Hierachical clustering

Tilani Gunawardena
Algorithms: Clustering

• Grouping of records ,observations or cases into
classes of similar objects.
• A cluster is a collection of records,
– Similar to one another
– Dissimilar to records in other clusters
What is Clustering?

• There is no target variable for clustering
• Clustering does not try to classify or predict the
values of a target variable.
• Instead, clustering algorithms seek to segment
the entire data set into relatively homogeneous
subgroups or clusters,
– Where the similarity of the records within the
cluster is maximized, and
– Similarity to records outside this cluster is
minimized.
Difference between Clustering and
Classification

• Identification of groups f records such that
similarity within a group is very high while the
similarity to records in other groups is very low.
– group data points that are close (or similar) to
each other
– identify such groupings (or clusters) in an
unsupervised manner
• Unsupervised: no information is provided to
the algorithm on which data points belong to
which clusters
• In other words,
– Clustering algorithm seeks to construct clusters of
records such that the between-cluster variation(BCV)
is large compared to the within-cluster
variation(WCV)
Goal of Clustering

Between-cluster variation:
Within-cluster variation:
Goal of Clustering
between-cluster variation(BCV) is
large compared to the within-
cluster variation(WCV)
(Intra-cluster distance) the sum of distances
between objects in the same cluster are
minimized
(Inter-cluster distance) while the distances
between different clusters are maximized

• Clustering techniques apply when there is no
class to be predicted
• As we've seen clusters can be:
– disjoint vs. overlapping
– deterministic vs. probabilistic
– flat vs. hierarchical
• k-means Algorithm
– k-means clusters are disjoint, deterministic, and
flat
Clustering

Issues Related to Clustering
• How to measure similarity
– Euclidian Distance
– City-block Distance
– Minkowski Distance
• How to recode categorical variables?
• How to standardize or normalize numerical
variables?
– Min-Max Normalization
– Z-score standardization ( )
• How many clusters we expect to uncover?
m,s

Type of Clustering
• Partitional clustering: Partitional algorithms
determine all clusters at once. They include:
– K-Means Clustering
– Fuzzy c-means clustering
– QT clustering
• Hierarchical Clustering:
– Agglomerative ("bottom-up"): Agglomerative
algorithms begin with each element as a
separate cluster and merge them into successively
larger clusters.
– Divisive ("top-down"): Divisive algorithms begin
with the whole set and proceed to divide it into
successively smaller clusters.

k-Means Clustering
• Input: n objects (or points) and a number k
• Algorithm
1) Randomly assign K records to be the initial
cluster center locations
2) Assign each object to the group that has the
closest centroid
3) When all objects have been assigned,
recalculate the positions of the K centroids
4) Repeat steps 2 to 3 until convergence or
termination

Termination Conditions
• The algorithm terminates when the centroids
no longer change.
• The SSE(sum of squared errors) value is less
than some small threshold value 
• Where p є Ci represents each data point in
cluster i and mi represent the centroid of
cluster i.
SSE = d(p- mi )2
pÎci
å
i=1
k
å

Example 1:
• Lets s suppose the following points are the
delivery locations for pizza

• Lets locate three cluster centers randomly

• Find the distance of points as shown

• Assign the points to the nearest cluster center
based on the distance between each center
and the point

• Re-assign the cluster centres and locate
nearest points

• Re-assign the cluster centres and locate
nearest points, calculate the distance

Example 2:
• Suppose that we have eight data points in
two-dimensional space as follows
• And suppose that we are interested in
uncovering k=2 clusters.

Point Distance from m1 Distance from m2 Cluster
membership
a
b
c
d
e
f
g
h


n
i
ii yxYXD
1
2
)(),(

membership
a 2.00 2.24
b 2.83 2.24
c 3.61 2.83
d 4.47 3.61
e 1.00 1.41
f 3.16 2.24
g 0.00 1.00
h 1.00 0.00


n
i
ii yxYXD
1
2
)(),(

membership
a 2.00 2.24 C1
b 2.83 2.24 C2
c 3.61 2.83 C2
d 4.47 3.61 C2
e 1.00 1.41 C1
f 3.16 2.24 C2
g 0.00 1.00 C1
h 1.00 0.00 C2
SSE=22+2.242+2.832+3.612+12+2.242+02+02=36
d(m1,m2)=1

Centroid of the cluster 1 is
[(1+1+1)/3,(3+2+1)/3]
=(1,2)
[(3+4+5+4+2)/5,(3+3+3+2+1)/5]
=(3.6,2.4)

membership
a
b
c
d
e
f
g
h
m1=(1,2)
m2=(3.6,2.4)

membership
a 1.00 2.67
b 2.24 0.85
c 3.61 0.72
d 4.12 1.52
e 0.00 2.63
f 3.00 0.57
g 1.00 2.95
h 1.41 2.13
m1=(1,2)
m2=(3.6,2.4)

membership
a 1.00 2.67 C1
b 2.24 0.85 C2
c 3.61 0.72 C2
d 4.12 1.52 C2
e 0.00 2.63 C1
f 3.00 0.57 C2
g 1.00 2.95 C1
h 1.41 2.13 C1
SSE=12+0.852+0.722+1.522+02+0.572+12+1.412=7.88
d(m1,m2)=2.63
m1(1,2)
m2(3.6,2.4)

[(1+1+1+2)/4,(3+2+1+1)/4]
=(1.25,1.75)
[(3+4+5+4)/4,(3+3+3+2)/4]
=(4,2.75)

membership
a
b
c
d
e
f
g
h
m1(1.25,1.75)
m2(4,2.75)

membership
a 1.27 3.01
b 2.15 1.03
c 3.02 0.25
d 3.95 1.03
e 0.35 3.09
f 2.76 0.75
g 0.79 3.47
h 1.06 2.66
m1(1.25,1.75)
m2(4,2.75)

membership
a 1.27 3.01 C1
b 2.15 1.03 C2
c 3.02 0.25 C2
d 3.95 1.03 C2
e 0.35 3.09 C1
f 2.76 0.75 C2
g 0.79 3.47 C1
h 1.06 2.66 C1
m1(1.25,1.75)
m2(4,2.75)
SSE=1.272+1.032+0.252+1.032+0.352+0.752+0.792+1.062=6.25
d(m1,m2)=2.93

How to decide k?
• Unless the analyst has a prior knowledge of
the number of underlying clusters, therefore,
– Clustering solutions for each value of K is
compared
– The value of K resulting in the smallest SSE being
selected

Summary
K-means algorithm is a simple yet popular method for
clustering analysis
• Low complexity :complexity is O(nkt), where t =
#iterations
• Its performance is determined by initialisation and
appropriate distance measure
• There are several variants of K-means to overcome its
weaknesses
– K-Medoids: resistance to noise and/or outliers(data that do
not comply with the general behaviour or model of the data
)
– K-Modes: extension to categorical data clustering analysis
– CLARA: extension to deal with large data sets
– Gaussian Mixture models (EM algorithm): handling
uncertainty of clusters

42
Hierarchical Clustering
• Build a tree-based hierarchical taxonomy
(dendrogram)
• One approach: recursive application of a partitional
clustering algorithm.
animal
vertebrate
fish reptile amphib. mammal worm insect crustacean
invertebrate

Hierarchical clustering and
dendrograms
• A hierarchical clustering on a set of objects D is a set
of nested partitions of D. It is represented by a binary
tree such that :
– The root node is a cluster that contains all data points
– Each (parent) node is a cluster made of two subclusters
(childs)
– Each leaf node represents one data point (singleton ie
cluster with only one item)
• A hierarchical clustering scheme is also called a
taxonomy. In data clustering the binary tree is called
a dendrogram.

44
• Clustering obtained
by cutting the
dendrogram at a
desired level: each
connected
component forms a
cluster.
Dendogram: Hierarchical Clustering

Hierarchical clustering: forming
clusters
• Forming clusters from dendograms

Hierarchical clustering
• There are two styles of hierarchical clustering algorithms
to build a tree from the input set S:
– Agglomerative (bottom-up):
• Beginning with singletons (sets with 1 element)
• Merging them until S is achieved as the root.
• In each steps , the two closest clusters are aggregates into a new
combined cluster
• In this way, number of clusters in the data set is reduced at each step
• Eventually, all records/elements are combined into a single huge cluster
• It is the most common approach.
– Divisive (top-down):
• All records are combined in to a one big cluster
• Then the most dissimilar records being split off recursively partitioning
S until singleton sets are reached.
• Does not require the number of clusters k in advance

Two types of hierarchical clustering algorithms :
Agglomerative : “bottom-up”
Divisive : “top-down

48
Hierarchical Agglomerative Clustering (HAC)
Algorithm
Start with all instances in their own cluster.
Until there is only one cluster:
Among the current clusters, determine the two
clusters, ci and cj, that are most similar.
Replace ci and cj with a single cluster ci  cj
• Assumes a similarity function for determining the
similarity of two instances.
• Starts with all instances in a separate cluster and then
repeatedly joins the two clusters that are most similar
until there is only one cluster.

Classification of AHC
We can distinguish AHC algorithms according to the type of
distance measures used. There are two approaches :
Graph methods :
• Single link method
• Complete link method
• Group average method (UPGMA)
• Weighted group average method (WPGMA)
Geometric :
• Ward’s method
• Centroid method
• Median method

Lance –Williams Algorithm
Definition(Lance-Williams formula)
In AHC algorithms, the Lance-Williams formula
[Lance and Williams, 1967] is a recurrence
equation used to calculate the dissimilarity
between a cluster Ck and a cluster formed by
merging two other clusters Cl ∪Cl′
where α
I
, α
I’
,β, γ are real numbers

AHC methods and the Lance-Williams
formula

Single link method
• Also known as the nearest neighbor method,
since it employs the nearest neighbor to
measure the dissimilarity between two
clusters

Cluster distance measure
• Single link
– Distance between closest elements in clusters
• Complete link
– Distance between farthest elements in clusters
• Centroids
– Distance between centroids(means) of two clusters

Single-link clustering
Nested Clusters Dendrogram
1
2
3
4
5
6
1
2
3
4
5
3 6 2 5 4 1
0
0.05
0.1
0.15
0.2

Example 02-Single link method
• x
1
= (1, 2)
• x
2
= (1, 2.5)
• x
3
= (3, 1)
• x
4
= (4, 0.5)
• x
5
= (4, 2)

• x
1
= (1, 2)
• x
2
= (1, 2.5)
• x
3
= (3, 1)
• x
4
= (4, 0.5)
• x
5
= (4, 2)
Merge X1 and X2

• x
1
= (1, 2)
• x
2
= (1, 2.5)
• x
3
= (3, 1)
• x
4
= (4, 0.5)
• x
5
= (4, 2)
Merge X1 and X2
Example 3-Complete link method

Summary
 Hierarchical Clustering
• For a dataset consisting of n points
• O(n2) space; it requires storing the distance matrix
• O(n3) time complexity in most of the cases(agglomerative
clustering)
• Advantages
– Dendograms are great for visualization
– Provides hierarchical relations between clusters
• Disadvantages
– Not easy to define levels for clusters
– Can never undo what was done previously
– Sensitive to cluster distance measures and noise/outliers
– Experiments showed that other clustering techniques
outperform hierarchical clustering
• There are several variants to overcome its weaknesses
– BIRCH: scalable to a large data set
– ROCK: clustering categorical data
– CHAMELEON: hierarchical clustering using dynamic modelling

Hierachical clustering

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