I was reading about Maximal Margin Classifiers in "Introduction to Statistical Learning" and could not understand how is the perpendicular distance of an observation (which is a vector) from the hyperplane calculated ? I do know how it is done for a 2-D and 3-D space, but the formula (9.11) makes no sense to me. Shouldn't it use the MAGNITUDE ? Can anyone help me with this please ?
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The equation $(9.11)$ is after the dot product of the hyperplane and point $x_i$. The author has skipped vector normalization. In maximization problems, normalization does not matter because the magnitude of the normalization vector is a constant. The perpendicular distance of of a point $x_i (x_{i1}, x_{i2}, ....,x_{1p})$ from hyperplance is: $\beta_0 + \beta_1x_{i1} + \beta_2x_{i2} + ... + \beta_px_{ip}$
Probably, you are studying the cost minimization of SVR. The author is trying to explain how to find the MAXIMAL hyperplane
- observation point $y_i$ and calculated point (calculation is explained in image above) lie on same side of plane. This will ensure that our prediction is correct in terms of sign
- maximize the distance from hyperplane: This will make regression robust. The points closer to hyperplane will be predicted correctly.
