Votingbased consensus clustering for combining multiple clusterings of chemical structures
 Faisal Saeed^{1, 2}Email author,
 Naomie Salim^{1} and
 Ammar Abdo^{3, 4}
DOI: 10.1186/17582946437
© Saeed et al.; licensee Chemistry Central Ltd. 2012
Received: 24 September 2012
Accepted: 11 December 2012
Published: 17 December 2012
Abstract
Background
Although many consensus clustering methods have been successfully used for combining multiple classifiers in many areas such as machine learning, applied statistics, pattern recognition and bioinformatics, few consensus clustering methods have been applied for combining multiple clusterings of chemical structures. It is known that any individual clustering method will not always give the best results for all types of applications. So, in this paper, three voting and graphbased consensus clusterings were used for combining multiple clusterings of chemical structures to enhance the ability of separating biologically active molecules from inactive ones in each cluster.
Results
The cumulative votingbased aggregation algorithm (CVAA), clusterbased similarity partitioning algorithm (CSPA) and hypergraph partitioning algorithm (HGPA) were examined. The Fmeasure and Quality Partition Index method (QPI) were used to evaluate the clusterings and the results were compared to the Ward’s clustering method. The MDL Drug Data Report (MDDR) dataset was used for experiments and was represented by two 2D fingerprints, ALOGP and ECFP_4. The performance of votingbased consensus clustering method outperformed the Ward’s method using Fmeasure and QPI method for both ALOGP and ECFP_4 fingerprints, while the graphbased consensus clustering methods outperformed the Ward’s method only for ALOGP using QPI. The Jaccard and Euclidean distance measures were the methods of choice to generate the ensembles, which give the highest values for both criteria.
Conclusions
The results of the experiments show that consensus clustering methods can improve the effectiveness of chemical structures clusterings. The cumulative votingbased aggregation algorithm (CVAA) was the method of choice among consensus clustering methods.
Background
Chemoinformatics, as defined by Brown [1], is the collection, representation and organisation of chemical data in order to create chemical information, which is applied to create chemical knowledge. It has been used for the process of drug discovery and design, especially in the lead identification and optimisation process, which is known as HighThroughput Screening (HTS).
According to Brown and Martin [2], the advent of highthroughput biological screening methods has given pharmaceutical companies the ability to screen many thousands of compounds in a short time. However, there are many hundreds of thousands of compounds available both inhouse and from commercial vendors. Whilst it may be feasible to screen many, or all, of the compounds available, this is undesirable for reasons of cost and time and may be unnecessary if it results in the production of some redundant information. Therefore, there has been a great deal of interest in the use of compound clustering techniques to aid in the selection of a representative subset of all the compounds available [3]. Given a clustering method, which can group structurally similar compounds together, and application of the similar property principle[4], which states that structurally similar molecules will exhibit similar physicochemical and biological properties, implies that the selection, or synthesis, and testing of representatives from each cluster produced from a set of compounds should be sufficient to understand the structureactivity relationships of the whole set, without the need to test them all. An appropriate clustering method will, ideally, cluster all similar compounds together whilst separating active and inactive compounds into different sets of clusters [2].
The main objective of clustering is to subdivide data objects into smaller groups known as clusters so that each group exhibits a high degree of intracluster similarity and intercluster dissimilarity [5]. Many different types of clustering techniques for chemical structures have been used in the literature [6–13].
Brown and Martin [2] considered the Ward’s clustering to be the most efficient method in clusterbased compound selection. However, as it is known, there is no clustering method capable of correctly finding the underlying structure for all data sets. So, the idea of combining different clustering results (consensus clustering) is considered as an alternative approach for improving the quality of the individual clustering algorithms [14].
Consensus clustering involves two main steps: (i) partitions generation and (ii) consensus function. In the first step, many partitions will be generated (the collection of partitions is called ensemble). There are no constraints about how the partitions must be generated. In the partitions generation step, many mechanisms can be applied including the using of: (i) different data representations, (ii) different individual clustering methods, (iii) different parameters initialisation for clustering methods and (iv) data resampling. In the second step, there are two main approaches, i.e. the objects cooccurrencebased and the median partitionbased approaches. Voting and graph based consensus clusterings are widely used for the first approach.

Robustness: The combination process must have better average performance than the single clustering algorithms.

Consistency: The result of the combination should be somehow, very similar to all combined single clustering algorithm results.

Novelty: Cluster ensembles must allow finding solutions unattainable by single clustering algorithms.

Stability: Results with lower sensitivity to noise and outliers.
In chemoinformatics, it is most unlikely that any single method will yield the best classification under all circumstances, even if attention is restricted to a single type of application [17]. Furthermore, the consensus scoring (data fusion) has been successfully used in chemoinformatics and, in particular, for virtual screening [18–25].
Over the last few years, data fusion has become accepted as a simple way of enhancing the performance of existing systems for ligandbased virtual screening by combining the results of two or more screening methods. In some cases, the fused search may even be better than the best individual screening method when averaged over large numbers of searches [20].
Chu et al.[17] used consensus clustering methods on sets of chemical compounds represented by 2D fingerprints (ECFP_4), and concluded that consensus methods can outperform the Ward's method, the current standard clustering method for chemoinformatics applications. However, based on the implemented methods, it was not the case if the clustering is restricted to a single consensus method. In this paper, we examined the use of voting and graphbased consensus clustering methods for combining multiple clusterings of chemical structures with different distance measures in order to improve the effectiveness of chemical structures clustering.
Experimental
Molecular fingerprints
For the clustering experiments, two molecular fingerprints were developed by Scitegic’s Pipeline Pilot software [26]. The first one was 120bit ALOGP, which includes octanolwater partitioning coefficient based on Ghose and Crippen’s method [27, 28]. ALOGP atom type code is generated based on the molecular hydrophobicity (lipophilicity), usually quantified as log P (the logarithm of 1octanol/water partition coefficient), which is an important molecular characteristic in drug discovery [28]. The second descriptor was the Scitegic extendedconnectivity fingerprints (1024 ECFP_4). The first character E in the fingerprint name denotes the atom abstraction method used to assign initial atom code which is derived from the number of connections to an atom, the element type, the charge and the atomic mass [29].
Dataset
MDDR dataset activity classes
Activity Index  Activity class  Active molecules  Pairwise similarity 

Mean  
31420  Renin Inhibitors  1130  0.290 
71523  HIV Protease Inhibitors  750  0.198 
37110  Thrombin Inhibitors  803  0.180 
31432  Angiotensin II AT1 Antagonists  943  0.229 
42731  Substance P Antagonists  1246  0.149 
06233  Substance P Antagonists  752  0.140 
06245  5HT Reuptake Inhibitors  359  0.122 
07701  D2 Antagonists  395  0.138 
06235  5HT1A Agonists  827  0.133 
78374  Protein Kinase C Inhibitors  453  0.120 
78331  Cyclooxygenase Inhibitors  636  0.108 
Ensemble generation
Every consensus clustering method is made up of two steps: (i) partitions generation and (ii) consensus functions. For the purposes of this paper, the partitions were generated by a single run of multiple individual clustering algorithms (singlelinkage, complete linkage, average linkage, weighted average distance, Ward and Kmeans methods). Every individual clustering used six distance measures in order to generate different ensembles. The thresholds of 500, 600, 700, 800, 900 and 1000 were used to generate partitions with different sizes (number of clusters). The same process was done for each 2D fingerprint in order to study the effectiveness of consensus clusterings on different molecular representations. The distance measures that were used with each clustering technique were Correlation, Cosine, Euclidean, Hamming, Jaccard and Manhattan.
Methods
Graphbased consensus clustering
Two graphbased consensus clustering algorithms, proposed by Strehl and Gosh [34], were used to obtain the consensus partition from ensembles generated in the previous step. The two algorithms were developed based on transforming the set of clusterings into a hypergraph representation. The first algorithm is Clusterbased Similarity Partitioning Algorithm (CSPA) in which a clustering signifies a relationship between objects in the same cluster and can thus be used to establish a measure of pairwise similarity. Because of this similarity measure, CSPA is also categorized under consensus similarity matrix methods. The second algorithm is the HyperGraph Partitioning Algorithm (HGPA) in which the cluster ensemble problem is formulated as partitioning the hypergraph by cutting a minimal number of hyperedges. Both algorithms were coded by the published cluster ensemble package that is available on (http://www.strehl.com).
For CSPA, the similarity matrix is generated so that each two objects have a similarity of 1 if they are in the same cluster and 0 otherwise. The process is repeated for each clustering method. A n× n binary similarity matrix S is created where n is the total number of objects in the dataset. The entries of S are divided by r, which is the number of clustering methods. Then, the similarity matrix is used to recluster the objects using any reasonable similaritybased clustering algorithm. Here, we view the similarity matrix as graph (vertex = object, edge weight = similarity) and cluster it using graph partitioning algorithm METIS [35] because of its robust and scalable properties in order to obtain the consensus partition.
The HGPA portions the hypergraph directly. This is done by removing the lower number of hyperedges. All hyperedges have the same weight and are searched by cutting the minimum possible number of hyperedges that partition the hypergraph in k connected components of approximately the same dimension. For the implementation of this method, the hypergraphs partitioning package HMETIS [36] was used.
Votingbased consensus clustering
The cumulative votingbased aggregation algorithm consists of two steps; the first one is to obtain the optimal relabeling for all partitions, which is known as the voting problem. Then, the votingbased aggregation algorithm is used to obtain the aggregated (consensus) partition. The votingbased aggregation algorithm described by Ayed and Kamel [37, 38] is modified to be used in this paper.
Let χ denote a set of n data objects, and let a partition of χ into k clusters be represented by an n×k matrix U such that ∑ _{q = 1}^{ k }ujq = 1, for ∀ j. Let u = {U_{ i }}_{i = 1}^{ b } denote an ensemble of partitions. The votingbased aggregation problem is concerned with searching for an optimal relabeling for each partition V^{ i } with respect to representative partition U^{ 0 } (with k^{0} clusters) and for a central aggregated partition denoted as Ū that summarises the ensemble partitions. The matrix of coefficients W^{ i }, which is a k^{ i } × k^{0} matrix of w_{ lq }^{ i } coefficients, is used to obtain the optimal relabeling for ensemble partitions.
In this paper, the fixedreference approach is used, whereby an initial reference partition is used as a common representative partition for all the ensemble partitions and remains unchanged throughout the aggregation process. Instead of selecting random partition, the partition that is generated by the method, which showed high ability to separate active from inactive molecules in our experiments, is suggested to be the reference partition U^{0}; and this method is the Ward’s clustering (the current standard clustering method for Chemoinformatics applications). The cumulative votingbased aggregation algorithm is described as follows:
 1.
select a partition U^{ i } ∈u which is generated by the Ward’s method and assign to U^{0}
 2.
for i=1 to b do
 3.
W^{ i } = (U^{iT}U^{ i })^{− 1}U^{ i }^{T}U^{0}
 4.
V^{ i } = U^{ i }W^{ i }
 5.
${\mathrm{U}}^{0}=\frac{i1}{i}{\mathrm{U}}^{0}+\frac{1}{i}{\mathrm{V}}^{i}$
 6.
end for
 7.
Ū = U ^{0}.
Performance evaluation
This calculation is carried out on each cluster and the Fmeasure is the maximum value across all clusters.
Results and discussion
The ensembles were generated by running the six individual clusterings, each with the six distance measures. Then, the ensembles were combined using voting and graphbased consensus clustering methods: CVAA, CSPA and HGPA. This process was repeated for each fingerprint (ALOGP and ECFP_4).
Effectiveness of clustering of MDDR dataset using FMeasure: ALOGP Fingerprint
Clustering method  No. of clusters  

500  600  700  800  900  1000  
Consensus clustering  CVAA  Correlation  26.80  21.96  18.96  18.49  17.6  15.45 
Cosine  24.79  21.72  19.01  18.19  16.46  14.81  
Euclidean  27.96  23.75  22.68  24.30  21.17  19.95  
Hamming  24.02  20.48  16.31  16.85  14.95  14.68  
Jaccard  23.58  21.96  18.01  18.46  16.72  15.35  
Manhattan  27.03  25.23  21.16  20.36  19.10  19.05  
CSPA  Correlation  5.06  4.65  4.16  3.56  3.35  3.04  
Cosine  5.17  4.65  4.08  3.62  3.37  3.05  
Euclidean  5.12  4.64  4.04  3.61  3.38  3.00  
Hamming  5.30  4.74  4.16  3.62  3.54  3.13  
Jaccard  5.31  4.82  4.15  3.77  3.48  3.13  
Manhattan  5.33  4.80  4.21  3.62  3.45  3.05  
HGPA  Correlation  7.13  5.48  5.45  4.65  4.35  4.37  
Cosine  8.06  6.04  5.03  4.52  4.45  4.08  
Euclidean  7.08  6.55  5.65  4.67  4.56  4.60  
Hamming  8.37  5.73  4.94  5.29  4.97  4.93  
Jaccard  7.63  6.22  5.98  4.53  5.24  3.92  
Manhattan  7.72  6.48  5.23  5.35  4.90  4.12  
Individual clustering  Ward's method  9.93  9.19  8.19  7.17  6.67  6.44 
Effectiveness of clustering of MDDR dataset using FMeasure: ECFP_4 Fingerprint
Clustering method  No. of clusters  

500  600  700  800  900  1000  
Consensus clustering  CVAA  Correlation  33.58  29.81  24.44  20.09  18.41  17.43 
Cosine  34.75  31.32  24.97  20.26  18.46  17.73  
Euclidean  25.43  23.34  20.51  19.13  16.47  14.64  
Hamming  25.48  24.04  20.23  19.62  17.31  14.73  
Jaccard  35.71  33.17  28.66  21.8  19.63  18.86  
Manhattan  25.41  23.98  20.30  19.53  17.25  14.65  
CSPA  Correlation  5.53  4.88  4.23  3.85  3.6  3.18  
Cosine  5.43  4.88  4.28  3.91  3.55  3.10  
Euclidean  5.47  4.87  4.17  3.79  3.53  3.33  
Hamming  5.45  4.82  4.23  3.87  3.58  3.19  
Jaccard  5.51  4.99  4.25  3.99  3.62  3.20  
Manhattan  5.44  4.85  4.23  3.89  3.62  3.20  
HGPA  Correlation  7.01  6.2  5.21  4.5  4.16  3.68  
Cosine  6.83  5.95  5.29  4.47  4.21  3.93  
Euclidean  7.29  5.82  5.29  4.39  4.48  3.94  
Hamming  7.01  5.83  5.29  4.50  4.37  3.69  
Jaccard  6.87  5.91  5.31  4.81  4.80  3.66  
Manhattan  7.81  5.17  5.38  4.61  4.66  3.68  
Individual clustering  Ward's method  11.61  10.71  9.04  8.29  7.64  7.02 
Effectiveness of clustering of MDDR dataset using QPI: ALOGP Fingerprint
Clustering method  No. of clusters  

500  600  700  800  900  1000  
Consensus clustering  CVAA  Correlation  43.84  47.38  48.72  50.70  53.41  54.06 
Cosine  45.60  46.08  47.56  50.46  53.79  54.50  
Euclidean  44.43  45.54  47.95  48.65  52.68  54.86  
Hamming  53.13  56.08  59.07  60.58  64.02  67.76  
Jaccard  57.86  60.62  64.07  66.49  70.68  73.53  
Manhattan  56.01  58.10  60.99  61.86  64.56  65.97  
CSPA  Correlation  46.81  50.04  51.72  51.78  54.23  56.36  
Cosine  46.04  49.49  51.42  52.11  54.48  55.92  
Euclidean  46.20  49.86  51.05  51.88  54.36  56.33  
Hamming  54.67  58.50  60.27  61.78  62.33  65.66  
Jaccard  55.03  59.13  60.84  61.03  63.73  67.44  
Manhattan  55.08  59.00  59.10  60.84  61.78  64.61  
HGPA  Correlation  47.59  49.51  52.39  54.45  56.86  58.56  
Cosine  45.58  48.44  52.78  54.42  56.36  58.70  
Euclidean  46.92  51.41  53.20  54.75  57.00  58.97  
Hamming  55.24  58.48  60.30  63.99  68.21  69.22  
Jaccard  55.71  59.89  64.10  65.15  70.48  71.60  
Manhattan  54.84  58.98  62.73  63.58  65.85  69.97  
Individual clustering  Ward's method  52.33  54.86  56.90  59.00  61.33  63.17 
Effectiveness of clustering of MDDR dataset using QPI: ECFP_4 Fingerprint
Clustering method  No. of clusters  

500  600  700  800  900  1000  
Consensus clustering  CVAA  Correlation  74.86  78.02  82.39  84.16  85.71  87.04 
Cosine  74.79  78.12  81.85  84.78  85.91  87.18  
Euclidean  71.04  74.92  78.41  81.91  84.47  86.80  
Hamming  70.99  74.36  78.47  81.68  84.24  86.28  
Jaccard  83.48  87.01  88.72  90.98  90.67  92.05  
Manhattan  70.74  74.26  78.52  81.74  84.12  86.09  
CSPA  Correlation  70.58  73.29  74.86  76.86  79.17  82.03  
Cosine  71.23  71.85  76.43  76.55  78.06  81.21  
Euclidean  65.33  67.09  72.49  72.73  74.50  78.75  
Hamming  64.68  66.82  69.88  71.25  74.17  76.64  
Jaccard  69.91  71.73  74.20  76.01  77.72  79.26  
Manhattan  63.07  65.77  68.83  71.50  74.06  77.33  
HGPA  Correlation  72.61  74.85  76.4  78.32  80.22  82.26  
Cosine  72.06  74.25  77.21  79.54  81.02  83.31  
Euclidean  70.71  72.82  75.02  76.80  80.50  82.66  
Hamming  69.45  72.21  74.08  77.71  79.67  82.36  
Jaccard  67.88  70.58  73.93  76.56  77.65  79.67  
Manhattan  72.74  72.14  75.68  77.94  81.42  82.97  
Individual clustering  Ward's method  75.83  79.88  83.34  84.25  86.49  88.25 
Visual inspection of Fmeasure and QPI values in Tables 2, 3, 4 and 5 enables comparisons to be made between the effectiveness of three consensus clustering methods and the Ward’s method. In addition, two fingerprints were used for the experiments in order to study the effectiveness of consensus clustering on different representations of molecular dataset.
For clustering of MDDR dataset which is represented by ALOGP fingerprint (Tables 2 and 4), the performance of votingbased consensus method (CVAA) outperformed the Ward’s method and the graphbased consensus methods (CSPA and HGPA) using the two criteria: Fmeasure and QPI. The highest Fmeasure values were obtained by using Euclidean distance measure with individual clustering methods in the ensemble generation step. While, using the QPI measure, the highest values were obtained by using the Jaccard distance measure. Moreover, the ensembles generated by the other distance measures showed a better performance of CVAA than Ward and graphbased consensus clustering methods using both criteria.
Similarly, the results in Tables 3 and 5 show that, when ECFP_4 fingerprint is used, the CVAA consensus clustering performed very well and outperformed Ward and graphbased consensus clustering methods using F and QPI measures. The Jaccard distance measure was the method of choice to generate the ensembles, which gives the highest values for both criteria.
T test statistical significance testing using Fmeasure
Paired differences  Sig. (2tailed)  

Mean  Std. deviation  Std. error mean  95% Confidence interval of the difference  
Lower  Upper  
a) ALOGP:  
Pair 1: CVAA  Wards  15.37  1.77  0.72  13.50  17.23  0.000004 
Pair 2: CVAA CSPA  19.16  2.11  0.86  16.95  21.38  0.000003 
Pair 3: CVAA  HGPA  17.24  1.75  0.71  15.40  19.08  0.000002 
b) ECFP_4:  
Pair 1: CVAA  Ward  17.25  5.48  2.24  11.49  23.01  0.000589 
Pair 2: CVAA  CSPA  22.01  6.41  2.62  15.27  28.75  0.000391 
Pair 3: CVAA – HGPA  20.84  5.95  2.43  14.58  27.09  0.000357 
Ttest statistical significance testing using QPI measure
Paired differences  Sig. (2tailed)  

Mean  Std. deviation  Std. error mean  95% Confidence interval of the difference  
Lower  Upper  
a) ALOGP:  
Pair 1: CVAA  Wards  7.61  1.92  0.78  5.58  9.63  0.000199 
Pair 2: CVAA CSPA  4.20  2.08  0.85  2.02  6.39  0.004290 
Pair 3: CVAA  HGPA  1.62  1.06  0.43  0.49  2.74  0.013842 
b) ECFP_4:  
Pair 1: CVAA  Ward  5.81  1.60  0.65  4.12  7.49  0.000301 
Pair 2: CVAA  CSPA  12.31  1.49  0.61  10.74  13.88  0.000005 
Pair 3: CVAA – HGPA  10.77  1.50  0.61  9.20  12.34  0.000010 
Tables 6 and 7 display a number of parameters: mean value, standard deviation, standard error and significance values for the pairs of the best Fmeasure and QPI values of clustering methods which are ((CVAA, Ward’s method), (CVAA, CSPA) and (CVAA, HGPA)) compared in the paired samples ttest procedure. The pairedsamples ttest procedure compares the means of two variables that represent the same group at different cluster size. Since the paired samples ttest compares the means for the two variables, it is quite useful to know what the mean values are. A low significance value for the ttest (typically less than 0.05) indicates that there is a significant difference that was satisfied between two variables. In Tables 6 and 7, it was noted that the significance field (Sig. (2tailed)) in terms of Fmeasure for AlogP is: CVAAWard’s method (0.000004), CVAACSPA (0.000003) and CVAAHGPA (0.000002), and for ECFP_4: CVAAWard’s method (0.000589), CVAACSPA (0.000391) and CVAAHGPA (0.000357). Similarly, the significance field (Sig. (2tailed)) in terms of QPI values for AlogP is: CVAAWard’s method (0.000199), CVAACSPA (0.004290) and CVAAHGPA (0.013842), and for ECFP_4 is: CVAA Ward’s method (0.000301), CVAACSPA (0.000005) and CVAAHGPA (0.000010). In addition, the significance value is low in Fmeasure and QPI values and the confidence interval for the mean difference does not contain zero. It is then concluded that the consensus clustering method, CVAA, obtained significant results by using Fmeasure and QPI values compared to Ward and graphbased consensus clustering methods. Moreover, the CVAA method is more efficient because the computational complexity of CVAA is O (n) which is better than other consensus clustering methods such as CSPA that has complexity of O (n^{2}), where n is the number of data objects [14].
Conclusions
The experiments results show that consensus clustering methods can improve the effectiveness of chemical structures clusterings. The cumulative votingbased aggregation algorithm CVAA was the method of choice among consensus clustering methods. The performance of CVAA consensus clustering significantly outperforms Ward and graphbased consensus clustering methods (CSPA and HGPA) using F and QPI measures for both ALOGP and ECFP_4 fingerprints, while the graphbased consensus methods outperform the Ward’s method only for ALOGP using QPI measure. The experiments reported here suggest that votingbased consensus clustering can perform very well when the partitions are generated by a single run of multiple individual clusterings that use Jaccard and Euclidean distance measures in the ensemble generation process.
Declarations
Acknowledgements
This work is supported by the Ministry of Higher Education (MOHE) and Research Management Centre (RMC) at the Universiti Teknologi Malaysia (UTM) under Research University Grant Category (VOT Q.J130000.7128.00H72). We also would like to thank MISMOHE for sponsoring the first author.
Authors’ Affiliations
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