The visualization of cluster solutions in gene expression data analysis gives practitioners an understanding of the cluster structure of their data and makes it easier to interpret the cluster results. Neighborhood graphs allow for visual assessment of relationships between adjacent clusters. The number of clusters in gene expression data is for biological reasons rather large. As a linear projection of the data into 2 dimensions does not scale well in the number of clusters there is a need for new visualization techniques using non-linear arrangement of the clusters. The new visualization tool is implemented in the open source statistical computing environment R. It is demonstrated on microarray data from yeast

Leisch, Friedrich

Scharl, Theresa

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Theresa Scharl & Friedrich LeischVisualizing Gene Clustersusing Neighborhood Graphs in RTechnical Report Number 16, 2008Department of StatisticsUniversity of Munichhttp://www.stat.uni-muenchen.deVisualizing Gene Clusters using NeighborhoodGraphs in RTheresa Scharl1 and Friedrich Leisch21 Department of Statistics and Probability Theory, Vienna University ofTechnologyWiedner Haupstraße 8-10/1071, 1040 Vienna, Austria,theresa.scharl@ci.tuwien.ac.at2 Department of Statistics, University of MunichLudwigstraße 33, D-80539 Mu¨nchen, Germany,friedrich.leisch@stat.uni-muenchen.deAbstract. The visualization of cluster solutions in gene expression data analy-sis gives practitioners an understanding of the cluster structure of their data andmakes it easier to interpret the cluster results. Neighborhood graphs allow for vi-sual assessment of relationships between adjacent clusters. The number of clustersin gene expression data is for biological reasons rather large. As a linear projectionof the data into 2 dimensions does not scale well in the number of clusters there is aneed for new visualization techniques using non–linear arrangement of the clusters.The new visualization tool is implemented in the open source statistical computingenvironment R. It is demonstrated on microarray data from yeast.Keywords: Cluster analysis, graphs, microarray data, R1 IntroductionGene expression microarray experiments yield large and complex multivariatedatasets that consist of several thousands of genes at multiple states. A typicalquestion during the analysis is to find groups in the data. Cluster analysis iscommonly used to reduce the complexity of the data from multidimensionalspace to a single nominal variable, the cluster membership. In the analysisof microarray data clustering is used as vector quantization and the set ofgenes is divided into artificial subsets. Genetic interactions are so complexthat the definition of gene clusters is not clear. Additionally microarray dataare very noisy and co–expressed genes can end up in different clusters. Asno clear density clusters exist in the data the relationship between clusters isvery important.Clusters of co–expressed genes can help to discover potentially co–regulatedgenes or association to conditions under investigation. Usually cluster anal-ysis provides a good initial investigation of microarray data before actuallyfocusing on functional subgroups of interest. In the literature numerous clus-ter algorithms for clustering gene expression data have been proposed. Be-sides traditional methods like hierarchical clustering , K–means, partitioning2 Scharl, T. and Leisch, F.around medoids (PAM, K–medoids) or self–organizing maps there are severalalgorithms dealing with time–course gene expression data (e.g., Heyer et al.,1999, De Smet et al., 2002, Ben-Dor et al., 1999).The display of cluster solutions particularly for a large number of clus-ters is very important in exploratory data analysis. Visualization methodsgive practitioners an understanding of the relationships between segmentsof a partition and make it easier to interpret the cluster results. Neighbor-hood graphs (Leisch, 2006) can be used for visual assessment of the clus-ter structure of centroid–based cluster solutions. A linear projection of thedata into 2 dimensions using for example linear discriminant analysis (LDA)does not scale well in the number of clusters. Gene expression data arehigh–dimensional data and are usually separated into a lot of clusters (e.g.,over 25 clusters in Heyer et al., 1999). As the vast amount of informationcannot be shown in the plane there is a need for new visualization tech-niques. Using non–linear arrangement of the clusters the cluster structurecan be displayed up to a very large number of clusters. In this work thelayout algorithms implemented in the open source graph visualization soft-ware Graphviz are used for non–linear arrangement of the clusters. The newvisualization tool is currently available at the homepage of the first author(http://www.ci.tuwien.ac.at/ scharl/Software/) and will be released as an Rpackage (R Development Core Team, 2007, http://www.R-project.org) soon.The functionality is demonstrated on a publicly available data set from yeast.2 Methods2.1 Cluster AlgorithmsIn this work we focus on centroid–based cluster algorithms like K–meansand PAM or others where clusters can be represented by centroids (e.g.,QT–Clust, Heyer et al., 1999). For a given data set XN = {x1, . . . , xN} thedistance between points x and y is given by d(x, y), e.g., the Euclidean orabsolute distance. CK = {c1, . . . , cN} is a set of centroids and the centroidclosest to x is denoted byc(x) = argminc∈CKd(x, c).The set of all points where ck is the closest centroid is given byAk = {xn|c(xn) = ck}.Minimizing the average distance between each data point and its closestcentroidD(Xn, CK) =1NN∑n=1d(xn, c(xn)) → minCKis the task of most cluster algorithms.Visualizing Gene Clusters using Neighborhood Graphs in R 32.2 Neighborhood GraphsNeighborhood graphs (Leisch, 2006) use the idea of topology–representingnetworks (TRNs, Martinetz and Schulten, 1994) to count the number of datapoints a pair of centroids is closest and second–closest. In TRNs the countsare used as weights for the edges of the graph. Silhouette plots (Rousseeuw,1987) are diagnostic plots revealing the goodness of a partition. The distancefrom each point to the points in its own cluster is compared to the distanceto points in the second closest cluster. The larger the silhouette values thebetter a cluster is separated from the other clusters. But silhouette plotsdo not show the proximity of clusters. They only give an indicator how well-separated single points are from other clusters. Neighborhood graphs combinethese two approaches and use the mean relative distances as edge weights inorder to measure how separated pairs of clusters are. Hence they display thedistance between clusters. In the graph each node corresponds to a clustercentroid and two nodes are connected by an edge if there exists at least onepoint that has these two as closest and second–closest centroid.As described above the centroid closest to x is denoted by c(x) and thesecond closest centroid to x is denoted byc˜(x) = argminc∈CK\{c(x)}d(x, c).Now the set of all points where ci is the closest centroid and cj is second–closest is given byAij = {xn|c(xn) = ci, c˜(xn) = cj}.For each observation x we defines(x) =2d(x, c(x))d(x, c(x)) + d(x, c˜(x)).s(x) is small if x is close to its cluster centroid and close to 1 if it is almostequidistant between the two cluster centroids. The average s–value of allpoints where cluster i is closest and cluster j is second closest can be used asa proximity measure between clusters and as edge weight in the graph.sij ={|Ai|−1∑x∈|Aij | s(x), Aij 6= ∅0, Aij = ∅|Ai| is used in the denominator instead of |Aij | to make sure that a small setAij consisting only of badly clustered points with large shadow values doesnot induce large cluster similarity.3 DataIn this work a publicly available dataset from yeast was investigated, theseventeen time point mitotic cell cycle data (Cho et al., 1998) available at4 Scharl, T. and Leisch, F.http://genome-www.stanford.edu. The dataset was preprocessed adapt-ing the instructions given by Heyer et al. (1999). The outlier time points 10and 11 were removed from the original 17 variables. The gene vectors werestandardized to have median 0 and MAD 1. Finally genes that were eitherexpressed at very low levels or did not vary significantly over the time pointswere removed. This procedure yields gene expression data on N = 2832 genes(observations) for T = 15 time points (variables). The data was clustered us-ing the K–means algorithm. In this example 15 clusters were selected.4 Software and ImplementationAll cluster algorithms and visualization methods used are implemented in thestatistical computing environment R. R package flexclust (Leisch, 2006) isa flexible toolbox to investigate the influence of distance measures and clus-ter algorithms. It contains extensible implementations of the K–centroids andQT–Clust algorithm and offers the possibility to try out a variety of distanceor similarity measures as cluster algorithms are treated separately from dis-tance measures. New distance measures and centroid computations can eas-ily be incorporated into cluster procedures. The default plotting method forcluster solutions in flexclust is the neighborhood graph.The visualization of partitioning cluster solutions is commonly accom-plished by linear projection of the data into 2 dimensions using for exampleLDA. In Figure 1 the best possible separation using LDA is shown. Therelationships between the centroids of 15 clusters can hardly be displayedin the plane. Therefore a new visualization tool is presented using non-linear arrangement of the nodes. Infrastructure for creating, manipulating,and visualizing graphs is provided in Bioconductor (Gentleman et al., 2005,http://www.bioconductor.org) package graph. The package contains func-tionality for data structure, classes and methods to manipulate graphs andenables efficient representations of very large graphs. An interface to the opensource graph visualization software Graphviz (http://www.graphviz.org/) isprovided in Bioconductor package Rgraphviz which returns the layout in-formation for a graph object, x- and y–coordinates of the graph’s nodes aswell as the parameterization of the trajectories of the edges. Several layoutalgorithms can be chosen.dot: hierarchical layout algorithm for directed graphsneato and fdp: layout algorithms for large undirected graphstwopi: radial layoutcirco: circular layoutAdditionally global and local properties (e.g., labels, shape, color, . . . ) canbe assigned to both nodes and edges.Using non–linear arrangement of clusters clearly improves the visualiza-tion of the cluster solution (Figure 2). In the new visualization method relatedVisualizing Gene Clusters using Neighborhood Graphs in R 5−8 −6 −4 −2 0 2 405101512 345678910 1112131415Fig. 1. Projection of neighborhood graph of a K–means cluster solution of the yeastdata into 2 dimensions.clusters are not forced to lie next to each other. For example cluster 11 lo-cated at the bottom end of the graph is related to cluster 1 located at thetop end of the graph. Additionally the graph is simplified by only drawingedges between nodes if the similarity of a cluster to another cluster is at least10%. In Figure 2 the cluster structure of the cluster solution can easily beinvestigated. Cluster 15 is very different from the remaining clusters as noedge is drawn to cluster 15 and the similarities to connected clusters are verysmall. This indicates that the genes in cluster 15 are very different from theremaining genes. As cluster 4 is similar to clusters 10 and 14 which are alsostrongly connected the 3 clusters seem to be highly related.4.1 Edge MethodsThe neighborhood graph is a directed graph as the similarity of cluster 1to cluster 2 is different from the similarity of cluster 2 to cluster 1. Besidesplotting the original directed graph there are several possibilities how to plotedges taking into account for instance the mean, minimum or maximum ofthe similarities between two clusters.6 Scharl, T. and Leisch, F.k1k2k3k4k5k6k7k8k9k10k11k12k13k14k15>= 0.40.30.20.10Fig. 2. Neighborhood graph of a K–means cluster solution for the yeast data usingthe dot layout algorithm.4.2 Node MethodsIn the simple visualization of a neighborhood graph one single kind of nodesymbol is used for all nodes. By just looking at the graph no informationabout the different clusters is revealed. There are several possibilities howto include additional information in the representation of nodes. The mostsimple method is to use color coding, e.g., to color nodes by size or tightnessof the corresponding clusters. Another possibility is to use different shapesor symbols for nodes representing clusters with specific properties.In Figure 3 the cluster solution is shown with tight clusters highlighted,i.e., clusters with small average distance of the genes to the cluster centroid.In this example the tightest clusters are numbers 15 and 8 indicating genesvery different from the rest of the genes (colored darkgrey) and numbers 13and 4 (lightgrey).4.3 Other FeaturesThe new visualization tool offers various possibilities for the analysis of mi-croarray data which cannot be shown here due to space constrains. The neigh-borhood graph is implemented in an interactive way and gene clusters can beVisualizing Gene Clusters using Neighborhood Graphs in R 7k1k2k3k4k5k6k7k8k9k10k11k12k13k14k15lllll>= 0.40.30.20.10Fig. 3. The neighborhood graph with tight clusters highlighted.investigated by clicking on the nodes. Plots of the expression profiles of thecorresponding genes pop up as well as tables giving further information aboutthe genes. Additional information about the gene clusters can be included inthe neighborhood graph. External information from differential expressionanalysis or functional grouping can easily be included in the node represen-tation, e.g., the accumulation of gene ontology (GO) categories in certaingene clusters.5 SummaryCluster analysis is commonly used in the analysis of gene expression datato find groups of co–expressed genes. But the definition of gene clusters isnot very clear as genetic interactions are extremely complex. For this reasonthe relationship between clusters is very important as co–expressed genescan end up in different clusters. The neighborhood graph is a useful tool tovisualize the underlying cluster structure. The visualization of partitioningcluster solutions is commonly accomplished by linear projection of the datainto 2 dimensions. However, this is not recommended for high–dimensionaldata like microarray data and a large number of clusters. In our new visual-8 Scharl, T. and Leisch, F.ization method layout algorithms for non–linear arrangement of the clustersare used to display the relationships between clusters. Our interactive soft-ware tools for the analysis of gene expression data is very helpful not onlyfor statisticians but also for practitioners.AcknowledgementThis work was supported by the Austrian Kind/Knet Center of Biopharma-ceutical Technology (ACBT).ReferencesBEN-DOR, A., SHAMIR, R. and YAKHINI, Z. (1999): Clustering gene expressionpatterns. Journal of Computational Biology, 6 (3–4), 281–297.BICKEL, D.R. (2003): Robust cluster analysis of microarray gene expression datawith the number of clusters determined biologically. Bioinformatics, 19 (7),818–824.CAREY, V.J., GENTLEMAN, R., HUBER, W. and GENTRY, J. (2005): Biocon-ductor Software for Graphs. In: R. Gentleman, V.J. Carey, W. Huber, R.A.Irizarry and S. Dudoit (Eds.): Bioinformatics and Computational Biology So-lutions Using R and Bioconductor. Springer, New York.CHO, R.J., CAMPBELLl, M.J., WINZELER E.A., STEINMETZ, L., CONWAY,A., WODICKA, L., WOLFSBERG, T.G., GABRIELIAN, A.E., LANDSMAN,D., LOCKHART, D.J. and DAVIS, R.W. (1998): A genome-wide transcrip-tional analysis of the mitotic cell cycle. Mol Cell, 2/1, 65-73.DE SMET, F., MATHYS, J., MARCHAL, K., THIJS, G., DE MOOR, B. andMOREAU, Y. (2002): Adaptive quality-based clustering of gene expressionprofiles. Bioinformatics, 18 (5), 735–746.GENTLEMAN, R., CAREY, V.J., HUBER, W., IRIZARRY, R.A. and DUDOIT,S. (Eds.) (2005): Bioinformatics and Computational Biology Solutions UsingR and Bioconductor. Springer, New York.HEYER, L.J., KRUGLYAK, S. and YOOSEPH, S. (1999): Exploring ExpressionData: Identification and Analysis of Coexpressed Genes. Genome Research, 9,1106-1115.LEISCH, F. (2006): A Toolbox for K-Centroids Cluster Analysis. ComputationalStatistics and Data Analysis 51 (2), 526–544.MARTINETZ, T. and SCHULTEN, K. (1994): Topology representing networks.Neural Networks, 7 (3), 507–522.R DEVELOPMENT CORE TEAM (2007): R: A Language and Environment forStatistical Computing. R Foundation for Statistical Computing, Vienna, Aus-tria, http://www.R-project.org.ROUSSEEUW, P.J. (1987): Silhouettes: A graphical aid to the interpretation andvalidation of cluster analysis. Journal of Computational and Applied Mathe-matics, 20, 53–65.

A genome-wide transcriptional analysis of the mitotic cell cycle.

A Toolbox for K-Centroids Cluster Analysis.

Adaptive quality-based clustering of gene expression proﬁles.

Bioconductor Software for Graphs. In:

Clustering gene expression patterns.

Exploring Expression Data: Identiﬁcation and Analysis of Coexpressed Genes.

R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing,

Robust cluster analysis of microarray gene expression data with the number of clusters determined biologically.

Silhouettes: A graphical aid to the interpretation and validation of cluster analysis.

Visualizing Gene Clusters using Neighborhood Graphs in R

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