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[[Image:ParCorFisherIris.png|right|400px|Parallel coordinates]]
[[File:Ggobi-flea2.png|right|400px|alt=Ggobi-flea2|Parallel coordinate plot of the flea data in [[GGobi]].]]
'''Parallel coordinates''' are a common way of visualizing and analyzing [[multivariate data|high-dimensional datasets]] consisting of multiple variables, or attributes, per entry.
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]] axes lines are drawn over the background, typically verticalvertically oriented and equally spaced. A point in ''n''-dimensional space is represented as a single [[polyline]] with ''n'' [[vertex (geometry)|vertices]] placed on the parallel axes; thevertices position of the vertex on the ''i''-th axis correspondscorrespond to the ''i''-theach [[coordinate]] of the n-dimensional point.
This visualization is closely relatedsimilar to [[time series]] visualization, except that itParallel isCoordinates are applied to data where the axeswhich do not correspond towith points inchronological time, and therefore do not have a natural order. Therefore, different axisaxes arrangements maycan be of interest, including translating axes horizontally, or inverting.
== History ==
==Higher dimensions==
On the plane with an xyXY cartesianCartesian coordinate system, adding more [[dimensions]] in parallel coordinates (often abbreviated ||-coords, PCP, or PCPPC) 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'' − 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). By 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.
==Statistical considerations==
== Reading ==
Inselberg ({{harvnb|Inselberg|1997|p= }}) made a full review of how to visually read out parallel coords'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 axis 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.
== Limitations ==
In parallel coordinates, each axis can have at most two neighboring axes (one on the left, and one on the right). For a d''n''-dimensional data set, at most d''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 heuristicsexperimentation and experimentationfeature engineering. In order toTo explore more complex relationships, axes mustmay be reordered or restructured.
ByOne arrangingapproach thearranges axes in 3-dimensional space (however, still in parallel, like nails informing a nail[[Lattice bedgraph]]), an axis can have more than two neighbors in a circle around the central attribute, and the arrangement problem getscan easier (forbe exampleimprove by using a [[minimum spanning tree]]).<ref name="sigmod13">{{cite book| author=Elke Achtert, [[Hans-Peter Kriegel]], Erich Schubert, Arthur Zimek
| title=Proceedings of the 2013 ACM SIGMOD International Conference on Management of Data
| chapter=Interactive data mining with 3D-parallel-coordinate-trees
== Other visualizations for multivariate data ==
* [[Radar chart]] – a visualization with coordinate axes arranged radially.
* [[Andrews plot]] – theA Fourier transform of athe parallelParallel coordinatesCoordinates graph.
== References ==
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