INTERPRETATION, INTERACTION, AND SCALABILITY

FOR STRUCTURAL DECOMPOSITION TREES

René Rosenbaum

, Daniel Engel

, James Mouradian

, Hans Hagen

and Bernd Hamann

1,2

Institute for Data Analysis and Visualization (IDAV), Department of Computer Science, University of California, Davis,

CA 95616-8562, U.S.A.

International Research Training Group (IRTG) 1131, University of Kaiserslautern, Kaiserslautern, Germany

Keywords:

High-dimensional Data, Projections, Value Visualization, Relation Visualization, Interaction, Scalability.

Abstract:

Structural Decomposition Trees (SDTs) have been proposed as a completely novel display approach to tackling

the research problem of visualizing high-dimensional data. SDTs merge the two distinct classes of relation and

value visualizations into a single integrated strategy. The method is promising; however, statements regarding

its meaningful application are still missing, constraining its broad adoption. This paper introduces solutions

for still-existing issues in the application of SDTs with regard to interpretation, interaction, and scalability.

SDTs provide a well-designed initial projection of the data to meaningfully represent its properties, but not

much is known about how to interpret this projection. We are able to derive the data’s properties from their

initial representation. The provided methods are valid not only for SDTs, but also for projections based on

principal components analysis, addressing a frequent problem when applying this technology. We further show

how interactive exploration based on SDTs can be applied to visual cluster analysis as one of its application

domains. To address the urgent need to analyze vast and complex amounts of data, we also introduce means for

scalable processing and representation. Given the importance and broader relevance of the discussed problem

domains, this paper justiﬁes and further motivates the usefulness and wide applicability of SDTs as a novel

visualization approach for high-dimensional data.

1 INTRODUCTION

The visualization of high-dimensional data sources is

a common but still unsolved problem. Structural de-

composition trees (SDTs) (Engel et al., 2011) rep-

resent a novel approach to this challenge. SDTs

combine value and relation visualizations into one

approach and thus provide a variety of beneﬁts not

available in existing visualization technology. Means

for interaction based on the novel representation add

unique options for data exploration. Research con-

cerning SDTs, however, is constrained to the intro-

duction and description of the construction and fun-

damental aspects of this novel displaying approach.

Statements concerned with its utilization and appli-

cation to concrete problems have not yet been pub-

lished. This results in long learning efforts and may

also lead to misinterpretations and failure when the

technique is applied.

This paper aims to provide guidance for differ-

ent aspects important for the successful application of

SDTs in data visualization. After reviewing related

work in the area of high-dimensional data visualiza-

tion (Section 2), the ﬁrst part of this paper (Section 3)

is particularly concerned with the interpretation of

an SDT. Thereby, we focus on alignment and length

of the different dimensional anchors and derive their

meaning and mathematical properties for gaining ﬁrst

insight quickly. Practical implications of these state-

ments provide the users with an understanding of

the capabilities of SDTs. Introducing guidelines for

visual cluster analysis, the second part (Section 4)

is concerned with appropriate interactive data explo-

ration. Based on a common data browsing approach,

strategies for gaining further insight into the data are

provided. Addressing the need for visualization tech-

nology able to deal with large and high-dimensional

data sets, the third part (Section 5) of the paper dis-

cusses different means to reduce the complexity of the

data and representation. Empirical results show that

much time can be saved when methods to introduce

scalability with regard to the number of data points

and dimensions are applied. We show that ambigui-

ties in the association of data points to data clusters

636

Rosenbaum R., Engel D., Mouradian J., Hagen H. and Hamann B..

INTERPRETATION, INTERACTION, AND SCALABILITY FOR STRUCTURAL DECOMPOSITION TREES.

DOI: 10.5220/0003841106360647

In Proceedings of the International Conference on Computer Graphics Theory and Applications (IVAPP-2012), pages 636-647

ISBN: 978-989-8565-02-0

 2012 SCITEPRESS (Science and Technology Publications, Lda.)

can be signiﬁcantly reduced by a scalable visual rep-

resentation.

The main contributions of this paper to the current

state of research are the following

• We provide statements and practical implications

to the geometric interpretation of SDTs and re-

lated projections.

• We introduce means for an interactive exploration

of a high-dimensional data set in the context of

visual cluster analysis.

• We introduce and justify means for a scalable pro-

cessing and representation of large and complex

data sets.

We conclude (Section 6) that SDTs are a valid

means for the visualization of high-dimensional data,

but must be understood and applied in the right way

in order to gain insight quickly and without misinter-

pretation and wrong conclusions. This paper aims to

provide the necessary information to accomplish this

objective successfully.

2 RELATED WORK

2.1 Visualization of High-dimensional

Data

As a result of most data acquisition tasks today, high-

dimensional data has always been of strong interest

to the visualization community. Many different ap-

proaches and techniques have been proposed. Ac-

cording to (Engel et al., 2011) they can be categorized

into value or relation visualizations.

By focussing on the conveyance of data coordi-

nate values for every data point, value visualizations

allow for a detailed analysis of the data. The paral-

lel coordinates plot (see Figure 1, top/left) is a typical

representative of this category. Due to their focus on

value representation for each data point, a common

problem with all associated techniques is that they

are often not scalable with regard to the amount and

dimensionality of the data. As a result this usually

leads to clutter and long processing times as the num-

ber of dimensions and amount of data points increase.

In order to overcome these issues, cluster-based ap-

proaches (Johansson et al., 2005; Zhou et al., 2008;

Artero et al., 2004), appropriate means for interaction

(Elmqvist et al., 2008; Hauser et al., 2002), and bet-

ter utilization of the available screen space (McDon-

nell and Mueller, 2008) have been proposed. Clut-

ter reduction is also achieved by dimension ordering

arranging the dimensions within the visual represen-

tation based on correlations within the data. The re-

search conducted by ANKERST ET AL.(Ankerst et al.,

1998) was the ﬁrst to formally state this problem and

has later been successfully expanded in (Yang et al.,

2003a) and (Peng et al., 2004). Although these meth-

ods are great improvements to reduce clutter, the dis-

played information is often too detailed and a mean-

ingful representation can generally not be obtained for

large data sets.

Instead of aiming at communicating individual

values, relation visualizations are designed to con-

vey relationships within the data. They are mostly

point mappings projecting the m-dimensional (m-D)

data into the low-dimensional presentation space. As

relations within the data may be too complex to be

completely conveyed in presentation space, projec-

tions are usually ambiguous. A prominent and widely

applied point projection approach is principal compo-

nents analysis (PCA) conveying distance relations in

m-D space by projecting into a plane that is aligned

to capture the greatest variance data space without

distorting the data (see Figure 1, top/right). Con-

trary, multi-dimensional scaling (MDS) commonly

uses general similarity measures to represent the data,

but leads to distortion and a visualization that may

be difﬁcult to interpret. Interpretation of the rep-

resentation is a general issue with point projections

as long as no means to comprehend the parameters

used for the projection are available. One such option

are dimensional anchor (DA) visualizations (Hoffman

et al., 1999) projecting and displaying the basis vec-

tors along the data points (see Figure 1, top/right).

These DAs are also an appropriate means to adjust the

projection interactively (Kandogan, 2001). Relation

visualizations usually lead to a meaningful overview

of the data. Their effectiveness, however, strongly de-

pends on the quality of the initial projection and the

means provided to interpret and interact with it. Cur-

rent research mainly focuses on improved representa-

tion of speciﬁc data structures, e.g., scientiﬁc point

cloud data (Oesterling et al., 2010), a better incor-

poration of domain-appropriate analysis techniques,

e.g., brushing and ﬁltering (Jänicke et al., 2008), or

computational speed gains (Ingram et al., 2009).

Due to the rather diverse properties of value and

relation visualizations, they each have distinct appli-

cation domains. Thus, they are often used simultane-

ously in exploratory multi-view systems, such as in

(Paulovich et al., 2007). Few publications exist that

tackle the problem of combining both classes into a

single approach. Most of them have been proposed

for value visualizations, such as the technology de-

scribed in (Yang et al., 2003a; Yang et al., 2003b;

INTERPRETATION, INTERACTION, AND SCALABILITY FOR STRUCTURAL DECOMPOSITION TREES

637

Figure 1: The two classes of visualizations for high-

dimensional data (value (top/left) and relation (top/right)

visualizations) are brought together by SDTs (bottom). All

visualizations represent the well-known ”cars” data set. The

SDT highlights the ﬁve distinct clusters by its branch struc-

ture also conveying the respective differences in the data

values.

Yang et al., 2004), (Johansson and Johansson, 2009),

or (Yuan et al., 2009). SDTs represent a completely

different approach that promises to bridge the existing

gap between both classes.

2.2 Structural Decomposition Trees

SDTs are founded on a sophisticated data projection,

but provide additional means to represent the dimen-

sion contributions for each data point (see Figure 1,

bottom). This is mainly achieved by introducing a

tree structure showing the projection path for each

displayed data point and thus its individual dimen-

sion values. The projection paths also allow for an

unambiguous identiﬁcation and interpretation of data

points that reside at different locations in m-D space,

but have been projected in close proximity in the pro-

jection space.

A main problem in showing the different pro-

jection paths is the introduced clutter. SDTs over-

come this issue by introducing a multi-stage process-

ing pipeline. Hierarchical clustering is used to iden-

tify, aggregate, and bundle common line segments of

m-D data points. The resulting tree has minimal over-

all branch length reducing the redundancies consid-

erably. Appropriate representation of the individual

dimension contributions is accomplished by a well-

designed drawing order. The tree itself is represented

by colored lines whereby the number of elements

within this subtree is encoded by branch thickness

(see Figure 1, bottom). The initial SDT projection

thereby maximizes the space between tree paths and

thus allows for a better interpretation of the visualiza-

tion (Engel et al., 2011).

Different means for interaction either on individ-

ual or groups of data points or the whole representa-

tion make possible for further exploration of the data

(see Figure 2). The projection can be signiﬁcantly

changed by a re-arrangement of the end points of the

DAs. These dimension vectors can be independently

modiﬁed in their lengths and angles relative to each

other. Thereby, so-called variance points are placed

along the unit circle in order to indicate angles that

lead to other promising projections. Means for high-

lighting dimensions and projection paths allow one to

emphasize all line segments corresponding to the co-

ordinates of a dimension or to investigate the struc-

tural decomposition of the data. Data ﬁltering can be

achieved by collapsing and hiding subtrees. After col-

lapsing, the main value contributions of all associated

data points are still visible and can be used and inter-

preted, e.g., for comparison with other subtrees.

Published research concerned with SDTs mainly

focusses on the technical foundations of the approach.

Semantic aspects, interpretation, and the use of SDTs

to visualize large data sets are not provided.

Figure 2: The main interactions provided by SDTs: Repo-

sitioning of DAs (green arrow) allows for an intuitive ad-

justment of the projection. Dimension highlighting (yellow

DA) conveys the individual contributions of a dimension in

the data (yellow tree segments). Path highlighting (pur-

ple tree branch) is intended to emphasize interesting tree

branches and substructures.

IVAPP 2012 - International Conference on Information Visualization Theory and Applications

638

3 INTERPRETATION OF THE

INITIAL LAYOUT

Projections are a powerful means to convey relations

in high-dimensional data. Due to the characteristics

of dimension reduction, however, they are often difﬁ-

cult to interpret. In previous work it was shown that

SDTs are speciﬁcally suited to depict data coordinates

in a way that aims at intuitive interpretation. Experi-

mental studies of PCA-based projections showed that

the projection conveys properties of the data by the

length and relation of the DAs to each other. This,

however, has never been explicitly quantiﬁed.

In this section, we investigate in full detail how the

initial arrangement of DAs in SDTs relates to the cor-

responding variables in the data and how the user can

interpret this arrangement to infer knowledge about

the data. Due to the use of a PCA-related projec-

tion method for SDTs, the given statements apply to

all PCA-based projections. We shortly recall dimen-

sional anchors and their arrangement, after which

we are linking the properties of DAs to those of the

data. We ﬁrst outline why DAs are used to reﬂect

a PCA projection and how their initial arrangement

is deﬁned. This is expressed by latent features in

the data, i.e., the eigenvectors and eigenvalues of the

data’s covariance matrix indicating the information

content within the different data dimensions. In order

to understand which data properties are visually en-

coded in a projection, we investigate how the projec-

tion is deﬁned by these features and what information

is thereby depicted. This is expressed by a deriva-

tion of the spectral decomposition of the covariance

matrix. After these steps, we show that the speciﬁc

DA arrangement allows one to derive conclusions and

data properties that are of keen interest to the user

but not depicted by the common plotting of princi-

pal components. Finally, statements to implications

of these properties aim for a better understanding of

an arbitrary PCA-based projection avoiding its misin-

terpretation.

DAs and their Arrangement. Since SDTs can

be computed and visualized both in 2D or 3D

space, the following considerations are made for an

arbitrary display dimensionality p. We assume that

n m-dimensional data points are stored row-wise

in X ∈ IR

(n×m)

. By the projection, X is mapped to

display point coordinates

X ∈ IR

(n×p)

The linear mapping of an m-D data point X

in p-

D display coordinates

is computed by the linear

combination of DAs a

∈ IR

, 1 ≤ j ≤ m, with the

corresponding coordinate of X

∑

1≤ j≤m

i, j

. (1)

This technique, the mapping in star coordinates

(Kandogan, 2001), can be understood as a general-

ization of drawing 3D objects on paper to arbitrary

dimensions. In the original work, however, the DAs

are initially arranged in a uniform distribution along a

unit circle. In general, this leads to a non-orthogonal

projection. This can be misleading because the dis-

tance in display space does not reﬂect distance in IR

To avoid this, a projection is designed to minimize

this mapping error. This error is commonly expressed

as the sum of squared pairwise distance differences

arising from the mapping from m to p dimensions,

∑

1≤i, j≤n

(D(X

)− d

(

))

, where d

is the Eu-

clidean distance metric and D is an appropriate dis-

tance metric of the application domain. This error

can be minimized, for example, by PCA in the case

D = d

. Instead of expressing the data by the orig-

inal unit vectors, PCA computes new orthogonal di-

rections (principal components) in which the data has

maximal variance and re-expresses all data points in

coordinates of these principal components. The pro-

jection is then deﬁned by the p principal components

that capture the highest variance in the data. Although

distance relations between data points are captured

well in this projection, the interpretation of principal

components is not intuitive. In almost all applications,

the link to the original data is essential for analysis.

Therefore, the depiction of the original data coordi-

nates and relations between the original data dimen-

sions is an important aspect for a projection.

Linking Properties of DAs to those of the Data.

In previous work (Engel et al., 2011), both approaches

have been combined and the initial arrangement of

DAs has been deﬁned to reﬂect a (weighted) PCA

projection into p-dimensional display coordinates.

We utilize DAs to make possible a better inter-

pretation and more intuitive understanding of the

underlying projection without losing any of the

underlying projection’s beneﬁt. In this research, we

investigate the properties of this DA projection in

more detail and deduct which properties of the DAs

link to which properties in the data. The following

considerations are based on the data’s covariance

matrix. Without loss of generality, we assume X to

be centered and, since the used weighting scheme in

previous work changes the covariance matrix (to be

weighted) a priori, we can neglect the weighting in

the following. We also neglect the global scaling by

−1

that does not inﬂuence relations in the data.

INTERPRETATION, INTERACTION, AND SCALABILITY FOR STRUCTURAL DECOMPOSITION TREES

639

The underlying PCA projection

X of X is deﬁned

X = X

Γ, with

Γ = (γ

(1)

,...,γ

(p)

) ∈ IR

(m×p)

being the matrix storing

column-wise the eigenvectors of the corresponding p

largest eigenvalues of the covariance matrix S of X.

Equation (1) implies that the linear mapping of DAs

A = (a

,...,a

)

∈ IR

(m×p)

is deﬁned as

= X A.

In order to initially arrange the DAs such that their

mapping is equivalent to that of the PCA, we deﬁne

each DA as a row vector of

Γ:



(1)

,...,γ

(p)



. (2)

This step is equivalent to the projection of the orig-

inal unit vectors 1

∈ IR

to IR

subject to the same

rotation, i.e., a

= 1

Γ.

It is important to note that PCA projects X by re-

ducing its dimensionality to p in an optimal variance-

preserving way. Thus, the information that is actually

displayed by this projection is that of the inherently

deﬁned best rank-p approximation

X of X.

Spectral Decomposition of the Covariance Matrix.

The process of dimensionality reduction by maxi-

mizing variance becomes clear when considering the

spectral decomposition of S. That is the decomposi-

tion of the combined variances of all elements in X

into successive contributions of decreasing variance:

S = λ

(1)

+ ... + λ

(r)

, with λ

being the

k highest eigenvalue of S and γ

(k)

the corresponding

eigenvector for 1 ≤ k ≤ r = rank(X).

Each contribution S

(k)

= λ

(k)

thereby in-

creases the rank of the matrix summation by one.

holds the variance of the contribution, whereas

(k)

deﬁnes the mixing of this variance, i.e., how

this contributes to S. Consequently, the covariance

matrix of the PCA’s p-dimensional best rank-p ap-

proximation

X of X equals the sum over the ﬁrst p

contributions, where usually p  rank(X ). The co-

variance between dimensions i and j of the projected

data

X is therefore

i, j

∑

1≤k≤p

(k)

. (3)

Similarly,

X can be deﬁned by

X = X

. For the

dimensions (columns) in

X the following equation

holds:

•,i

∑

1≤ j≤m

•, j

(

)

i, j

•,i

is built from

X by the linear combination of all X

•, j

with coef-

ﬁcients (

)

i, j

∑

1≤k≤p

(k)

. Consequently,

these coefﬁcients deﬁne the orthogonal projection

of the data and account for the similarities between

columns in

X, i.e., for rank(

X).

Conclusions. With the above considerations in

mind, we show in the following that the length of each

DA and the angles between them reﬂect speciﬁc prop-

erties of the projection and of the projected data

The mixing matrix

holds normalized contribu-

tions to

S and relates to the DA’s arrangement in the

sense that (

)

i, j

∑

1≤k≤p

(k)

i, j

/λ

i, j

, whereas

i, j

= cos ∠(a

) ||a

||a

. We can draw the fol-

lowing conclusions:

1. The length of DAs equals the standard deviation

of the respective dimension in

X, normalized for

each contribution

(k)

by its variance λ

||a

(2)

∑

1≤k≤p

(γ

(k)

)

(3)

i,i

= ˜s

2. The cosine of the angle between two DAs equals

the correlation of the respective dimensions in

X, where both covariance and standard deviation

are normalized for each contribution

(k)

by its

variance λ

cos∠(a

) =

||a

(2)

∑

1≤k≤p

(k)

˜s

(3)

(k)

i, j

˜s

= ˜r

i, j

Figure 3 shows an SDT as an example for DA vi-

sualization highlighting these means for visualization.

Implications. It is important to emphasize that

does not represent the whole data X but only its best

rank-p approximation. That is,

X is the approxima-

tion of X that can be optimally depicted in p dimen-

sions with regard to its variance. Therefore,

X is

the orthogonally projected data on the subspace IR

which is spanned in a way that the projection re-

ﬂects the dominant trends in X. However, IR

can

only cover the most important information in the data.

While other subspaces that are left out globally ac-

count for less variance in the data, relations therein

may still be of importance for the user. Unfortunately,

this information cannot be captured in a single projec-

tion and, consequently, parts of the relations between

the original data dimensions in IR

are lost. The user

has to be aware of this dilemma because it may lead to

IVAPP 2012 - International Conference on Information Visualization Theory and Applications

640

Figure 3: A 13-dimensional air quality data set visualized

by the SDT approach: Dimensions exhibiting a large vari-

ance are represented by long DAs. Correlations within the

data are indicated by DAs that are placed close to each other

(colored axes). The variance points associated to these di-

mensions indicate that these correlations hold for the whole

high-dimensional data space (parallel coordinates plot pro-

vided for illustration purposes only).

possible misinterpretations stemming from the visual

assessment of the DAs’ properties.

Because principal components are mutually or-

thogonal, it is possible that the depicted standard devi-

ation of certain dimensions is lower in the initial pro-

jection than in other projections (globally ”less op-

timal” combinations of principal components). This

depends on the overall information content of this di-

mension in the subspaces collapsed by dimensionality

reduction. Thus, the knowledge derived from the DAs

can only be a subset of the hidden information and

usually represents a high-level view only. To avoid

misinterpretation, they must be further evaluated.

Indicators for that information loss, e.g., the prin-

cipal components, are often not part of the projection.

The quality of the approximation

X, with regard to

one dimension, is directly reﬂected by the amount of

lost variance in this dimension. A single projection

cannot convey this information. To overcome this

drawback, an SDT display provides variance points

for each dimension. The number of variance points

associated to a DA is equal to the dimensionality of

the data set. Each variance point consists of two cir-

cles. The outer circle’s radius equals s

, the dimen-

sion’s standard variation. The inner circle equals

the part of the dimension’s standard variation that is

shown in the projection. The ratio between both cir-

cles reﬂects the actual variance of this dimension in

the projection (

X) in relation to the dimensions’ total

variance, i.e., (

)

. and thus, allows one to infer

knowledge about the importance of the data dimen-

sion. Variance points also provide guidance for inter-

active exploration. As their positions along the unit

circle hold the position of this DA in other orthogonal

projections, they can be found and explored quickly.

The position of the i’th variance point thereby reﬂects

the position of the corresponding DA in the orthogo-

nal projection deﬁned by

Γ = (γ

(1)

,γ

(i)

), the combi-

nation of the ﬁrst and i’th principal component. As

shown in Figure 3 and 4, a wide range of the vari-

ance is usually captured by a few large variance points

strongly reducing the data space that must be ex-

plored.

Figure 4: Variance points help to ﬁnd other orthogonal pro-

jections of the data. Large variance points (left) indicate

projections most suited to convey the variance in the data.

Opposite variance points (right), even if not accounting for

much variance, often lead to strongly different projections

helping to identify unexpected data properties.

Commonly, the user is aware of the fact that pro-

jections have an inherent information loss. Projec-

tions that map different points in IR

to the same loca-

tion in IR

make this fact clear. This ambiguity is of-

ten a severe problem and also stems from the principal

illustration of “lost” subspaces. These points differ in

those subspaces that are disregarded by dimension-

ality reduction and are therefore projected onto the

same location. By visualizing the projection path of

each data point, SDTs prevent possible misinterpre-

tations by assuring the user that data points are only

equal when they share the same path. The display

of an SDT, however, requires additional processing

power and introduces further graphical primitives into

the data representation. This leads to occlusion prob-

lems, clutter, and long processing times when large

data sets are to be displayed. How to solve these is-

sues by proper interactive exploration and means for

a scalable processing and representation is discussed

in the following sections.

INTERPRETATION, INTERACTION, AND SCALABILITY FOR STRUCTURAL DECOMPOSITION TREES

641

Figure 5: Moving a DA in circles activates the motion paral-

lax effect of the human visual system letting the tree and the

data points appear more ”plastic”. By providing many dif-

ferent coordinated projections, potential point clusters can

be identiﬁed or veriﬁed. Best results are obtained by using

a DA corresponding to a dimension with high variance.

4 INTERACTION FOR VISUAL

CLUSTER ANALYSIS

Unconsidered subspaces, visual ambiguities, and oc-

clusion issues within the initial representation can

be addressed via interactive data exploration. This

section introduces an exploration strategy for inter-

active visual cluster analysis as a common repre-

sentative for the various goals in high-dimensional

data visualization. The strategy was developed to

cope with the large and complex data sets resulting

from mass spectrography in air quality research (Bein

et al., 2009). We follow the information visualization

mantra (Shneiderman, 1996) starting with providing

an overview to the data, then ﬁltering data that are of

minor interest, and eventually applying a drill-down

step to uncover interesting details. As this involves

many common exploration tasks, the given statements

are broadly applicable to a variety of use cases.

A ﬁrst overview about the data and its properties

is provided by the initial projection of the data as de-

scribed in the previous section. Subspaces hidden by

dimension reduction, however, can contain further in-

formation important for the analyst. They are made

available by a successive exploration of individual

dimensions via their respective variance points. To

achieve this, DAs are moved to other associated vari-

ance points leading to a different but still orthogonal

projection of the data. To explore most important in-

formation ﬁrst, it is meaningful to use large variance

points indicating a strong inherent information con-

tent. We also propose to use variance points placed at

opposite positions on the unit circle. Although posi-

tion has no meaning regarding the amount of informa-

tion content, this leads to strong changes in the pro-

jection and may reveal unexpected and important in-

sight (see Figure 4). Switching between close points

does not signiﬁcantly change the projection and can

usually be skipped even for large variance points.

Figure 6: Cluster identiﬁcation and veriﬁcation by a stretch-

ing of a DA: While clutter hinders the detection of point

clusters in the initial projection (left), this interaction re-

veals individual clusters and even sub-clusters in the data

(right). Modiﬁcation of a DA also allows the users to reveal

the contribution and inﬂuence of the associated dimension

to an examined cluster. The two clusters at the bottom of

the representations are not affected by the interaction.

Even when subspace exploration is facilitated by

variance points, usually there are still visual ambigu-

ities resulting from dimension reduction and an over-

plotting of the points and the SDT. To overcome this,

we introduce a novel exploration technique based on

the motion parallax effect of the human visual sys-

tem. By selecting and moving an appropriate DA,

this effect creates a pseudo three-dimensional impres-

sion of the two-dimensional representation (see Fig-

ure 5) letting the points and the tree appear more

”plastic”. During this interaction, point clusters can

be identiﬁed by their constant grouping. In our ex-

periments, we revealed that continuously moving the

DA in circles with varying diameter was particularly

helpful to emphasize point clustering. By moving in

circles, projections revealing point relations are dis-

played multiple times helping the human visual sys-

tem to memorize this insight. Continuously chang-

ing diameter stretches or compresses potential clus-

ters allowing for improved identiﬁcation or veriﬁca-

tion. Not every dimension is equally suited to achieve

this. We propose to select a dimension that does

strongly contribute to higher tree branches, e.g., one

that has a high variance in data values. As the move-

ment strongly affects the top of the SDT leaving its

stem nearly unchanged, it can increase motion paral-

lax. Appropriate dimensions can easily be found by

IVAPP 2012 - International Conference on Information Visualization Theory and Applications

642

Figure 7: The orthogonal placement of two DAs (red and

yellow color) in the presentation can simplify the evalua-

tion of correlations between two-dimensions. Small con-

tributions of points near the origin and large contributions

at the top/right corner for both dimensions indicate a linear

correlation for this pair of dimensions.

dimension highlighting emphasizing all line segments

corresponding to the coordinates of a dimension and

thus conveying their distribution. Sometimes only the

orientation of the tree or of large branches is to be

changed, e.g., to overcome visibility and occlusion is-

sues. To support this, we propose to ﬁnd and relocate

a dimension with strong contribution to the stem of

the SDT, e.g., a dimension with low variance. This

leaves the initial crone structure of the SDT widely

unaltered for further analysis.

Besides the described circle movement, further in-

sight into the structure of the data can be gained by

modifying the lengths of DAs only. Dimensions caus-

ing a visual separation of data points are most likely to

contribute to clustering. Enlarging the respective DA

separates data points and can help identifying clusters

or verifying assumptions (see Figure 6). All points

of a potential cluster show a similar behavior during

length changes. Path highlighting can be used for fur-

ther veriﬁcation. In case of a valid m-D cluster, all as-

sociated points must share the same projection path.

Length modiﬁcation is also particularly useful to visu-

ally emphasize value contributions in the tree. Large

contributions can easily be identiﬁed by their strong

response to length changes.

Potential correlations in the initial projection are

indicated by DAs that are placed close together. As

shown in the previous section, this is only true for the

current rank-p approximation and must not hold for

the whole data space. Correlations can be veriﬁed by

an individual examination of the variance points for

each involved dimension. If a dimension has only a

single large and many small variance points, it can be

concluded that most of its information is encoded in

the current position of the associated DA. When true

for all involved dimensions, they are correlated not

only in the current presentation but also in the data

space. In the contrary, when all variance points of

a dimension are of same size, the variance is scat-

tered and the current anchor position represents just

a subset of this information. In this case, it cannot be

concluded that the depicted correlation holds in data

space and further examination is required. One means

to achieve this is to limit further exploration to the in-

volved dimensions using ﬁltering.

In case correlation has to be evaluated for two di-

mensions only, they may also be placed orthogonally

to each other. Their visual emphasis by dimension

highlighting could then reveal typical patterns in the

corresponding SDT branches (see Figure 7).

Figure 8: Interactive complexity and clutter reduction tak-

ing advantage of the capabilities of SDTs: dimension ﬁlter-

ing (left) reduces the number of branch segments in the tree,

node collapsing (right) the number of displayed subtrees.

Once an overview and ﬁrst insights have been ob-

tained, it is meaningful to ﬁlter out less interesting di-

mensions or subtrees to reduce clutter. Due to the fact

that the visual contribution of a dimension is deter-

mined by the length of the corresponding DA, dimen-

sions can be fully removed by placing their anchors

at the center of the projection (see Figure 8, left).

Appropriate candidates are dimensions that are cor-

related or show similar characteristics. They can be

substituted by a single super-DA, whereby its angle is

determined by the average and the length by the sum

of all associated DAs. This changes the point projec-

tions only slightly. We further propose to remove di-

mensions having (1) very small variance points or (2)

many, very small branches of similar length at high

tree levels indicating little structure in the data. Less

interesting subtrees can be removed from the visual-

ization by node collapsing. The subtree is then repre-

sented by a single point (see Figure 8, right).

Once the data representation has been ”cleaned”

INTERPRETATION, INTERACTION, AND SCALABILITY FOR STRUCTURAL DECOMPOSITION TREES

643

Figure 9: Means for zoom and pan interactions allow the

users to drill-down into the presentation and data to obtain

momentary detail.

be removing less interesting dimensions and data

points, there is space for analysis on a more granu-

lar level. Details to identiﬁed clusters, such as their

properties and potential sub-clusters, can be obtained

by panning and zooming into the representation (see

Figure 9).

For meaningful interaction, data display in real-

time is mandatory. How to speed-up data processing

and to improve the representation in case of many data

points and dimensions is discussed in the next section.

5 SCALABLE PROCESSING AND

REPRESENTATION

Large data sets with a high level of complexity lead

to long processing times and visual clutter. This

decreases the usability and general applicability and

usefulness of a technology. These issues can be over-

come by scalable processing and representation of

the data discussed in this section.

5.1 Scalable Processing

5.1.1 Complexity

The processes required to compute an initial SDT rep-

resentation are: hierarchy-building, conﬁguring axes,

and rendering. Complexity mainly depends on the

number of data points, N, and dimensions, D.

Most of the processing power required to calcu-

late and provide SDTs is used to establish the data

hierarchy and the initial axis layout. Common hierar-

chical clustering techniques are quadratic in complex-

ity with respect to the number of data points, because

they require comparing





, or O(N

) point pairs;

naive methods can be as complex as O(N

) because

they perform each set of O(N

) comparisons every

time two clusters are aggregated. Performing these

comparisons usually involves computing a difference

along each dimension, adding a complexity of O(D).

The processing power needed to provide an appro-

priate initial data projection strongly depends on the

applied approach. The PCA-based method used for

SDTs has a complexity of O(N

) (Engel et al., 2011).

Rendering the SDT is the least expensive compu-

tation. Given the fact that the tree hierarchy consists

of 2N or O(N) nodes and rendering each node is lin-

ear in the number of dimensions, or O(D), the render-

ing takes O(ND) time.

Summarizing, the complexity of the processes re-

quired to compute an initial SDT representation is

O(N

D) + O(N

) + O(ND) = O(N

D) (4)

5.1.2 Computation

In order to reduce the computation time of the ini-

tial SDT and axis layout, we propose N as well as D.

Based on equation (4), reduction of N has the greatest

impact on the complexity of the corresponding pro-

cesses. This especially applies to the initial hierarchi-

cal clustering and the calculation of the projection.

Many different methods to meaningfully reduce

the number of data points have been proposed (El-

lis and Dix, 2007). A simple and often used strat-

egy that can also be applied to SDTs is uniform sub-

sampling (see Figure 10). More sophisticated meth-

ods can be used depending on known characteristics

of the data, the purpose of the visualization, or cur-

rent user demands. SDTs do not have any special re-

quirements for the selection of an appropriate point

reduction strategy.

While point reduction should have the greatest im-

pact on initial computation times, data sets of high di-

mensionality can require signiﬁcant computation and

be difﬁcult to visualize effectively. Similar to point re-

duction, numerous methods to meaningfully decrease

D have been proposed. They range from simple meth-

ods, such as PCA, to more sophisticated strategies,

such as the use of an importance hierarchy based on

the dimensions (Yang et al., 2003c), to manual meth-

ods in case automated technology fails. In any case,

Figure 10: Example of an SDT for all data (left, 1000 data

points) and a uniformly sub-sampled subset (right, 100 data

points). Emphasized properties of the data are well con-

veyed in both representations. The large colored dots indi-

cate end points of DAs.

IVAPP 2012 - International Conference on Information Visualization Theory and Applications

644

such approaches reduce computation time proportion-

ally to the fraction of dimensions eliminated. As long

as the applied method does not remove or cover data

properties the analyst might be interested in, SDTs

do not demand any special dimension reduction strat-

egy. The initial SDT layout before and after reduc-

tion, however, is usually different.

Point and dimension ﬁltering also reduce the com-

putation power needed to render and interact with the

visualization in real time. Since rendering the SDT is

linear with respect to both N and D, point reduction

is not as superior a method with respect to reducing

computation time as it is with respect to the initial

computations. However, the fact that users can ex-

pect to have far more data points than dimensions un-

derpins the signiﬁcance of prioritizing point over di-

mension reduction. In order to justify the theoretical

assumptions stated we conducted a number of exper-

iments to assess the complexity of SDTs with regard

to a changing number of N and D.

In the experiments we used a prototypical imple-

mentation of SDTs as described in (Engel et al., 2011)

and a 255-dimensional air quality data set consist-

ing of 70,000 points provided from our collaborators

from the UC Davis Air Quality Research Center. For

complexity reduction we applied uniform point sub-

sampling and PCA. Figure 11 shows the initial SDT

computation time for various subsets of the data with

either 13, 50, and all 255 dimensions. As shown, the

computation time is a non-linear polynomial function

of N, and a linear function of D. The differences

between the original and reduced data sets are sig-

niﬁcant. Speciﬁcally, computing the SDT for 1,000

points takes approximately 80% less time for a 50-

dimensional subset of the data than for all 255 dimen-

sions. Computations involving all 255 dimensions

were 100 times faster for 200 points instead of 1,000

points, showing the tremendous growth in complexity

as a function of N. This demonstrates the signiﬁcance

of point reduction. Clustering and initial projection,

however, are computed in a pre-processing step and

may be stored for multiple uses in cases where point

reduction is not applicable or meaningful.

5.2 Scalable Representation

A major problem of all projection-based visualization

techniques is the low-dimensionality of the presenta-

tion space. With too many dimensions, projection-

based visualizations can become difﬁcult to under-

stand; SDTs in particular can become cluttered and

busy, especially by rendering DAs. However, since

SDTs aggregate common dimensional components of

clusters into single branches, they are visually orga-

Figure 11: Initial SDT computation time as a function of N

and D. The gold points represent 255-dimensional subsets

of the data, the bisque points the 50-dimensional subsets,

and the dark points the 13-dimensional subsets. When N

is small, the computation time is almost identical for all ex-

amined subsets. The timing results reﬂect well the predicted

theoretical behavior.

nized and can be easily interpretable for data sets with

upwards of twenty dimensions. Depending on various

properties of the data, such as the presence of distinct

clusters, even higher-dimensional data sets containing

thirty or even forty dimensions can be clearly inter-

pretable. Beyond ﬁfty dimensions, however, the sole

presence of the shown DAs makes the display over-

crowded and difﬁcult to understand.

Another common issue of projection-based vi-

sualization techniques is that projecting data into a

space with fewer dimensions can lead to a close po-

sitioning of data points that are much more distant in

high-dimensional space. SDT’s attempt to overcome

this problem by means of a tree structure, but illustrat-

ing this structure might not remove all ambiguities in

the representation for very large data sets. In order

to overcome this, we propose to highlight and differ-

entiate data points below certain tree levels. In order

to achieve this, we enclose the data points and tree

branches below a certain level in uniquely colored iso-

surfaces covering the associated points; the SDT itself

remains unchanged. Any isosurface-rendering algo-

rithm, such as the well-known Marching Cubes (MC)

algorithm (Lorensen and Cline, 1987), is suitable for

this task. As shown in Figure 12, this highlights the

association and distribution of points belonging to the

same cluster even when points associated with differ-

ent clusters are located nearby or even at identical po-

sitions in presentation space.

When data sets are dense and many points occlude

one another, the relative densities of clusters are am-

biguous to the user. Isosurfaces can also be used

to overcome this problem by conveying cluster den-

INTERPRETATION, INTERACTION, AND SCALABILITY FOR STRUCTURAL DECOMPOSITION TREES

645

(a) Data set with ambiguous cluster projections.

(b) Closeup of ambiguous clusters.

Figure 12: Disambiguating point positions in SDTs: The

green and purple isosurfaces correspond to two distinct

subtrees of data. Their intersection indicates that these

branches represent data points that reside at different loca-

tions in the high-dimensional data space but are positioned

near each other in the presentation space.

sity. To achieve this, isosurfaces are rendered around

sub-trees of the SDT in regions of high point density,

quickly highlighting sub-trees that might require fur-

ther analysis. Some isosurface rendering algorithms,

including the MC algorithm, can directly or indirectly

rely on point density in display space as a form of in-

put and only draw isosurfaces where relevant. Isosur-

faces can also be color- or opacity-coded to provide

the users with more speciﬁc information on cluster

density in a given region (see Figure 13).

Isosurfaces are also a valid means to represent

data points that have been ﬁltered by one of the ap-

proaches described above or by interactive folding.

Instead of the associated data points, only correspond-

ing isosurface are shown. To visually de-emphasize

such sub-clusters, however, they should be uniquely

colored and highly transparent.

Figure 13: Disambiguating cluster density in SDTs: The

front-most cluster in this visualization is signiﬁcantly

denser than the other clusters shown. This is conveyed by

the green isosurface.

6 CONCLUSIONS

SDTs are a valid means to visualize and explore high-

dimensional data. However, several questions im-

portant for a broad adoption still remain to be an-

swered. Our paper addressed several of these ques-

tions. We were particularly interested in the insight

that can be gained from an interpretation of the ini-

tial projection of the data. We delivered proof that the

length and relation of DAs allow one to draw mean-

ingful conclusions about the information content of a

single and correlations between multiple dimensions

of the data. We also provided means and guidelines

for their interactive exploration in visual cluster anal-

ysis. In order to deal with large data sets, we pro-

posed and empirically justiﬁed options for scalable

processing and representation. All addressed prob-

lems and the introduced solutions showed that SDTs

can be successfully used in a variety of application

domains to cope with the challenging problem of

high-dimensional data analysis, visualization, and in-

teractive exploration.

The general abilities of SDTs have not yet been

evaluated empirically. Future work can be directed at

the design and implementation of a user study com-

paring SDTs to value visualizations, such as the paral-

lel coordinates plot, and relation visualizations, such

as PCA. When based on common real-world scenar-

ios, this evaluation can also justify the usefulness of

SDTs for a broad variety of application domains. The

results we already obtained from our experiments and

user feedback are promising and provide evidence for

the true value of the novel approaches presented here.

IVAPP 2012 - International Conference on Information Visualization Theory and Applications

646

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