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Unformatted text preview: spanned by µ∗ and lose nothing ˆk 31 ESL Chapter 4 — Linear Methods for Classiﬁcation Trevor Hastie and Rob Tibshirani 32 ESL Chapter 4 — Linear Methods for Classiﬁcation Trevor Hastie and Rob Tibshirani Can project onto even lower dimensions, using the principal components of µ∗ : ˆk • compute MK ×p , the matrix of centroids, and its sphered version ˆ M ∗ = M Σ−1/2 • Reduce M ∗ by principal components; i.e. compute B ∗ = covariance matrix of M ∗ (with mass πk on each row): B ∗ = M ∗ T Dπ M ∗ . Then compute the eigen-decomposition of B ∗ : B ∗ = V ∗ DB V ∗ T • z = v T x is the th discriminant (or canonical) variable, with ˆ v = Σ−1/2 v ∗ 33 ESL Chapter 4 — Linear Methods for Classiﬁcation Trevor Hastie and Rob Tibshirani 4 Linear Discriminant Analysis o oo o oo oo o oo 0 -2 -4 Coordinate 2 for Training Data 2 o o oo o o oo oo o o oo o o oo ooo oo o oo o o ooo ooo o oo o o o o o o oo o oo o o o o • oo oo oo oo o • o o ooo o o o oo o oo o oo o o o o ooo o o o o ooo ooo o o o o o o o o oo o oo o o • o oo oo o oo o oo oo o o •o o o o o oo o o o o o oo o o oo oo o oo o o o o o o o o o o o o o oo o oo oo o o o o oo o o o o o o oo o o oo oo o o oo ooo ooo o o oo oo oo oo •o o o o o ooo o o oo o o o o oo o oo o o o o o • ooo o o oo oo o o o o o o • o oo oo o o ooo o oo oo o oo o o oo o o oo o o o oo oo o o oo o oo o o o o o o o o oo o o oo o oo o o oo o oo o o oo o ooo o oo o o oo o o o o o o oo o oo o oo ooo o oo o oo oo o o o o o• oo o oo o oo o o o o o o •o o oo o o o o o o oo oo o o ooo o o o ooo o o o o o oo • o o oo o oo o o o ooo oo oo o o oo o o oo oo o o o oo o o oo o o o oo o oo oo o o o oo o •o o o o o oo oo o oo o oo o o ooo oo oo o o o o • • • • • • • • • • • oo o -6 o o -4 -2 o 0 2 4 Coordinate 1 for Training Data A two-dimensional plot of the vowel training data. There are eleven classes with X ∈ R10 , and this is the best view in terms of a LDA model. The heavy circles are the projected mean vectors for each class. The class overlap is considerable. 34 ESL Chapter 4 — Linear Methods for Classiﬁcation Trevor Hastie and Rob Tibshirani Projections onto pairs of discriminant variates Linear Discriminant Analysis o o o o o • • • • -4 -2 2 • ••• •• • • •• • • •• o o o o o 0 2 4 o -6 • • • 1 2 • o o o o -2 • 0 •• • • ••• • • -1 o o oo oo o o o o o oo oo o o o o o oo oo o oo o o o o...
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