Data Mining: Practical Machine Learning Tools and Techniques, Second Edition

Figure 6.14 shows the implicit boundaries that are formed between two rec-

tangular classes if the distance metric is adjusted to measure distance to the

nearest point of a rectangle. Even in two dimensions the boundary contains a

total of nine regions (they are numbered for easy identification); the situation

will be more complex for higher-dimensional hyperrectangles.

Proceeding from the lower left, the first region, in which the boundary is

linear, lies outside the extent of both rectangles—to the left of both borders of

the larger one and below both borders of the smaller one. The second is within

the extent of one rectangle—to the right of the leftmost border of the larger

rectangle—but outside that of the other—below both borders of the smaller

one. In this region the boundary is parabolic, because the locus of a point that

is the same distance from a given line as from a given point is a parabola.The

240 CHAPTER 6| IMPLEMENTATIONS: REAL MACHINE LEARNING SCHEMES

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Figure 6.14A boundary between two rectangular classes.