## Introduction

Clustering can be
considered the most important *unsupervised
learning* problem; so,
as every other problem of this kind, it deals with finding a *structure* in a collection of unlabeled data.
A loose definition of clustering could be “the process of organizing objects
into groups whose members are similar in some way”.
A *cluster* is therefore a collection of objects which are coherent
internally, but clearly dissimilar to the objects belonging to other clusters.

In this case
we easily identify the 3 clusters into which the data can be divided; the
similarity criterion is *distance*:
two or more objects belong to the same cluster if they are “close” according to
a given distance (in this case geometrical distance). This is called *distance-based clustering*, here I’m
going to deal with is distance-based clustering.
Another kind of clustering is *conceptual
clustering*: two or more objects belong to the same cluster if this
one defines a concept *common* to all that objects. In other words,
objects are grouped according to their fit to descriptive concepts, not
according to simple similarity measures.

### Classification

*Clustering algorithms may be classified as listed below*

**Flat
clustering** (Creates a set of clusters without any explicit
structure that would relate clusters to each other; It’s also called exclusive
clustering)**Hierarchical
clustering** (Creates a hierarchy of clusters)**Hard
clustering** (Assigns each document/object as a member of exactly
one cluster)**Soft
clustering** (Distribute the document/object over all clusters)

**Algorithms **

- Agglomerative
(Hierarchical clustering)
- K-Means (Flat clustering,
Hard clustering)
- EM Algorithm (Flat
clustering, Soft clustering)

Hierarchical Agglomerative Clustering (HAC) and
K-Means algorithm have been applied to text clustering in a straightforward way.
Typically it usages normalized, TF-IDF-weighted vectors and cosine similarity.
Here, I have illustrated the k-means algorithm using a set of points in
n-dimensional vector space for text clustering.

### K-Means Algorithm

The k-means clustering algorithm is
known to be efficient in clustering large data sets. This clustering algorithm
was developed by MacQueen , and is one of the simplest and the best known
unsupervised learning algorithms that solve the well-known clustering problem. The K-Means
algorithm aims to partition a set of objects, based on their
attributes/features, into *k* clusters, where *k* is a predefined or user-defined
constant. The main idea is to define *k* centroids, one for each cluster. The
centroid of a cluster is formed in such a way that it is closely related (in terms
of similarity function; similarity can be measured by using different methods
such as cosine similarity, Euclidean distance, Extended Jaccard) to all objects in that
cluster.

**Basic K-Means Algorithm **

- Choose k number of clusters to be determined
- Choose k objects randomly as the initial cluster center
- Repeat

3.1. Assign each object to their closest cluster

3.2. Compute new clusters, i.e. Calculate mean points.

4. Until

4.1. No changes on cluster centers (i.e. Centroids do not change
location any more) OR

4.2. No object changes its cluster (We may define stopping criteria as
well)

##

**Residual Sum of Squares **

RSS is a measure of how well the
centroids represent the members of their clusters, the squared distance of each
vector from its centroid summed over all vectors.

The algorithm then moves the cluster centers around in space in order to minimize RSS

**Termination Condition**

We can apply one of the following termination conditions.

- A fixed
number of iterations I has been completed. This
condition limits the runtime of the clustering algorithm, but in some
cases the quality of the clustering will be poor because of an
insufficient number of iterations.
- Assignment
of documents to clusters does not change between iterations. Except for
cases with a bad local minimum, this produces a good clustering, but
runtimes may be unacceptably long.
- Centroids
do not change between iterations. This is equivalent to not Terminate when RSS falls below a threshold. This
criterion ensures that the clustering is of a desired quality after
termination. In practice, we need to combine it with a bound on the number
of iterations to guarantee termination.
- Terminate
when the decrease in RSS falls below a threshold
*t* . For small *t*,
this indicates that we are close to convergence. Again, we need to combine
it with a bound on the number of iterations to prevent very long runtimes.

###

**Bad choice of initial seed**

In K-Means algorithm there
is unfortunately no guarantee that a *global minimum* in the objective
function will be reached, this is a particular problem if a document set
contains many *outliers* , documents that are far from
any other documents and therefore do not fit well into any cluster. Frequently,
if an outlier is chosen as an initial seed, then no other vector is assigned to
it during subsequent iterations. Thus, we end up with a *singleton
cluster* (a cluster with only one document) even though there is probably a
clustering with lower RSS.

*Effective heuristics for seed selection include:*

- Excluding outliers from the seed set
- Trying out multiple starting points and choosing
the clustering with the lowest cost; and
- Obtaining seeds from another method such as
hierarchical clustering.

## Diving into the code

**Document Representation **

Each document is
represented as a vector using the** vector
space model**. The vector space model also called term vector model is an
algebraic model for representing text document (or any object, in general) as
vectors of identifiers. For example, TF-IDF weight.

Here I have defined
`DocumentVector`

class whose instance holds the document and its corresponding
representation on vector space.

public class DocumentVector
{
public string Content { get; set; }
public float[] VectorSpace { get; set; }
}

And instance of `DocumentCollection`

represtnts all
the documents to be clustered

class DocumentCollection
{
public List<String> DocumentList { get; set; }
}

##### TF-IDF

*TF-IDF* stands for term frequency-inverse document frequency, is a numerical statistics which reflects how
important a word is to a document in a collection or corpus, it is the most
common weighting method used to describe documents in the Vector Space Model,
particularly on IR problems.

The number of times a term occurs in a
document is called its *term
frequency*. *We
can calculate the ***term
frequency** for a word as the
ratio of number of times the word occurs in the document to the total number of
words in the document.

The **inverse
document frequency** is a
measure of whether the term is common or rare across all documents. *It is obtained by dividing the total number
of **documents by the number of documents containing the term, and then
taking the logarithm of that quotient. *

The **tf*idf** of term t in document d is calculated as:

private static float FindTFIDF(string document, string term)
{
float tf = FindTermFrequency(document, term);
float idf = FindInverseDocumentFrequency(term);
return tf * idf;
}
private static float FindTermFrequency(string document, string term)
{
int count = r.Split(document).Where(s => s.ToUpper() == term.ToUpper()).Count();
return (float)((float)count / (float)(r.Split(document).Count()));
}
private static float FindInverseDocumentFrequency(string term)
{
int count = documentCollection.ToArray().Where(s => r.Split(
s.ToUpper()).ToArray().Contains(term.ToUpper())).Count();
return (float)Math.Log((float)documentCollection.Count() / (float)count);
}

##### Finding Similarity Score

I have used cosine similarity to identify the similarity score of a document. The method `FindCosineSimilarity`

takes two argument `vecA`

and `vecB`

as parameter
which are vector representation of document A and B, and returns the
similarity score which lies between 1 and 0, indicating that document A and B
are completely similar and dissimilar respectively.

public static float FindCosineSimilarity(float[] vecA, float[] vecB)
{
var dotProduct = DotProduct(vecA, vecB);
var magnitudeOfA = Magnitude(vecA);
var magnitudeOfB = Magnitude(vecB);
float result = dotProduct / (magnitudeOfA * magnitudeOfB);
if (float.IsNaN(result))
return 0;
else
return (float)result;
}

##### K-Means Algorithm Implementation

To implement K-Means
algorithm I have defined a class `Centroid`

in which documents are assigned during
the clustering process.

public class Centroid
{
public List<DocumentVector> GroupedDocument { get; set; }
}

**Preparing Document Cluster **

public static List<Centroid> PrepareDocumentCluster(int k,
List<DocumentVector> documentCollection,ref int _counter)
{
globalCounter = 0;
List<Centroid> centroidCollection = new List<Centroid>();
Centroid c;
HashSet<int> uniqRand = new HashSet<int>();
GenerateRandomNumber(ref uniqRand,k,documentCollection.Count);
foreach(int pos in uniqRand)
{
c = new Centroid();
c.GroupedDocument = new List<DocumentVector>();
c.GroupedDocument.Add(documentCollection[pos]);
centroidCollection.Add(c);
}
Boolean stoppingCriteria;
List<Centroid> resultSet;
List<Centroid> prevClusterCenter;
InitializeClusterCentroid(out resultSet, centroidCollection.Count);
do
{
prevClusterCenter = centroidCollection;
foreach (DocumentVector obj in documentCollection)
{
int index = FindClosestClusterCenter(centroidCollection, obj);
resultSet[index].GroupedDocument.Add(obj);
}
InitializeClusterCentroid(out centroidCollection, centroidCollection.Count());
centroidCollection = CalculateMeanPoints(resultSet);
stoppingCriteria = CheckStoppingCriteria(prevClusterCenter, centroidCollection);
if (!stoppingCriteria)
{
InitializeClusterCentroid(out resultSet, centroidCollection.Count);
}
} while (stoppingCriteria == false);
_counter = counter;
return resultSet;
}

**Initializing cluster center **

Cluster center is
initialized for the next iteration, here the `count`

variable holds the value of
user defined initial cluster center.

private static void InitializeClusterCentroid(out List<Centroid> centroid,int count)
{
Centroid c;
centroid = new List<Centroid>();
for (int i = 0; i < count; i++)
{
c = new Centroid();
c.GroupedDocument = new List<DocumentVector>();
centroid.Add(c);
}
}

**Finding closest cluster center **

This function returns the
index of closest cluster center for each document, I have used cosine
similarity to identify the closeness of document. The array `similarityMeasure`

holds the similarity score for the
document `obj`

with each cluster center, the index which has maximum score is taken as the closest cluster center
of the given document.

private static int FindClosestClusterCenter(List<Centroid> clusterCenter,DocumentVector obj)
{
float[] similarityMeasure = new float[clusterCenter.Count()];
for (int i = 0; i < clusterCenter.Count(); i++)
{
similarityMeasure[i] =
SimilarityMatrics.FindCosineSimilarity(
clusterCenter[i].GroupedDocument[0].VectorSpace, obj.VectorSpace);
}
int index = 0;
float maxValue = similarityMeasure[0];
for (int i = 0; i < similarityMeasure.Count(); i++)
{
if (similarityMeasure[i] >maxValue)
{
maxValue = similarityMeasure[i];
index = i;
}
}
return index;
}

##### Identifying the new position of the cluster center

After each document being assigned to its closest cluster center we recalculate the mean of each cluster
center which indicates the new position of cluster center (centroid).

private static List<Centroid> CalculateMeanPoints(List<Centroid> _clusterCenter)
{
for (int i = 0; i < _clusterCenter.Count(); i++)
{
if (_clusterCenter[i].GroupedDocument.Count() > 0)
{
for (int j = 0; j < _clusterCenter[i].GroupedDocument[0].VectorSpace.Count(); j++)
{
float total = 0;
foreach (DocumentVector vSpace in _clusterCenter[i].GroupedDocument)
{
total += vSpace.VectorSpace[j];
}
_clusterCenter[i].GroupedDocument[0].VectorSpace[j] =
total / _clusterCenter[i].GroupedDocument.Count();
}
}
}
return _clusterCenter;
}