Parallel Coordinate Plots

Parallel coordinate plots were first introduced by Inselberg and by Wegman. The main idea of parallel coordinate plots is to switch from Cartesian coordinates, where points are plotted along orthogonal axes,[Page 728] to a projective geometry, where axes are plotted side by side.

Figure 1 Relationship Between Scatterplot and Parallel Coordinate Plots in a 2D Data Sketch

Figure 2 Relationship Between a 3D Scatterplot and Parallel Coordinate Plots in 3D Data

Figure 3 Scatterplot and Parallel Coordinate Plot of 10 Points on a Line (Left)

Note: Left line translates to the common intersection point on the right.

Figures 1 and 2 show this principle for three data points in two and three dimensions. On the left-hand side of Figure 1, the scatterplot shows the three points (2, 8), (7, 7), and (8, 2) plotted as +, x, and o, respectively. The same data points are plotted on the right-hand side: The first observation (2,8) is shown by first putting a “+” at the 2 of the x-axis, then a “+” at the value 8 of the y-axis. Both of these markers are then connected by a line. This way, each observation corresponds to a single line.

The advantage of this approach becomes visible in Figure 2. Measurements in a third variable are added to our previous example. Although we would need interactive tools, such as rotation, to be able to see the relationship between the three points in the three-dimensional scatterplot on the left, the parallel coordinate plot on the right is extended naturally to three dimensions by putting another axis alongside the other two, drawing markers for each of the measurements and connecting them by lines to the corresponding values of the neighboring axis.

Obviously, this principle is applicable far beyond the (usually up to three) dimensions of Cartesian displays.

The disadvantage of parallel coordinate plots is that we lose our familiar coordinate system. Because the two systems show the same information, it is a matter of a little practice to get used to the new coordinate system.

Example

Parallel coordinate plots are used mostly to find and explore high-dimensional clusters in data. Figure 4[Page 729] shows a parallel coordinate plot of the Swiss bank note data set by Flury and Riedwyl. Measurements of 100 genuine and 100 forged Swiss bank notes were taken, with the goal of identifying forgeries. Six different measurements are available to us: the horizontal length of a bill at the top (Length), left and right vertical width (Lwidth, Rwidth), vertical width of the margin at center bottom (Bmargin) and center top (TMargin), as well as the diagonal length of the image on the bill. Marked in grey are genuine bills. From the parallel coordinate plot, we can see that the forged bills tended to have larger margins at the bottom and top, and the diagonal length of the image seemed to be smaller than those measurements of genuine bills.

Figure 4 Parallel Coordinate Plot of the Swiss Bank Note Data

Note: Forged bank notes are marked in black.

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