The discriminant direction minimizes this overlap for

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Unformatted text preview: + + + Although the line joining the centroids defines the direction of greatest centroid spread, the projected data overlap because of the covariance (left panel). The discriminant direction minimizes this overlap for Gaussian data (right panel). 9 ESL Chapter 4 — Linear Methods for Classification Trevor Hastie and Rob Tibshirani Linear boundaries and their projections 1 2 1 2 1 1 2 1 2 3 2 2 1 22 2 3 22 2 3 3 22 2 3 33 2 33 12 3 22 3 2 2 2 22 2 33 2 3 2 222 2 1 2 2 3 22 2 1 2 2 2 33 2 2 22 2 2 2 2 2 22 2 22 22 22 22 2 22 22 22 2 22 3 3 3 33 2 2 2 222 2 22 22 2 1 3 33333 3 22 2 2 22 2 3 33 3 3 3 3 3 2 2 22 2 2 2 2 3 3 33 2 22 2 2 2 22 2 2 2 2 3 2 3 2 2 22 2 2 22 2222 22 2 13 3 2 2 22 2 2 2222 2 3 333 3 3 2 22 1 2 2 222 22 1 22 2 2 2 2 2 1 33 33 3 33 212 2 22 1 2 2 1 1 2 1 11 1 3 33 3 22 2 2 2 1 22 22 2 2 1 1 1 1 1 33 33 3 22 12 2 2 1 1 33 3 3 3 2 3 11 1 2 1 11 1 11 2 1 21 1 1 33 33 1 2 1 2 21 1 2 33 33 11 1 1 1 1 11 3 33 22 12 1 1 11 22 1 1 1 1 21 3 33 3 1 1 1 1 1 11 1 1133 3 3 3 3333 3 1 33 2 111 2 1 2 1 1 3 3 33 1 1 111 1 33 3 11 33 33 3 1 1 1 11 1 1 1 1 1 11 3 1 3 33 3 1 11 1 11 1 1 11 1 11 11 11 1 1 11 11 1 111 33 133 3 13 333333 3 11 1 3 11 1 11 3 1 1 1 1 11 1 1 1 1 1 33 3333 3 3 1333 1 1 1 11 1 1 1 33 3 33 11 1 1 1 1 11 1 1 1 33 1 11 1 11 1 33 3 3 3333 11 3 3 3333 33 3 1 111 1 1 1 3 333 111 33 33 1 11 333 3 3 33 2 1 1 2 1 2 3 2 2 1 22 2 3 22 2 3 3 22 2 3 33 2 33 12 3 22 3 2 2 2 22 2 33 2 3 2 222 2 1 2 2 3 22 2 1 2 2 2 33 2 2 22 2 2 2 2 2 22 2 22 22 22 22 2 22 22 22 2 22 3 3 3 33 2 2 2 222 2 22 22 2 1 3 33333 3 22 2 2 22 2 3 33 3 3 3 3 3 2 2 22 2 2 2 2 3 3 33 2 22 2 2 2 22 2 2 2 2 3 2 3 2 2 22 2 2 22 2222 22 2 13 3 2 2 22 2 2 2222 2 3 333 3 3 2 22 1 2 2 222 22 1 22 2 2 2 2 2 1 33 33 3 33 212 2 22 1 2 2 1 1 2 1 11 1 3 33 3 22 2 2 2 1 22 22 2 2 1 1 1 1 1 33 33 3 22 12 2 2 1 1 33 3 3 3 2 3 11 1 2 1 11 1 11 2 1 21 1 1 33 33 1 2 1 2 21 1 2 33 33 11 1 1 1 1 11 3 33 22 12 1 1 11 22 1 1 1 1 21 3 33 3 1 1 1 1 1 11 1 1133 3 3 3 3333 3 1 33 2 111 2 1 2 1 1 3 3 33 1 1 111 1 33 3 11 33 33 3 1 1 1 11 1 1 1 1 1 11 3 1 3 33 3 1 11 1 11 1 1 11 1 11 11 11 1 1 11 11 1 111 33 133 3 13 333333 3 11 1 3 11 1 11 3 1 1 1 1 11 1 1 1 1 1 33 3333 3 3 1333 1 1 1 11 1 1 1 33 3 33 11 1 1 1 1 11 1 1 1 33 1 11 1 11 1 33 3 3 3333 11 3 3 3333 33 3 1 111...
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