Geology Reference
In-Depth Information
X
N
y
1
N
y
y ¼
x
y
ð
3
:
46
Þ
i¼1
where D
cen
(X, Y) is the maximum object to object linkage distance. N
x
and N
y
are
the number of objects in clusters X and Y, respectively and
;
x
y are the mean vectors
of the
first and the second clusters, respectively.
3.4.1.5 Ward
'
is Linkage Clustering
Ward
is linkage clustering is a common analysis technique with analysis of variance
(ANOVA). Ward
'
s method [
79
] is a popular method which is not based on the
distance matrix. The method begins with N single member groups and, as with
other hierarchical methods, merges two groups at each step. All combinations of
two groups are considered. The chosen pair is the one which minimizes the
information loss, namely the sum of the squared distances between the points and
the centroids of their respective groups summed over the resulting groups. In this
approach, the linkage function estimates distance between the two clusters, the
increase in the error sum of squares (ESS) after fusing two clusters into a single
cluster. The mathematical expression for Ward
'
'
is linkage clustering is as shown
below.
ESS for cluster X can be expressed as
2
N
x
X
X
N
x
N
x
1
ESS
ð
X
Þ
¼
x
i
x
i
ð
3
:
47
Þ
i¼1
i¼1
and Ward
'
s linkage function can be expressed as
D
w
X
ð
;
Y
Þ
¼
ESS
ð
XY
Þ
½
ESS
ð
X
Þþ
ESS
ð
Y
Þ
ð
3
:
48
Þ
where
jj
is the absolute value of a scalar value or the norm (the
“
length
”
)ofa
vector and XY is combined vector obtained by fusion of clusters X and Y.
In theory, hierarchical CA should be achievable by divisive methods,
(agglomerative methods in reverse) by starting with all N observations and splitting
the cluster into two clusters of the most similar groups and so on until reaching
n clusters of one element. These divisive methods are computationally intensive and
impractical in some ways, unless for small sample sizes. To choose the closest
groups for merging, agglomerative hierarchical clustering requires all S(S
1)/2
possible pairs of S groups to be examined. As for divisive clustering, for each group
of size
n
s
, the number of possible ways to make a split is 2
n
s
1
−
1[
9
,
81
].
