Вычислительные технологии

Том 15, № 2, 2010

Document clustering using a modified differential

evolution algorithm

R. M. Aliguliyev

Institute of Information Technology of National Academy of Sciences of Azerbaijan, Baku e-mail: a.ramizSscience .az, aramizSiit.ab.az

Document clustering algorithms play important role in helping users to get relevant information, navigate, summarize and organize an enormous amount of textual document available on the Internet, and in digital libraries, news sources, and company-wide intranets. The objective of our work is to develop a high quality criterion functions for partitional document collection. In present paper we introduce two weighted criterion functions. We've also proposed a modified differential evolution algorithm to optimize the criterion functions. The proposed methods experimentally were evaluated on WebKb dataset. Experiments showed that the weighted criterion functions outperform the results obtained by the unweighted criterion functions.

Keywords: document clustering, weighted k-means, modified differential evolution.

Introduction

Clustering is useful for organization, summarization and navigation of semi-structured web pages and has been investigated as a fundamental operation in many areas such as data mining [1], information retrieval [2, 3], topic detection [4, 5], and as a preprocessing step for other algorithms such as document summarization [6-12]. For instance, in [6-9] the authors show that using clustering techniques for preprocessing in document summarization is a viable and effective technique. Unlike document classification, no labeled documents are provided in clustering; hence, clustering is also known as unsupervised learning. The methods used for document clustering covers several research areas, such as database, information retrieval, and artificial intelligence including machine learning and natural language processing [1-3, 13, 14].

High quality document clustering algorithms play important role in helping users to get relevant information, navigate, summarize and organize an enormous amount of textual document available on the Internet, and in digital libraries, news sources, and company-wide intranets. The objective of our work is to develop a high quality criterion functions for partitional document collection. In present paper we introduce two weighted criterion functions. We've also proposed a modified differential evolution algorithm to optimize the criterion functions. The proposed methods experimentally were evaluated on WebKb dataset.

1. Related Work

Clustering methods have been addressed in many contexts and disciplines such as data mining, document retrieval, image segmentation and pattern classification The prevalent

© IIBT CO PAH. 2010.

clustering algorithms have been categorized in different ways depending on different criteria. As with many clustering algorithms, there is trade-off between speed and quality of results. The existing clustering algorithms can be simply classified into the following two categories: hierarchical clustering and partitional clustering. A comprehensive survey of the various clustering algorithms can be found in [15, 16].

From the complexity of the problem to find the best clustering or the least very good clusterings for a given set of n objects, it should be clear, that a hierarchical method can only be feasible in very specific cases. But such a method could be used for an initial clustering. More important are the so-called partitioning algorithms where contrary to the hierarchical methods each object will be attached individually to a cluster. Partitional clustering algorithms assign each object to a partition. In recent years, it has been recognized that the partitional clustering technique is well suited for clustering a large document database due to their relatively low computational requirements [17].

The k-means method [15, 16, 18] is a commonly used partitional clustering method in information retrieval and other related research areas. Given from n objects, the method first select k objects as initial k clusters. It then iteratively assigns each object to the most similar cluster based on the mean value of the objects in each cluster. There are many variations of the k-means method [19, 20]. In [21] presented a novelty-based document clustering method by proposing a new algorithm based on the k-means method. The novelty-based clustering method is based on a novelty-based similarity measure, which is an extension of a traditional approach in information retrieval, the cosine similarity measure in the VSM [2]. Novelty-based document clustering is a document clustering technique that puts high weights on recent documents and low weights on old ones. In [20] is proposed a generic iterative clustering scheme that, coupled with some particular reweighting scheme, may indeed bring improvements over "classical" clustering from the theoretical standpoint. This iterative clustering scheme can be specialized to bring weighted variants of k-means, fuzzy k-means, Expectation Maximization, and harmonic means clustering, among others [16, 18]. The experimental results clearly display differences in the benefits of the weighting scheme depending on the original clustering algorithms.

A major problem of using the existing methods is the selection of variables. The existing algorithms cannot select variables automatically because they treat all variables equally in the clustering process. In [22] is presented a new k-means type algorithm called W — k-means that can automatically weight variables based on the importance of the variables in clustering. W — k-means adds a new step to the basic k-means algorithm to update the variable weights on the current partition of data. Based on the current partition in the iterative k-means clustering process, the algorithm calculates a new weight for each variable based on the variance of the within cluster distances. The new weights are used in deciding the cluster memberships of objects in the next iteration. The weights can be used to identify important variables for clustering and the variables which may contribute noise to the clustering process and can be removed from the data in the future analysis.

2. Document Clustering

The standard clustering technique consists of the following steps: 1) feature selection and data representation model, 2) similarity measure selection, 3) clustering model, 4) clustering algorithm that generates the clusters using the data model and the similarity measure, 5) validation [3].

2.1. Document Representation Model

Let given the collection of documents D = (D\,D2,... ,Dn), where n is the number of documents in the collection. Let T = (T\,T2,..., Tm) represent all the terms occurred in the document collection D. Here m is the number of unique terms in the document collection. In the most clustering algorithms, the dataset to be clustered is represented as a set of vector, where each vector corresponds to a single object and is called the feature vector. The feature vector should include proper features to represent the object.

We represent each document using the VSM. In this model each document Di is located as a point in a m dimensional vector space, Di = (wi\,wi2,..., wim), i = 1,..., n, where the dimension is the same as the number of terms in the document collection. The component wij is defined using the scheme tf-idf. The tf-idf scheme combines the definitions of term frequency and inverse document frequency, to produce a composite weight for each term in each document. This weighting scheme assigns to term a weight in document given by

where nij is the term frequency (i. e. denotes how many term Tj occurs in document Di), nj denotes the number of documents in which term Tj appears. The term log (n/nj), which is very often referred to as the idf factor, accounts for the global weighting of term Tj. The idf factor has been introduced to improve the discriminating power of terms in the traditional information retrieval.

In other words, tf-idf assigns to term Tj a weight in document Di that is

• highest when Tj occurs many times within a small number of documents (thus lending high discriminating power to those documents);

• lower when the term occurs fewer times in a document, or occurs in many documents (thus offering a less pronounced relevance signal);

• lowest when the term occurs in virtually all documents. 2.2. Similarity Measure

The cosine measure has been one of the most popular document similarity measures due to its sensitivity to document vector pattern. The cosine measure computes the cosine of the angle between two feature vectors and is used frequently in text mining where vectors are very large but sparse. The cosine similarity between two documents Di and Di calculate as:

(1)

m

E wij wij j=i

i l = 1, ... ,n.

(2)

2.3. Weighted Clustering

A hard clustering C is a partition of the dataset D into mutually disjoint subsets C\,C2,...,Ck called clusters. Formally, C = (C\, C2,..., Ck) such that Cpf] Cq = 0,

k

p = q (i.e., two different clusters should have no documents in common) and (J Cp = D

p

(i.e., each document should definitely be belonged to a cluster). We also assume that for all q = 1,... ,k Cq = 0 and Cq G D, i. e. each cluster should have at least one document assigned and it must not contain all documents. In other words, k represents the number of non-empty clusters.

Our study involves two weighted clustering criterion functions that are given in below. These criterion functions have been proposed in the context of partitional clustering algorithms:

v^ cos(Di,Dl) . .

= > > -V ; ->■ max, 3

1 ^ ^ aial

p=l Di,Di£Cp i l

T« ^ V^ cos (Di,OP) (A\ ---->max- (4)

p=1 DiSCp 1

In these clustering methods to each document is assigned a weight defining its position in a document collection [23, 24]:

cos (Dj,Q) . . ,

ai = —-, i = l,...,n, (5)

Ecos (Di,O) i=i

where O is the center of the document collection D.

1 n

The jth coordinate Oj of the center O calculate as: Oj = — ^^Wij (j = 1,... ,m). The

n i=1

weight (5) defines the degree of relative similarity of the document Di to the centre of the D, i.e., this weight defines the position of the document Di relative to the center of the entire

n

collection D. It is not difficult to see that Y1 ai = 1.

i=1

The Ff criterion function (3) maximizes the sum of the pairwise similarities between the documents assigned to each cluster. The Fa criterion function (4) is the weighted version of the k-means algorithm.

To show efficiency of weight assignment to documents the criterion functions (3), (4) we shall compare with the following criterion functions

k

F1 = ^^ ^^ cos (Di, Dl) — max, (6)

p=1 Di,Di&Cp

F2 = ^^ cos (Di, Op) — max. (7)

p=1 DiGCp

The criterion functions (6), (7) can be received from the functions (3), (4) at the assumption ai = a > 0 for any i (i = 1,..., n). The criterion function (6) maximizes the sum of the average pairwise similarity between the documents assigned to each cluster. The criterion function (7) is the vector-space variant of k-means method [17].

k

2.4. Clustering using Differential Evolution

The evolutionary algorithms differ mainly in the representation of parameters (usually binary strings are used for genetic algorithms while parameters are real-valued for evolution strategies and differential evolution) and in the evolutionary operators [14].

2.4-1- Chromosome Encoding

For representing the ath chromosome of the population at the current generation (at time t) here the following notation has been used:

Xa (t) = [Xa,i (t) , Xa,2 (t) , . . . , Xa,mk (t)] , (8)

where mk = m • k, a = 1,... ,N, N is the size of the population.

According to this encoding the first m positions represent the m dimensions of the first cluster centre, the next m genes represent those of the second cluster centre, and so on. For example, let m = 2 and k = 4. Then X = [0.16; 0.37; 0.23; 0.75; 0.82; 0.26; 0.94; 0.68] represents the four cluster centers (0.16; 0.37), (0.23; 0.75), (0.82; 0.26) and (0.94; 0.68).

2.4-2. Population Initialization

The k cluster centers encoded in each chromosome are initialized to k randomly chosen points from the n points D = {Di, D2,..., Dn}. This process is repeated for each of the N chromosomes in the population.

2.4-3. Fitness Computation

The fitness functions according to objective functions (3), (4) we define as follows:

f°{x) = W(xr <9)

= <10)

so that minimization of these fitness functions (9), (10) lead to maximization of criterion functions (3), (4), respectively. Fitness functions, corresponding to the criterion functions (6), (7), define in a similar way.

2.4-4- Modified Crossover

The proposed version for the chromosome of the best current solution Xb (t) randomly chooses two other chromosomes Xa (t) and Xc (t) (b = a = c) from the same generation. Then it calculates the scaling difference (1 — Acr) (xc,s (t) — xa,s (t)) and creates a trial offspring chromosome by adding the result to the chromosome Xb (t) scaled by the factor Acr. Thus for the sth gene (s = 1, 2,..., mk) yb,s (t + 1) of child chromosome Yb (t +1), we have:

, s _ J MaxMin (Acrxb,s (t) + (1 — A cr) (xc,s (t) xa,s (t))) if 0S < pTcr, /-.-.N yb,s ( ) \ xb,s (t), otherwise. ( )

The scaling factor, Acr G [0.5,1.0] and the crossover constant, prcr G [0,1], are control parameters, which set by the user. 9S is the uniformly distributed random number within the range [0,1] chosen once for each s G {1,..., mk}.

The function MaxMin (xs (t)) in formula (11) defines as:

xfn (t) + is (t) , if xs (t) < xfn (t),

MaxMin (xs (t)) = { xs (t), if xfn (t) < xs (t) < xrx (t) , (12)

x^ax (t) - (t), if xs (t) > x^ (t) ,

xmax(t) _ xmin(t)

where xfn(t) = min {xas (t)} , x™ax = max {xas(t)}, 5S (t) = 3 V ;-

a€{1,2,...,N} ae{1,2,...,N} n

Differential evolution uses the principle of "survival of the fittest" in its selection process which may be expressed as:

X (t + 1) = { Y> (t + 1), if fa (Y, (t + 1)) <fza (Xb (t)), (13)

[ Xb (t), otherwise,

where fitness functions fa (x) (z = 1, 2) are defined by formulas (9), (10).

2.4.5. Modified Mutation

The mutation operation for target chromosome Xb (t) is performed according to the following formulation:

/..IN I MaxMin (Amtxb,s (t) + (1 — Amt) (x b,r (t) — xb,q (t))) if ns < prmt, yb,s (t + 1) I xb,s (t), else, (14)

with random indexes s, q, r G {1,..., mk}, integer, mutually different, s = q = r. The control parameters — the scaling factor, Amt G [0.5,1.0] and the mutation constant, prmt G [0,1], select by the user. ns is the uniformly distributed random number within the range [0,1] chosen once for each s G {1,..., mk}. If the mutant chromosome yields a better value of the fitness function, it replaces its parent in the next generation; otherwise the parent is retained in the population.

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It is obvious that direct application of the basic differential evolution algorithm can lead to the case that the obtained solution can be infeasible. The infeasible solution is that solution which coordinates are beyond interval (xmin,xmax). Therefore for prevention occurrence of the infeasible solution we modify the crossover (11) and mutation (14) operators by introducing the function MaxMin (x (t)).

2.4-6. Termination Criterion

The algorithm terminates when a maximum number of fitness calculation is achieved. 2.5. Validation

Cluster validation refers to quantitatively evaluating the quality of a clustering solution.

2.5.1. Internal Validity Indices

In this subsection we present three indices using different definitions of inter and intra-cluster connectivity. Many validity measures have been proposed for evaluating clustering results. Most of these popular validity measures do not work well for clusters with different densities and/or sizes. They usually have a tendency of ignoring clusters with low densities. In [14] proposed a validity measure that can deal with this situation. This measure is a function of the ratio of the sum of within-cluster scatter to between-cluster separation:

S

CS i(fc) = ^7-—-(15)

max {sim (Op,Oq)}

p=1 {q=1,...,k,q=p

This cluster validity index is inspired by the work reported in [25] and has been suitably modified. The denominator term in (15) computes the largest similarity between cluster centers. The numerator in (15) measures the average smallest similarity between two documents lying in the same cluster, whereas the latter uses the smallest similarity between two documents lying in the same cluster to measure the scatter volume. Similarly, is defined the second measure:

k

E min {sim (Di,Op)}

p=1 DiGCp

CS2 = -, (16)

E max {sim (Op,Oq)}

p

= 1 q=1,...,k,q=p

the third measure we define as:

k1

? Jc~\ zJ sim(A,Op)

p=1 I pI D- cC

CS 3(fc) =-^-. (17)

k sim (Op, O) p=i

The numerator in (17) measures the average similarity of documents to the cluster centers. The denominator term in (17) computes the sum of similarity between cluster centers and the centre of the entire collection.

The validity indexes (15)-(17) simultaneously take care of the compactness and separation factors into account while dealing with complex structure datasets. The largest value of these measures (15)-(17) indicates a valid optimal partition.

2.5.2. External Validity Indices

Our second set of validation will focused on comparing of the clustering results produced by the proposed criterion functions with the "ground truth" results. The quality of a clustering solution was measured by using different metrics. These metrics measure the matching of clusters computed by each method to the "ground truth" clusters, meaning they measure how close each clustering method is to the "correct" clustering that would be produced manually by a human. However, in situations where documents are already labeled, we can compare the clusters with the "true" class labels.

Assume that the dataset D is composed of the classes C+ = ..., C++j (true

clustering) and we apply a clustering procedure for finding clusters C = (Ci,...,Ck) in this dataset. We present various indices to compare two partitions C = (C1,..., Ck) and

C+ = (C+,...,C++).

Important classes of criteria for comparing clustering solution are based on counting the pairs of points on which two clustering agree/disagree. The best-known clustering distances based on point pairs is the F-measure [26].

F-measure. If we want to compare a clusters C to a classes C+, a simple approach would be to calculate the precision (P), recall (R) and the F-measure, used widely in the information retrieval literature to measure the success of the retrieval task.

The F value of the cluster Cp and the class C++ is just the harmonic mean of the precision and the recall [26]:

F (CP,C})= V \J \-P-L. (18)

Precision is calculated as the portion of cluster Cp that is the documents of class C++, thus measuring how homogenous cluster C is with respect to class C +:

/ , \ Cp fl Cp+ / s

|Cp |

Similarly, recall is calculated as the portion of documents from class C++ that are present in cluster Cp, thus measuring how complete cluster Cp is with respect to class C++ :

/ , \ Cp fl C++ / s

R(Cp,C++) =-(20)

v p J C++

The F-measure of cluster Cp is the maximum F value attained at any class in the entire classes C+ = (C+,..., C++) . That is,

F (Cp )= max F(Cp,Cp++), p =1, 2,...,k. (21)

The F-measure of the entire collection is defined to be the sum of the individual cluster specific F-measures weighted according to the cluster size. That is,

F(c) = ^K^F(c,p)_ (22)

n

p=i

In general, the higher the F-measure values, the best clustering solution is.

Now we propose information-based methods to compare partitions C = (C1,..., Ck) and C+ = (C+,..., C++). The commonly used external validity indices based on information are partition coefficient, and variation of information. These measures represent plausible ways to evaluate the homogeneity of a clustering solution.

Partition Coefficient. The partition coefficient (PC) was introduced by Bezdek [27]. The partition coefficient measures the amount of overlap between clusters. Considering a

cluster Cp, the PC defines as follows:

k+ / PC (CP)= £

p+=i \

Cp n C++

|Cp|

(23)

1

PC (Cp) is a value between — and 1. If almost all documents of Cp belong to a same cluster in C++, then PC (Cp) is close to 1. Now, if documents of Cp are randomly divided into all

clusters of C+ then PC (Cp) is close to —. A global partition coefficient is computed:

k

PC (C,C+) =-J]PC(Cp)

1

p=i

k k+ ? I

p=ip+=i \

( CpQ^ |Cp|

(24)

1

1

PC (C, C+) is also a value between —— and 1. Now, if PC (C, C+) is close to —, C and

v y v y k+

C+ are almost independent. Moreover, if PC (C, C+) is close to 1, then C is close to C+.

Variation of Information. Another information-based clustering measure is variation of information (VI ) [28]:

1

k k+

p=ip+=i

|Cpl|C++l \cPnc+f

(25)

The maximum value of the variation of information is log n, which is achieved when the partitions are as far apart as possible, which in this case means that one of them places all the documents together in a single cluster while the other places each document in a cluster on its own. The maximum value increases with n because larger datasets contain more information, but if this property is undesirable one can simply normalize by log n, as we do in the calculations presented here:

1 k k+ / 6 p=ip+=i \

|Cpl|C++l

|cPnc+r

(26)

0=1p+ = i \ I p ' ' ~ p

In general, the smaller the variation of information values, the best clustering solution is.

3. Experiments

For our experiments we have used subset the WebKb dataset, as it is currently the most widely used benchmark in document clustering research. The subset contains 3.907 pages were manually classified into 7 categories. This dataset is available from [29].

In all the experiments discussed in this paper, stopwords were removed using the stoplist provided in [30] and the terms were stemmed using Porter's scheme [31], which is a commonly used algorithm for word stemming in English.

2

2

3.1. Simulation Strategy and Parameters

The optimization procedure used here is stochastic in nature. Hence, for each criterion function it has been run several times. The results reported in this section are averages over 50 runs for each criterion functions. Finally, we would like to point out that algorithm was developed from scratch in Delphi 7 platform on a Pentium Dual CPU, 1.6 GHz PC, with 512 kB cache, and 1 GB of main memory in Windows XP environment. For establishment of DE parameters the WebKb dataset has been broken in two parts: train and test. The train dataset contains 979 documents and the test dataset contains 2928 documents. After training of DE on train dataset for parameters following values are established: N (population size) = 1000, tmax (number of fitness evaluation) = 500, Xcr (the scaling factor for crossover) = 0.9, prc(the crossover constant) = 0.8, Xmt (the scaling factor for mutation) = 0.7, prm(the mutation constant) = 0.6. After setting the parameters of differential evolution experiments have been spent on test dataset.

3.2. Results and Analysis

We report experiments comparing our weighted versions of clustering algorithms with their unweighted versions. Table gives the comparative analysis of the results of weighted and unweighted criterion functions. As can be seen from Table 1 the weighted criterion functions produced the best results than unweighted criterion functions. Highlighted entries represent the best performing criterion function in terms of average validity indices. In brackets it is shown improvement in percentage. Here, we've used relative improvement

(■weighted method — unweighted method) unweighted method

for the indexes CSi, CS2, CS3, F-measure, and PC. For the index VI we've used relative improvement

(■unweighted method — weighted method)

weighted method

x 100.

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Comparison the weighted and unweighted criterion functions

Criterion Validity indices

functions CSi CS2 cs3 F-measure VI PC

Ff 7.7527 9.9513 6.8687 0.8294 0.1267 0.8040

(66.55%) (98.94 %) (53.57%) (1.53%) (3%) (8.99 %)

F 4.6548 5.0021 4.4726 0.8169 0.1305 0.7377

spa 2 0.0011 0.0004 0.4667 0.6835 0.3136 0.6533

(57.14%) (100%) (7.88%) (17.80%) (5.26%) (20.42 %)

F2(k-means) 0.0007 0.0002 0.4326 0.5802 0.3301 0.5425

The experiments presented in this subsection showed interesting trends. The quality of the solutions produced by some seemingly similar criterion functions are often substantially different. For instance, both Ff and F^ find clusters by maximizing a particular within cluster similarity function. However, F^ performs substantially better than F2a. This is also true for unweighted criterion functions Fi and F2. Finally, from the analysis it is possible to draw a conclusion, that internal validity indexes are more sensitive to weighing (assigning of weight to document) than external validity indexes.

Conclusion

In our study a simple weighed approach for document clustering is proposed. In this approach to each document assigned the weight defining its relative positions in a collection of documents. Experiments have shown that such approach improves clustering solution. To be convinced of it we compared the performance of the k-means method with the weighted variant of the k-means method proposed by us. The experimental results were shown that our criterion function outperforms the k-means method. For estimation of clustering solution have been used six validity indexes, three from them are internal validity indexes, other three — external validity indexes. In this paper to optimize the criterion functions we developed a modified DE algorithm. The proposed modification prevents the occurrence of infeasible solution in the work process of DE. Acknowledgment

Author acknowledges the anonymous referee for fruitful comments, which did improve quality of the paper.

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Received for publication 24 February 2009