k-means算法的简单示例
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α , β as the centre and K=2.
Step 3 again.
The new centers of the two clusters are P(1.6,4.8) and Q(4.8,1.6)
Example
A
d PA 1 d PB 0.89
B
E C D P(1.6,4.8) A B
P , Q as the centre and K=2.
Step 3 again.
The new centers of the two clusters are equal to the original P(1.6,4.8) and Q(4.8,1.6)
Final
E C D
cluster 2
A B F I J G H
d AE 2.24 d AF 3.61
< > < > > > > > > >
d BA 1 d BB 0
d BC 1.41 d BD 1
d BE 2 d BF 2.83
d AG 4.47
d AH 5.39 d AI 4.24
H I J
d BG 3.61 d BH 4.47
Algorithm Procedure
1. Randomly select K points from complete samples as the initial center.(That's what k means in K-means) 2. Each point in the dataset is assigned to the closed cluster,based upon the Euclidean distance between each point and each cluster center. 3. Each cluster's center is recomputed as the average of the points in that cluster. 4. Iterate step 2 or more until the new center of cluster equals to the original center of cluster or less than a specified threshold,then clustering finished.
Example
x y center ( ,
i
E C α(1,4.5) A B D
j
i
j
)
PA, B,C , D, E (1.6,4.8)
β(3.75,2.875)
F I
cluster 1
new center
H
G J
QF ,G,H ,I , J (4.8,1.6)
cluster 2
B
C
D
E
β(3.75,2.875)
F I
F G
dE 1.8 dF 3.91 dG 4.72 dH 5.59 dI 4.61 dJ 5.32
G J
H
H I J
< < < < < > > > > >
d A 2.97 d B 2.08
dC 3.48 dD 2.75
Example
x y center ( ,
i
E
j
i
j
)
cluster 2
C D
A(1,4)
B(2,4)
ቤተ መጻሕፍቲ ባይዱ
A ,C
F I G J H
11 4 5 ( , ) (1,4.5) 2 2
new center
cluster 1
B,D,E ,F ,G,H ,I , J (3.75,2.875)
C D E F G
F
G H
d PC 0.63 d PD 0.45
d PE 1.26 d PF 3.69
d PG 4.40
d PH 5.22 d PI 4.49
H
Q(4.8,1.6) I J
I
J
d PJ 5.10
< < < < < > > > > >
dQA 4.49
cluster 1
Clustering finished !
Disadvantages
one of the main disadvantages to k-means is the fact that you must specify the number of clusters(K) as an input to the algorithm.As designed,the algorithm is not capable of determining the appropriate number of clusters and depends upon the user to identify this in advance.
Example
E
C D B
A
F I
G J
H
How to cluster A,B...H,J into two clusters?
Example
A
E C D
d AA 0 d AB 1
B C D E F G
F I G J H
d AC
A(1,4)
d BC
B(2,4)
d AC 1 d AD 1.41
dQB 3.69 dQC 5.10 dQD 4.4
dQE 5.22 dQF 0.89 dQG 0.45
dQH 1.26
d QI 1 dQJ 0.63
Step 2 again.
So,we classify A,B,C,D,E as a cluster and F,G,H,I,J as another cluster.
Example
x y center ( ,
i j
i
j
)
E C D P(1.6,4.8) A B
cluster 1
M A,B,C ,D,E (1.6,4.8)
NF ,G,H ,I , J (4.8,1.6)
F H Q(4.8,1.6) J G
new center
cluster 2
I
K-means Clustering
K-means Clustering
K-means clustering is a sort of clustering algorithm and it is a m et hod of vector quantization, originally from signal processing, that is popular for cluster analysis in data mining. K-means clustering aims to partition n observations into k clusters in which each observation belongs to the cluster with the nearest mean, serving as a prototype of the cluster. --From Wikipedia
d BI 3.61 d BJ 4.24
d AJ 5
d AB means distance A→B
Randomly choose A,B as the centre and K=2.
Step 1 and 2.
So,we classify A,C as a cluster and B,E,D,F,G,H,I and J as another cluster.
The new centers of the two clusters are (1,4.5) and (3.75,2.875)
Randomly choose A,B as the centre and K=2.
Step 3.
Example
A
E C α(1,4.5) A B D
dA 0.5
dB 1.12 dC 0.5 dD 1.12
K=2
K=3
d E 3.58 d F 0.91 dG 1.53 dH 2.41 d I 1.89 d J 2.25
α , β as the centre and K=2.
Step 2 again.
So,we classify A,B,C,D,E as a cluster and F,G,H,I,J as another cluster.
Step 3 again.
The new centers of the two clusters are P(1.6,4.8) and Q(4.8,1.6)
Example
A
d PA 1 d PB 0.89
B
E C D P(1.6,4.8) A B
P , Q as the centre and K=2.
Step 3 again.
The new centers of the two clusters are equal to the original P(1.6,4.8) and Q(4.8,1.6)
Final
E C D
cluster 2
A B F I J G H
d AE 2.24 d AF 3.61
< > < > > > > > > >
d BA 1 d BB 0
d BC 1.41 d BD 1
d BE 2 d BF 2.83
d AG 4.47
d AH 5.39 d AI 4.24
H I J
d BG 3.61 d BH 4.47
Algorithm Procedure
1. Randomly select K points from complete samples as the initial center.(That's what k means in K-means) 2. Each point in the dataset is assigned to the closed cluster,based upon the Euclidean distance between each point and each cluster center. 3. Each cluster's center is recomputed as the average of the points in that cluster. 4. Iterate step 2 or more until the new center of cluster equals to the original center of cluster or less than a specified threshold,then clustering finished.
Example
x y center ( ,
i
E C α(1,4.5) A B D
j
i
j
)
PA, B,C , D, E (1.6,4.8)
β(3.75,2.875)
F I
cluster 1
new center
H
G J
QF ,G,H ,I , J (4.8,1.6)
cluster 2
B
C
D
E
β(3.75,2.875)
F I
F G
dE 1.8 dF 3.91 dG 4.72 dH 5.59 dI 4.61 dJ 5.32
G J
H
H I J
< < < < < > > > > >
d A 2.97 d B 2.08
dC 3.48 dD 2.75
Example
x y center ( ,
i
E
j
i
j
)
cluster 2
C D
A(1,4)
B(2,4)
ቤተ መጻሕፍቲ ባይዱ
A ,C
F I G J H
11 4 5 ( , ) (1,4.5) 2 2
new center
cluster 1
B,D,E ,F ,G,H ,I , J (3.75,2.875)
C D E F G
F
G H
d PC 0.63 d PD 0.45
d PE 1.26 d PF 3.69
d PG 4.40
d PH 5.22 d PI 4.49
H
Q(4.8,1.6) I J
I
J
d PJ 5.10
< < < < < > > > > >
dQA 4.49
cluster 1
Clustering finished !
Disadvantages
one of the main disadvantages to k-means is the fact that you must specify the number of clusters(K) as an input to the algorithm.As designed,the algorithm is not capable of determining the appropriate number of clusters and depends upon the user to identify this in advance.
Example
E
C D B
A
F I
G J
H
How to cluster A,B...H,J into two clusters?
Example
A
E C D
d AA 0 d AB 1
B C D E F G
F I G J H
d AC
A(1,4)
d BC
B(2,4)
d AC 1 d AD 1.41
dQB 3.69 dQC 5.10 dQD 4.4
dQE 5.22 dQF 0.89 dQG 0.45
dQH 1.26
d QI 1 dQJ 0.63
Step 2 again.
So,we classify A,B,C,D,E as a cluster and F,G,H,I,J as another cluster.
Example
x y center ( ,
i j
i
j
)
E C D P(1.6,4.8) A B
cluster 1
M A,B,C ,D,E (1.6,4.8)
NF ,G,H ,I , J (4.8,1.6)
F H Q(4.8,1.6) J G
new center
cluster 2
I
K-means Clustering
K-means Clustering
K-means clustering is a sort of clustering algorithm and it is a m et hod of vector quantization, originally from signal processing, that is popular for cluster analysis in data mining. K-means clustering aims to partition n observations into k clusters in which each observation belongs to the cluster with the nearest mean, serving as a prototype of the cluster. --From Wikipedia
d BI 3.61 d BJ 4.24
d AJ 5
d AB means distance A→B
Randomly choose A,B as the centre and K=2.
Step 1 and 2.
So,we classify A,C as a cluster and B,E,D,F,G,H,I and J as another cluster.
The new centers of the two clusters are (1,4.5) and (3.75,2.875)
Randomly choose A,B as the centre and K=2.
Step 3.
Example
A
E C α(1,4.5) A B D
dA 0.5
dB 1.12 dC 0.5 dD 1.12
K=2
K=3
d E 3.58 d F 0.91 dG 1.53 dH 2.41 d I 1.89 d J 2.25
α , β as the centre and K=2.
Step 2 again.
So,we classify A,B,C,D,E as a cluster and F,G,H,I,J as another cluster.