36 esl chapter 4 linear methods for classication

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Unformatted text preview: oo o o oo o o o oo oo oo ooo o oo ooooo o o o oo oo o o o o o o o o oo o oo o o oo o o o o o o o ooo o o ooooo o o o o o oo o o o o oo oo o o o ooo o o oo o ooo o o o o oo o o ooo oo o o o o o oo o o oo ooo o o o oooo oo o o o oo oo o o o o oo ooo o o o o o oo o o o o o oooo o o o o o o o o o o o o oo oo o o o oo oo o oo o o o o o oo ooo o o o oo ooo o oo oo o o o oo o o o o oo o o o oo o o oo o o o o o oo o o oo oo o oo o o o o o o o o o oo o o o o o o oo o o oo o o o o oo o oo o oo o o o oo o o oo oo o o o oo o o o o o o o o o o o o o o ooo o o o oo o o o • o -2 •• Coordinate 3 •• • ••• • • •• • 0 o 0 o o o oo o oo o o o oo o oo oo oo o o o o o oo o o oo oo o o oo o o o ooo o o oo oo o o o o oo o o oo oo o oo o o o o o o oo o ooo o oo o o oo o oo oo o oo o ooo o o o o o ooooo o ooo o o o ooo o ooooo o o o o oo o oo oooo o oo o o o ooo o oo o o oo oo oo o o ooo o o o ooo o o o o oo o o oo o o oo o oo oo o o oo ooo oo o o oo oo o o oo o o o oo o ooo o o o o o oooo o o ooo oo o oo o o o o oo o o oooo oo o o oo o oo o o o o o o oo o o o oo o o o oooo ooo ooo o oo oo o o oo o o o o ooo o o oo o o o o oo oo o o o o oo o o o o oo o o o o o oo oo ooo o oo o o o o ooo o o o oo o o o oo o o o oo o o o o oo oo o o oo o o o oo ooo o oo o o oo o oo o oo o oo o o oo o oo o o o oo o o ooo o o o o oo o oo o o o oo o o o oo o o o o oo o o o oo oo o o o oo o o o o o o o o o oo oo o o oo o o o o o o o oo oo o o o o oo o o o oo o • -2 Coordinate 3 2 o -2 -1 0 1 2 o o oo o 3 Coordinate 9 35 ESL Chapter 4 — Linear Methods for Classification Trevor Hastie and Rob Tibshirani + + + + 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). 36 ESL Chapter 4 — Linear Methods for Classification Trevor Hastie and Rob Tibshirani Fisher’s formulation of discriminant analysis • Find z = aT x such that the between-class variance is maximized ˆ relative to within-class variance W = Σ: aT Ba max a∈IRp aT W a (Rayleigh quotient) or max aT Ba subject to aT W a = 1 p a∈IR • gives v1 = a; then find next direction orthogonal to first max aT Ba subject to aT W a = 1, aT W v1 = 0 p a∈IR • gives v2 = a etc This equivalent to the PCA of standardized centroi...
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