The right plot shows quadratic decision boundaries

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Unformatted text preview: 1 1 1 3 333 111 33 33 1 11 333 3 3 33 2 The left plot shows some data from three classes, with linear decision boundaries found by linear discriminant analysis. The right plot shows quadratic decision boundaries. These were obtained by finding linear 2 2 boundaries in the five-dimensional space X1 , X2 , X1 X2 , X1 , X2 . Linear inequalities in this space are quadratic inequalities in the original space. 10 ESL Chapter 4 — Linear Methods for Classification 1 33 3 3 + + Trevor Hastie and Rob Tibshirani + 3 3 3 3 1 33 22 13 2 11 3 33 2 33 32 2 1 33 2 3 33 3 22 1 2 33 1 2 1 2 22 2 3 2 3 11 1 1 1 3 2 1 31 1 3 1 11 2 22 1 1 22 1 1 2 21 2 1 1 1 12 1 2 2 2 2 The left panel shows three Gaussian distributions, with the same covariance and different means. Included are the contours of constant density enclosing 95% of the probability in each case. The Bayes decision boundaries between each pair of classes are shown (broken straight lines), and the Bayes decision boundaries separating all three classes are the thicker solid lines (a subset of the former). On the right we see a sample of 30 drawn from each Gaussian distribution, and the fitted LDA decision boundaries. 11 ESL Chapter 4 — Linear Methods for Classification Trevor Hastie and Rob Tibshirani Quadratic discriminant analysis 1 1 δk (x) = − log |Σk | − (x − µk )T Σ−1 (x − µk ) + log πk k 2 2 1 2 1 2 1 1 2 1 3 22 1 2 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 2 222 22 2 13 3 2 2 2 22 2 2 2 2 2 22 3 333 3 3 2 1 2 22 1 2 22222 2 2 22 2 1 33 33 3 33 22 2 22 12 1 2 2 1 12 1 3 33 3 2 2 22 22 1 1 1 2 1 2 1 22 1 1 1 33 33 3 22 1 2 2 2 1 22 1 1 1 33 3 3 3 2 11 1 11 2 11 1 1 1 21 1 1 3 33 3 33 2 1 2 21 1 2 3 1 11 1 2 2 1 1 1 11 1 11 1 1 11 1 1 1 33 3 333 12 22 21 31333 3 11 1 1 3 33 1 11 2 33 3 2 111 1 1 2 1 1 33333 33 1 11 11 1 33 3 1 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 1 33 33 3 33 3 11 1 11 1 11 33 1 1 1 1 11 1 1 1 1 1 33 3333 3 3 1333 1 1 11 1 1 1 33 3 33 11 1 1 1 1 11 1 1 1 1 33 1 11 1 11 1 33 3 3 3333 11 3 3 3333 3 33 3 1 111 1 1 1 3 33 111 33 33 1 11 333 3 3 33 2 1 1 2 1 3 22 1 2 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 2...
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