1.7 Parallel Coordinates Plots

Parallel coordinates plots (PCP) constitute a technique that is based on a non-Cartesian coordinate system and therefore allows one to ``see'' more than four dimensions.

Figure: Parallel coordinates plot of observations 96-105. MVAparcoo1.xpl

Figure: The entire bank data set. Genuine bank notes are displayed as black lines. The counterfeit bank notes are shown as red lines. MVAparcoo2.xpl

The idea is simple: Instead of plotting observations in an orthogonal coordinate system, one draws their coordinates in a system of parallel axes. Index $j$ of the coordinate is mapped onto the horizontal axis, and the value $x_j$is mapped onto the vertical axis. This way of representation is very useful for high-dimensional data. It is however also sensitive to the order of the variables, since certain trends in the data can be shown more clearly in one ordering than in another.

EXAMPLE 1.4   Take once again the observations 96-105 of the Swiss bank notes. These observations are six dimensional, so we can't show them in a six dimensional Cartesian coordinate system. Using the parallel coordinates plot technique, however, they can be plotted on parallel axes. This is shown in Figure 1.22.

We have already noted in Example 1.2 that the diagonal $X_6$ plays an important role. This important role is clearly visible from Figure 1.22 The last coordinate $X_6$ shows two different subgroups. The full bank note data set is displayed in Figure 1.23. One sees an overlap of the coordinate values for indices 1-3 and an increased separability for the indices 4-6.

Summary

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Parallel coordinates plots overcome the visualization problem of the Cartesian coordinate system for dimensions greater than 4.

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Outliers are visible as outlying polygon curves.

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The order of variables is still important, for example, for detection of subgroups.

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Subgroups may be screened by selective coloring in an interactive manner.