Model Data Selection and Data Pre-processing Approaches - Hydrological Data Driven Modelling - page 55

this is illustrated in Fig. 3.4 b. Assume the clusters are X and Y, x i , is an object
belonging to the first cluster and y j an object belonging to the next.

Then the mathematical expression for linkage function is

ð ; Þ ¼

x 2 X ; y 2 Y ds x i ;

ð 3 : 42 Þ

D max X Y max y j

where ds (x i ,y j ) is the distance between the objects x i and y j , and D max X ; Y ð Þ is

maximum object to object linkage distance.

3.4.1.3 Average Linkage Clustering

The distance between groups using group average clustering is the average of the
distances between all pairs of elements (one from both groups under consideration)
and is depicted in Fig. 3.4 c. The mathematical expression for average linkage
clustering is as follows:

X X

N y

N x

1
N x

D mean X ð ; Y Þ ¼ ds x i ; y j ð 3 : 43 Þ

N y
x 2 X ; y 2 Y

i¼1 j¼1

where ds(x, y) is the distance between the objects x and y and D mean X ; ð Þ is the
mean object to object linkage distance. N x and N y are the number of objects in
clusters X and Y, respectively.

3.4.1.4 Average Group Linkage Clustering (Centroid Clustering)

Average group linkage clustering is also popularly known as centroid clustering. In
the case of centroid clustering, inter-group distance is de ned as the mean vector of

all the variables already within the group. In each iteration, the two clusters whose
centroids are the closest are merged. A problem with this method is that, if the sizes
of the two groups differ vastly, when the groups are fused together the centroid of
the new group will lie very close to the large group. The mathematical expression
for average group linkage clustering is as follows:

D cen X ð ; Y Þ ¼ ds x ðÞ ; y 3 : 44 Þ

where

X

N x

1
N x

x ¼ x i ð 3 : 45 Þ

i¼1