidx ==2 ,1) ,Y( cidx ==2 ,2) , ’b . ’ , c t r s ( : , 1 ) , c t r s ( : , 2 ) , ’ kx ’ ) ;
30
31 hold on
32 plot (Y( cidx ==3 ,1) ,Y( cidx ==3 ,2) , ’y . ’ ,Y( cidx ==4 ,1) ,Y( cidx ==4 ,2) , ’g . ’ )
;
33
34 hold on
35 plot (Y( cidx ==5 ,1) ,Y( cidx ==5 ,2) , ’ c . ’ ,Y( cidx ==6 ,1) ,Y( cidx ==6 ,2) , ’m. ’ )
;
36
37 hold on
38 plot (Y( cidx ==7 ,1) ,Y( cidx ==7 ,2) , ’k . ’ ) ;
39
40 figure
5
19.5. MATLAB SOURCE CODE
19.5 Matlab Source Code

41 [ silk , h]= s i l h o u e t t e (Y, cidx , ’ sqEuclidean ’ ) ;
42 mean( s i l k )
6
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K-means Clustering Algorithm with Matlab Source code

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K-means Clustering Algorithm with Matlab Source code