Parallel coordinates: Difference between revisions

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==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). ByIn 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==

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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.

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

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

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

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== 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 ==