Parallel coordinates

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Parallel coordinates[1] is a common way of visualizing high-dimensional geometry and analyzing multivariate data.

To show a set of points in an n-dimensional space, a backdrop is drawn consisting of n parallel lines, typically vertical and equally spaced. A point in n-dimensional space is represented as a polyline with vertices on the parallel axes; the position of the vertex on the i-th axis corresponds to the i-th coordinate of the point.

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[edit] Higher dimensions

Adding more dimensions in the parallel coordinates (often abbreviated ||-coords or PCs) 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 polylines 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 rather than Euclidean space. Also known are the patterns corresponding to (hyper)planes, curves, several smooth (hyper)surfaces, proximities, convexity and recently non-orientability. [2].

[edit] Statistical considerations

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[citation needed].

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. More details can be found at [3].

The necessity of scaling stems directly from the fact that the plot is based on interpolation (linear combination) of each consecutive variable[3]. Therefore, the variables must be in common scale, and there are many scaling methods to be considered as part of data perpetration process that can reveal more informative views.

[edit] Extensions and related plots

The generalized parallel coordinate plot (GPCP) has been proposed by (Moustafa and Wegman 2002) [4] as a generalisation of parallel coordinates plots, based on parameter transformation. In this design, instead of plotting the raw data, it is transformed in some way first. If the interpolation function is piecewise lagrange, this cooresponds to the traditional PCP. If splines are used as the interpolation function, then the smooth parallel coordinate plot (SPCP) is achieved. 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. [3]. This SPCP design gives a clear quantization level of each data attribute, that can best describe its distribution in complex situations, even with large data sets. Finally, if one uses the fourier interpolation of degree equals to the data dimensionality, then Andrews plot (Andrews 1972) is achieved. The GPCP design gives opportunities to researchers to explore alternative interpolation functions that best suited for particular application.

[edit] References

  1. ^ Alfred Inselberg (1985). "The Plane with Parallel Coordinates". Visual Computer 1 (4): pages 69-91. doi:10.1007/BF01898350. 
  2. ^ A. Inselberg (2007). Parallel Coordinates: VISUAL Multidimensional Geometry and its Applications. Springer. 
  3. ^ a b c . R. Moustafa, E. Wegman (2006). "Multivariate continuous data - Parallel Coordinates". In: Unwin, A., Theus M., Hofmann, H. (Eds.), Graphics of Large Datasets: Visualizing a Million, Springer: 143-156. 
  4. ^ R. Moustafa, E. Wegman (2002). "One Some Generalization to Parallel Coordinate Plot". Seeing a million, A Data Visualization Workshop, Rain am Lech (nr.), Germany. 

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