## Thursday, December 27, 2012

### Trying to revive the C3M algorithm

C3M is so much unlike other hierarchical  or k-means category of algorithms for clustering. The C3M (Cover-Coefficient Clustering Method) is a single-pass clustering algorithm built by researchers in IR (Information Retrieval). Hence, it is primarily meant for documents. Because IR researchers built this algorithm, they take advantage of the fact that term-document matrices are usually sparse and can be represented well by an inverted index. The main advantage of the algorithm is that it provides a way to estimate/compute the number of clusters given a document-term matrix.

Here are some of the claims made by their paper " Concept and Effectiveness of the Cover Coefficient Based Clustering Methodology for Text Databases by Fazli Can and Esen OZkarahan".
a) Clusters are stable
b) Algorithm is independent of order of documents, hence, we will always a unique clusters
c) The memory overhead is really low
d) The algorithm distributes the documents evenly among cluster. In other words, it does not create fat clusters or singletons.

The algorithm starts of by examining the document-term matrix D of a collection of size m by n. From this it computes C ( m by m) and C' (n by n), where C[i,j] indicates the extent to which i covers j. The higher c[i,j] the higher the similarity between the two documents. Similarly, C' records how much one terms covers the other. From C and C' matrices it is possible to compute a document's seed power. The higher the seed power of a document, the higher the chance that it being selected to be a seed of a cluster.Once the cluster seeds are selected. For each document (not assigned yet) we find the closest cluster by simply examining c[i,j], where i corresponds to the unassigned document and j corresponds to each of the cluster seeds.

If m denotes the total number of documents, xd denotes the average number of words in each document and, tg denotes the average of documents in which a term appears, then the complexity of this clustering approch is O(xd X m X tg)

Creating the C matrix

initialize C[m,m] <- 0="0" p="p">for each document i
for each term t_i in that document
Compute P(t_i) = D(i,t)/ sum_t {D(i,t)}
Get the inverted index for term t_i .. Lets call it inv(t_i)
compute p(d_j|inv(t_i)) for each document in the inverted list
for each document j in inv(t_i)
c[i,j] = c[i,j] + p(t_i)*p(d_j|inv(t_i))

Similarly one can compute c'

Compute number of clusters n_c = \sum_i c[i,i]

Computing Cluster seed power
for each document i
P(i) = c[i,i]*(1-c[i,i])* sum_j  { d[i,j] c'[j,j] (1-c'[j,j])}

Pick the top n_c documents that have highest P(i) and set them as cluster seeds  {D_cluster_Seeds}

Assigning documents to cluster seeds
for each of the document i (that is not a cluster seed)
assigned = argmax(j) {c[i,j] where j < {D_cluster_Seeds}}

Cluster Validity

This paper presents a unique way of gauging the cluster structure as it measures this in terms of the query. Given two cluster structures -one random assignment and second the cluster structure under test, a set of queries with the corresponding set of relevant documents, the cluster validity is computed as follows. Suppose a query has k relevant documents, then we compute the value $n_t$ and $n_tr$ corresponding to the test cluster structure and random clustering. The authors use $n_t$ or $n_tr$ to denote the number of target clusters. The target cluster is a cluster that contains at least 1 relevant document to the query. If the clustering is any good, then $n_t$ should be less than $n_tr$. The average of $n_t$ is computed with $n_tr$ over all the queries and may be additionally subject to significance testing.

Now let me explain how $n_t$ is computed. For a given query, the authors first find the size of the relevant set (from ground truth). Lets denote this by k. Given $n_c$ number of clusters, for each cluster, $P_i$ is computed. $P_i$ computes the probability that a cluster $C_i$ will be selected given that randomly k documents from the set of m documents were picked. There is a direct formula in the paper that computes this. Computing $P_1+P_2+...P_n_c$ yields $n_t$ for the test cluster. The procedure is repeated for the cluster structure obtained by random clustering.

Although it makes mathematical sense, I wonder what would be the actual number of target clusters for the queries based on the actual location of the relevant documents. I personally feel that this is a far better and more accurate representation of the reality. Of course, we can compare the number of target clusters with a random cluster structure or with another clustering algorithm.

Querying

Querying with different term weighting has been extensively covered in this paper and I will not be able to cover all of it here.

I am going to implement this algorithm and update this post with some results.