Parallel coordinates: Difference between revisions

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{{short description|Chart displaying multivariate data}}

In [[mathematics]], '''Parallel coordinates'''<ref name="pc">

[[Image:ParCorFisherIris.png|right|400px|Parallel coordinates]]

{{cite journal|author=A. Inselberg, B. Dimsdale|title=Parallel coordinates: A tool for visualizing multi-dimensional geometry|journal=In Proc. Visualization 90|___location=San Francisco, CA|pages=pages 361–370|year=1990}}

[[File:Ggobi-flea2.png|right|400px|alt=Ggobi-flea2|Parallel coordinate plot of the flea data in [[GGobi]].]]

</ref> is a common technique for representing [[high-dimensional]] [[data]].

'''Parallel Coordinates''' plots are a common method of visualizing [[multivariate data|high-dimensional datasets]] to analyze multivariate data having multiple variables, or attributes.

To showplot, or visualize, a set of [[point (geometry)|points]] in an [[n-dimensional space|''n''-dimensional space]], a backdrop is drawn consisting of ''n'' [[parallel (geometry)|parallel]] [[linelines segment|lineare segments]],drawn typicallyover the background representing [[verticalcoordinate]] andaxes, typically oriented vertically with equal spacing between them. Then a given pointPoints in [[''n''-dimensional space]] isare represented as aindividual [[polyline]]s with ''n'' [[vertex (geometry)|vertices]] placed on the parallel lineaxes segments;corresponding theto positioneach of[[coordinate]] theentry vertex onof the i''n''-thdimensional segmentpoint, correspondsvertices toare theconnected iwith ''n-th1'' [[coordinate]]polyline of the pointsegments.

This data visualization is similar to [[time series]] visualization, except that Parallel Coordinates are applied to data which do not correspond with chronological time. Therefore, different axes arrangements can be of interest, including reflecting axes horizontally, otherwise inverting the attribute range.

==History==

In 1990, the parallel coordinate plot (PCP) was introduced as a method for visualizing [[multivariate]] data [[Wegman (1990)]].Since that time, the technique has become an important tool for those who attempt exploratory [[data analysis]] and visual [[data mining]].

== History ==

The concept of Parallel Coordinates is often said to originate in 1885 by a French mathematician [[Philbert Maurice d'Ocagne]].<ref>Ocagne, M. (1885). Coordonnées Parallèles et Axiales: Méthode de transformation géométrique et procédé nouveau de calcul graphique déduits de la considération des coordonnées parallèlles. Gauthier-Villars. [https://archive.org/details/coordonnesparal00ocaggoog archive.org]</ref> d'Ocagne sought a way to provide graphical calculation of mathematical functions using alignment diagrams called [[nomogram]]s which used parallel axes with different scales.

For example, a three-variable equation could be solved using three parallel axes, marking known values on their scales, then drawing a line between them, with an unknown read from the scale at the point where the line intersects that scale.

The use of Parallel Coordinates as a visualization technique to show data is also often said to have originated earlier with [[Henry Gannett]] in work preceding the Statistical Atlas of the United States

for the 1890 Census, for example his "General Summary, Showing the Rank of States, by Ratios, 1880", <ref name="hg">{{cite book |first=Henry |last=Gannett |title=Scribner's statistical atlas of the United States |section=General Summary Showing the Rank of States by Ratios 1880 |url=https://www.davidrumsey.com/luna/servlet/detail/RUMSEY~8~1~32803~1152181}}</ref>

that shows the rank of 10 measures (population, occupations, wealth, manufacturing, agriculture, and so forth) on parallel axes connected by lines for each state.

However, both d'Ocagne and Gannet were far preceded in this by [[André-Michel Guerry]],<ref>Guerry, A.-M. (1833). Essai sur la Statistique Morale de la France. Paris: Crochard.</ref> Plate IV, "Influence de l'Age",

where he showed rankings of crimes against persons by age along parallel axes, connecting the same crime across age groups.<ref>Friendly, M. (2022). The life and works of André-Michel Guerry, revisited. Sociological Spectrum, 42(4-6), 233–259. https://doi.org/10.1080/02732173.2022.2078450</ref>

Parallel Coordinates were popularised again 87 years later by [[Alfred Inselberg]]<ref name="pc">{{cite journal |first=Alfred |last=Inselberg |title=The Plane with Parallel Coordinates |journal=Visual Computer |volume=1 |issue=4 |pages=69–91 |year=1985 |doi=10.1007/BF01898350 |s2cid=15933827 }}</ref> in 1985 and systematically developed as a coordinate system starting from 1977. Some important applications are in [[Traffic collision avoidance system|collision avoidance algorithms]] for [[air traffic control]] (1987—3 USA patents), [[data mining]] (USA patent), [[computer vision]] (USA patent), Optimization, [[process control]], more recently in [[Intrusion detection system|intrusion detection]] and elsewhere.

==Higher dimensions==

On the plane with an XY Cartesian coordinate system, adding more [[dimensions]] in parallel coordinates (often abbreviated ||-coords, PCP, or PC) involves adding more axes. The value of parallel coordinates is that certain geometrical properties in high dimensions transform into easily seen 2D patterns. For example, a set of points on a line in ''n''-space transforms to a set of [[polyline]]s in parallel coordinates all intersecting at ''n''&nbsp;&minus;&nbsp;1 points. For ''n'' = 2 this yields a point-line duality pointing out why the mathematical foundations of parallel coordinates are developed in the [[Projective space|projective]] rather than [[Euclidian space|euclidean]] space. A pair of lines intersects at a unique point which has two coordinates and, therefore, can correspond to a unique line which is also specified by two parameters (or two points). In contrast, more than two points are required to specify a curve and also a pair of curves may not have a unique intersection. Hence by using curves in parallel coordinates instead of lines, the point line duality is lost together with all the other properties of projective geometry, and the known nice higher-dimensional patterns corresponding to (hyper)planes, curves, several smooth (hyper)surfaces, proximities, convexity and recently non-orientability.<ref name="pc2">{{cite book |first=Alfred |last=Inselberg |title=Parallel Coordinates: VISUAL Multidimensional Geometry and its Applications |publisher=Springer |year=2009 |isbn=978-0387215075 }}</ref> The goal is to map n-dimensional relations into 2D patterns. Hence, parallel coordinates is not a point-to-point mapping but rather a ''n''D subset to 2D subset mapping, there is no loss of information. Note: even a point in nD is not mapped into a point in 2D, but to a polygonal line—a subset of 2D.

Adding more [[dimensions]] to the parallel coordinate plot simply involves adding more axes to the right of the plot and extending the line to join up with the new points forming a polygonal line from the first to the last parallel co-ordinate axis.

==Statistical considerations==

Part of the value of parallel coordinates is that certain geometrical properties in high

[[File:Parallel coordinates-sample.png|thumb|Representative sample for parallel coordinates.]]

dimensions translate into easily seen 2D properties. For example, a set of points that lie on a line in n-space will translate to a set of polylines in parallel coordinates that all intersect at a common point. It is worth noting that the PCP is developed based on the projective geometry view point.

When used for statistical data visualisation there are three important considerations: the order, the rotation, and the scaling of the axes.

The order of the axes is critical for finding features, and in typical data analysis many reorderings will need to be tried. Some authors have come up with ordering heuristics which may create illuminating orderings.<ref>{{cite journal |first1=Jing |last1=Yang |first2=Wei |last2=Peng |first3=Matthew O. |last3=Ward |first4=Elke A. |last4=Rundensteiner |year=2003 |url=http://davis.wpi.edu/~xmdv/docs/tr0313_osf.pdf |title=Interactive Hierarchical Dimension Ordering Spacing and Filtering for Exploration of High Dimensional Datasets |pages=3–4 |journal=IEEE Symposium on Information Visualization (INFOVIS 2003) }}</ref>

==Generalized Parallel Coordinate Plot==

'''Generalized Parallel Coordinate Plot (GPCP)''' [[Moustafa and Wegman (2001)]] is a more recent

The rotation of the axes is a translation in the parallel coordinates and if the lines intersected outside the parallel axes it can be translated between them by rotations. The simplest example of this is rotating the axis by 180 degrees.<ref name="Gpc2" />

design that is based on [[parameter]] [[transformation (mathematics)|transformation]] processes. In this

design, data is transformed into the parameter space through

Scaling is necessary because the plot is based on interpolation (linear combination) of consecutive pairs of variables.<ref name="Gpc2">{{cite book |first1=Rida |last1=Moustafa |first2=Edward J. |last2=Wegman |chapter=Multivariate continuous data – Parallel Coordinates |editor1= Unwin, A. |editor2=Theus, M. |editor3=Hofmann, H. |title=Graphics of Large Datasets: Visualizing a Million |publisher=Springer |pages=143–156 |year=2006 |isbn=978-0387329062 }}</ref> Therefore, the variables must be in common scale, and there are many scaling methods to be considered as part of data preparation process that can reveal more informative views.

[[interpolation]] functions (basis). If the interpolation function is set

to be the Piecewise Lagrange of degree one, then the traditional

A smooth parallel coordinate plot is achieved with splines.<ref name="Gpc1">{{cite journal |first1=Rida |last1=Moustafa |first2=Edward J. |last2=Wegman |title=On Some Generalizations of Parallel Coordinate Plots |journal=Seeing a Million, A Data Visualization Workshop, Rain Am Lech (Nr.), Germany |year=2002 |url=http://herakles.zcu.cz/seminars/docs/infovis/papers/Moustafa_generalized_parallel_coordinates.pdf |archive-url=https://web.archive.org/web/20131224111246/http://herakles.zcu.cz/seminars/docs/infovis/papers/Moustafa_generalized_parallel_coordinates.pdf |url-status=dead |archive-date=2013-12-24 }}</ref> In the smooth plot, every observation is mapped into a parametric line (or curve), which is smooth, continuous on the axes, and orthogonal to each parallel axis. This design emphasizes the quantization level for each data attribute.<ref name="Gpc2" />

parallel coordinate plot (TPCP) is achieved, i.e. every data record

is mapped into a broken line crossing the parallel axes in the

== Reading ==

parameter space. If the interpolation function is set to be the

Inselberg ({{harvnb|Inselberg|1997|p= }}) made a full review of how to visually read out parallel coordinates relational patterns.<ref>{{citation|last1=Inselberg |first1=A.|year=1997 |chapter=Multidimensional detective |title=Information Visualization, 1997. Proceedings., IEEE Symposium on |isbn=0-8186-8189-6|pages=100–107|doi=10.1109/INFVIS.1997.636793|s2cid=1823293 |citeseerx=10.1.1.457.3745 }}</ref> When most lines between two parallel axes are somewhat parallel to each other, it suggests a positive relationship between these two dimensions. When lines cross in a kind of superposition of X-shapes, it's a negative relationship. When lines cross randomly or are parallel, it shows there is no particular relationship.

[[spline (mathematics)|Splines]], then the smooth parallel coordinate plot is achieved. In

the smooth plot, every observation is mapped into a parametric

== Limitations ==

line/curve, which is smooth, continuous on the axes, and

In parallel coordinates, each axis can have at most two neighboring axes (one on the left, and one on the right). For a ''n''-dimensional data set, at most ''n''-1 relationships can be shown at a time without altering the approach. In [[time series]] visualization, there exists a natural predecessor and successor; therefore in this special case, there exists a preferred arrangement. However, when the axes do not have a unique order, finding a good axis arrangement requires the use of experimentation and feature engineering. To explore more relationships, axes may be reordered or restructured.

orthogonal to each parallel axis. This unprecedent design gives a

clear quantization level of each data attribute, that can best

One approach arranges axes in 3-dimensional space (still in parallel, forming a [[Lattice graph]]), an axis can have more than two neighbors in a circle around the central attribute, and the arrangement problem can be improve by using a [[minimum spanning tree]].<ref name="sigmod13">{{cite book |author=Elke Achtert |author2=[[Hans-Peter Kriegel]] |author3=Erich Schubert |author4=Arthur Zimek

describe its distribution in complex situations, even with large

| title=Proceedings of the 2013 ACM SIGMOD International Conference on Management of Data

data sets. The GPCP design gives opportunities to researchers to

| chapter=Interactive data mining with 3D-parallel-coordinate-trees

explore alternative interpolation functions and a quick interpretation

| pages=1009–1012

of the outcome.

| publisher=Association for Computing Machinery

| ___location=New York City, NY | year=2013 | doi=10.1145/2463676.2463696| isbn=9781450320375

| s2cid=14850709

}}</ref> A prototype of this visualization is available as extension to the data mining software [[ELKI]]. However, the visualization is harder to interpret and interact with than a linear order.

== Software ==

While there are a large number of papers about parallel coordinates, there are only a few notable software publicly available to convert databases into parallel coordinates graphics.<ref>{{cite web|url=http://eagereyes.org/techniques/parallel-coordinates|title=Parallel Coordinates|last=Kosara|first=Robert|year=2010}}</ref> Notable software are [[ELKI]], [[GGobi]], [[Mondrian data analysis|Mondrian]], [[Orange (software)|Orange]] and [[ROOT]]. Libraries include [[Protovis.js]], [[D3.js]] provides basic examples. D3.Parcoords.js (a D3-based library) specifically dedicated to parallel coordinates graphic creation has also been published. The [[Python (programming language)|Python]] data structure and analysis library [[Pandas (software)|Pandas]] implements parallel coordinates plotting, using the plotting library [[matplotlib]].<ref>[https://pandas.pydata.org/pandas-docs/version/0.21.0/visualization.html#parallel-coordinates Parallel Coordinates in Pandas]</ref>

== Other visualizations for multivariate data ==

* [[Radar chart]] – A visualization with coordinate axes arranged radially.

* [[Andrews plot]] – A Fourier transform of the Parallel Coordinates graph.

* [[Sankey diagram]] - A visualization that emphasizes flow/movement/change from one state to another.

== References ==

<references />

==Further reading==

* Heinrich, Julian and Weiskopf, Daniel (2013) ''[https://diglib.eg.org/handle/10.2312/conf.EG2013.stars.095-116 State of the Art of Parallel Coordinates]'', Eurographics 2013 - State of the Art Reports, pp.&nbsp;95–116

1. Insleberg, A. (1985) " The plane with parallel coordinate," Visual

* Moustafa, Rida (2011) '' Parallel coordinate and parallel coordinate density plots'', Wiley Interdisciplinary Reviews: Computational Statistics, Vol 3(2), pp.&nbsp;134–148.

Computer, 1, 69--91.

* Weidele, Daniel Karl I. (2019) ''[https://doi.org/10.1109/VISUAL.2019.8933632 Conditional Parallel Coordinates]'', IEEE Visualization Conference (VIS) 2019, pp.&nbsp;221–225

2. Wegmn, E. J. (1990) "Hyperdimensional data analysis using parllel coordinate" JASA, 85,664-675.

==External links==

* [http://www.math.tau.ac.il/~aiisreal Alfred Inselberg's Homepage], with Visual Tutorial, History, Selected Publications and Applications

* [http://catt.okstate.edu/jones98/parallel.html A small, easy introduction] by Christopher V. Jones

* [http://www.agocg.ac.uk/reports/visual/casestud/brunsdon/abstract.htm An Investigation of Methods for Visualising Highly Multivariate Datasets] by C.Brunsdon, A.S.Fotheringham & M.E.Charlton, [[University of Newcastle upon Tyne|University of Newcastle]], [[UK]]

* [http://www.amitgoel.com/pcoord/ Parallel Coordinates Visualization Applet]

* [http://www.dcs.napier.ac.uk/~marting/parCoord/GrahamKennedyParallelCurvesIV03.pdf Using Curves to Enhance Parallel Coordinate Visualisations] by Martin Graham & Jessie Kennedy, [[Napier University]], [[Edinburgh]], [[UK]]

* [http://www.galaxy.gmu.edu/stats/syllabi/inft979/GeneralizedParallelCoordinates.pdf On Some Generalization of Parallel Coordinate Plots] by Rida E. A. Moustafa and Edward J. Wegman (2001), [[George Mason University]], [[Fairfax, VA]]

*[http://s92417348.onlinehome.us/software/dataloom/index.html Data Loom — a parallel coordinates visualisation tool for the Mac]

*[http://home.subnet.at/flo/mv/parvis/index.html parvis — a parallel coordinates tool] licensed under the [[GNU GPL]] - implemented in [[Java programming language|Java]]

==External links==

[[Category:Visualization]]

* [http://www.cs.tau.ac.il/~aiisreal Alfred Inselberg's Homepage], with Visual Tutorial, History, Selected Publications and Applications

* [http://www.agocg.ac.uk/reports/visual/casestud/brunsdon/abstract.htm An Investigation of Methods for Visualising Highly Multivariate Datasets] by C. Brunsdon, A. S. Fotheringham & M. E. Charlton, [[University of Newcastle upon Tyne|University of Newcastle]], [[UK]]

*[http://www.dcs.napier.ac.uk/~marting/parCoord/GrahamKennedyParallelCurvesIV03.pdf Using Curves to Enhance Parallel Coordinate Visualisations] {{Webarchive|url=https://web.archive.org/web/20070315191533/http://www.dcs.napier.ac.uk/~marting/parCoord/GrahamKennedyParallelCurvesIV03.pdf |date=2007-03-15 }} by Martin Graham & Jessie Kennedy, [[Napier University]], [[Edinburgh]], [[UK]]

*[http://eagereyes.org/techniques/parallel-coordinates Parallel Coordinates], a tutorial by Robert Kosara

*[https://github.com/IBM/conditional-parallel-coordinates Conditional Parallel Coordinates] – Recursive variant of Parallel Coordinates, where a categorical value can expand to reveal another level of Parallel Coordinates.

[[Category:Data and information visualization]]

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[[Category:Multi-dimensional geometry]]

[[Category:Statistical charts and diagrams]]