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Chapter 12 Clustering, Distance Method, and
Ordination
12.2 Similarity Measures
Commonly used distance:
Euclidean Distance:
d  x, y  
x1  y1 2  x2  y 2 2    x p  y p 2

x  y t x  y 
Statistical Distance:
d x, y  
x  y t S 1 x  y  ,
where S is the sample variance-covariance matrix.
Minkowski Distance:
 p
m
d  x, y    xi  yi 
 i1

1
m
.
Canberra Metric:
p
d  x, y   
i 1
xi  yi
xi  yi  .
Czekanowski coefficient:
p
d  x, y   1 
2 min  xi , yi 
i 1
p
 x  y 
i 1
i
.
i
12.3 Hierarchical Clustering Methods
Agglomerative Hierarchical Clustering Algorithm (Grouping N
Objects):
1. Start with N clusters, each containing a single entity and an N  N
symmetric matrix of distances (or similarities) D  d ik .
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2. Search the distance matrix for the nearest (most similar) pair of
clusters. Let the distance between “most similar” clusters U and
V be dUV .
3. Merge clusters U and V. Label the newly formed cluster (UV).
Update the entries in the distance matrix
(a) deleting the rows and columns corresponding to clusters U
and V and
(b) adding a row and column giving the distances between
cluster (UV) and the remaining clusters.
4. Repeat Steps 2 and 3 a total of N-1 times. (All objects will be in a
single cluster after the algorithm terminates.) Record the
identity of clusters that are merged and the levels (distances) at
which the merges take place.
There are 3 linkage methods. The main differences among these
methods are the distances between (UV) and any other cluster W.
(I) Single Linkage:
d UV W  min d UW , d VW .
(II) Complete Linkage:
d UV W  max dUW , dVW .
(III) Average Linkage:
d UV W 
where
d ik
 d
i
ik
k
N UV  NW ,
is the distance between object i in the cluster (UV) and
object k in the cluster W, and N UV  and N W are the number of items
in clusters (UV) and W, respectively.
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