Voting-based consensus clustering for combining multiple clusterings of chemical structures

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 graph-based 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 voting-based aggregation algorithm (CVAA), cluster-based similarity partitioning algorithm (CSPA) and hyper-graph partitioning algorithm (HGPA) were examined. The F-measure 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 voting-based consensus clustering method outperformed the Ward’s method using F-measure and QPI method for both ALOGP and ECFP_4 fingerprints, while the graph-based 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 voting-based 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 High-Throughput Screening (HTS).
According to Brown and Martin [2], the advent of high-throughput 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 in-house 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 structure-activity 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 intra-cluster similarity and inter-cluster dissimilarity [5]. Many different types of clustering techniques for chemical structures have been used in the literature [6][7][8][9][10][11][12][13].
Brown and Martin [2] considered the Ward's clustering to be the most efficient method in cluster-based 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 co-occurrence-based and the median partition-based approaches. Voting and graph based consensus clusterings are widely used for the first approach.
Topchy et al. [15] and Fred and Jain [16] summarised the main advantages of using consensus clustering in the following terms: 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][19][20][21][22][23][24][25].
Over the last few years, data fusion has become accepted as a simple way of enhancing the performance of existing systems for ligand-based 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 graph-based consensus clustering methods for combining multiple clusterings of chemical structures with different distance measures in order to improve the effectiveness of chemical structures clustering.

Molecular fingerprints
For the clustering experiments, two molecular fingerprints were developed by Scitegic's Pipeline Pilot software [26]. The first one was 120-bit ALOGP, which includes octanol-water 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 1-octanol/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
Experiments were conducted over the most popular chemoinformatics databases: the MDL Drug Data Report database [30]. This database consists of 102516 molecules which was described by Hert et al. [21] and used for consensus scoring that combined the results of different similarity searches of a chemical database. Also, it has been used for many virtual screening experiments [31][32][33]. According to Hert et al. [26], the subset dataset was chosen from MDDR database which is quite disparate in nature, some of the molecules being structurally homogeneous (e.g., rennin and HIV-1 protease inhibitors) while others were structurally diverse (e.g., cyclooxygenase and protein kinase C inhibitors); the diversity was estimated by the mean pairwise Tanimoto similarity across each set of active molecules (activity class). The calculations of pairwise Tanimoto similarity were conducted using Pipeline Pilot software. The MDDR dataset contains eleven activity classes (8294 molecules) and the details of this dataset are listed in Table 1. Each row in the table contains an activity class, the number of molecules belonging to the class and the diversity of the class.

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 (single-linkage, complete linkage, average linkage, weighted average distance, Ward and K-means 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.

Graph-based consensus clustering
Two graph-based 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 hyper-graph representation. The first algorithm is Cluster-based 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 Hyper-Graph Partitioning Algorithm (HGPA) in which the cluster ensemble problem is formulated as partitioning the hyper-graph by cutting a minimal number of hyper-edges. Both algorithms were coded by the published cluster ensemble package that is available on (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 re-cluster the objects using any reasonable similarity-based 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 hyper-graph directly. This is done by removing the lower number of hyper-edges. All hyper-edges have the same weight and are searched by cutting the minimum possible number of hyper-edges that partition the hyper-graph in k connected components of approximately the same dimension. For the implementation of this method, the hyper-graphs partitioning package HMETIS [36] was used.

Voting-based consensus clustering
The cumulative voting-based 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 voting-based aggregation algorithm is used to obtain the aggregated (consensus) partition. The voting-based 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 P denote an ensemble of partitions. The voting-based 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 fixed-reference 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: Cumulative Voting-based Aggregation Algorithm 1. select a partition U i ∈u which is generated by the Ward's method and assign to U 0

Performance evaluation
The results were evaluated based on the effectiveness of the methods to separate active from inactive molecules using two measures: the F-measure [39] and Quality Partition Index (QPI) measure [40]. As defined by [17], if the cluster contains n compounds, that a of these are active and that there is a total of A compounds with the chosen Activity. The precision, P, and the recall, R, for that cluster are: This calculation is carried out on each cluster and the F-measure is the maximum value across all clusters.
In addition, according to [13], an active cluster can be defined as a non-singleton cluster for which the percentage of active molecules in the cluster is greater than the percentage of active molecules in the dataset as a whole. Let p be the number of actives in active clusters, q be the number of inactives in active clusters, r be the number of actives in inactive clusters (i.e., clusters that are not active clusters) and s be the number of singleton actives. The high value occurs when the actives are clustered tightly together and separated from the inactive molecules. The QPI is defined to be:

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 graph-based consensus clustering methods: CVAA, CSPA and HGPA. This process was repeated for each fingerprint (ALOGP and ECFP_4). The mean of F-measure and QPI values were averaged over the eleven activity classes of the dataset. Tables 2, 3, 4 and 5 show the effectiveness of MDDR dataset clustering for ALOGP and ECFP_4 fingerprints. In all tables, the best values for F-measure and QPI of consensus clustering methods for each column were bold-faced for ease of reference.
Visual inspection of F-measure 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 voting-based consensus method (CVAA) outperformed the Ward's method and the graph-based consensus methods (CSPA and HGPA) using the two criteria: F-measure and QPI. The highest F-measure 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 graph-based 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 graph-based 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.
Some statistical significance tests (T-tests) were performed to show the improvements achieved by the consensus clustering methods, as shown in Tables 6 and 7. It was found that the performance of voting-based consensus method is statistically significant when using both criteria.
Tables 6 and 7 display a number of parameters: mean value, standard deviation, standard error and significance values for the pairs of the best F-measure and QPI values of clustering methods which are ((CVAA, Ward's     In addition, the significance value is low in F-measure 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 F-measure and QPI values compared to Ward and graph-based 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 voting-based aggregation algorithm CVAA was the method of choice among consensus clustering methods. The performance of CVAA consensus clustering significantly outperforms Ward and graph-based consensus clustering methods (CSPA and HGPA) using F and QPI measures for both ALOGP and ECFP_4 fingerprints, while the graph-based consensus methods outperform the Ward's method only for ALOGP using QPI measure. The experiments reported here suggest that voting-based consensus clustering can perform very well when the partitions are generated by a single run of multiple individual clusterings