Eurographics Conference on Visualization (EuroVis) 2019M. Gleicher, H. Leitte, and I. Viola(Guest Editors)

Volume 38 (2019), Number 3

ClustMe: A Visual Quality Measure for RankingMonochrome Scatterplots based on Cluster Patterns

Mostafa M. Abbas1, Michaël Aupetit1, Michael Sedlmair2, and Halima Bensmail1

1 QCRI, HBKU, Doha, Qatar2 VISUS, University of Stuttgart, Germany

AbstractWe propose ClustMe, a new visual quality measure to rank monochrome scatterplots based on cluster patterns. ClustMe isbased on data collected from a human-subjects study, in which 34 participants judged synthetically generated cluster patternsin 1000 scatterplots. We generated these patterns by carefully varying the free parameters of a simple Gaussian Mixture Modelwith two components. and asked the participants to count the number of clusters they could see (1 or more than 1). Based onthe results, we form ClustMe by selecting the model that best predicts these human judgments among 7 different state-of-the-artmerging techniques (DEMP). To quantitatively evaluate ClustMe, we conducted a second study, in which 31 human subjectsranked 435 pairs of scatterplots of real and synthetic data in terms of cluster patterns complexity. We use this data to compareClustMe’s performance to 4 other state-of-the-art clustering measures, including the well-known Clumpiness scagnostics. Wefound that of all measures, ClustMe is in strongest agreement with the human rankings.

CCS Concepts•Human-centered computing → Visual analytics; Empirical studies in visualization; • Computing methodologies → Clus-ter analysis; Mixture modeling;

1 Introduction

In visual analytics, users often need to analyze many differentviews of multidimensional data with the goal to identify interesting“patterns” and unforeseen relations. The most typical way to do sois by using scatterplots, in which the axes are either defined by pairsof the original data dimensions, or as new dimensions synthesizedby some dimension reduction technique [SMT13] [NA18].

During the visual analytics process, hundreds of such scatter-plots might be generated, and the user needs to visually skimthrough them one-by-one looking for visual patterns. When an in-teresting pattern is spot, often a more thorough analysis follows,with advanced interaction and connections with other views orknowledge to explain the identified pattern. The initial skimmingtask is an exploration task as per Brehmer and Munzner’s typol-ogy [BM13]. It is a perceptual task focusing on patterns with-out interpretation, but it is tedious when the number of views tobe skimmed gets large. To better support this process, the visual-ization community has proposed various Visual Quality Measures(VQM) [BTK11,BBK∗18] that help guiding the users towards per-ceptually interesting patterns. To do so, VQMs quantify certainvisual patterns, which then allow ranking visualizations based ontheir potential interest for the user. Time and cognitive effort can besaved for later analytical tasks. According to Bertini et al. [BS06],VQMs should thus ideally be based on perceptual models rather

than heuristics and computational approaches. A perceptual VQMis then a measure that imitates how humans would score viewsbased on perceived visual patterns and that can be used to accu-rately predict human perceptual judgments in the skimming task.

There has been a considerable amount of work on computa-tional and perceptual VQMs [BBK∗18], including some whichcharacterize different patterns in color-coded [SA15, AS16] andmonochrome scatterplots [WAG05,MTL18]. So far, however, therehas been very little work on modeling the human perception ofgrouping patterns in monochrome scatterplots and using thesemodels to design VQMs. This is surprising as it is a very typicaltask in visual analytics [BSIM14] as these groups in 2D scatter-plots can reveal important relationships between multidimensionaldata or dimensions.

In this work, we set out to fill this gap by developing for the firsttime a VQM based on perceptual data, to rank monochrome scatter-plots depending on their grouping patterns. To do so, we collectedperceptual data from a human subject experiment and selected thebest clustering model of this data to form a perception-based VQMof grouping patterns. We thus call our new VQM: ClustMe (shortfor Clustering Measure).

Closest to our work is the Clumpiness Scagnostic [TT85,WAG05]. However, Clumpiness is not based on perceptual data;

c© 2019 The Author(s)Computer Graphics Forum c© 2019 The Eurographics Association and JohnWiley & Sons Ltd. Published by John Wiley & Sons Ltd.

Abbas et al. / ClustMe: A Visual Quality Measure for Ranking Monochrome Scatterplots based on Cluster Patterns

hence, it is unclear how well it matches with human judgments.Also, in theory any clustering technique could be used to form acomputational VQM and to rank monochrome scatterplots. How-ever, most clustering techniques [XW05] come with certain as-sumptions and parameters that need to be set by the user. In theend, such a process would be tedious and require in-depth knowl-edge on clustering techniques from the user.

To overcome these issues, we propose the new ClustMe measure,which is based on three contributions:

(1) In the Experiment 1, we collected 34000 human judgmentsfrom 34 subjects tasked to count clusters in 1000 monochromescatterplots. Following a generative approach [SNE∗16], the scat-terplots were generated from systematically varying parameters ofa mixture of two Gaussian distributions. This allowed us to pre-cisely control the visual appearance and collect low-level percep-tual judgments [SA15], fulfilling the precision criterion describedby the McGrath’s model of research methodology [Mcg95].

(2) Through modeling these data, we developed the novelClustMe VQM as a Gaussian Mixture Model (GMM) [BC96,BCRR97, FR02] trained using the Bayesian Information Crite-rion (BIC) [Sch78] and refined with the Demp merging technique[Hen10]. We arrived at ClustMe, by systematically exploring howwell different merging techniques model the data collected in Ex-periment 1. In doing so, we bridge between the perception of sim-ple, measurable mixture of two Gaussian clusters from Experiment1, and complex cluster structures that can be found in the real worldand modeled by a GMM plus a merging technique.

(3) In the Experiment 2, we collected 13485 human judgmentsfrom 31 subjects ranking 435 pairs of monochrome scatterplots ofreal and synthetic data. We compare these human-made rankingswith ClustMe, with the Clumpiness Scagnostic [TT85, WAG05],and with 3 standard clustering techniques: X-means [PM00],CLIQUE [AGGR98] and DBSCAN [EKSX96]. The main goal ofthis study was to evaluate how ClustMe and other approaches per-form under realistic conditions, fulfilling the realism and generaliz-ability criteria of the McGrath’s model [Mcg95]. The main resultsshow that ClustMe best agrees with the human raters.

2 Related Work

We discuss related work on visual quality measures based on per-ceptual data, as well as on visual and algorithmic clustering tech-niques.

2.1 Visual Quality Measures based on perceptual dataMany empirical studies on perceived patterns in scatterplotshave been conducted. Closest to our work are studies regardingmonochrome cluster patterns. Here, a statistical model of clusterperception in scatterplots based on proximity, density change, andconcentration of the points has been proposed [Sad97], based on acontrolled study involving 36 subjects and 24 stimuli. Another con-trolled study explored the perceived principal axis of a set of pointswhen manipulating the separation between a small subset of pointsand a main ellipsoidal cluster [CSM08]. The human perception ofhom*ogeneous dot patterns has been studied varying densities andgaps between two square-shaped clusters [O’C74]. The study in-

volved 8 subjects and 24 patterns. Human perception of the num-ber of dots in scatterplots is also evaluated to analyze how thesepatterns are perceived as texture shapes or single dots dependingon the density [ACB16]. All these studies investigate a small sam-ple of possible cluster shapes, and the data was not collected toserve the design of visual quality measures of cluster patterns.

The survey of Bertini et al. [BTK11] gives an overview over Vi-sual Quality Measures (VQMs). VQMs build on research in imagequality metrics [LK11] but are more specifically designed to quan-tify patterns in data visualization. They can support human explo-ration by automatically filtering, ranking, and mapping visual en-codings based on patterns that might be of interest to a human ob-server [TMF∗12, DW14]. Most VQMs are based on heuristics, buta recent trend focuses on designing measures based on perceptualstudies. A study by Pandey et al. [PKF∗16] explored how humansubjects perceive and name different patterns in monochrome scat-terplots, giving evidence that VQMs should align with visual per-ception to reproduce human judgments accurately, as it was positedby Bertini et al. [BS06,BTK11]. An expansive taxonomy and qual-itative analysis more specific to grouping patterns in color-codedand monochrome scatterplots gives insight on the many factorsat play in cluster perception [STMT12]. Beyond the number ofclusters, other factors are proposed like the shape of the groups,their relative size or their density to name a few. This taxonomycan guide the design of a visual quality measure based on per-ceptual data for cluster patterns. The first line of work to cre-ate such perception-based measures was on correlation in scatter-plots [RB10,HYFC14,KH16] and parallel coordinates [LMvW10].Matute et al. [MTL18] proposed a skeletonization process whichcaptures both shape and orientation of the scatterplot to measuresimilarity between scatterplots, and evaluate it with a human sub-ject experiment.

A more data-driven approach was proposed by Demiralp et al.[DBH14]. From human subjects, they learned the perceptual simi-larity between marks with different shape, size and colors and ap-plied this similarity measure to support the choice of mark designin scatterplots. Albuquerque et al. [AEM11] describe a perception-based metric derived from pairwise comparison judgment of differ-ent reference views by human subjects, and an eigenface decompo-sition of the images to build a perceptual space based on the ref-erence views. A new view can be projected in the same perceptualspace and its similarity to the reference views can be measured. Ithas been tested on a correlation task in monochrome scatterplotsand a class-separation task in color-coded scatterplots. Sedlmairand Aupetit [SA15] proposed a machine learning framework touse data collected from perceptual human subject studies to selectcomputational VQMs that best match the human visual perception.Based on this framework, they designed 2002 new measures opti-mizing the match between patterns and human perception [AS16].In both cases, these VQMs were designed to quantify class sepa-ration in color-coded scatterplots. In this work, we also follow adata-driven process, but focus on quantifying the grouping patternsin monochrome scatterplots [BSIM14].

2.2 Quantifying Clusters in Scatterplots

We found only two VQMs that directly quantify (monochrome)grouping patterns. Most closely to our work is Clumpiness [TT85]

c© 2019 The Author(s)Computer Graphics Forum c© 2019 The Eurographics Association and John Wiley & Sons Ltd.

Abbas et al. / ClustMe: A Visual Quality Measure for Ranking Monochrome Scatterplots based on Cluster Patterns

a

1. Generate Gaussian Mixture Models 3. Task Human Subjects

4. Run Merging Techniques

1000 (+20) 34

Q(Q,X)1X

(Q,X)146

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5. Compare2. Generate Scatterplots

(Qi), X i

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6. ClustMe process

M * sep

…

GMM+BIC

(C,MBIC)

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(5,9)

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GMM“1”

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M 1 sep i

M 7 sep i

Experiment 1: collecting simple cluster separation judgments

Experiment 2: collecting pairwise judgments on how much scatterplots are clustered

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Which is more structured/clustered/clumpy?

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“Right”“Left”

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30 pictures - 435 pairs

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257 pictures

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…

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Other VQMs (Clumpiness, X-means, CLIQUE, DBSCAN)

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Use: Compare ClustMe and other VQMs vs Human pairwise ranks

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(8,9) (7,9) (7,7) (6,9) (5,5) (4,5) (3,7) (2,8) (1,3)(2,6)

0.87 0.79 0.78 0.67 0.54 0.52 0.43 0.42 0.39 0.23

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Figure 1: Experiment 1: (1) The space of possible grouping patterns Θ is defined by a bivariate Gaussian Mixture Model (GMM) with 2components. (2) 1000 scatterplots Xi are generated by the GMM Θi. (3) 34 human subjectsH are asked to judge if they see 1 or more than 1cluster for each Xi giving a probability of separationHi

sep. (4) 7 merging techniquesMm are run to evaluate each scatterplot (Θi,Xi) givinga separation decisionMi

sep. (5) Both, human and machine decisions are compared to find the merging techniqueM∗sep that best matchesthe human judgment with the Mathew Correlation Coefficient and the Vanbelle Kappa index. The result of this process is our VQM ClustMe.(6) ClustMe can then be used to automatically quantify grouping patterns in new scatterplots (red frame): it runs a GMM for which wefind the optimal number of components MBIC according to the Bayesian Information Criterion (BIC). Then, we use the best mergerM∗sepapproximating the human perception to decide about merging pairs of the MBIC components to get the final number of clusters C. The pair(C,MBIC) defines the ClustMe Visual Quality Measure of the scatterplot. ClustMe VQM allows ranking scatterplots based on the complexityof their grouping patterns.

first proposed by Tukey in 1985. It is a computational VQM part ofthe Scagnostics framework proposed by Wilkinson et al. [WAG05].Clumpiness relies on a statistic of edge lengths of a minimum span-ning tree of the data points. It is parameter-free and thus does notrequire hand-tuning of parameters. But it was not intended orig-inally to resemble human perception. Hence, it has limitations inthe kind of visual grouping patterns it can detect, and is often notconsistent with the human perception of clusters (see Supplemen-tary Material Figure 3).

CLIQUE is a heuristic, density-based clustering technique whichpartitions the data space into a grid and searches for rectangularareas of high density [AGGR98]. It has been used for interactivedimensionality reduction [JJ09], however it is not based on percep-tual data and requires setting the number of cuts per dimensions todefine the grid, and the density threshold parameters.

In Section 6, we will compare ClustMe with both, Clumpinessand CLIQUE, as well as with other standard clustering techniques:X-means [PM00], and DBSCAN [EKSX96].

3 Design Considerations

We first briefly outline the methodological process we follow in thiswork, and then discuss our choice to take Gaussian Mixture Modelsas the general modeling framework.

3.1 Design Process: Data-DrivenOur work follows the data-driven approach for designing VQMsbased on perceptual data as outlined by Sedlmair and Au-petit [SA15]. This process includes several critical steps. (1) Asa first step, we need to gather a large set of scatterplots containinga sufficient variation of the pattern of interest. (2) We then need totask human subjects to judge if (or how) they perceive this patternin these scatterplots. (3) Afterwards, we look for an automatic tech-nique that can detect this pattern, and compare it with the collected

human judgments using statistical approaches. (4) The best of thesetechniques is declared a “model” of the human perception of thispattern and can be used as a VQM for it.

In this work, we use this approach to build a VQM for visualclustering tasks. Specifically, we will assume a Gaussian mixturemodel and learn from human data which merging technique to use.This choice allows us to combine the generic nature of GMM tomodel complex cluster shapes, and the simplicity of merging de-cision to accurately model the collected human judgments. In thefollowing, we describe the reasoning behind these components inmore detail. The overall process is summarized in Figure 1.

3.2 Modeling Approach: Gaussian Mixture Models (GMMs)

The main goal of our work is to design an algorithm to quantifycluster/grouping patterns in monochrome scatterplots on par withhuman perception. Clustering techniques [XW05] assign points toclusters without supervision. So, a way to design a VQM for group-ing patterns could be to use these techniques to group data first bymimicking human perception, and then to quantify the character-istics of these well-defined clusters to get the VQM. However, asexplained by Von Luxburg et al. [vLWG12], there is not a singleway to define a cluster and each clustering technique can detecta formally different family of grouping patterns. Moreover, we donot want to cluster data per se in a way that a human would men-tally assign points to groups, but we are more flexible in aiming fora score that would allow ranking scatterplots based on how muchthey are clustered according to human visual perception. Therefore,the model we need requires three key characteristics:

• Being generative: In order to collect data from human subjectstudies, we need a model that is able to generate interesting stim-uli, i.e. grouping patterns of various shapes.• Being universal: We need a model that can be trained to accu-

rately represent a large range of realistic grouping patterns from

c© 2019 The Author(s)Computer Graphics Forum c© 2019 The Eurographics Association and John Wiley & Sons Ltd.

Abbas et al. / ClustMe: A Visual Quality Measure for Ranking Monochrome Scatterplots based on Cluster Patterns

the most simple to the most complex, likely to appear in unseenscatterplots.• Being parameter-free: We need a model which could relieve

the user from arbitrary hand-tuning of parameters.

Gaussian Mixture Models (GMMs) [BC96,BCRR97,FR02] andtheir extension with merging techniques [Hen10] have all thesecharacteristics at once. In general, GMMs can model—with de-sired accuracy—any simple or complex data density distribution bya mixture of multivariate Gaussian distributions. Data points are as-signed to Gaussian cluster components with some probability basedon their distance to the cluster center. A hard assignment (partition)can be obtained by assigning points to the cluster from which theyget maximum probability. Most importantly, however, GMMs havethe three key characteristics that are important for our design goals:

GMMs are generative models: GMMs can be used to generatesample points in a scatterplots with the desired statistical proper-ties. Thus, they give natural handles for a controlled human sub-jects experiment by letting us vary the volume, orientation, densityand overlap of the generated Gaussian clusters. In doing so, we cancontrol the complexity of the resulting grouping patterns. No otherclustering technique allows this control.

GMMs are universal models: GMMs can model any smooth datadistribution [PS91]. We can infer their parameters based on sampledata points using statistical inference (learning from data) [FR02].Each component of a GMM identifies a single Gaussian cluster.However, in practice it is possible for a grouping pattern with acomplex non-Gaussian distribution to be modeled by several over-lapping components of the GMM. Yet, the GMM—being a den-sity function—does not encode explicitly the fact that several ofits Gaussian components can represent points from what would beperceived as the same single non-Gaussian cluster (points form-ing a banana shape, for instance in Figure 3 right). Merging tech-niques [Hen10] have been proposed to solve this issue based onthe natural assumption that a wide density gap between two com-ponents call for leaving them separated, and a strong overlap formerging.

Indeed, merging techniques can automatically decide if pairs ofGMM components shall be merged or remain separated. Merg-ing would mean that the components represent data from the samecluster (all within the banana shape, for instance), while separatedwould typically indicate a density gap between them. These merg-ing techniques extend the GMM framework, making it a muchmore generic model (GMM+Merging) for clustering. Specifically,it allows to identify and quantify non-trivial grouping patterns inmany practical cases. On top of that, deciding to merge clusterswith an algorithm is directly linked to a simple cluster-countingtask that we can ask to a human, as we analyze in the next section.Hennig studies 7 of these merging techniques [Hen10] that we willevaluate in our experiments.

GMMs are parameter-free models: GMMs allow determiningthe geometrical characteristics of a cluster (volume, density andorientation), and the optimal number of clusters based on statis-tical criteria. This process runs fully automatic, making GMMs aparameter-free model. A typical approach in GMMs is to use theBayesian Information Criterion (BIC) [Sch78] to select the model

which best compromises between high likelihood and low com-plexity (a regularization technique to avoid over-fitting, as recom-mended for any data-driven model).

In summary, GMM+Merging allows us to easily gather largeamounts of human judgments on simple patterns, while at the sametime the resulting VQM will be able to independently depict com-plex cluster patterns. In the sequel, we will describe each step of ourprocess in more detail: (1) We fist present the collection process ofhuman judgments. (2) Based on this data, we find the best mergingtechnique to model human cluster perceptions, forming the founda-tions of ClustMe. And (3) we evaluate ClustMe in a second humansubjects experiment.

4 Experiment 1: Gathering Data for Modeling

In this section, we describe Experiment 1, in which we collecteddata from a cluster-counting task. The resulting data will later beused to find the best merging technique to model human judgmentsforming our VQM.

4.1 TaskWe asked participants to perform a simple and fast perceptual taskthat requires to observe a single scatterplot at a time and decide ifit contains 1 or more-than-1 clusters.

Picking this task was based on a careful consideration of differ-ent alternatives. Different tasks regarding pattern perception havebeen proposed either to judge how two patterns are similar, or tojudge some characteristics of a single pattern. The similarity be-tween patterns can be measured by relative ranking using someLikert scale [TBB∗10] or by manual grouping of similar scatter-plot based on their thumbnail images [DBH14, PKF∗16]. Clusterpatterns can also be characterized by identifying their elements us-ing a lasso selection, or the overall pattern evaluated by countingthe number of clusters [EMdSP∗15].

In contrast, we decided to focus on even simpler patterns, charac-terizing the perception of two clusters only. The key element of ourproposition is to use a counting approach with a GMM of 2 Gaus-sian components as it is amenable to very fast decisions by humans.This allows us to collect amounts of data, which are beneficial forrobust learning approaches. Moreover, the collected data can be di-rectly modeled by the merging stage of a GMM+Merging cluster-ing technique: indeed, for a merging technique to decide to mergeor not to merge two Gaussian components of a GMM is equiva-lent to a human subject deciding if she can see 1 or more-than-1groups respectively, in a scatter plot generated by such a GMMwith 2 components. We call this 2D-Gaussian-Clusters-merging-perception task, the 2DGCMP task for short. Hence, our proposal isthat a GMM+Merging model for clustering, where the merging partbest mimics the way human judge on the 2DGCMP task, would bea good candidate for a VQM of more complex grouping patterns.

4.2 Generating Scatterplot StimuliWe generated 1000 scatterplots varying the parameters of a Gaus-sian mixture of 2 components. Using a stratified sampling ap-proach, we ensured to cover the large space of possible shapes gen-erated by the GMM, including well-separated clusters and moreoverlapping ones (see Figure 2). In the following, we provide moredetails on each of these steps.

c© 2019 The Author(s)Computer Graphics Forum c© 2019 The Eurographics Association and John Wiley & Sons Ltd.

Table 1: Selected values of the GMM parameters.

Param. Description ValuesN Number of points {100, 1000}τ Prior probability of

component u.{0.2, 0.3, 0.4, 0.5}

µ Distance betweencomponents u and v

{0, 1, 2, 3, 5, 8, 13, 21}

σa,bu,v Scaling factors {0.5, 1, 1.5, 2, 2.5, 3}

θu, θv Rotation angles {0, π/8, π/4, 3π/8, π/2 }

Generating data points from a Gaussian Mixture Model:Each scatterplot displays a set of N points x ∈ R2 drawn fromthe distribution generated by a mixture model with M bivariateGaussian components gi with distribution gi(x) = g(x|µi,Σi) =

det(2πΣ)−12 exp(− 1

2 (x−µi)>

Σ−1i (x−µi)). µi is the 2-dimensional

mean vector of the component gi determining the center positionof the Gaussian cluster, and Σi is its 2× 2 covariance matrix de-ciding about the elliptic shape of the cluster (orientation, width andlength). Σi is further decomposed into Σi = RiSiR>i where Si is adiagonal scaling matrix with independent scales σ

ai and σ

bi along

a and b orthogonal axes respectively. This gives an elliptic shapeto the cluster with width and length driven by a and b, wheneverσ

ai 6= σ

bi , and Ri is a rotation matrix of angle θi which orients the

elliptic shape with respect to the a-axis. Figure 1 (stage 1) visuallyillustrates these parameters.

Si =

(σ

ai 0

0 σbi

)Ri =

(cosθi −sinθisinθi cosθi

)In our case M = 2 and the density function of the GMM is:

p(x|Θ) =M

∑i=1

πigi(x) = πugu(x)+πvgv(x) withM

∑i=1

πi = 1 (1)

where τ = πu = 1− πv is the prior probability that component ugenerates a data point and Θ = (πu,πv,µu,µv,Σu,Σv) is the param-eter vector. In order to generate a point of the scatterplot, first thegenerating component gi is chosen at random with probability τ

to select gu, then a point is drawn at random from its bivariateGaussian distribution with parameters µi and Σi. We keep track ofthe generating component u or v of each point x ∈ X in a scatter-plot (X ,Θ) forming the sets U and V respectively (U ∪V = X andU ∩V = ∅).

Parameters Setting: Our goal is to cover most of the parameterspace, so that a large variety of shapes that could be generated bya Gaussian mixture is covered. To do so, we used the sets of val-ues given in table 1. We set µa

u = µbu = µa

v = 0 only varying µ = µbv

the Euclidean distance between the two components u and v alongthe b-axis, which plays a crucial role to make the two componentsoverlap. All combinations of parameter values result in 2,073,600unique 9-dimensional parameter vectors, each generating a theoret-ically distinct grouping pattern (Stage (1) in Figure 1), except forpatterns generated with σ

ai = σ

bi which are invariant to any θi.

Selecting Diverse Scatterplot Stimuli: To reach a manageablenumber of stimuli for the study, we sampled 1020 scatterplots fromthe 2,073,600, 20 for training and 1000 for evaluation (Stage (2)

in Figure 1). We empirically estimated the total time required tocomplete the experiment to be about 45 minutes. In order to getvarious cluster patterns that cover the full space of possible shapes(Figure 2), we could not simply uniformly sample the GMM pa-rameter space, as these parameters are not straightforwardly relatedto the resulting shape’s perceived complexity. Instead, we used astratified sampling approach similar to the one used by Pandey etal. [PKF∗16]. Section 2 in the supplementary material providesmore details about this procedure.

4.3 Experimental Design

Series and task: We split the 1020 scatterplots into 11 series. Thefirst series contained 20 scatterplots (one per K-means group) forthe participants to get used to the diversity of grouping patterns andthe task, before the actual recording started. For the remaining 10series, each series contains 100 scatterplots randomly sampled fromthe 1000 scatterplots without replacement. We showed the 11 seriesto each participant independently and randomized the order of scat-terplots to account for learning effects and other biases [VZS18].The participants had an optional break of 2 minutes between eachseries. The task was to push key “1” or “2” of the keyboard depend-ing on the number of clusters perceived in the scatterplot (Stage (3)in Figure 1). When participants perceive 2 or more than 2 clustersthey were instructed to hit key “2”. For each (participant identifier,scatterplot) pair we recorded the hit key and the time to hit it.

Human Subjects: We recruited 34 adult participants (27 males /7 females), aged between 19 and 56 (average 31.9), 27 of themwith a university degree (BSc, MSc, PhD). All participants reportednormal or corrected-to-normal vision, and the experiment was con-ducted in a lab with daylight on a standard desktop computer andkeyboard. Participants also reported about having experience withdata analysis applications (1 never, 4 rarely, 27 often). The averagecompletion time was 24 min (stdev: 8 min) excluding breaks.

Process: Before the experiment, we explained the perceptual natureof the task and encouraged the participants not to take more than2 seconds per task. We also explained that we would not give aclear definition of “cluster” as the very objective of the task was tounderstand what forms a single cluster for human subjects.

Practical observations: When a key was hit, it was not possible tocome back to change the choice. Several participants reported hav-ing felt frustrated of pushing the key too fast sometimes, mostlyduring the first series. This might generate some noise in the col-lected judgments although we do not expect it to be significantwhen integrated over 34 participants with randomized plot order.

4.4 Results and Findings

For each scatterplot (X ,Θ), we count the number n1 and n2 out of34 participants that perceived 1 or more clusters respectively. Wesummarize these human judgments as the human separation scorewhich is the probability Hsep = n2

n1+n2that a scatterplot contains

2 distinct clusters. The human separation score is high when 2 ormore clusters are more frequently perceived than 1 cluster. Figure2 displays a sample of the scatterplots generated in this experimenttogether with their human separation score.

2.9% 2.9% 2.9% 5.9% 5.9%

8.8% 17.6% 20.6% 29.4% 32.4%

32.4% 58.8% 64.7% 85.3% 85.3%

88.2% 88.2% 91.2% 97.1% 100%

Figure 2: Subset of the 1000 scatterplots judged by the 34 humansubjects with the percentage of human raters (Hsep) judging theydisplay more than a single cluster.

5 ClustMe

According to our process outlined in Sec. 3.1, we now use the judg-ments from Experiment 1 to design ClustMe. To do so, we comparethis data to automatic decisions from different merging techniqueswith the goal to find the best one at mimicking human judgments.

5.1 Selecting Merging TechniquesWe consider the 7 merging techniques analyzed by Hennig [Hen10](Stage (4) in Figure 1). These merging techniques are all based onfirst fitting a GMM to the distribution of the points in the scat-terplot. The standard approach for estimating the parameters ofa GMM is to fix a number of components M (See Equation 1),and then maximize the likelihood using Expectation Maximiza-tion [BCRR97]. Several numbers M of components are tested, andthe optimal number is selected as the one maximizing the BayesianInformation Criterion [Sch78] (BIC). Then each pair of compo-nents is tested for merging, and the merging decisions are aggre-gated to get the final clusters. The 7 merging techniques work asfollows:

• Bhat merges components whose Bhattacharyya [Bha43] dis-tance is lower than a threshold;• Demp merges two components if the misclassification probabil-

ity between them is higher than a threshold;• RidgeUni merges components if their mixture is unimodal ac-

cording to Ray and Lindsay’s criterion [RL05];• RidgeRatio expand RidgeUni merging nearly-unimodal pairs of

components up to some threshold;• Dipuni uses RidgeRatio to decide which clusters to merge but

stops merging according to the p-value of the dip statistical test;• DipTantrum is similar to Dipuni but the p-value is computed as

in Tantrum et al. [TMS03];• Predictive is based on the pairwise prediction strength that a

clustering model, computed on one half of the data, has aboutthe other half.

Table 2: Vanbelle Kappa κV index

DipTan Demp Dipu Bhat RidgeU RidgeR Pred.788 .788 .786 .785 .750 .696 .529

We use the Rmixmod R-package [LIL∗15] to implement theGMM and BIC, and then the fpc R-package [Hen15] proposedby Hennig to merge the components of the GMM and output thefinal clustering. In our experiment we always used the default pa-rameters given by the fpc functions (See Section 1 of the supple-mentary material).

5.2 Comparison with Human Judgment

We now compare the merging techniques with the human judg-ments of the 1000 scatterplots (Stage (5) in Figure 1).

We use Vanbelle’s Kappa κV index [VA09], which quantifies thedegree of agreement between an isolated rater (one of the mergingtechniques) and a group of raters (the 34 subjects) on a nominalscale (the single merging decision and the 34 individual categoricaljudgments). It ranges from -1 (systematic opposed decisions) to 1(full agreement). The group of raters is regarded as a whole, a refer-ence or gold-standard group with its own heterogeneity. κV comesdown to standard Cohen’s Kappa κ [Coh60] inter-rater agreementwhen there is a single rater in the group. Table 2 gives the κV in-decies for each of the merging techniques, settling Demp on parwith DipTantrum among the best ones with κV = 0.788 . As perthe scale proposed by Landis and Koch [LK77], values between0.6 and 0.8 are interpreted to be in substantial agreement with thehuman raters.

As we can see in Table 2, the first four merging techniques are al-most equal as per κV index. As we want to pick one of them to builda VQM, we need to find another measure that could further distin-guish between them. We thus use classification theory [BDA13] toquantify how much the merging technique (predicted class) agreeswith the averaged human decision (actual class). For this evalua-tion we first transform the human separation score into a humandecision Hmrg by thresholding the probability Hsep at 0.5. Foreach of the 1000 scatterplots, we end up with a human decisionHmrg(X ,Θ)∈{1,2}, and the GMM combined with a merging tech-nique gives a merging decisionMmrg(X ,Θ) ∈ {1,2} for the sameplot (X ,Θ). We then use Matthews Correlation Coefficient (MCC),which is recommended by Bekkar et al. [BDA13] to account forclass imbalance as 81.5% of the 1000 scatterplots are classified 1by humans (a single cluster is perceived). Figure 3 (top) shows theMCC for the 7 merging techniques, visualized as a boxplot encod-ing the distribution of 10000 bootstrap samples estimating the vari-ance of this score. Bootstrap sampling [ET93] is used to estimatethe sensitivity of the accuracy score to the current sample of 1000scatterplots and evaluates how different it could be on new samplesdrawn from the same distribution of scatterplots. Demp followed byBhat, stands out again among the best techniques (69.5% medianaccuracy).

As Demp is the only merging technique to rank both timesamong the first, we select it as the best merging techniques to modelthe human decision in the 2DGCMP task, to be used in ClustMe.

5.3 Definition of ClustMeClustMe is a multi-scale VQM made of two numbers C and MBICcoming out of a 2-step procedure (Stage (6) in Figure 1):

1. A bivariate GMM is trained with the Expectation Maximiza-tion (EM) technique on the 2-dimensional points X of thescatterplot to find the optimal set of parameters Θ

∗ for eachM ∈ M = {1, . . . ,Mmax}. The Bayesian Information Criterion(BIC) is used to select the optimal number MBIC ∈M of com-ponents. Setting Mmax = N ensures to find the optimal MBIC,but in practice, M is incremented until BIC reaches a maximumvalue which typically occurs at some low value M << N.

2. Then, the Demp merging technique, which performed best onmodeling human judgments in Experiment 1, is applied to allpairs of components of the BIC-optimal GMM. In doing so, itdecisions about merging some of them can be made, ending upin the points being assigned to a set of C ≤MBIC clusters.

Johansson and Johansson [JJ09] proposed to use the number ofclusters found by the CLIQUE clustering technique as a VQM forcluster patterns. Following the same idea, we propose that ClustMeprovides the number C of clusters as a primary complexity mea-sure of grouping patterns, and MBIC as a secondary measure fortie-breaking scatterplots with equal C. For instance, an elliptic pat-tern and a banana shape pattern can both end quantified by ClustMeas a single C = 1 cluster, but the banana shape will likely requireseveral elliptic Gaussian components (MBIC > 1) to cover it (seeFigure 3 Right), while the elliptic pattern only needs a single com-ponent (MBIC = 1).

Therefore, we define the ClustMe VQM of a scatterplot X as:

ClustMe(X) = (C,MBIC)X

Figure 3 (Right) illustrates the different stages of the ClustMeprocess on a scatterplot with a simple grouping pattern, theClustMe VQM is (C,MBIC) = (3,9). Also, if applied in Experi-ment 1 with only 2 components in the GMM and an ideal exactmatch between Demp and the human judgments, ClustMe wouldbe (1,2) when the human perceives a single cluster (merging deci-sion) and (2,2) otherwise.

5.4 How to use ClustMe as a VQMThe ranking based on ClustMe(X) is done first by decreasing C,and then for each equal C by decreasing MBIC. In other words, thehigher the value of C and MBIC (for same C), the more complexthe grouping pattern would be perceived by a human in the scatter-plot X . ClustMe VQM ordered by increasing values look like thefollowing series and quantify less complex (left) to more complex(right) grouping patterns:

Simple patterns← (1,1)≺ (1,2)≺ ·· · ≺ (1;MBIC)≺ (2,2)≺ . . .

· · · ≺ (2,MBIC)≺ (3,3)≺ ·· · ≺ (MBIC,MBIC)→ Complex patterns

6 Experiment 2: Evaluation of ClustMe

In this section, we discuss the second study. We compared ClustMeto the state-of-the-art Clumpiness [DW14], as well as 3 other clus-tering techniques, for which we used the number of clusters foundas the VQM for grouping patterns complexity (Figure 4):

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Figure 3: Left: median and variance bootstrap estimate of theMatthews Correlation Coefficient (MCC) accuracy of each merg-ing technique decisionMmrg to predict the human merging deci-sionHmrg from Experiment 1. Demp is the most accurate model ofthe human merging decision. Right: from top to bottom, example ofan original scatterplot, points assigned to color-coded componentsof the BIC-optimal GMM, and ClustMe output after merging withDemp.

• X-means [PM00] (RWeka R-package [HBZ09]) is a parameter-free clustering technique similar to K-means [MJ66] where thenumber K of clusters is selected automatically, but which canonly provide convex-shaped clusters;• CLIQUE [AGGR98] (subspaceR-package [HH15]) is a grid-

based clustering technique which has been used as a VQM forgrouping patterns [JJ09], but it requires arbitrary hand-tuning ofgrid size and density threshold parameters;• DBSCAN [EKSX96] (dbscan R-package [HP18]) is a density-

based clustering technique which can detect clusters with non-convex shapes, but requires arbitrary hand-tuning of ε andminPts parameters.

6.1 TaskFollowing previous evaluation approaches for VQMs [TBB∗10],we set out to compare ClustMe and the other VQMs in terms ofhow they rank a set of scatterplots in comparison to human rank-ings. We thus first need a human ranking of scatterplots in terms ofthe perceived complexity of grouping patterns. A straightforwardway to get such a human ranking is to task subjects with pairwisecomparisons.

6.2 Selected Scatterplot StimuliTo increase realism and generalizability as compared to Experi-ment 1, we this time considered a set of 257 unseen real and syn-thetic scatterplots from previous work [SMT13]. From this set, wesampled 435 pairs of scatterplots to run a pairwise comparison ex-periment. We then use these pairwise human rankings as a baselineto assess the quality of the above VQMs.

As comparing all possible pairs of 257 scatterplots wouldamount to an infeasible number of 32,896 trials per participant,we selected a random subsample of 35 scatterplots from the initialset, 5 for training and 30 for actual evaluation (Figure 5). This leads

a

1. Generate Gaussian Mixture Models 3. Task Human Subjects

4. Run Merging Techniques

1000 (+20) 34

Q(Q,X)1X

(Q,X)146

(Q,X)15

(Q,X)1000

2073600

Qi, (X i)

7

5. Compare2. Generate Scatterplots

(Qi), X i

(Qi, M m sep i)

“2”

“2”“1”

“2”

… M * sep

6. ClustMe process

M * sep

…

GMM+BIC

(C,MBIC)

MergerM * sep H sep

?

(5,9)

ClustMeC=5MBIC=9

Scatterplot

GMM“1”

“1”“2”

“2”“2”

Sampling

H sep i(Qi, H sep i)

M 1 sep i

M 7 sep i

Experiment 1: collecting simple cluster separation judgments

Experiment 2: collecting pairwise judgments on how much scatterplots are clustered

…

Which is more structured/clustered/clumpy?

31

“Left”

“Right”“Left”

“Same”

30 pictures - 435 pairs

Judge

257 pictures

Evaluate

…

…

ClustMe

Other VQMs (Clumpiness, X-means, CLIQUE, DBSCAN)

>

>

Use: Compare ClustMe and other VQMs vs Human pairwise ranks

Use: Find best Merging and create ClustMe

>

>=

>

>

vs

vs +0.8

-0.6

VanbelleKappa

(8,9) (7,9) (7,7) (6,9) (5,5) (4,5) (3,7) (2,8) (1,3)(2,6)

0.87 0.79 0.78 0.67 0.54 0.52 0.43 0.42 0.39 0.23

Sampling

>

>=

>

>=

Figure 4: Experiment 2: 31 human subjects decide which one of two scatterplots is more structured/clustered/clumpy for 435 pairs built from30 out of 257 real and synthetic yet unseen scatterplots. The agreement between ClustMe, Clumpiness, X-means, CLIQUE, and DBSCANrankings of these 435 pairs to the human rankings is evaluated with Vanbelle’s Kappa index.

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Figure 5: The 30 scatterplots randomly picked from the 257 bench-mark data used for gathering human judegments in Experiment 2.

to a feasible set of 10 training and 435 evaluation pairs that we canshow to each participant, and a total of 13,454 human judgmentscollected.

6.3 Experimental designSeries and task We first generated the 10 and 435 pairs of picturesfor the training and evaluation sets. We then showed these two se-ries to each participant independently, and randomized the pairs ineach series as well as the left-right position of the pictures in eachpair. After the training series, participants were given a short pause.For each pair, the task was to press the “left” or “right” arrow keybased on whether participants perceived the left or right scatterplotas more “structured, clustered or clumpy”. Participants could pressthe “space” bar when they thought both scatterplots were on par.

Human Subjects We recruited 31 participants (19 males and 9 fe-males), 30 with a university degree, aged between 23 and 43 (av-erage 31.1) and conducted the study in the same environment as

study 1. 29 participants reported normal or corrected-to-normal vi-sion. 1 reported to be color-blind, and 1 to have a lazy-eye, neitherof which is a problem for our study of monochrome 2D scatter-plots though. Participants reported about having experience withdata analysis applications (4 rarely, 26 often). The average comple-tion time per trial was 1.89 sec (stdev: 1.65 sec) excluding breaks.

Process We followed the same process as in Experiment 1, that is,encouraging participants not to take more than 2s per task, and jus-tifying the reason for not giving an exhaustive definition for “struc-tured, clustered or clumpy”. We also decided again not to includea “back” button. Despite some potentially false hits, this choicefosters fast, perceptual judgments, which we deemed much moreimportant for our study design.

6.4 Results and FindingsAs in Experiment 1, we use Vanbelle’s kappa κV to evaluate thedegree of agreement between the judgments of the 31 human ratersand the results of the 5 different VQM rankings on each of the 435pairs. We assumed 3 categories of values for the pairwise rankingsgiven by the raters: “left < right” (right arrow key pressed), “left >right” (left arrow key), “left = right” (space bar). Over the 13,454human judgments, 25% (3,413) were “left = right”, the distribu-tion of human raters selecting this option is displayed in Figure 6(Top). Vanbelle’s kappa is the measure best adapted to our settingof categorical judgments, as it takes into account both the agree-ment between the group of human raters and the isolated VQMrater, together with the group inter-rater agreements.

ClustMe VQM provides 27 “left = right” ranks over the435 pairs (6.2%), X-means 406 (93.3%), the best CLIQUE 198(45.5%), the best DBSCAN 110 (25.3%), but Clumpiness providedno equal rank. Indeed, all clustering-based VQM and ClustMe canonly take a finite set of possible values based on the number of clus-ters found. For instance, with a maximum MBIC = 9 in this experi-ment, and given C ≤MBIC, the ClustMe measure (C,MBIC) couldonly take 45 distinct values. In contrast, Clumpiness is based on aratio of lengths of minimum spanning tree edges [WAG05] rangingfrom 0 to 1, making equal values even for very similar scatterplotsunlikely. Hence, Clumpiness might be in disagreement with humanraters only because it gives slightly different scores to scatterplotsjudged as “equal” by the human raters. For the sake of fairness, wedefined an additional Coarse-Clumpiness score.

Coarse-Clumpiness: We allow Coarse-Clumpiness to give ‘equal’

02468

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Figure 6: Top: Number of subjects selecting equal complexity fora given proportion of the 435 trials. Bottom: Vanbelle Kappa in-dex results from Experiment 2 with agreement interpretation as perLandis and Koch [LK77].

ranks for pairs with sufficiently similar Clumpiness scores. Howmuch similarity is enough is difficult to decide, so we quantifyClumpiness scale into q levels, q ranging from 1 to 45, the numberof ClustMe levels in this experiment. For each number q of quan-tification levels, we consider as ‘equal’ (category ‘left = right’) thepairs of the 435 scatterplots whose Clumpiness values differencewas less than 1/q (1 being the maximum value for Clumpiness),and compute the Vanbelle’s Kappa index κV for that q. We reportthe best Coarse-Clumpiness score κ

∗V found among these 45 val-

ues, which occurred for q∗ = 24 levels, to be compared with thestandard ClustMe and the other clustering-based VQMs.

All resulting Vanbelle’s Kappa scores are displayed in Figure 6(Bottom) together with the interpretation scale proposed by Lan-dis and Koch for the Kappa index [LK77]. ClustMe has the highestκV = 0.671 (substantial agreement) while Coarse-Clumpiness isonly in moderate agreement with κV = 0.444. Still the quantizationprocess helped to improve significantly over the standard Clumpi-ness which is in slight agreement with κV = 0.112. X-means isamong the worst VQM only in slight agreement with κV = 0.022which can be explained by its inability to model non-convex clustershapes. For DBSCAN and CLIQUE, Figure 6 shows the distribu-tion of all scores that we obtained through a systematic grid searchfor optimal clustering parameters. The median score for DBSCAN(0.001) and CLIQUE (0.062) is only in slight agreement with hu-man raters. The best DBSCAN reached only a moderate agree-ment (κV = 0.491) with ε = 0.02 and minPts = 7. While the bestCLIQUE reached a substantial agreement (κV = 0.651) with 5 gridintervals and 0.18 density threshold, on a par with ClustMe. How-ever, this systematic exploration is not realistic: the non-expert usercannot spend time in this setting unless transforming the perceptualskimming task into a cognitive task. Moreover, the best setting wefound for each clustering technique is biased towards the set ofscatterplots we used in this experiment (overfitting), i.e. it is un-likely to be the best setting for any new scatterplot. By contrast, thechoice of the merging technique for ClustMe has been done on a to-tally different set of scatterplots in the Experiment 1, demonstratingthe generalization capacity of ClustMe to rank new scatterplots.

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Figure 7: ClustMe computing time in seconds per number of pointsN in the scatterplot for all 257 benchmark scatterplots (Left) andzoom in on the ones with less than 4000 points (Right). The com-puting time is roughly linear with the number of points

Qualitative analysis: Figure 8 shows the top 7 and bottom 7 scat-terplots from the ranking of the whole set of 257 scatterplots or-dered by decreasing VQM for all five approaches. CLIQUE andClustMe rankings appear consistent with most complex patterns ontop and least complex ones at the bottom. Other VQMs are notso consistent: strongly structured patterns appear at bottom rankof Clumpiness, DBSCAN and X-means, which would be missedby the user if she used these VQMs to rapidly focus on top-ranked views as they are expected to be the most interesting ones.This qualitative analysis confirms the quantitative agreement scoresfound in the experiment above.

6.5 ClustMe ScalabilityFigure 7 shows the computing time for ClustMe with respect to thenumber of points for the benchmark scatterplots. All experimentsran on a single core desktop computer (MacOS, 3.4 GHz Intel Corei5 processor, 8 GB 1600 MHz DDR3 RAM). ClustMe takes ap-proximately 15 sec for a scatterplot with 5000 points and about 3ms per point (333 pts/sec). This means that Clumpiness, X-means,CLIQUE, and DBSCAN, which only need a few miliseconds perscatterplot, are by far faster to compute than ClustMe. A detailedanalysis of ClustMe shows that the training of the initial GMM isthe bottleneck, while the merging process is extremely fast rely-ing only on the parameters of the GMM but not on the number ofpoints. This is an obvious computational drawback of our method.However, at the same time we must consider its benefits as a VQMthat can better mimic human perception of grouping pattern com-plexity. While there is no obvious way to increase the accuracy ofcomputational VQMs like Clumpiness, X-means, CLIQUE or DB-SCAN, there are several possibilities to speed-up ClustMe. Com-putations of the ClustMe VQM are independent for each scatter-plot and could be distributed on multiple processors. Speed-up isalso possible using in-memory Myria DBMS [MHT∗15] or spatialindexing structures [Moo99, vdM14]. The number of data pointscould also be reduced by sub-sampling or by using a small set ofcore representative data built prior to the GMM training [FFK11].

7 Discussion

ClustMe and Experiment 1: The 2DGCMP task of Experiment1 is actually the simplest setting to assess ClustMe (i.e. GMM +BIC +Demp). It corresponds to the optimal and ideal case where

ClusteMe Clumpiness CLIQUE DBSCAN X-means

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