Download Clustering Formulation using Constraint Optimization

optimize a criterion, such as minimizing the intra-cluster distances or maximizing the inter-cluster distances.
We formally define a clustering C as a set of clusters C1 , . . . , Ck that represent
a partition of a set of data points P = {p1 , . . . , pn } obtained
byToptimizing a
S
specific criterion; so we have C ⊆ P , ∀C ∈ C : C ⊆ D, C = D, C = ∅. Note
that we consider non-overlapping clusters. Each point pi is represented by an
m-dimensional vector. In order to determine the similarity between data objects
we consider a distance function d(pi , pj ) between two data objects. The data
objects are described by attributes, which may be qualitative or quantitative. In
case of quantitative attributes, we consider the euclidean distance between two
m-dimensional vectors.
The optimization criterion determines the quality of the clustering. There
are many different ways to measure the goodness of a clustering:
Sum of squared inter-cluster distances. Given some distance function d(·, ·) over
points, e.g., the Euclidean distance,
we
P
P can measure the sum of squared distances
over each cluster as follows: C∈C i,j∈|C|,i<j d2 (pi , pj ).
Sum of squared error to centroid. A more common measure is the “error” of each
cluster, that is, the distance of each point in the cluster to the mean (centroid)
of that cluster.
Considering the centroid
P of a cluster as the mean of the data points that
p∈C p
then, the sum of squared error is measured as:
belong to it, i.e., mC =
|C|
P
P
2
C∈C
p∈C d (p, mC ).
Sum of squared error to medoids. Instead of using the mean (centroid) of the cluster, we can also use the medoid of the cluster, that is, the point that is most representative of the cluster. Let the medoid of a cluster bePthe point with smallest
d2 (p,y)
. The sum of
average distance to the other points: mC = argminy∈C p∈C|C|
P
P
squared error to the medoids is then measured as follows: C∈C p∈C d2 (p, mC ).
Cluster diameter. Another measure of coherence is to measure the diameter of
the largest cluster, where the diameter is the largest distance between any two
points of a cluster. This leads to the following measure of maximum cluster
diameter: maxC∈C maxi,j∈|C|,i<j d(pi , pj ).
Inter-cluster margin. The margin between two clusters is the minimal distance
between any two points that belong to the different clusters. The margin gives
an indication of how different the clusters are from each other (e.g. how far
apart they are). This can be optimized using the following measure of minimum
inter-cluster margin: mini,j∈|C|,i<j minp1 ∈Ci ,p2 ∈Cj d(p1 , p2 ).
2.1
Prototype-based Clustering
Among the most studied and applied clustering methods there are the prototypebased ones that require the number of clusters to be known in advance. These
techniques create a partitioning of the data where each cluster (partition) is