Data reduction for spectral clustering to analyze high ...take 2 years and 5 terabytes of memory to analyze a typical flow cytometry sample with 300,000 events. We developed a novel

METHODOLOGY ARTICLE Open Access

Data reduction for spectral clustering to analyzehigh throughput flow cytometry dataHabil Zare1,2, Parisa Shooshtari1,2, Arvind Gupta3, Ryan R Brinkman2,4*

Abstract

Background: Recent biological discoveries have shown that clustering large datasets is essential for betterunderstanding biology in many areas. Spectral clustering in particular has proven to be a powerful tool amenablefor many applications. However, it cannot be directly applied to large datasets due to time and memorylimitations. To address this issue, we have modified spectral clustering by adding an information preservingsampling procedure and applying a post-processing stage. We call this entire algorithm SamSPECTRAL.

Results: We tested our algorithm on flow cytometry data as an example of large, multidimensional data containingpotentially hundreds of thousands of data points (i.e., “events” in flow cytometry, typically corresponding to cells).Compared to two state of the art model-based flow cytometry clustering methods, SamSPECTRAL demonstratessignificant advantages in proper identification of populations with non-elliptical shapes, low density populationsclose to dense ones, minor subpopulations of a major population and rare populations.

Conclusions: This work is the first successful attempt to apply spectral methodology on flow cytometry data. Animplementation of our algorithm as an R package is freely available through BioConductor.

BackgroundHigh throughput data analysis is a crucial step inresearch endeavours involving gene expression, proteinclassification, and flow cytometry. A classical approachfor analysing biological data is to first group individualdata points based on some similarity criterion, a processknown as clustering, and then compare the outcome ofclustering with the biological hypotheses. An example ofthis approach is in the analysis of flow cytometry datawhere populations of cells that express specific intracel-lular or surface proteins are identified. Flow cytometryis a technique for measuring physical, chemical andbiological characteristics of individual microscopic parti-cles such as cells and chromosomes. It has many appli-cations in molecular and cell biology for both clinicaldiagnosis and research purposes [1]. In cytometers, cellsare individually passed through a laser beam and thescattered light is captured to measure up to 19 charac-teristic of each cell [2]. As thousands of cells can beanalyzed per second, cytometers can generate large-

sized datasets. Recently, sophisticated methods havebeen developed for automatic analysis of flow cytometrydata [3-5]. The proposed clustering techniques include:mixture modeling approach [6], model-based clusteranalysis [7], feature-guided clustering [8], density-basedclustering [9], combining the curvature information withdensity information [10], and image processing [11].The automatic techniques are useful in clinical andresearch applications such as: application of high-content flow cytometric screening (FC-HCS) to theproblem of cellular signature definition for acute graft-versus-host-disease [12], vaccine trials [13], visualizingdata in stem cell research [14], and immunophenotypiccharacterization of B-cell chronic lymphoproliferativedisorders (B-CLPD) [15].

Problem StatementAutomated identification of flow cytometry cell popula-tions is complicated by overlapping and adjacent popula-tions, especially when low and high density populationsare close to each other. Analysing such data requiresclustering methods that can separate these populations.Non-parametric methods include density clustering [16],real-time adaptive clustering [17], and Kohonen

* Correspondence: [email protected] Fox Laboratory, BC Cancer Agency, 675 W 10th Ave., Vancouver, BC,CanadaFull list of author information is available at the end of the article

Zare et al. BMC Bioinformatics 2010, 11:403http://www.biomedcentral.com/1471-2105/11/403

© 2010 Zare et al; licensee BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative CommonsAttribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

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self-organizing maps [18]. The application of these meth-ods is restricted since the first two are subjective due to adependency on user-defined thresholds, and the latterone requires the number of clusters to be determined bythe user. While accurately determining the number ofclusters may not be a key issue in some clinical cytometryanalysis [19], this requirement can be a critical obstaclefor other analyses such as identifying novel populationsfor biomarker discovery [3].Model-based clustering techniques such as FLAME [20],

flowClust [21] and flowMerge [22] have been developed toimprove results. flowMerge uses the flowClust frameworkto identify clusters based on a t-mixture model methodol-ogy, followed by a merging step to account for overestima-tion of the number of clusters by the Bayesian informationcriterion. FLAME uses a skew t-mixture model, which isin theory more robust to skew, because unlike t-distribu-tions, skew t-distributions can be asymmetric [20]. How-ever, the running time of this algorithm increases with thefourth degree of the number of dimensions. In practicethis tends to make the algorithm impractical for morethan five dimensions, while flow cytometry data can haveup to 19 dimensions. Overall, the major drawback of theseparametric methods is the requirement for assumptionson either the size of the clusters or the cluster distribu-tions and shapes [23], which could result in incorrect iden-tification of biologically interesting populations. Inaddition, one challenge for existing approaches is the iden-tification of rare populations. Spectral clustering is a non-parametric clustering method that avoids the problems ofestimating probability distribution functions by using aheuristic based on graphs [24]. It has proved useful inmany pattern recognition areas [25-28]. Not only does itnot require a priori assumptions on the size, shape or dis-tribution of clusters, but it has features that make it parti-cularly well-suited to clustering biological data:

• It is not sensitive to outliers, noise or shape ofclusters;• It is adjustable so that biological knowledge can beutilized to adapt it for a specific problem or dataset;• There is mathematical evidence to guarantee itsproper performance [29].

Two main challenges in applying spectral clusteringalgorithm on large data sets are the computationallyexpensive steps of constructing the normalized matrix andcomputing its eigenspace. For instance, for high through-put biological data containing one million data points (i.e.,vertices), it requires computing eigenspace of a million bymillion matrix, which is infeasible in terms of memory andtime. Although there are some approximation methods forspeeding up this computation [30,31], these could produceundesired errors in the final results. The problem of

applying this algorithm on large datasets has been studiedin [32] using Nyström’s method. They suggest a strategyof sampling data uniformly, clustering the sampled pointsand extrapolating this solution to the full set of points.However, sampling data uniformly can miss low-densitypopulations entirely when the density of adjacent popula-tions varies considerably, a situation that often arises forbiologically interesting populations in flow cytometry data.Appendix 3 includes an experiment to explain the effect ofuniform sampling in such cases.Data reduction schemes have been developed to reduce

the complexity of the flow cytometry data while preservingthe information [33,34]. These methods reduce the dimen-sionality but not the size of the dataset, the latter being themore important bottleneck for spectral clustering.

Our ApproachWe hypothesized that spectral clustering could signifi-cantly improve high throughput biological data analysis.However, serious empirical barriers are encounteredwhen applying this method to large data sets. Specifi-cally, for n data points, the running time is O(n3),requiring O(n2) units of memory. For instance, it wouldtake 2 years and 5 terabytes of memory to analyze atypical flow cytometry sample with 300,000 events. Wedeveloped a novel solution for this problem through ournon-uniform information preserving sampling. Ourheuristic approach is specific to cytometry applicationsand made it possible for the first time, to apply spectralclustering method on flow cytometry data.

ResultsIn this paper, we distinguish between the terms biologi-cal populations, clusters and components as follows. Apopulation is a set of cells with similar functionality ormolecular content. By a cluster, we mean a set of datapoints that are grouped together by spectral clusteringalgorithm. We incorporate a post-processing stage onspectral clusters to find the connected componentsintended to estimate the biological populations.

AlgorithmSpectral ClusteringThe first step is to build a graph. The vertices representthe n data points (e.g., cells in flow cytometry data), andthe edges between the vertices are weighted based onsome similarity criterion. The adjacency matrix of thegraph is then normalized using the following formula:

A D AD= − −12

12 , (1)

where A is the adjacency matrix of the graph and D isa diagonal matrix where the (i, i) entry is equal to the


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sum of the weights on the edges that are adjacent tovertex i.The next step is to compute eigenspace of the normal-

ized matrix. That is, all vectors Vi and values li satisfy-ing the following equation are computed:

AV Vi i= . (2)

In order to find k clusters, an n by k matrix is builtusing the k eigenvectors with highest eigenvalues. Therows of this matrix are normalized and finally k-meansis used to cluster the rows.However, the above method cannot be directly

applied to flow cytometry data due to large number ofdata points (cells) per sample. Our solution for this pro-blem is a data reduction scheme developed specificallyfor this purpose. This reduces the number of verticessignificantly, but in a way such that biological informa-tion can be preserved by updating the weights on theedges.Data Reduction SchemeWhile data size can be reduced by known sampling meth-ods [35], a very delicate method should be used to pre-serve biologically important information. From a high-level perspective, our data reduction scheme (Figure 1)consists of two major steps; first we sample the data in arepresentative manner to reduce the number of vertices ofthe graph (Figure 1b). Sample points cover the whole dataspace uniformly (Figures 2b), a property that aids in theidentification of both low density and rare populations. Inthe second step as described below, we define a similaritymatrix that assigns weights to the edges between thesampled data points. Higher weights are assigned to theedges between nodes in dense regions so that informationabout the density is preserved (Figure 1c).Faithful Sampling Algorithm

1. Label all data points as unregistered.2. repeat3. Pick a random unregistered point p {the repre-sentative of a new community}

4. Label all unregistered data points within dis-tance h from p as registering5. Put registering points in a set called community p6. Relabel registering points as registered7. until All points are registered8. return All communities

After faithful sampling is completed, the set of allrepresentatives can be regarded as a sample from thedata. Reducing the value of parameter h will increasethe number of sample points, resulting in increasedcomputation time and required memory. Conversely,increasing h will result in fewer sample points that maylead to too low a resolution. In such a case, the com-puted spectral clusters may fail to estimate the real cellpopulations appropriately. In our implementation, weuse an iterative procedure (explained in the overview ofour algorithm) to adjust h automatically such that thenumber of representatives will be in range 1500-3000.As a result of this adjustment, the following two objec-tives are achieved. First, computing the eigenspace of agraph with a number of points in this range is feasible,(it takes less than one minute by a 2.7 GHz processor.)Second, the communities are “small” (Figure 2) and theresulting resolution is high enough such that no biologi-cally interesting information is lost.In the sampling stage, there is no preference in picking

up the next data point, therefore, the final distribution ofthe sampled points will be uniform in the “effective” space.That is, the representatives are distributed almost uni-formly in the space where data points were present (Figure2). As a consequence, by repeating sampling procedure thefinal results of clustering will not change significantly. Thisobservation is confirmed quantitatively in Appendix 1. Byconsidering just the representatives, density information iseffectively ignored so working directly with these represen-tatives results in improper outcome. On the other hand,some biological information from the original data is pre-served by the above algorithm that can be retrieved toguide the clustering algorithm. More precisely, for eachsample point, we know the list of all points in its

Figure 1 Data reduction scheme. (a) Running spectral clustering is impractical on data that contains thousands of points. (b) Faithful samplingpicks up a reasonable subset of points such that running spectral clustering is possible on them. However, all information about the localdensity is lost by considering only these sample points. (c) We assign weights to the edges of the graph; the edges between the nodes indenser regions are weighted considerably higher. The information about the local density is retrieved in this way.


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neighbourhood (i.e., the members of the correspondingcommunity). In the next stage, we use this information todefine the similarity between two sample points to modifythe behaviour of spectral clustering. In this sense, oursampling scheme is faithful, meaning that the valuablebiological information from the original data points is pre-served even after sampling. We call the overall procedure,which consists of faithful sampling, computing modifiedsimilarity matrix and spectral clustering, SamSPECTRALclustering.Similarity MatrixIn this study, we use the following heat kernel formula[36] to compute the similarity between two vertices iand j:

s e

pi p j

i j,

( )

,

,

=−2

2 2(3)

where (pi, pj) is the Euclidean distance betweenthem. s is a scaling parameter that controls howrapidly similarity between pi and pj falls off withincreasing distance. We define the similarity betweentwo communities c and c’ as the sum of all pairwisesimilarities between all members of the first commu-nity and all members of the second community. Thatis,

S sc c i j

j ci c

, , ,′∈ ′∈

= ∑∑ (4)

where i and j are members of c and c’ respectively.We do not normalize the similarity by dividing the

above sum by the size of communities because wewould lose valuable biological information that is sup-posed to be preserved. In short, the size of the commu-nities determines the local density of the data points,which is biologically of great importance.The above definition is motivated by the following

intuition from potential theory that explains how bio-logical information is preserved after faithful samplingby assigning similarities in this way. The eigenvectorsof a graph are interpreted as potential functions on theelectric network modeled by the graph [37]. Assumingthe radius of each community is small enough, thepotential values of the community members are almostthe same. On the other hand, in potential theory, theequivalent conductance between a group of nodes {vi}with equal potential values and another group of nodes{wj} that also have equal potential values is computedby the summation of pairwise conductance betweennodes vi and wj for all i and j. Since in our model, thesimilarity between two vertices is equivalent to theconductance between the corresponding electricalnodes, it is reasonable to sum up pairwise similarities

Figure 2 Faithful sampling. (a) Original data from telomere data set before sampling. (b) The distribution of representatives is almost uniformin the space after faithful sampling.


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to estimate the equivalent similarity between commu-nities (Figure 3a).Number of ClustersThe number of clusters must be determined before run-ning the spectral clustering algorithm [38]. To find thisnumber automatically and in an efficient manner, wepropose a method that is motivated by the followingobservation from spectral graph theory:Theorem [39]: The number of connected partitions of

a graph is equal to the number of eigenvectors witheigenvalue 1.We observed that typically for flow cytometry data, if s

is adjusted properly as explained in the SamSPECTRALpackage vignette [40], the first few eigenvalues are close toone and at a point we call knee point they start to decreasealmost linearly. We compute the knee point by applyinglinear regression to the eigenvalues curve (Figure 3b) anduse the horizontal coordinate of this point as a rough esti-mate for the number of spectral clusters.Combining ClustersApplying spectral clustering on sampled data results ingraph partitioning, which is almost optimum in thesense of having minimum normalized cut [41,42]. How-ever, in some cases, a biologically interesting populationmight be split into two or more smaller clusters bySamSPECTRAL. We addressed this issue by adding a

post-processing stage wherein the partitions of a popula-tion are combined based on known properties of flowcytometry cell populations. Typically, biologically mean-ingful cell populations in flow cytometry data have theirhighest density at the centre, and their density decreasestowards the border of the population. Since higher den-sity regions indicate communities with relatively moremembers, the conductance between them is expected tobe relatively higher (Equation 4). Thus, similaritybetween communities is higher in regions with higherdensities and the highest similarity is expected to be atthe centre of the biological population. This observationforms the basis for our criterion for combining clusters.Specifically, similarity between communities determinesthe weight on graph edges and we define the maximumweight of the edges of a spectral cluster as within simi-larity of that cluster. Also, the maximum weight of theedges between two different spectral clusters is definedas between similarity. If the ratio of between similarityto within similarity is greater than a predefined thresh-old (separation factor), we conclude that these clustersare partitions of a single population, and should com-bine them to form a component. We repeat this stageuntil no two components can be combined. The finalcomponents computed in this way are called connectedcomponents of the data, and estimate the real biological

Figure 3 Defining the similarity between two communities and identifying the number of clusters. (a) We define the similarity betweentwo communities c and c’ as the sum of pairwise similarities between the members of c and the members of c’. (b) This figure shows thelargest eigenvalues of a sample from the stem cell dataset. The number of clusters is estimated according to the knee point of eigenvaluescurve. This point is defined as the intersection of the above regression line and the line y = 1. The horizontal coordinate of the knee pointestimates the number of spectral clusters.


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populations. With smaller separation factors, spectralclusters tend to combine more often.Overview of SamSPECTRAL AlgorithmIn summary, the stages of our algorithm are as follows,assuming the data contains n points in a d dimensionalspace of volume V, and the parameters m (max numberof communities), s (scaling parameter), and separationfactor are set properly.

1. Sampling:(a) Let h=12Vmd.(b) Repeat:

• Run faithful (biological information preser-ving) sampling algorithm. Suppose m′ com-munities are built.• Update: h h m

md= ( )′ .

Until m m m2 ≤ ′ ≤ .2. Compute the similarities between communities byadding pairwise similarities si, j defined by 3:

S sc c i j

j ci c

, , .′∈ ′∈

= ∑∑ (5)

3. Build a graph wherein each community is a ver-tex. Put edges between all pairs of vertices andweight them by similarity between correspondingcommunities.4. Analyze the spectrum of the above graph to findthe clusters;

(a) Normalize the adjacency matrix of the graphaccording to Equation 1.(b) Compute the eigenspace of the graph and setk, number of clusters, according to the kneepoint of eigenvalues curve.(c) Run classical spectral clustering algorithm tofind k clusters.

5. Combine the clusters to find connectedcomponents:

(a) Initiate the list of components equal to thelist of spectral clusters.(b) Repeat:

• For any pairs of components Ci, Cj , set:

separation ratiobetween simi arity

within simi ari_ :

( , )_

_=

1

1

Ci C jtty( , )Ci C j

(6)

• For each component Ci, compute:

M i C Cj i

i j( ) : max( _ ( , ))=≠

separation ratio

• If for all i, M(i) ≤ separation_factor, break.• Pick an i such that M(i) > separation_factorand let:

j C Cj i

i j=≠

arg separation ratiomax( _ ( , ))

• Combine Ci and Cj , then update list ofcomponents.

Until number of components > 1.

In the sampling stage, we start with the initial valueh V

md= 1

2 for the neighbourhood. m is a parameterthat controls m′, the final number of sample points suchthat m m m2 ≤ ′ ≤ . Since in our implementation, we useManhattan metric to measure the distance betweenpoints, the volume of a community can be estimated by( )2h d V

m= . Therefore if the the data points were dis-tributed uniformly in the space, we would get m samplepoints in the first run. However, in practice, we need torepeat the procedure after updating the neighbourhoodvalue. According to our experiments, a few iterationsare enough to fulfil the terminating conditionm m m2 ≤ ′ ≤ . As the running time of this part of Sam-SPECTRAL is O(nm), which is negligible compared toeigenspace computation time, we did not attempt tooptimize the sampling loop.Modified Markov Clustering Algorithm (MCL)Step 4 in the above algorithm is the classic spectral clus-tering method. This step potentially could be substitutedby any clustering algorithm for weighted graphs. To ver-ify that our approach is extensible in this sense, we sub-stituted classic spectral clustering with MarkovClustering (MCL) [43] keeping the rest of our algorithm,sampling and post-processing steps, unchanged.MCL finds the partitions of a graph by simulating flow

on the nodes. Simulation is done by iteratively multiply-ing two type of matrices that correspond to expansionand inflation operations [43]. Because flow and eigen-space of a graph are strongly related1, the outcome ofthis approach tends to be similar to spectral clusteringthrough computing eigenspace.

TestingWe implemented our algorithm with R, and applied iton four different datasets. We were able to identifysome types of biologically interesting populations thatwere previously known to be hard to distinguish, includ-ing:

1. Overlapping populations (Figure 4a-c).2. Subpopulations of a major population (Figure 4d-f).3. Non-elliptical shaped populations (Figures 5 andFigure 6a-c).4. Low density populations close to dense ones (Fig-ures 6d-f and Figure 7).5. Rare populations comprising less than 2% of alldata points (Figure 8).


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Figure 4 Comparative clustering of the telomere dataset. (a-c) Proper identification of overlapping populations. Although two populationsshown by red and blue contours are overlapping in all bi-variant plots of this 3-dimensional sample, SamSPECTRAL can properly distinguishthem by considering multiple parameters simultaneously.(d) SamSPECTRAL can also identify two major subpopulations of granulocytes correctly,as verified by expert analysis. (e) flowMerge does not distinguish between two populations of interest, and (f) FLAME improperly splits the samesample into several clusters.


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Here, we demonstrate the capabilities of SamSPEC-TRAL in identifying biological populations in thesecases and compare our results with two state ofthe art methods for clustering flow cytometry data,flowMerge (version 0.4.1) and FLAME (version 3),respectively obtained through BioConductor andGenePattern.Overlapping PopulationsTraditionally, identifying cell populations in flow cyto-metry data is accomplished by visualizing the multidi-mensional data as a series of bivariate plots, andseparating interesting sections manually, in a processtermed gating. Gating becomes challenging for highdimensional data since when the data is mapped totwo dimensions, some clusters may overlap, resultingin the mixing of different populations. Consequently,even a trained operator cannot identify overlappingpopulations properly in all cases. However, our algo-rithm prevents this undesired error by considering alldata dimensions together (Figure 4a-c). Model basedmultidimensional techniques also perform generallywell in this regard.Subpopulations of a PopulationFigure 4d-f shows a major blood population (granulo-cytes) formed from two distinct subpopulations as veri-fied by expert manual analysis. SamSPECTRAL couldclearly distinguish between two subpopulations. flow-Merge merged these two populations into one, whileFLAME split both subpopulations.Non-elliptical Shaped PopulationsWhile most model based techniques have a prioriassumptions on the shape of populations that resultedin mixing or splitting populations, our method workedrelatively well on the samples with arbitrary shape popu-lations. In Figure 5, the PI positive population (bluediagonal one) was clearly identified despite its non-

elliptical shape. flowMerge could also distinguish thispopulation, but it incorrectly split the PI negative popu-lation into two parts. FLAME did not correctly distin-guish the two populations. Figure 6a-c shows the outputof the three algorithms on a four dimensional samplefrom GvHD dataset. While the red population has acomplex shape, it could be identified with high accuracyby SamSPECTRAL. While FLAME produced a satisfac-tory result, flowMerge mixed this population with theone below it.Low Density Populations Close to Dense PopulationsFigure 6d-f shows a sample from GvHD dataset contain-ing a relatively low density and a high density popula-tion close together. SamSPECTRAL clearly distinguishedthe red population in the centre of the plot from theyellow dense population to its left. Moreover, it did notmix the red population with the other low density popu-lation to its right. FlowMerge also clustered this samplerelatively well, requiring five times more processingtime. The performance of FLAME was not satisfactoryfor this sample due to mixing the desired populationwith the other low density ones.Figure 7 depicts a sample from the stem cell dataset

containing a relatively low density population shown inblue. In each row, three 2-dimensional plots of the 3-dimensional data sample are presented. SamSPECTRALcould distinguish the blue population although it wassurrounded by three relatively denser populations (theyellow, green and red ones). FlowMerge mixed thispopulation with the yellow one, while FLAME mixed itwith the red one.Rare PopulationsIdentifying rare populations has many significant appli-cations in flow cytometry experiments including distin-guishing cancer stem cells, hematopoietic stem celltransplantation, detection of fetal cells in maternal

Figure 5 Comparative clustering of dead cells (PI positive) and live cells (PI negative) in the viability data. (a) SamSPECTRAL coulddistinguish between dead cells (blue) and live cells (red) properly. (b) flowMerge identified dead cells correctly, but split live cells into twoclusters. (c) FLAME did not distinguish between these two population.


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Figure 6 Comparative clustering of the GvHD dataset. (Left) Identification of non-elliptical shaped populations. (a) SamSPECTRAL couldproperly identify the red, non-elliptical population, while (b) flowMerge mixed this population with the one below it. (c) FLAME producedsatisfactory results in identifying this population. (Right) Identification of low density populations close to dense populations. (d) SamSPECTRALand (e) flowMerge could identify the low density population shown in red at the centre of the figure correctly, while (f) FLAME merged thispopulation with the other ones surrounding it.


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Figure 7 Comparative identification of a low density population surrounded by much denser populations in the stem cell data set. (a-c) SamSPECTRAL correctly identified the blue, low density population, while (d-f) flowMerge merged it to the yellow, high density population.(g-i) FLAME merged it to the red population. (j-l) The outcome of our modified MCL was similar to that obtained by SamSPECTRAL using classicspectral clustering. This shows that SamSPECTRAL is extensible by substituting classic spectral clustering with other clustering algorithms forweighted graph.


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blood, detection of leukocytes in leukocyte-depleted pla-telet products, detection of injected cells for biotherapyand malaria diagnosis [44].Figure 8 shows a typical sample from the stem cell data

set that contains a rare population in red. This popula-tion is positive for all the three markers and in each sam-ple, it comprises between 0.1% to 2% of total cells. Weperformed an experiment on 34 samples from the stemcell data set and compared the performance of Sam-SPECTRAL, flowMerge and FLAME. This rare popula-tion was distinguished manually and the result of manualgating was considered as the basis for our comparison.FLAME and flowMerge could identify this populationonly in 11 (32%) and 9 (26%) of samples, respectively.SamSPECTRAL could distinguish this population in

27 (79%) samples including all the ones that were iden-tified by FLAME and flowMerge. In the 7 (21%) samplesthat SamSPECTRAL failed, the rare population of inter-est contained less than 0.15% of all data points.To measure the accuracy of SamSPECTRAL, we

define sensitivity and specificity as follows. For eachsample, we call a cell positive if it belongs to the rarepopulation of interest, and it is negative otherwise. Sen-sitivity is defined to be the number of truly identifiedrare cells divided by the total number of rare cells.Accordingly, specificity is the number of cells identifiedas negative divided by the total number of truly negativecell. The 27 (79%) cases where SamSPECTRAL correctlyidentified the rare population, had a 0.83 mean sensitiv-ity with a 0.26 standard deviation. The median sensitiv-ity was .99. Specificity was 1 except for one sample. Ifwe consider the samples with a rare population biggerthan 0.2% of the total data, we obtained median = 1,mean = 0.93 and standard deviation of 0.15 for sensitiv-ity. A detailed report of the results of this experiment isprovided as a table in additional file 1.SamSPECTRAL with MCLFigure 7j-l depicts the output of MCL on a sample fromstem cells dataset. We ran MCL on the sampled data

obtained by our faithful sampling algorithm and thenthe post-processing step was applied to the resultingclusters. This experiment showed there was no signifi-cant difference for SamSPECTRAL in clustering eitherthrough computing eigenvectors (Figure 7a-c) or byMCL (Figure 7j-l)2.

DiscussionAlthough spectral clustering algorithm is a powerfultechnique, it can not be directly applied to large datasetsas it is computationally expensive both in time andmemory. In this study, we developed a sampling methodand combined it with spectral clustering by modifyingthe similarity matrix based on potential theory. As aresult, for the first time, analysing flow cytometry datausing spectral methods becomes possible and practical.We applied SamSPECTRAL to four different flow cyto-metry datasets to demonstrate its applicability on abroad spectrum of flow cytometry data, and comparedits performance to two state of the art model-basedclustering methods optimized for flow cytometry data.Detecting rare populations is a challenging problem

and in spite of its significant applications in medical andbiological research, little progress has been achieved inautomatic identification of such populations. Our datareduction scheme is delicate enough not to miss rarepopulations comprising between 0.2% to 2% of the totaldata. SamSPECTRAL can identify populations of relativesize in this range with acceptable accuracy.Since our method, SamSPECTRAL, is a multidimen-

sional clustering approach, it can identify overlappingpopulations that are generally hard to identify by man-ual gating that uses sequential two dimensional visuali-zations of the data. SamSPECTRAL is the first methodthat has demonstrated the ability to correctly identifysubpopulations of major flow cytometry cell populations.An important challenge in analysing flow cytometry

data is in clustering data files that contain populationsthat significantly differ in density. Model-based

Figure 8 Rare population in the stem cell data set. (a-c) This is a typical sample from the stem cell data set that contains a rare population.In these three dimensional plots, the red dots represent the cells that are positive for all three markers. Only 23/9721 (0.24%) events belong tothis population in this sample. SamSPECTRAL could properly identify the rare population in 27/34 (79.4%) samples from the stem cell data set.


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techniques can produce errors in identifying a low den-sity population close to denser populations because theytypically make assumptions on the density of clusters[23]. Our experiments demonstrated that SamSPEC-TRAL can properly tackle this problem. Besides thepractical observations, this capability is justified by thefollowing observation. Spectral methodology clusters thegraph such that the normal cut is “almost” optimum[41]. Now, assume that it can distinguish between twoclusters when their densities are comparable. Then, ifthe size of the smaller cluster is reduced without changein its shape or distribution, the normal cut betweenthem remains similar because the number of verticesand edges reduces almost proportionally to each other.Therefore, the clusters remain distinguishable. Thisexplains why the overall performance of SamSPECTRALis independent of cluster densities as long as theirshapes are preserved.Since parametric methods such as FLAME and flow-

Merge make a priori assumptions on the distributionor shape of the clusters [23], they may fail in identify-ing populations with arbitrary shapes. Although flow-Merge attempts to solve this issue by finding moreclusters than needed and then merging them together,it still does not produce satisfactory results when theshape of the cluster is complex. SamSPECTRAL hasthe capability of identifying arbitrary shape clusterssince it is a non-parametric approach that makes noassumptions on the shape and distribution of clusters,and clusters data based only on similarity between datapoints. Compared to other non-parametric methods,our algorithm has the advantages of automaticallyidentifying the number of clusters and having low sen-sitivity to the predefined thresholds. Therefore, userscan adjust the parameters only once by running Sam-SPECTRAL on one or two random samples from aflow cytometry data set. Then, the algorithm can berun on the rest of data set without changing theparameters.Not only does our sampling scheme increase the

speed of spectral clustering without losing importantbiological information, but the resulting algorithm is fas-ter than other methods considered in this study. Moreprecisely, the running time of SamSPECTRAL is O(dmn) + O(m3) where O(dmn) is the running time forbuilding m communities from n points in d dimensionand O(m3) is the running time for computing the eigen-space. After this step, the k-means clustering runs veryfast in time O(k m t) to find k clusters using eigenvec-tors by t iterations. In comparison, the time complexityof the original MCL method is O(nr2) with no guaranteeon upper bound for number of iterations r, other thann. Practically, for our model of flow cytometry datawhere all pairs of data points are connected, we could

not run MCL before applying our modification to it.Moreover, SamSPECTRAL running time is significantlyless than model-based techniques. The running time offlowMerge is O(d2k2nt) and FLAME runs in time O(d4klnt) where l is the number of times it runs to findthe optimal number of clusters. In practice, we can keepm as small as 1500-3000 without loosing important bio-logical information, and consequently SamSPECTRALran at least 5-10 times faster than flowMerge andFLAME on the studied datasets. Furthermore, the timeefficiency of our algorithm is more noticeable for higherdimensional data such as the one provided as additionalfile 2. This sample contains 100,000 events in 23 dimen-sions and SamSPECTRAL can analyze it in less than25 minutes by a 2.7 GHz processor.

ConclusionsFaithful sampling is based on potential theory. It reducesthe size of input for spectral clustering algorithms andconsequently they can now be efficiently applied onflow cytometry data in spite of its large size. Practically,our approach demonstrated significant advantages inproper identification of populations with non-ellipticalshapes, low density populations close to dense ones,minor subpopulations of a major population, rare popu-lations, and overlapping populations. No state of the artmethod can solve the challenges in identifying popula-tions with the above properties simultaneously. More-over, applying SamSPECTRAL to other biological datasuch as microarrays and protein databases may result insignificant improvements in gene expression and proteinclassification.Besides, our faithful sampling algorithm can have

interesting applications by itself. For instance, it can beused appropriately to reduce the size of input for otherclustering algorithms that are based on spectral graphtheory such as Markov Clustering Algorithm (MCL),electrical circuit based clustering, and agent based graphclustering [45]. We have shown the extendibility of ourapproach in this sense by substituting classic spectralclustering with MCL, a method that has many applica-tions in bioinformatics.Other directions for future work include applying

other schemes for estimating similarities between com-munities, combining clusters based on other combina-torial algorithms or biological criteria, and repeating thealgorithm several times to obtain a more stableoutcome.

MethodsTo run flowMerge and FLAME optimally, we used sev-eral settings for their parameters, finally selecting thosethat gave us the best results. For SamSPECTRAL algo-rithm, we set m = 3000 to keep the running time bellow


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1 minute by a 2.7 GHz processor and the obtainedresults remained satisfactory for all samples we analyzed.The separation factor and scaling parameter (s) are twomain parameters that needed to be adjusted. Decreasings and increasing the separation factor will result inidentifying more populations. In particular, if s isdecreased, then according to the heat kernel formula,the weights on the edges of the graph will decreaseexponentially. Therefore, the graph will be sparser andtends to obtain more partitions. In consequence, thealgorithm identifies more spectral clusters. This phe-nomenon can be useful in identifying rare populations.On the other hand, if separation factor is too high, asingle population may be split into parts. In our experi-ments, we applied SamSPECTRAL on one or two ran-dom data samples of a data set and tried differentvalues. Then, the selected parameters were fixed andused to apply SamSPECTRAL on the rest of data sam-ples. The parameters values for the data sets presentedin this paper are provided in additional file 3, 4, 5, and6. The reader is referred to the SamSPECTRAL Biocon-ductor package vignette for more explanation on howto adjust parameters for a given data set3.

DatasetsWe tested our algorithm on four different flow cytome-try datasets as explained briefly here. The GvHD datasetis available in flowCore package through BioConductor,and the rest are available upon request.Stem CellsTo investigate heterogeneity in the differentiation beha-viour of hematopoietic stem cells, a subpopulation ofadult mouse bone marrow was isolated and then eachsingle stem cell was transplanted into one of 352 recipi-ents [46]. 16 blood samples were taken from the recipi-ents in biweekly intervals and were studied in acytometer. The investigation contained hundreds of datafiles that needed to be analyzed to count the frequencyof each subtype of white cells they contained.TelomereIn all vertebrates, telomeres consist of tandem DNArepeats of the sequence d(TTAGGG) and associatedproteins. Telomere length is known to be crucial ele-ments in ageing and various diseases including cancerand it can be estimated by flow cytometry [47]. Sincetelomere length is different for various cell populations,these need to be distinguished before calculating telo-mere length.GvHDAcute graft versus host disease (GvHD) is a commonoutcome after bone marrow transplantation. It is diffi-cult to diagnose in its early stages in order to providetimely treatment. To investigate how flow cytometry canhelp predict the development of GvHD, and to study its

advantages over microarrays, peripheral blood samplesfrom 31 patients undergoing allogeneic blood and mar-row transplant were analyzed [48]. The samples weretaken at progressive time points post-transplant andwere stained with four appropriate lymphocyte phenoty-pic and activation markers defining 121 different popu-lations using six markers.ViabilityPropidium iodide (PI) is a widely used marker for deter-mining viability of mammalian cells [49] because it hasthe capability of passing through only damaged cellmembranes. However, depending on the complexity ofthe data, identifying dead cells automatically might stillbe difficult even if this marker is used. We tested thecapability of our algorithm in identifying dead cellsusing PI marker on a dataset from the Terry FoxLaboratory.

Appendix 1In the results section, we explained that the resolutionof the sample points (Figure 2) is high enough such thatby repeating the randomized faithful sampling proce-dure, the outcome of SamSPECTRAL does not vary sig-nificantly. The following experiment is performed toconfirm this observation quantitatively. In this experi-ment we used F-measure, which is known to be appro-priate for comparing clustering results of flow cytometrydata [50]. F-measure varies in range 0-1 and reaches itsbest value at 1 when the two clustering results are iden-tical. We ran SamSPECTRAL on a sample from thestem cell data set 20 times and compared the finalresults. The F-measure values obtained by pairwise com-parison between the final results had mean = 0.98, med-ian = 0.98 and standard deviation 0.0097.

Appendix 2We performed the following experiment to show theeffect of edge weights on performance of spectral clus-tering. As shown in Figure 9, we produced syntheticdata containing one normal distribution with relativelyhigh density surrounded by four relatively small clus-ters with lower densities. The number of points ineach small cluster is less than 0.01% of the wholedata and noise is added to the data space uniformly(Figure 9a). For the central dense distribution, we setsxx = syy = 2,sxy = syx = 0 and the surrounding clusters are normal

distributions with xx yy xy yx

1 1 1 10 08 0 30 0= = = =. , . , , xx yy xy yx

2 2 2 20 07 0 08 0= = = =. , . , , xx yy xy yx

3 3 3 30 50 0 10 0= = = =. , . , , xx yy xy yx

4 4 4 40 10 0 70 0= = = =. , . , .The R code to produce this synthetic data and run

SamSPECTRAL on it is provided in additional file 7.


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After faithful sampling is done (Figure 9b), the sam-ple points are distributed almost uniformly, and theinformation about the local density of original data islost. However, faithful sampling provides us with moreinformation than only the sample points. It will alsoreturn the members of each community and our datareduction scheme uses this information to assign

weight to the edges. According to formulas 3 and 4,the more populated and closer two communities are,the higher the weight between them will be (Figure1c). According to Figure 9c, this strategy is successfulin retrieving information about local density as all thefive clusters are distinguished properly bySamSPECTRAL.

Figure 9 Performance of SamSPECTRAL on synthetic data. (a) This synthetic two dimensional data consists of a normal distribution with30,000 points, four normal distribution each with 300 points and a uniform background noise with 4000 points. (b) Around 3000 sample pointsare picked up by faithful sampling. These are distributed almost uniformly in the space, therefore, almost all information about density will belost if one considers only the samples points. (c) The final outcome of SamSPECTRAL confirms that the information about density could beretrieved by properly assigning weights to the edges of the graph. The high density cluster is shown in red and the surrounding sparser clustersare shown in yellow, light blue, green and black.

Figure 10 Comparing Uniform sampling with faithful sampling. Directly applying classical spectral clustering is not efficient on this sampleof the stem cell dataset which contains 48000 cytometry events in 3 dimensions. (a) Although only 2115 data points were selected by faithfulsampling, each population has a considerable number of representatives in the selected points. (b) 3000 points were selected by uniformsampling. The low density population in the middle of the plot consists of only 55 sample points resulting in mixing this population with a highdensity one incorrectly (d). (c) The result of SamSPECTRAL on the original data is satisfactory because the low density red population and otherhigh density populations are identified properly.


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Appendix 3We observed that some low density populations disap-peared entirely when simple uniform sampling wasemployed. To investigate the effect of this phenomenon onthe final clustering results, we performed an experiment ona sample of the stem cell dataset that contained 48,000events in 3 dimensions. First, 3,000 data points wereselected uniformly at random. Then, we assigned a label toeach of these selected points by applying classical spectralclustering on them. Finally, for each original data point, thelabel of the closest selected point was considered as its clus-ter label. Figures 10d and 10c show the results of thisapproach and SamSPECTRAL, accordingly. The red popu-lation that was distinguished by SamSPECTRAL correctlyin Figure 10c consists of only 4% of the data. This popula-tion could not be distinguished properly by any setting ofthe parameters after uniform sampling (Figure 10d).

Footnotes1 Cheeger inequality is an example of such a relation[41].

2 The CD45+ cells that are considered as outliers byMCL are not plotted in Figure 7j-l.

3 The vignette is located at: http://bioconductor.org/packages/devel/bioc/vignettes/SamSPECTRAL/inst/doc/Clustering_by_SamSPECTRAL.pdf

Additional material

Additional file 1: Report on identification of rare population. Thetable contains the full detailed report on our comparative experiment foridentifying rare populations.

Additional file 2: High dimensional flow cytometry data. This datafile contains a matrix with 100,000 rows and 23 columns that representsa flow cytometry sample with 100,000 events. It can be directly loaded inR and analyzed by SamSPECTRAL. It takes less than 12 minutes toperform faithful sampling on this 23 dimensional data.

Additional file 3: Parameters for GvHD data set. These values areappropriate for running SamSPECTRAL on GvHD data set.

Additional file 4: Parameters for stem cell data set. These values areappropriate for running SamSPECTRAL on stem cell data set.

Additional file 5: Parameters for telomere data set. These values areappropriate for running SamSPECTRAL on telomere data set.

Additional file 6: Parameters for viability data set. These values areappropriate for running SamSPECTRAL on viability data set.

Additional file 7: Simulation with synthetic data. This R source codeproduces synthetic data with 5 clusters shown in Figure 5. The resultingdata is passed to SamSPECTRAL to be clustered.

AcknowledgementsThe authors would like to thank Adrian Cortes, Connie Eaves, Peter Lansdorpand Keith Humphries for providing data, Bari Zahedi and Irma Vulto for theirbiological insight and manual data analysis, Josef Spidlen for assistance inrunning FLAME, Aaron Barsky and Nima Aghaeepour for their editorialcomments, and Mani Ranjbar for programming guidance. This work wassupported by NIH grants 1R01EB008400 and 1R01EB005034, the Michael

Smith Foundation for Health Research, the National Science and EngineeringResearch Council and the MITACS Network of Centres of Excellence.

Author details1Department of Computing Science, University of British Columbia,Vancouver, BC, Canada. 2Terry Fox Laboratory, BC Cancer Agency, 675 W10th Ave., Vancouver, BC, Canada. 3Faculty of Science, University of BritishColumbia, Vancouver, BC, Canada. 4Department of Medical Genetics,University of British Columbia, Vancouver, BC, Canada.

Authors’ contributionsThe authors wish it to be known that, in their opinion, the first two authors,HZ and PS, should be regarded joint first authors. HZ designed andimplemented the faithful sampling algorithm, and the method forcomputing similarity between communities. PS developed the idea of non-uniform sampling for spectral clustering and experimentally verified thestability of the algorithm. HZ and PS jointly worked on the method forestimating the number of clusters, and post-processing steps. HZ and PSperformed the experiments. HZ, PS and RB wrote the paper. RB provideddata and computing facilities. AG studied the convergence of faithfulsampling. AG and RB supervised the project. All authors read, edited andapproved the final manuscript.

Received: 21 December 2009 Accepted: 28 July 2010Published: 28 July 2010

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doi:10.1186/1471-2105-11-403Cite this article as: Zare et al.: Data reduction for spectral clustering toanalyze high throughput flow cytometry data. BMC Bioinformatics 201011:403.

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Data reduction for spectral clustering to analyze high ...take 2 years and 5 terabytes of memory to analyze a typical flow cytometry sample with 300,000 events. We developed a novel

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