Ensemble Clustering of High Dimensional Data with FastMap Projection

  • Imran Khan
  • Joshua Zhexue Huang
  • Nguyen Thanh Tung
  • Graham Williams
Conference paper

DOI: 10.1007/978-3-319-13186-3_43

Part of the Lecture Notes in Computer Science book series (LNCS, volume 8643)
Cite this paper as:
Khan I., Huang J.Z., Tung N.T., Williams G. (2014) Ensemble Clustering of High Dimensional Data with FastMap Projection. In: Peng WC. et al. (eds) Trends and Applications in Knowledge Discovery and Data Mining. PAKDD 2014. Lecture Notes in Computer Science, vol 8643. Springer, Cham

Abstract

In this paper, we propose an ensemble clustering method for high dimensional data which uses FastMap projection to generate subspace component data sets. In comparison with popular random sampling and random projection, FastMap projection preserves the clustering structure of the original data in the component data sets so that the performance of ensemble clustering is improved significantly. We present two methods to measure preservation of clustering structure of generated component data sets. The comparison results have shown that FastMap preserved the clustering structure better than random sampling and random projection. Experiments on three real data sets were conducted with three data generation methods and three consensus functions. The results have shown that the ensemble clustering with FastMap projection outperformed the ensemble clusterings with random sampling and random projection.

Keywords

Ensemble clustering FastMap Random sampling Random projection Consensus function 

1 Introduction

The emergence of new application domains results in very high dimensional big data such as text data, microarray data and smart phone user behavior data. Such high dimensional data with thousands of features present a big challenge to current data mining techniques [10]. Curse of dimensionality and sparsity are two main problems, among others, that handicap many clustering algorithms to find strongly cohesive clusters from very high dimensional big data.

Ensemble clustering [14, 15] is a new approach for clustering that integrates results of different clusterings from the same original data generated by different clustering algorithms or from different data sets sampled from the original data. The ensemble clustering can have a higher accuracy than individual component clustering results. A large number of research results have accelerated this field [6]. Different ensemble clustering methods have been proposed to ensemble different types of clustering results like clustering results of one algorithm from the same data with different parameters initialization [13], results from the same data by using different algorithms [1], results of multiple component data sets from the same data set [8]. Subspace ensemble clustering has become a useful strategy to find robust clusters from such sparse and high dimensional data.

In high dimensional data, clusters often exist in different subspaces. Ensemble clustering based on full space clustering algorithms fail to cluster such data. The innovation of subspace ensemble clustering techniques is promised to resolve this problem. Recently, two methods for generating low dimensional component data have been used to resolve the problem of subspace ensemble clustering of high dimensional data clustering. One method generates low dimensional data by randomly sampling different features. The other method generates low dimensional component data by using a random projection matrix to project the original high dimensional data onto a low dimensional space. We call the former random sampling method and the latter random projection method. In recent days, different flavours of random projection are available [11, 16] but for ensemble clustering former random projection has been used [4]. Both, random sampling and random projection benefit ensemble clustering for high dimensional sparse data. However, the drawback of these methods is that they cannot well preserve the clustering structure of the original data in their generated low dimensional component data, which increases discrepancy of clustering structures in component data sets, thus affecting the performance of ensemble clustering for high dimensional data.

In this paper, we present a new low dimensional component data generation method by FastMap [5], an algorithm that is used to generate low dimensional transformation of high dimensional data. Given a distance matrix of \(N\) objects, FastMap uses the well known Cosine Law to compute the coordinates of the \(N\) objects that are projected to the line of two pivot objects selected from the data set. By removing the distance component from the new generated dimension, a new set of coordinates are computed. This process repeats until \(k\) dimensional representation of the \(N\) objects is obtained. The advantage of FastMap projection in comparison with random sampling and random projection is that it can better preserve the clustering structure of the original data in its generated component data sets. Thus, the performance of ensemble clustering is improved significantly.

We propose two methods to measure preservation of clustering structure of original data in the generated component data sets. We used three real world data sets to analyze preservations by random sampling, random projection and FastMap projection. The comparison results have shown that FastMap preserved clustering structure better than other two methods. We also used the three data sets to conduct ensemble clustering experiments with three component data generation methods and three consensus functions to ensemble clustering results. \(k\)-means algorithm was used to generate component clustering. The results have shown that the ensemble clustering with FastMap projection outperformed the ensemble clusterings with random sampling and random projection on all three data sets. The overall performance of FastMap was the best among the three methods.

2 Framework for Subspace Ensemble Clustering

Ensemble clustering of a data set \(X\) is a process to integrate multiple clustering results produced by one or more clustering algorithms from component data sets sampled from X into a single clustering of \(X\) with a result that is usually much better than the results of individual clusterings on \(X\) [15]. The subspace ensemble clustering framework consists of the following steps.
  • \(\mathbf{Step 1:}\) Generate \(K\) different component data sets {\(C_1,C_2\), \(\cdot \)\(\cdot \)\(\cdot \) ,\(C_K\)} from \(X\) using a component generation method.

  • \(\mathbf{Step 2:}\) Cluster the \(K\) component data sets to produce \(K\) component clusterings {\(\pi _1\), \(\pi _2\), \(\cdot \)\(\cdot \)\(\cdot \), \(\pi _k\)} independently using one or more clustering algorithms.

  • \(\mathbf{Step 3:}\) Ensemble \(K\) component clusterings into a single clustering \(\pi \) using an ensemble method called a consensus function.

Figure 1 shows a generic framework of ensemble clustering.
Fig. 1.

Generic framework of ensemble component clustering.

2.1 Subspace Component Generation

In ensemble clustering of high dimensional data, we are interested to generating low dimensional component data sets that can better preserve the clustering structure of the original data so as to improve the performance of ensemble clustering on high dimensional data. Currently, random projection and random sampling are two widely used methods for low dimensional component data generation. We review these two method briefly below.

Random Projection. The random projection method projects high dimensional data into low dimensional space by using a random matrix [3]. A data set \(X_{N\times m}\) with \(m\) dimensions and \(N\) objects is projected into a low dimensional subspace data \(Y_{N\times p}\) with p\(\ll \)m, via,
$$\begin{aligned} Y_{N \times p} = X_{N \times m} \times R_{m\times p} \end{aligned}$$
(1)
Computationally, random projection cost is O(p \(\times \) m \(\times \) N). The value of each element \(r_{ij}\) of matrix \(R\) should follow Gaussian distribution.
$$\begin{aligned} r_{ij} = \sqrt{3} \left\{ \begin{array}{ll} +1 &{} with \,\,\,\,\, probability \,\,\,\,\, \frac{1}{6} \\ \,\,\,\, 0 &{} with \,\,\,\,\, probability \,\,\,\,\, \frac{2}{3} \\ -1 &{} with \,\,\,\,\, probability \,\,\,\,\, \frac{1}{6} \end{array} \right. \end{aligned}$$
(2)
Random Sampling. A number of methods have been proposed to generate low dimensional component data sets by randomly selection of features. These methods differ in the way of selection of dimensions from the original high dimensional data set. A commonly used random sampling is to use a threshold on a feature or set of features, and a feature is selected to add in the component data set if the corresponding value exceeds the threshold value. The features can be selected randomly by using any of the following proposed methods like Gini index, Quinlan’s information gain ratio or Mingers’s G statistic. In our experiments, we have used Quinlan’s information gain ratio for features extraction.

2.2 Component Data Clustering

Any clustering algorithm can be used to cluster a low dimensional component data. Popular clustering algorithms are \(k\)-means, subspace \(k\)-means and hierarchical clustering methods. The advantage of the \(k\)-means type algorithms is its efficiency in handling large data. In this work, we used \(k\)-means. Quite often, different clustering algorithms were used to generate different component clustering results for ensemble clustering. However, there is no clear guidance how the different clustering algorithms should be used. In practice, it is more convenient to use one clustering algorithm for ensemble clustering, rather than multiple clustering algorithms.

2.3 Ensemble Component Clusterings

An ensemble method is used to ensemble multiple component clusterings from different component data sets into a single clustering as the final clustering result. In ensemble clustering, ensemble method is also called consensus function. Several consensus functions have been proposed with different strategies and methods to ensemble component clustering results. Below, we briefly review three ensemble methods that were used in this work.

Similarity-Based Consensus Function. A clustering signifies a relationship between objects in the same cluster and can thus be used to establish a measure of pairwise similarity [15]. A similarity matrix for each component clustering is constructed. In the similarity matrix, the element indexed two objects in the same cluster is assigned value 1, otherwise, the element has value 0 if the two objects are in different clusters. After computation of \(K\) similarity matrices, a final matrix is obtained as the average of corresponding cells of all similarity matrices. The METIS algorithm [9] is then used to resultant similarity matrix to get final clustering ensemble.

Hyper Graph-Based Consensus Function. In Hyper Graph-based Consensus Function (HGPA), an ensemble problem is formulated as partitioning the hypergraph by cutting a minimal number of hyperedges [15]. The hyper graph is constructed by considering objects of a data set \(X\) as \(N\) vertices, and hyper-edges with the same weight are used to connect a set of vertices by using K component clusterings. The algorithm HMETIS [12] is used to partition the hyper-graph into unconnected components by cutting a minimum number of hyper-edges.

Meta Cluster-Based Consensus Function. This method was introduced in the meta-clustering algorithm (MCLA) [15]. Similar to HGPA, a hyper graph is constructed by considering clusters of component clusterings as vertices and edges are used to connect these vertices. The weight between two vertices (clusters) is computed by using the following binary Jaccard distance equation.
$$\begin{aligned} \mathrm J acSim (C_x,C_y) = \frac{{C^{'}_{x}} C_{y}}{ \Vert C_{x} \Vert _{2}^{2} + \Vert C_{y} \Vert _{2}^{2} - {C^{'}_{x}} C_{y} } \end{aligned}$$
(3)
where \(C^{'}_x\) and \(C_y\) are two vectors of \(N\) elements representing two clusters where \(N\) is the number of objects in the data set \(X\). Each element in the vector corresponds to one object. If the cluster contains the object, the corresponding element is assigned to 1. Otherwise, the element is 0. \(C^{'}_x\) is the transpose of \(C_x\). METIS is used to partition this hyper graph to identify \(K\) meta-clusters. Finally, a voting method is used to assign each data point to its most associated meta-cluster.

3 FastMap Projection for Component Data Generation

3.1 FastMap Projection

FastMap [5] is an efficient algorithm to generate \(k\) dimensional coordinates of \(N\) objects from a dissimilarity matrix of the \(N\) objects. Given a high dimensional data \(X\) of \(m\) dimensions and \(N\) objects, we use a distance function to compute the distance matrix \(S_{N\times N}\). We select two objects \(O_{a}\) and \(O_{b}\) with a large distance as pivot objects and take the straight line passing the two objects as the projection axis of the first dimension coordinate \(F_1\). The coordinate of object \(O_i\) in the first dimension is computed by using the following cosine equation.
$$\begin{aligned} \mathrm x _{i} = \frac{ d_{a,i}^{2} + d_{a,b}^{2} - d_{b,i}^{2}}{d_{a,b}} \end{aligned}$$
(4)
where \(d_{a,b}\) is the distance between the pivot objects \(O_a\) and \(O_b\), \(d_{a,i}\) is the distance between the pivot object \(O_a\) and object \(O_i\) and \(d_{b,i}\) is the distance between the pivot object \(O_b\) and object \(O_i\).
After the coordinates of all \(N\) objects are computed, we compute the reduced distance matrix \(S^{'}\) of \(N\) objects according to Lemma 1 in [5] as
$$\begin{aligned} \mathrm d ^{'}(O_i, O_j)^2 = d(O_i, O_j)^2 - (x_i - x_j)^2 \end{aligned}$$
(5)
where \(d^{'}\) is the reduced distance in \(S^{'}_{N\times N}\), \(d\) is the distance in \(S_{N\times N}\) and the last term on the right is the squared distance in the new dimension.

Given \(S^{'}_{N\times N}\), we can choose a new pair of pivot objects and use (4) to compute the coordinates of the second dimension. We repeat this process \(k\) times to generate a \(k\) dimensional data of \(X\).

We can also use Principal Component Analysis (PCA) to generate a low dimensional data of \(X\). However, given \(X\), we can only use PCA to generate one low dimensional data, thus not suitable for ensemble clustering which requires multiple component data sets. Using FastMap, we can use a random process to select different pairs of pivot objects to produce different projections of data as component data sets. Another advantage of FastMap is that it is efficient to handling large data.

3.2 Evaluation of Component Data Generation

In this section, we present two methods to evaluate component data generation for ensemble clustering, i.e., preservation of clustering structure of the original data in generated component data sets.

Intrinsic Dimensionality. Given a high dimensional data set \(X_{N\times m}\) of \(m\) dimensions and \(N\) objects, we use a component data generation function \(\varPhi (X,\theta )\) to generate a subspace data \(Y_{N\times p}\), i.e., \(\varPhi (X,\theta )=Y_{N\times p}\) where \(\theta \) are input parameters to produce different \(Y\)s from \(X\). Let \(\mathbf Y = \{Y_1,\dots ,Y_L\}\) be a set of \(L\) component data sets all in \(p\) dimensions and \(\mathbf D = \{D_1,\dots ,D_L\}\) the set of \(L\) distance matrices computed from \(\mathbf Y \). Given a distance matrix \(D_i\), we take the upper half mutual distances of \(D_i\) and plot the histogram of the mutual distances. Large mean and small variance of the histogram distribution of \(D_i\) represent a problem of curse of dimensionality. We use intrinsic dimensionality to measure curse of dimensionality of a data set as in [2].

Definition 1

The intrinsic dimensionality of a data set in a metric space is defined as \(\rho \) = \(\mu ^2\)/2\(\sigma ^2\) where \(\mu \) and \(\sigma \) are the mean and variance of its histogram of distances.

We use intrinsic dimensionality \(\rho \) to evaluate a method \(\varPhi (X,\theta )\). For each component \(Y_i\), we compute \(\rho _i\). Then, we compute the average \(\bar{\rho }\) of \(\rho _i\) of \(L\) component data sets in \(\mathbf Y \). The smaller \(\bar{\rho }\) the better the method \(\varPhi (X,\theta )\).

Distance Preservation. Let \(D\) be the distance matrix of high dimensional data \(X_{N\times m}\) and \(\{D_1,\dots ,D_L\}\) the set of \(L\) distance matrices computed from \(\mathbf Y \). Given \(D\) and \(D_i\) from \(Y_i\), we compute
$$\begin{aligned} Dist = \sqrt{\frac{\sum _{i=1,j=1}^{N}(d_{s}(o_{i},o_{j}) - d_{o}(o_{i},o_{j}))^{2}}{\sum _{i=1,j=1}^{N}d_{o}^{2}(o_{i},o_{j}) } } \end{aligned}$$
(6)
where \(N\) is the total number of objects in the data sets, \(d_{o}\) is the distance between two objects in data set \(X_{N\times m}\) and \(d_{s}\) is the distance between corresponding two objects in the subspace component data set \(Y_i\). \(Dist\) is a measure of distance preservation of component data set \(Y\) generated from \(X\). The smaller \(Dist\), the better the component data set \(Y\) in preserving the mutual distances of objects in \(X\). Dasgupta’s results [7] have shown that random projection preserves the separation among Gaussian clusters having large variances. Random projection can change the shape of highly eccentric clusters to more spherical shape [7].

4 Experiments

In this section, we present experiments on real world data sets to evaluate the performance of ensemble clusterings with FastMap projection. The FastMap results are compared with the results of ensemble clustering with random sampling and random projection. Three high dimensional data sets in different application domains were selected from the available Web sites of UCI machine learning repository and feature selection at Arizona state university. The characteristics of these data sets are listed in Table 1.
Table 1.

Real world data sets.

Data sets

\(\#\)Instances

\(\#\)Features

Source

\(\#\)Classes

Internet Ad

1000

1558

Multivariate

02

GLI-85

85

22283

Microarray

02

Orlraws10P

100

10304

Image

10

4.1 Experiment Settings

For each data set, we used three methods, random sampling (RS), random projection (RP) and FastMap projection (FP) to generate component data sets. We used the \(k\)-means clustering algorithm to cluster each component data set. The number of clusters \(k\) was given as the number of classes in the data set. For ensemble clustering, we used the three consensus functions discussed in Sect. 3, i.e., hyper graph based consensus function (HGPA), similarity-based consensus function (CSPA) and meta cluster-based consensus function (MCLA). By combining the three component data generation methods and the three consensus functions, we produced 9 ensemble clustering results from each data set. We denote these 9 ensemble clustering methods as RS-CSPA, RP-CSPA, FM-CSPA, RS-MCLA, RP-MCLA, FM-MCLA, RS-HGPA, RP-HGPA and FM-HGPA respectively. We have also shown the ensemble clustering results by using \(k\)-means (KM) upon original data.

In conducting the experiments for comparisons, the component data sets from the same data set were generated with the same number of dimensions by each component generation method. Each ensemble clustering was produced from 10 component clusterings which were produced with the \(k\)-means algorithm.

4.2 Evaluation Methods

We used four evaluation methods to evaluate the results of ensemble clustering with the 9 ensemble clustering methods. They were one unsupervised method and three supervised methods given below.

The unsupervised evaluation method is cluster compactness (CP) calculated as
$$\begin{aligned} CP = \frac{1}{n} \sum _{x=1}^{k} n_x \Bigg (\frac{\sum _{o_i,o_j \in C_x} d(o_i,o_j)}{n_x(n_x-1/2)} \Bigg ) \end{aligned}$$
(7)
where \(d(o_i,o_j)\) is the distance between the objects \(o_i\) and \(o_j\) in a cluster \(C_x\), \(n_x\) is the number objects in a cluster \(C_x\). The smaller the value of CP, the better the clustering result.
The three supervised evaluation methods are normalized mutual information (NMI), adjusted rand index (ARI) and clustering accuracy (CA), calculated as follows.
$$\begin{aligned} CA&= \frac{1}{n} \sum _{x=1}^{k} max_y n_{x,y} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \nonumber \\ ARI&= \frac{\sum _{x=1}^{k} \sum _{y=1}^{k} \left( {\begin{array}{c}n_{x,y}\\ 2\end{array}}\right) - s_3}{\frac{1}{2} (s_1+s_2)-s_3} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\\ NMI&= \frac{\sum _{x=1}^{k} \sum _{y=1}^{k} n_{x,y} log \frac{nn_{x,y}}{n_x n_y} }{\sqrt{ \sum _{x=1}^{k} n_x log\frac{n_x}{n} \sum _{y=1}^{k} n_y log\frac{n_y}{n}}} \nonumber \end{aligned}$$
(8)
where \(n_x\) and \(n_y\) are the total number of objects in cluster \(x\) and class \(y\) respectively, \(n_{x,y}\) is the total number of objects in cluster \(x\) and class \(y\), \(n\) is the total number of objects in the given data set, \(s_1 = \sum _{x=1}^{k} \left( {\begin{array}{c}n_{x}\\ 2\end{array}}\right) \), \(s_2 = \sum _{y=1}^{k} \left( {\begin{array}{c}n_{y}\\ 2\end{array}}\right) \) and \(s_3 = \frac{2s_1s_2}{n(n-1)}\). The larger the values of these measures, the better the clustering result.

4.3 Experimental Results

Table 2 lists the ensemble clustering results of three data sets produced with 9 ensemble clustering methods. These results were evaluated with 4 evaluation methods. The best results were marked in bold font. We can see that under clustering accuracy evaluation, the ensemble clustering with FastMap projection performed best in all data sets compared with the ensemble clusterings with other two methods with the same consensus function. In all 9 ensemble clustering methods, the highest clustering accuracy was obtained in all data sets by the ensemble clustering with FastMap projection. The improvements in data sets GLI85 and Internet Advertisements were significant compared with other two methods.
Table 2.

Clustering result comparison.

Methods

Internet Ad

GLI-85

Orlraws10P

-

CP

NMI

ARI

CA

CP

NMI

ARI

CA

CP

NMI

ARI

CA

RS-CSPA

171.5

0.01

0.50

0.54

561570

0.45

0.48

0.67

3711

0.67

0.81

0.78

RP-CSPA

163.1

0.51

0.53

0.62

542341

0.61

0.69

0.69

3686

0.69

0.83

0.80

FM-CSPA

145.9

0.55

0.61

0.74

529449

0.58

0.49

0.74

3011

0.71

0.84

0.85

RS-HGPA

271.1

0.01

0.49

0.53

581715

0.14

0.47

0.67

3616

0.69

0.83

0.80

RP-HGPA

202.3

0.24

0.49

0.52

567975

0.25

0.48

0.67

3624

0.70

0.84

0.81

FM-HGPA

202.3

0.68

0.50

0.54

525676

0.29

0.49

0.69

3523

0.72

0.85

0.82

RS-MLCA

171.5

0.02

0.50

0.54

570900

0.31

0.48

0.67

3619

0.72

0.85

0.82

RP-MLCA

184.0

0.62

0.54

0.65

547288

0.68

0.48

0.68

3608

0.73

0.86

0.83

FM-MLCA

143.3

0.59

0.61

0.77

522337

0.50

0.49

0.71

3575

0.75

0.87

0.84

KM-CSPA

170.6

0.47

0.56

0.68

614645

0.46

0.47

0.70

3798

0.59

0.81

0.70

KM-HGPA

202.3

0.47

0.49

0.54

645685

0.27

0.48

0.69

3705

0.65

0.84

0.77

KM-MCLA

149.9

0.48

0.62

0.71

608945

0.47

0.49

0.69

3755

0.61

0.85

0.71

The results of ensemble clustering with FastMap projection were also better than those with other two methods under ARI evaluation in all data sets. The majority best results were also obtained with FastMap method, except for one case of GLI-85. However, the difference was not very significant.

Under CP and NMI evaluations, the ensemble clustering with FastMap projection also outperformed the ensemble clusterings with other two methods in most data sets. The majority best results also occurred in the FastMap method. These results demonstrated that the FastMap projection for component data set generation improved the performance of ensemble clustering of high dimensional data.

4.4 Comparisons of FastMap Projection vs. Random Sampling and Random Projection

We used the three component data generation methods: random sampling, random projection and FastMap projection to generate component data sets from 3 real world data sets. The characteristics of these data sets are presented in Table 1 in the next section. We computed two measures on the component data sets. Table 3 shows the comparisons of three methods in intrinsic dimensionality. The component data sets are in three different dimensions. We can see that the intrinsic dimensionality values in FM columns are smaller than the values in RP and RS columns. These results indicate that FastMap can generate better component data sets than random sampling and random projection. One exception is data set \(\mathbf{{GLI85}}\) which is a fat Microarray data with only 85 objects but 22283 features. Although FastMap projection was a little worse than random projection in intrinsic dimensionality, we shown in the previous section that the ensemble clustering results of FastMap projection in this data set are still better than the results from other two methods.
Table 3.

Intrinsic dimensionality.

Data sets

5-dimensional space

10-dimensional space

15-dimensional space

-

FM

RP

RS

FM

RP

RS

FM

RP

RS

Internet Ad

3E-05

5E-05

0.526

2E-05

4E-05

0.485

1E-05

4E-04

0.390

GLI85

5E-11

2E-11

5E-09

5E-11

1E-11

2E-09

9E-11

1E-10

1E-10

Orlraws10P

1E-06

5E-05

1E-03

2E-06

9E-05

8E-03

1E-06

1E-04

7E-03

Table 4 shows the comparisons of three methods in distance preservation. The values of distance preservation in FM columns are much smaller than those in RP and RS columns. These results demonstrate that FastMap projection has better distance preservations than other two methods.
Table 4.

Distance preservation.

Data sets

5-dimensional space

10-dimensional space

15-dimensional space

-

FM

RP

RS

FM

RP

RS

FM

RP

RS

Internet Ad

0.681

0.739

1.812

0.690

0.901

1.001

0.671

0.719

1.009

GLI85

0.596

0.639

0.960

0.539

0.590

0.921

0.447

0.46

0.931

Orlraws10P

0.479

0.519

1.238

0.394

0.469

1.108

0.359

0.493

0.998

5 Conclusions

In this paper, we have presented the FastMap projection method to generate low dimensional component data sets for ensemble clustering. We have analyzed FastMap projection, random sampling and random projection and demonstrated that FastMap projection can better preserve the clustering structure of the original data than other two methods. Because of this property, the ensemble clustering with FastMap projection outperformed ensemble clusterings with other two methods in experiments on three real world high dimensional data sets. Beside better performance, another advantage of FastMap is that it is efficient in handling big data and flexible in component data generation.

Acknowledgment

This research is supported by Shenzhen New Industry Development Fund under Grant No. JC201005270342A.

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Imran Khan
    • 1
  • Joshua Zhexue Huang
    • 1
    • 2
  • Nguyen Thanh Tung
    • 1
  • Graham Williams
    • 1
  1. 1.Shenzhen Key Laboratory of High Performance Data Mining, Shenzhen Institutes of Advanced TechnologyChinese Academy of SciencesShenzhenChina
  2. 2.College of Computer Science and Software EngineeringShenzhen UniversityShenzhenChina

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