Parallel Coordinate Plots for Neighbor Retrieval

Jaakko Peltonen

1,2

and Ziyuan Lin

1,2

Helsinki Institute for Information Technology HIIT, Department of Computer Science, Aalto University, Espoo, Finland

School of Information Sciences, University of Tampere, Tampere, Finland

Keywords:

Parallel Coordinates, Visualization, Machine Learning, Dimensionality Reduction.

Abstract:

Parallel Coordinate Plots (PCPs) are a prominent approach to visualize the full feature set of high-dimensional

vectorial data, either standalone or complementing other visualizations like scatter plots. Optimization of

PCPs has concentrated on ordering and positioning of the coordinate axes based on various statistical criteria.

We introduce a new method to construct PCPs that are directly optimized to support a common data analysis

task: analyzing neighborhood relationships of data items within each coordinate axis and across the axes. We

optimize PCPs on 1D lines or 2D planes for accurate viewing of neighborhood relationships among data items,

measured as an information retrieval task. Both the similarity measurement between axes and the axis positions

are directly optimized for accurate neighbor retrieval. The resulting method, called Parallel Coordinate Plots

for Neighbor Retrieval (PCP-NR), achieves better information retrieval performance than traditional PCPs in

experiments.

1 INTRODUCTION

Exploration of high-dimensional data is challeng-

ing and numerous tools have been developed, in-

cluding summarization approaches like clustering and

component analysis, and numerous visualization ap-

proaches. Visualization of high-dimensional data lets

experts visually conﬁrm, reject or modify hypotheses

generated by analysts or automated methods; com-

municate information about data in a compact visual

way; and interactively explore data and features and

give feedback about models of data.

Scatter plots are a common visualization for mul-

tivariate data, but traditional plots are usually lim-

ited to 2D or 3D. For higher dimensional data each

plot must show a small subset of features (some-

times several plots are collected in a scatter plot ma-

trix) or a few new features computed by transforming

original features. Such transformed features can be

found by linear and nonlinear dimensionality reduc-

tion (Wism

uller et al., 2010).

We concentrate on visualization by the Parallel

Coordinate Plot (PCP) (Inselberg, 2009; Heinrich and

Weiskopf, 2013), prominent in data mining. In a PCP,

data axes are ordered vertical lines and each data point

corresponds to a piecewise linear path across the data

axes; sometimes interpolated curves are used in place

of piecewise linear paths. Axes are placed in par-

allel horizontally or vertically—alternative organiza-

tions include circular layouts. By representing each

data item as a line path, PCPs show all coordinates

(feature values) of all items, which helps gain a com-

prehensive view of data. Feature values are shown in

their original form, not needing dimensionality reduc-

tion or other transformation, helping interpretability if

features have intuitive meanings. PCP can be used as

standalone visualizations for high-dimensional data;

also, PCPs and scatter plots are often provided as al-

ternative complementary views of data in visualiza-

tion tools.

Ordering of PCP Axes. A main drawback of

PCPs is that the order of data axes greatly affects visu-

alization quality. In a traditional PCP axes are placed

onto a line (a 1D ordering) where each axis has at

most two neighbors: the order determines which axes

are neighbors. Any axis order shows all data coor-

dinates, but relationships between coordinates along

different axes are easiest to see between neighbor-

ing axes; the ability to see multi-dimensional struc-

ture such as grouping of data over several coordi-

nates depends on the axis order. It is hard and time-

consuming to get an overview of data and feature re-

lationships from an unorganized PCP with many di-

mensions. Some approaches aim to show all pairwise

relationships by paths over multiple copies of axes,

but often it is desired to show each axis only once; To

Peltonen J. and Lin Z.

Parallel Coordinate Plots for Neighbor Retrieval.

DOI: 10.5220/0006097400400051

In Proceedings of the 12th International Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications (VISIGRAPP 2017), pages 40-51

ISBN: 978-989-758-228-8

get good visual quality for PCPs of high-dimensional

data, optimizing the order is thus necessary. A further

generalization optimizes both the order of axes and

their relative axis positions onto a 1D line (yielding a

2D PCP) or a 2D plane (yielding a 3D PCP).

Coloring of Line Paths for Data Items. The abil-

ity to detect data relationships among PCP axes de-

pends not only on order and positioning of axes but

also on the ability of analysts to follow data lines

across axes. To help in this, lines of different data

items should be made easily distinguishable from one

another, especially when not only the data density but

also individual data items are important to analyze.

Typically line style, marker style, and in particular

color are used to distinguish line paths of different

data items.

Our Contribution: PCPs Directly Optimized

for a Low-level Analysis Task. In this paper we pro-

pose a PCP construction approach where data-space

criteria are used in a uniﬁed fashion for three aspects

of the PCP: axis ordering, axis positioning, and data

line coloring, and the construction is directly opti-

mized for a prominent data analysis task not well sup-

ported by previous orderings: analysis of neighbors.

We formulate this visualization task as a rigorous

information retrieval task. Our approach, called Par-

allel Coordinate Plots for Neighbor Retrieval (PCP-

NR), ﬁnds an optimized layout for the parallel coor-

dinate axes onto a 1D line or 2D plane. Our solution

is based on a simple principle: axes containing simi-

lar information about data neighborhoods should be

close-by on the display. Our approach yields a well-

deﬁned task for constructing parallel coordinate plots,

where success of each construction stage can be quan-

titatively measured and optimized.

• Given a high-dimensional data set, the ﬁrst stage

in our approach is to measure similarity between

axes. We evaluate similarity of two axes by an

information retrieval approach, that is, by differ-

ences in neighborhood information between axes:

two axes are similar if they show similar neighbor-

hoods between data items. We quantify the simi-

larity as an information retrieval cost of retrieving

neighbors seen in one axis from the other axis.

• The second stage is to use the similarities to po-

sition axes onto the PCP: for a traditional 2D

PCP this means positioning the axes onto a one-

dimensional line; for a 3D PCP this means po-

sitioning the axes onto a 2D plane. We again

optimize this stage for an analysis task: relating

feature axes by ﬁnding similar “neighbor” axes

for each axis. We again take an information re-

trieval approach and optimize positioning of axes

so that a user looking at the PCP can easily iden-

tify which axes are similar.

• The third stage is to pick which axes to connect by

line paths of data points. In a traditional 2D PCP

we naturally connect adjacent axes; in a 3D PCP

we use a minimum spanning tree approach in the

obtained axis layout.

• The fourth stage is to set colors to data items and

draw their line paths. We optimize colors to de-

note overall data similarity, as an information re-

trieval task of relating data items: items get sim-

ilar colors if their coordinates are overall similar

across all dimensions. A user can retrieve data

with similar overall features by retrieving lines of

similar color from the PCP.

In summary, we contribute, based on an informa-

tion retrieval approach, 1) a formalization of PCP con-

struction optimized for a low-level analysis task of

relating data items by retrieving their neighbors; 2)

a data-driven divergence measure between coordinate

axes; 3) a method to arrange axes onto a 2D or 3D

PCP, optimized for retrieval of related axes; 4) a data-

driven method to optimize line colors for data items,

optimized for retrieval of overall similar data items.

Our solution outperforms baselines in experiments.

To our knowledge, this is the ﬁrst paper to pro-

pose a PCP construction method fully optimized for

an analysis task. All construction stages, from eval-

uating axis similarity to positioning axes and color-

ing data lines, are optimized for the task of relating

data items to their neighbors. Unlike previous order-

ing approaches suited for correlation analysis, clus-

tering data or features, and outlier detection, the task

of analyzing neighborhood relationships has not been

well supported although it is a common need in data

analysis.

In the following, Section 2 discusses related work,

Section 3 introduces our information retrieval ap-

proach to construct the PCP-NR, Sections 4 describes

experiments that illustrate the method, compares it to

previous ones, quantiﬁes its advantage, and shows a

case study. Section 5 concludes.

2 BACKGROUND

We discuss user tasks in exploratory data analysis,

and previous work on PCPs and nonlinear dimension-

ality reduction.

User Tasks. Shneiderman (1996) created a tax-

onomy of tasks and data types that appear in visual

data analysis. He listed several tasks that users may

wish to perform during analysis: 1) overview–gain an

overview of the entire data collection; 2) zoom–zoom

Parallel Coordinate Plots for Neighbor Retrieval

in on items of interest; 3) ﬁlter–ﬁlter out uninteresting

items; 4) details-on-demand–select an item or group

and get details when needed; 5) relate–view relation-

ships among items; 6) history–keep a history of ac-

tions to support, undo, replay, and progressive reﬁne-

ment; 7) extract–allow extraction of sub-collections

and of query parameters. Most exploratory data anal-

ysis systems aim to provide facilities for each of these

tasks in some manner through the interface, how-

ever, typically it is not attempted to optimize individ-

ual data visualizations within a system to serve these

tasks although sometimes design choices (types of vi-

sualizations to be shown, etc.) may be motivated by

needs of such tasks. In this work we aim to directly

optimize a visualization of high-dimensional data for

an analysis task: we focus on creating PCPs opti-

mized for the ﬁfth task–viewing relationships among

data items. We formalize this task as optimizing PCPs

for information retrieval of related (neighboring) data

items from the PCP.

Previous Work on PCPs. PCP research includes

methods for arrangement of axes, interpolating and

bundling data lines, presenting density functions, and

interactivity approaches (Inselberg, 2009; Heinrich

and Weiskopf, 2013); PCPs and related approaches

have been integrated in visualization systems to show

relationships between multiple plots; for example,

Claessen and van Wijk (2011) integrate scatter plots,

PCPs, and histograms in regular arrangements, Viau

and McGufﬁn (2012) connect multivariate charts by

curves showing relations between feature tuples, and

scatter plots are sometimes shown between adjacent

PCP axes (Heinrich and Weiskopf, 2013). For brevity

we focus on methods for axis ordering and position-

ing.

Previous Work on Ordering PCP Axes. For R

axes there are R(R − 1)/2 symmetric pairwise rela-

tionships which cannot all be shown in a single or-

der of axes. Some approaches aim to show all (or

most) pairwise relationships by paths through multi-

ple copies of different axes, shown e.g. as a paral-

lel coordinate matrix (Heinrich et al., 2012); Insel-

berg (2009) notes dR/2e permutations of axes sufﬁce

to visualize all pairwise relationships. But even if a

good set of permutations was easy to ﬁnd the num-

ber of permutations needed could become excessive

for high-dimensional data; often a single PCP (or at

most a few) is desired, and we focus on methods to

ﬁnd a single order where each axis is shown once. To

get good visual quality for PCPs of high-dimensional

data, optimizing the order is needed.

Brute-force optimization of axis ordering for some

order-dependent cost function would need O(R!) cost

evaluations to ﬁnd the global optimum, where R is

the number of axes, thus more advanced optimiza-

tion methods are often used (Fua et al., 1999; Peng

et al., 2004; Makwana et al., 2012); see Heinrich and

Weiskopf (2013) for a review. Orderings are often

grouped into data-space and image-space approaches;

the former are based on data statistics and the latter

evaluate visual aspects of the PCP. Analysis tasks sup-

ported by current ordering approaches include corre-

lation analysis, clustering of data or dimensions, and

outlier detection (Heinrich and Weiskopf, 2013).

In traditional PCPs axes may be ordered in a data-

space approach by a statistical property such as skew-

ness (Schloerke et al., 2014) which evaluates each

axis alone rather than evaluating similarity of axes.

Such approaches might notice “interesting” feature

axes but may not be able to organize which interest-

ing features are similar. A basic measure to compare

similarity of axes is the correlation coefﬁcient of coor-

dinates between axis pairs: a highly positive or highly

negative coefﬁcient indicates the two axes are related.

We use the correlation coefﬁcient as a baseline sim-

ilarity measure as it is well understood and widely

used, and compare our proposed PCP construction to

a method using the correlation coefﬁcient measure.

Our method gets better results than the baseline in the

experiments of Section 4. Our approach is also well

justiﬁed—the similarity measure is directly optimized

for the low-level analysis task of analyzing neighbors,

which is not well served by simple correlation.

In this ﬁrst paper on information retrieval per-

spective to PCPs, we do not aim at a comprehensive

comparison among PCP ordering approaches, and fo-

cus on showing good performance with respect to the

well-understood baseline. As noted above, some ap-

proaches can be interpreted to support tasks like cor-

relation analysis, clustering, or outlier detection, but

to our knowledge none aim at analysis of neighbor-

hood relationships whereas our approach is directly

mathematically optimized for it. On some data sets, a

similarity measure not designed for a particular analy-

sis task might perform reasonably, but without a clear

connection to an analysis task it is hard for analysts

to choose an appropriate similarity. Our method pro-

vides a clear, optimized choice for analysts focusing

on neighbor relationships.

Generalizations of the Ordering Task. A gener-

alization of the axis ordering task is to optimize not

only the order of axes but also their relative positions:

intuitively, axes that show similar relationships be-

tween data items can be placed close to each other,

and axes showing very different relationships can be

placed farther off. Axis positions can indicate to the

analyst the overall axis similarities whereas data lines

provide the details; close-by positions of very similar

IVAPP 2017 - International Conference on Information Visualization Theory and Applications

axes could even suggest the analyst only needs to in-

spect one from each group of similar axes. The linear

arrangement problem has been shown to be NP-hard

(Ankerst et al., 1998). Guo (2003) has found linear ar-

rangements using a spanning tree based heuristic. A

further generalization is arrangement of axes not only

onto a 1D line but onto a 2D plane, yielding a 3D

PCP where axes are drawn along the third dimension.

Such 3D versions have been discussed for example

by Wegenkittl et al. (1997), and related constructions

have been suggested (Fanea et al., 2005; Johansson

et al., 2006). A recent work called 3DPC-tree plot

(Achtert et al., 2013) creates a 3D plot by positioning

axes onto a 2D plane based on the minimum spanning

tree of a user-chosen pairwise similarity measure be-

tween axes.

The choice of similarity measure affects similarity

based axis layouts. Similarities unrelated to the data

analysis task might yield misleading layouts; similari-

ties should be directly optimized for the analysis task,

which our method accomplishes for the neighborhood

analysis task.

Coloring of Line Paths for Data Items. Distin-

guishing lines of different data items is important both

for analyzing statistical relationships of data density

between axes and when individual data items are im-

portant to analyze. If two items have different val-

ues along the left-hand axes of a PCP, but have a very

similar coordinate along an axis midway through the

PCP, their line paths cross at the axis, and it is easy

to lose track which continuation of the line path be-

longs to which data item, sometimes called the line

tracing or linking problem. The goodness of axis or-

dering and visual presentation of data lines both affect

the line tracing. Typically line style, marker style, and

in particular color are used to distinguish line paths of

different items, so that each item uses a single color

for its line path. Color may be based on annotation

such as classes or outputs of a clustering algorithm

(Heinrich and Weiskopf, 2013); more generally ﬁnd-

ing a coloring for a large high-dimensional data set is

challenging. For large data assigning clearly separate

colors to all data items is not feasible but well-done

optimization can still assign similar colors to similar

items; our solution accomplishes this. Successful col-

oring can be combined with, e.g., bundling to further

help line tracing; we focus on coloring.

Information Retrieval based Visualization.

Nonlinear dimensionality reduction for scatter plot vi-

sualization has recently been formalized as an infor-

mation retrieval task (Venna et al., 2010), which has

yielded an information retrieval perspective to exist-

ing non-linear dimensionality reduction methods and

new well-performing methods (Venna et al., 2010;

Peltonen and Kaski, 2011; Yang et al., 2013). These

methods yield only dimensionality-reduced versions

of data; the resulting coordinates are not in any easy

way relatable to original data coordinates and there

is no obvious applicability of these methods to PCPs

of original data coordinates. (If original variables are

not desired, output coordinates after any dimension-

ality reduction mapping can be shown in a PCP.)

A recent extension considers “meta-visualization”

where individual scatter plots are arranged onto a

display (Peltonen and Lin, 2015); the technique in-

volves evaluating neighborhood similarity between

plots similarly to how we evaluate similarity between

PCP axes. However, there are several differences: 1)

The domain is entirely different, their work consid-

ers arrangement of several scatter plots onto a display

only, whereas our work constructs a single PCP of

a data set, including its axis locations and the color-

ing of its data lines. 2) Even setting aside domain

differences, their work considers pairwise differences

between scatter plots only, whereas in a PCP it is cru-

cial to support two different retrieval tasks: retrieval

of neighboring axes showing similar data (thus sup-

porting retrieval of neighbors across adjacent pairs of

axes) and retrieval of neighboring data items accord-

ing to overall similarity across all axes. Their work

does not involve anything analogous to the latter task

and is unsuitable for it. Our work solves this dual

information retrieval need by both optimizing an ar-

rangement of axes and optimizing a coloring for data

lines. 3) Their work only places scatter plots onto

a display separately, and does not create connections

among them, whereas in a PCP connecting axes by

data lines is crucial; we both organize PCP axes and

create their connections, in the 2D case by adjacency

of axis positions and in the 3D case by a spanning tree

approach. The connections, shown as data lines, vi-

sualize overall data item similarity by optimized col-

oring. Additionally, their evaluation of visualization

quality does not consider connections between plots

whereas we measure quality of visualization across

connected axes as is suitable for PCPs.

Our work in this paper is the ﬁrst approach tak-

ing an information retrieval perspective to optimiza-

tion of parallel coordinate plots, and is also the ﬁrst

neighbor embedding method organizing axes onto a

parallel coordinate plot.

3 PCP-NR: THE METHOD

We optimize PCPs for analysts who want to explore

data, and in particular want to perform the subtask

where they wish to relate data items. For example, in

Parallel Coordinate Plots for Neighbor Retrieval

a bioinformatics study data items can be tissue sam-

ples from patients described by features that are ac-

tivities of numerous genes; in ﬁnancial analysis items

can be companies described by ﬁnancial indicators

like liquidity; in an educational study items can be

students described by performance in various courses.

We assume the analyst is studying data through neigh-

borhood relationships, and wishes to relate data items

by ﬁnding their neighbors (similar other samples). We

model the analysis as two visual information retrieval

tasks:

Task 1: Retrieval of Neighboring Data Items

Across Axes. Suppose the analyst inspects an axis

of interest, showing particular similarity relationships

(neighborhoods) among data items along the axis.

The analyst may want to ﬁnd similar axes, that is,

axes showing similar relationships among the data.

We optimize the axis placement so the analyst can

easily retrieve similar axes as neighbors in the PCP.

In detail, from any axis the analyst can visually re-

trieve neighborhoods of data items: given an item of

interest, the analyst can see which other items have

similar values in the axis and can pick (retrieve) the

most similar (neighboring) items. Next, given many

axes in a PCP, the analyst can visually retrieve which

axes show similar neighborhoods as an axis she is in-

terested in, vs. which axes show different information.

In a well-organized PCP axes showing similar infor-

mation should be placed nearby, to make them easy

to retrieve. We quantify and optimize the PCP for this

retrieval task.

Task 2: Retrieval of Neighboring Data Items

based on the Whole PCP. Given all axes in a PCP, the

analyst may want to visually quickly retrieve which

data items are overall similar (neighbors) based on

their coordinates in all axes. To do this based on

shapes of data lines would require the analyst to

assess all data in all axes at once, which is time-

consuming, as data items may not be similar enough

to form clear bundles of lines across all axes. In-

stead, we support the task by optimized coloring: we

quantify neighborhoods of data items by their high-

dimensional coordinates, and optimize a coloring for

data lines so that similar data items get similar colors.

Retrieving overall similar data items is then simple for

the analyst and reduces to picking lines with similar

colors. We quantify the retrieval by comparing neigh-

borhoods in the high-dimensional original space and

in the color space, and optimize colors to minimize

retrieval errors.

Comparing Axes. In visual information retrieval

an analyst may look at a PCP axis and visually

retrieve neighbors for each data point of interest.

The analyst can visually compare neighborhoods be-

tween several axes: two axes showing similar neigh-

borhoods yield ﬁndings supporting each other; axes

showing different neighborhoods reveal different nov-

elties in data.

Neighborhoods of Data Items. Let {x

}

i=1

be a

set of R-dimensional input data samples (data items).

There are then R different axes in the PCP, and sam-

ples have positions {x

r,i

}

i=1

along the rth axis. Each

feature axis shows a different conﬁguration of the data

along the axis, thus each axis shows different neigh-

borhood relationships between data.

In the rth axis, we deﬁne a probabilistic neigh-

borhood for each data item i, as a distribution p

( j|i)} over the neighbors j 6= i. p

( j|i) is the prob-

ability that an analyst inspecting data item i on the

axis would retrieve item j as an interesting neighbor

for further study, so that samples j close to i on the

axis are more likely to be retrieved. We set

( j|i) =

exp(−(x

r,i

− x

r, j

)

/σ

r,i

)

∑

k6=i

exp(−(x

r,i

− x

r,k

)

/σ

r,i

)

. (1)

Intuitively, Eq. (1) says that the closest neighbors j

(whose values x

r, j

are closest to x

r,i

) have the greatest

chance to be picked for inspection next, but further-

away neighbors also have a chance to be picked. Eq.

(1) ﬁts our requirements of probabilistic neighbor-

hood: 1. it is normalized and thus a distribution; 2.

the numerator is a decreasing function of distance.

We allow the falloff rate of p

( j|i) be ﬂexibly con-

trolled by σ

r,i

. The σ

r,i

can be set in a data-driven

way to make the p

have entropy logk for a desired

rough number of relevant neighbors k as in Hinton

and Roweis (2002); Venna et al. (2010); we use an

even simpler choice and set σ

r,i

to 1/10 of the max-

imum pairwise Euclidean distance between points in

axis r which works well in experiments.

Comparing Two Axes. Consider an analyst who

moves from studying axis r to studying another axis

. The analyst previously picked out the neighboring

items for a particular item i from r, and now tries to

pick out the same items from r

. The data have differ-

ent values along the two axes, and two kinds of differ-

ences happen. Some points j were neighbors of i in

axis r (high probability p

( j|i)) but are far off in axis

(low p

( j|i)) and are missed when neighbors are

retrieved from r

. On the other hand, some points j

that were far from i in axis r (low p

( j|i)) look like

neighbors in r

(high p

( j|i)) and are novel neigh-

bors when neighbors are retrieved from r

. Misses

and novel neighbors are symmetric so that if a neigh-

bor from axis r is missed in r

, it is a novel neighbor

in r compared to r

. Figure 1 illustrates the setup for

simple hard neighborhoods where each data item is or

is not a neighbor to i; we generalize this to soft prob-

abilistic neighborhoods.

IVAPP 2017 - International Conference on Information Visualization Theory and Applications

missed

neighbor

novel

neighbor

missed

neighbor

Axis 1 Axis 2

Figure 1: Differences in visual information retrieval for

query point i, when neighbors in one parallel coordinate

plot axis (left) are retrieved from a second axis (right). At

left, the query point is shown as a large black circle, neigh-

bors of the query point are shown as green circles, and non-

neighbors are shown as red stars; the portion of the axis

where neighbors reside is also indicated with light green

background and portion of non-neighbors with light red

background. P

denotes the set of points with high neigh-

borhood probability p

( j|i) in the ﬁrst axis, and P

denotes

points with high p

( j|i) in the second axis. Missed neigh-

bors have high p

( j|i) but low p

( j|i); an analyst looking

at the second axis would miss them. Novel neighbors have

low p

( j|i) but high p

( j|i); they were not apparent in the

ﬁrst axis.

Recent work in neighbor embedding (Venna et al.,

2010) has shown comparisons of neighborhoods can

be quantiﬁed by the information retrieval measures

precision and recall which can be expressed as

Kullback-Leibler (KL) divergences between proba-

bilistic neighbor distributions. We use a similar in-

sight between axes: for query item i the amount of

differences in neighbor retrieval between axes r and

can be generalized as the KL divergence

, p

) =

∑

j6=i

( j|i)log

( j|i)

(2)

where in the case of hard neighborhoods the diver-

gence is proportional (proof equivalent to Appendix

A of Venna et al. (2010)) to the number of missed

neighbors from r to r

(recall from r to r

), or equiv-

alently, the number of novel neighbors from r

to r

(precision from r

to r). To compare full axes r and r

we sum over the query points, yielding

r,r

∑

, p

) =

∑

j6=i

( j | i) log

( j | i)

(3)

which is proportional to the average amount of misses

if retrieving neighbors based on axis r from axis r

, or

the average amount of novel neighbors if retrieving

neighbors based on axis r

from axis r due to symme-

try. Intuitively, D

r,r

will be small if axes have similar

overall trends or high-level clusters, but also if data

are similar at the local level; our experiments show

we detect neighborhood similarities even when large

global changes take place between axes.

Optimizing the Axis Layout. Given the differ-

ence measure (3) for comparing any two axes, we ex-

tend the concept of neighborhood to neighborhoods

of the whole axes themselves. Much like the data item

neighborhoods in (1), we deﬁne a probabilistic neigh-

borhood for axis r as

u(r

| r) =

exp



−D

r,r

/σ



∑

˜r6=r

exp(−D

r,˜r

/σ

)

(4)

where D

r,r

is the difference measure (3) and u(r

r) denotes the probability that an analyst who had

carefully studied data neighborhoods among the axes

would, after inspecting axis r, pick axis r

to inspect

next. In a good PCP, this neighborhood of a careful

analyst should match as well as possible with the ap-

parent neighborhood of axes based on their physical

locations on the PCP, which we deﬁne analogously as

v(r

| r) =

exp



−kz

− z



/σ

∑

˜r6=r

exp(−kz

− z

˜r

/σ

)

(5)

where z

is the on-screen location for axis r. Based on

the distributions u

= {u(r

| r)}, v

= {v(r

| r)}, we

optimize the locations z

to minimize the information

retrieval differences in retrieval of neighboring axes;

that is, we minimize the sum of Kullback-Leibler di-

vergences

E = λ

∑

) + (1 − λ)

∑

) (6)

with respect to axis locations z

with a nonlinear opti-

mization algorithm, here conjugate gradient, to obtain

the optimized axis layout. We set λ = 0.5, emphasiz-

ing misses and novel neighbors equally in retrieval of

neighboring axes.

Creating Lines for Data Points. With the axis

layout done, we draw piecewise linear paths between

axes for data points. When axes are placed on a

1D line (as a 2D PCP) we connect neighboring axes;

when placed on a 2D plane (as a 3D PCP), we connect

at most O(d) axes as the traditional PCP does, fol-

lowing a minimum spanning tree (MST) idea (Achtert

et al., 2013), but with W = (w

)

d×d

= kz

−z

as the edge weight matrix.

Optimizing the Data Line Colors. The axis lay-

out was optimized to show pairwise similarities be-

tween axes. We optimize colors of data lines to show

overall similarity of data items, as a neighbor retrieval

task. We deﬁne neighborhoods p

all

= {p

all

( j|i)} of

data i and neighbors j by overall similarity of the co-

ordinates x

, corresponding to a careful analyst study-

ing the whole PCP, as

all

( j|i) =

exp(−kx

− x

)

k/σ

all,i

)

∑

k6=i

exp(−kx

− x

/σ

all,i

)

, (7)

Parallel Coordinate Plots for Neighbor Retrieval

and deﬁne color similarity of data lines as neighbor-

hoods q

color

= {q

color

( j|i)} based on color coordi-

nates c

color

( j|i) =

exp(−kc

− c

/σ

all,i

)

∑

k6=i

exp(−kc

− c

/σ

all,i

)

(8)

and the falloff σ

all,i

are set as in Venna et al. (2010).

The amount of differences is again quantiﬁed by

divergences as

∑

all

color

) + D

color

, p

all

)

which is proportional to the sum of missed neighbors

and novel neighbors in the color space compared to

the original; we minimize both differences with re-

spect to the c

by conjugate gradient optimization.

Our experiments show this works well, and can reveal

interesting patterns of the data.

4 EXPERIMENTS

Comparison on Artiﬁcial Data. We demonstrate

our PCP-NR on 2D and 3D PCPs on an artiﬁcial

data set. In the 2D version, we compare to the PCP

implementation ggparcoord in R package GGally

(Schloerke et al., 2014), using Skewed as the dimen-

sion order option (similar results would be obtained

with other options). In the 3D version, we com-

pare to an implementation of 3DPC-tree plot (Achtert

et al., 2013), which creates an MST and connects tree

edges based on distances 1 −|ρ

r,r

| between axes r and

, where ρ

r,r

is the Pearson correlation coefﬁcient,

and places axes on a 2D plane with a balloon lay-

out (Herman et al., 2000). The artiﬁcial data matrix

M = (m

i j

)

160×15

is created as follows. We divide the

15 features evenly into 3 consecutive feature groups.

For each group of 5 features, we uniformly sample a

“group means” matrix G = (g

i j

)

8×5

from U[0,1]

8×5

and randomly assign each of the 160 data point to one

of the 8 data groups. Assume in the feature group

k (k = 1,2, or 3), m

i j

(5(k − 1) ≤ j ≤ 5k) is assigned

to l ∈ {1,·· · ,8}, we then set m

i j

= g

+ε (5(k−1) ≤

j ≤ 5k,m = j − 5(k − 1)) where ε ∼ U[0,0.05]. Fea-

ture groups with such multimodal densities are difﬁ-

cult for correlation based methods to ﬁnd. Figure 2

shows PCP-NR correctly places features in the same

group close-by, while the ggparcoord implementa-

tion or the 3DPC-tree plot fail to do so.

We quantitatively measure visualization quality in

the 3D version by average ratio of “sum of within-

group pairwise distances between axis positions” to

“sum of cross-group pairwise distances between axis

positions”. A good layout should yield a low ratio,

meaning well grouped features. We repeat PCP-NR

for 50 times with different randomizations, yielding a

ratio 0.1385 ± 0.0048. 3DPC-tree plot yields a ratio

0.4376, signiﬁcantly worse than PCP-NR.

The coloring of data lines is also better in PCP-

NR for data analysis: while the simple hue mapping

of mean coordinates in the baseline is bright and col-

orful, it fails to reveal several different groups in data

(only a few groups are visible), whereas our coloring

clearly shows the large number of groups of samples

each behaving overall similarly. Note that the overall

similarity used for PCP-NR coloring is based on all

axes: data items that behave similarly in most dimen-

sions will get similar colors, but will still show some

color variation if they differ along some dimensions.

Comparison on Real Data Sets. We show

the performance of PCP-NR on several data

sets on UCI Machine Learning Repository (Lich-

man, 2013): Breast Cancer Wisconsin (Diagnos-

tic),Cardiotocography,Parkinsons,QSAR biodegrada-

tion,Leaf,and Wine.We separate features and labels in

the data sets, and create PCPs only from data features,

labels are left for quantitative evaluation.

To measure the performance of the proposed

method on the real data sets, we assume the analyst

does retrieval of axes from the visualization: when the

analyst inspects one dimension, he then retrieves its

neighbors based on graph distances between vertices

in the MST. We quantify ground truth distances of di-

mensions by “differences” between k-nearest neigh-

bor predictions from individual dimensions: for the

data set X and its any dimension s, we create a matrix

∈ R

n×C

, with n the number of data points and C the

number of categories in the data set. We let K

be the

k-NN prediction probability of category c for x

, given

the values only from dimension s, and kK

− K

the ground truth distance between dimension s and s

We stress the distance deﬁned in this way does not

have bias towards our method, as we do not use label

information to create the PCP.

The performance in this information retrieval task

(retrieval of axes) is then naturally measured by the

precision-recall curve, a well-understood information

retrieval quality measure. The curve from a method

with better retrieval performance should be located

at the top-right in the ﬁgure, meaning the method

can achieve both better precision and recall. We set

k = 20 in the evaluations and again compare PCP-

NR with 3DPC-tree plot + balloon layout. Figure 3

and Figure 5 show the results, from 3D PCPs and 2D

PCPs, respectively. Figure 4 shows the 3-dimensional

PCP and 2-dimensional PCP for the Wine data set

we used. We also report the AUCs as summariza-

tions of the precision-recall curves, as shown in Table

1 and Table 2. The relative surplus is calculated as

(AUC

PCP−NR

− AUC

Baseline

)/AUC

Baseline

IVAPP 2017 - International Conference on Information Visualization Theory and Applications

Cross-group

connections that

do not preserve

neighborhood

relations are

placed far off

6 7 8

11 14

12 15 13

2 5

13 11 8 3 4 15 9 5 7 6 14 10 12 1 2

Figure 2: Results of the toy data set. Top left: 2D PCP-NR; bottom left: ggparcoord implementation; middle: 3D PCP-

NR; right: 3DPC-tree plot + Balloon layout. The feature groups are dimension 1–5, 6–10, and 11–15. In the 2D version,

PCP-NR arranges the features positions so that the feature groups are at the left, center, and right respectively. In the 3D

version, we highlight the groups by black ellipses. Meanwhile in the baseline methods, the connections between features do

not reﬂect the ground-truth grouping. Also, our coloring correctly shows the 8 groupings in the data points in most of the

case. High-resolution versions of this and other ﬁgures are available online at https://github.com/pcpnr/icdm16.

reast

ancer

isconsin

nostic

)

1.0

0.8

0.6

0.4

0.2

0.2 0.4 0.6 0.8 1.0

Baseline

PCP- NR

Precision

Recall

Brease Cancer Wisconsin (Diagnostic)

Baseline

PCP-NR

ardiotocography

0.5

0.4

0.3

- Baseline

- PCP- NR

0.2

0.1

0.2

0.4

0.6 0 .8 1.0

Recall

Precision

Recall

Cardiotocography

Baseline

PCP-NR

Recall

Precision

Recall

Parkinsons

Baseline

PCP-NR

Recall

recision

Precision

Recall

QSAR Biodegradation

Baseline

PCP-NR

Recall

Precision

Recall

Leaf

Baseline

PCP-NR

Recall

Precision

Recall

Wine

Baseline

PCP-NR

Figure 3: 3D PCP: the precision-recall curves from differ-

ent data sets, compared with the baseline 3DPC-tree plot +

balloon layout. In most of the data sets, the curves from our

method lie top-right to the curves from the baseline, indi-

cating better retrieval performance.

Table 1: AUCs of precision-recall curves in Figure 3. PCP-

NR outperforms on 4 of 6 data sets. When the baseline

outperforms, the relative surplus is not large.

PCP-NR Baseline Rel. sur.

Breast Cancer

Wisconsin (Diagnostic)

0.523 0.440 19.0%

Cardiotocography 0.290 0.265 9.56%

Parkinsons 0.536 0.570 -5.96%

QSAR biodegradation 0.273 0.233 17.3%

Leaf 0.560 0.561 -0.332%

Wine 0.349 0.301 15.2%

Case Study. We apply PCP-NR onto a human

gene expression collection from the ArrayExpress

database (Parkinson et al., 2009) for visual analy-

7 6 9 2

3 10

Figure 4: 3D PCP and 2D PCP for Wine data from PCP-NR.

sis. The data set contains D = 105 experiment re-

sults (gene expression activities) from comparisons

between healthy and diseased subjects. Additional la-

bels about the relevant diseases are available for each

experiment, which are “cancer”, “cancer-related”,

“malaria”, “HIV”, “cardiomyopathy”, or “other”. We

Parallel Coordinate Plots for Neighbor Retrieval

Precision

Recall

Baseline

PCP-NR

Breast Cancer Wisconsin (Diagnostic)

Recall

Precision

Baseline

PCP-NR

Precision

Recall

Cardiotocography

Recall

Precision

Recall

Parkinsons

Baseline

PCP-NR

Recall

Precision

QSAR Biodegradation

Baseline

PCP-NR

Recall

Precision

Recall

Leaf

Baseline

PCP-NR

Recall

Precision

Recall

Wine

Baseline

PCP-NR

Figure 5: 2D PCP: the precision-recall curves from different

data sets, compared with the baseline PCP implementation

ggparcoord from R package GGally. In most of the data

sets, the curves from our method lie top-right to the curves

from the baseline, indicating better retrieval performance.

Table 2: AUCs of the precision-recall curves in Figure 5.

PCP-NR outperforms 4 out of 6 data sets. When the base-

line outperforms, the relative surplus is not large.

PCP-NR Baseline Rel. sur.

Breast Cancer

Wisconsin (Diagnostic)

0.530 0.214 147%

Cardiotocography 0.240 0.242 -1.06%

Parkinsons 0.508 0.279 81.8%

QSAR biodegradation 0.201 0.250 -17.9%

Leaf 0.600 0.254 134%

Wine 0.336 0.201 63.6%

are interested in how the connection between the dif-

ference of the involved diseases and the activities from

different sets of gene pathways is reﬂected in the visu-

alization with PCP.

We preprocess the data set as in Caldas et al.

(2009): ﬁrst use gene set enrichment analysis (GSEA)

to measure W = 385 activities of known gene path-

ways, then train a topic model from the “experiment–

pathway activity” matrix with L = 50 topics, each of

which corresponds to a subset of pathways. It was

shown that the obtained topics act as different aspects

of biological activity across the experiments (Caldas

et al., 2009). See Figure 6 for an illustration of 13 se-

lected topics. We create an axis from each topic, with

50 axes in total as follows.

Let Y ∈ R

D×W

be the “experiment–pathway ac-

tivity” matrix, and Z ∈ R

L×W

be the “topic–pathway

activity” matrix derived from the topic model, with

the s-th row of Z, corresponding to the pathway

activities in topic s. For each topic s, we select W

pathways from Y with the largest activities in z

form a D ×W

matrix Y

, where W

is determined by

Figure 6: An illustration of 13 selected topics in the topic

model inferred from the “experiment–pathway activity”

matrix. The serial numbers of the selected topics are shown

in the small circles at the center. The top part lists the

experiments, and the bottom part lists the pathways. The

curves show the experiments-topics-pathways connection

in a way that the topics are active in the experiments they

link to, and the pathways are active in the topics they link

to. The curve widths correspond to the topic activities. The

revealed hidden structure is highlighted by the curve col-

oring. As an example, topic 50 is active in experiments

like “stk3 overexpression”, “aco1 overexpression”, etc. and

pathways like “hsa00190 oxidative phosphorylation”, “ox-

idative phosphorylation”, etc are active in this topic 50.

the discriminative power of the selected features: we

choose W

to be the smallest number that reaches the

highest leave-one-out accuracy of k-nearest neighbor

classiﬁcation over k and the number of the most active

features. With the constructed Y

, we create the s-th

axis as v

= Y

, with r

the one-dimensional linear

discriminant projection from the disease labels. After

juxtaposing the v

into V ∈ R

D×L

, we create the PCP

from PCP-NR for V , shown in Figure 7.

We focus our analysis on group A, B, and C

marked in Figure 7, because either the axes within

are closer, or there are axes of large degree in the tree.

The top pathways are listed in Figure 7 (bottom).

Group A is related to apoptosis. Besides the top

3 pathways in the group, the relation between NK

cells and apoptosis was also studied (Warren and

Smyth, 1999). In the large group B, it is known that

Aminoacyl-tRNA synthetases involve ATP (Rapaport

et al., 1987), and glycan degradation plays an impor-

tant role in the starch utilization system (Koropatkin

et al., 2012), which produces ATP. On the other hand,

TCR (T-cell receptor) and BCR (B-cell receptor) are

part of the immunity system, so we can say these two

pathways form a subgroup. Finally, group C is on

cell cycle: mTOR is related to cell growth (Laplante

and Sabatini, 2012); the enrichment of RACCYCD in

cell cycle was recently reported (Sanchez-Diaz et al.,

IVAPP 2017 - International Conference on Information Visualization Theory and Applications

Cluster A top pathways:

APOPTOSIS

APOPTOSIS_KEGG

APOPTOSIS_GENMAPP

NKCELLSPATHWAY

TH1TH2PATHWAY

Cluster C top pathways:

MTORPATHWAY

RACCYCDPATHWAY

FMLPPATHWAY

P38MAPKPATHWAY

HCMVPATHWAY

Cluster B top pathways:

AMINOACYL_TRNA_

BIOSYNTHESIS

HSA00500_STARCH_

AND_SUCROSE_

METABOLISM

HSA01032_GLYCAN_

STRUCTURES_

DEGRADATION

TCRPATHWAY

SIG_BCR_SIGNALING_

PATHWAY

Figure 7: Top: the 3D PCP from the derived feature ma-

trix V for the 105 experiments created by PCP-NR. The nu-

meric labels are topic serial numbers. We focus our analysis

on 1) the central part with more short links, and 2) the axes

with larger degrees in the MST, which potentially reveals

more relations between the axes or topics. We can also

notice some blue lines between axes differentiating them-

selves from other green-ish lines in majority. Those lines

may suggest label information about the experiment that are

not used in the creation of the PCP according to our color-

ing algorithm. Bottom: the top view of the axes, with the

groups marked by red circles, which are analyzed in the case

study. For each group, we list the top 5 pathways: we sort

pathways by their activity levels, then choose 5 from the top

if pathways appear in at least two topics (axes) in the group.

2013); p38 MAPK pathway, together with other path-

ways in MAPK families, regulates different stages of

cell cycle (Rubinfeld and Seger, 2005); and HCMV is

a pathogen that induces disease by affecting cell cycle

in different ways (Salvant et al., 1998).

17 43

38 6 27 2

36 20 9 10

12 45 41

39 21 47 3 8 23 44 42 16 34

37 19 4 33 26 14

5 40

29 49 35

Top pathways

APOPTOSIS

HSA00562 INOSITOL PHOSPHATE METABOLISM

AT1RPATHWAY

APOPTOSIS KEGG

APOPTOSIS GENMAPP

Figure 8: Top: The 2D PCP from the derived feature matrix

V for the 105 experiments created by PCP-NR. The numeric

labels are topic serial numbers. Topic 32 and topic 30 are

positioned at the two ends of the PCP, which suggests the

neighbor relationship (neighbor distributions on each query

data point) is very different on those two axes. The unusual

neighbor relationship can be visually checked at topic 32,

where a majority of data points concentrates on or near one

single value. The positioning of the two outliers are consis-

tent with Figure 7, in the sense that they are also leaves in

the MST. We focus on the smaller group at the right half of

the plot in the case study. Bottom: The top 5 pathways in

the smaller group at the right half of the ﬁgure, extracted in

the same way as described in Figure 7 (bottom).

A similar analysis can be conducted on the 2D

version of PCP-NR. Figure 8 shows the PCP. We fo-

cus the case study on the smaller group standing out

at the right half of the plot; the bottom of the ﬁgure

lists the top 5 pathways. Again we can see pathways

are apoptosis-related: besides the pathway relation al-

ready suggested by the pathway names, AT1R was re-

ported to have connection with breast cancer (Zhao

et al., 2010), and inositol (hexa)phosphate can help

cancer inhibition (Vucenik and Shamsuddin, 2006).

Thus these two pathways are related with apoptosis.

5 CONCLUSIONS

We presented Parallel Coordinate Plots for Neighbor

Retrieval (PCP-NR), a method for constructing paral-

lel coordinate plots (PCPs), either 2D or 3D PCPs,

where each design step is directly optimized for a

low-level component task of exploratory data analy-

sis. To our knowledge this is the ﬁrst method to di-

rectly optimize PCP construction for a quantiﬁable

low-level data analysis task.

In particular we optimize the PCP for the task of

relating data items, speciﬁcally for retrieving neigh-

Parallel Coordinate Plots for Neighbor Retrieval

borhood relationships between data items. All con-

struction stages are optimized for this task: 1) similar-

ity of feature axes is evaluated by similarity of neigh-

borhood relationships shown in each axis; 2) axis

placement is optimized so that similar axes (showing

similar neighbor relationships) are placed nearby in

the PCP, allowing the user to retrieve similar axes eas-

ily from looking at the PCP; 3) coloring of data lines

is optimized to show overall neighbor relationships

of data items across all features, allowing the user to

track relationships of similar data items over all axes.

We do not claim neighbor retrieval is the only task

PCPs should be optimized for—relating data items

(to neighbors) is one of the component tasks in ex-

ploratory data analysis, and methods can be created

to optimize PCPs for other component tasks. Fu-

ture work could ﬁnd theoretical connections describ-

ing earlier PCP works (such as axis ordering meth-

ods) as approximate optimization of other component

tasks of exploratory data analysis. In this sense, our

work is the ﬁrst in a research direction of optimizing

PCPs for subtasks of exploratory data analysis.

Our construction method is general and applicable

both to 2D and 3D PCPs. Resulting PCPs have a sim-

ilar form as traditional 2D and 3D PCPs, but the new

PCPs are optimized for an analysis task; the PCPs are

directly pluggable into visualization systems featur-

ing PCPs, potentially improving their ability to serve

data analysts. For reasonably-sized data construction

of the plots is already fast; for very large data sets

recent work in speedup of scatter plot optimizations

(Yang et al., 2013) may be adapted to PCP optimiza-

tion. As other further work, it is easy to add prefer-

ences about layouts as penalties to Eq. (6), such as a

repulsion term (Peltonen and Lin, 2015) keeping axes

a desired minimum apart if needed for readability.

ACKNOWLEDGEMENTS

We acknowledge the computational resources pro-

vided by the Aalto Science-IT project. Authors be-

long to the Finnish CoE in Computational Inference

Research COIN. The work was supported in part by

TEKES (Re:Know project). The work was also sup-

ported in part by the Academy of Finland, decision

numbers 252845, 256233, and 295694.

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