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吳育德 陽明大學放射醫學科學研究所 台北榮總整合性腦功能研究室 PowerPoint Presentation
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Introduction To Linear Discriminant Analysis. 吳育德 陽明大學放射醫學科學研究所 台北榮總整合性腦功能研究室. Linear Discriminant Analysis.

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吳育德 陽明大學放射醫學科學研究所 台北榮總整合性腦功能研究室


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slide1

Introduction To Linear Discriminant Analysis

吳育德

陽明大學放射醫學科學研究所

台北榮總整合性腦功能研究室

linear discriminant analysis
Linear Discriminant Analysis

For a given training sample set, determine a set of optimal projection axes such that the set of projective feature vectors of the training samples has the maximumbetween-class scatter and minimum within-class scatter simultaneously.

linear discriminant analysis1
Linear Discriminant Analysis

Linear Discriminant Analysis seeks a projection that best separate the data .

Sb : between-class scatter matrix

Sw : within-class scatter matrix

slide4
LDA

Fisher discriminant analysis

Sol:

slide5
LDA

Fisher discriminant analysis

where

, = k1+k2

and let

slide6
LDA

Fisher discriminant analysis

slide7
LDA

Generalized eigenvalue problem.....Theorem 2

Let M be a real symmetric matrix with largest eigenvalue

then

and the maximum occurs when , i.e. the unit eigenvector associated with .

Proof :

slide8
LDA

Generalized eigenvalue problem.....proof of Theorem 2

slide9
LDA

Generalized eigenvalue problem.....proof of Theorem 2

Cor:

If M is a real symmetric matrix with largest eigenvalue .

And the maximum is achieved whenever ,where is the unit eigenvector associated with .

slide10
LDA

Generalized eigenvalue problem…….. Theorem 1

Let Sw and Sb be n*n real symmetric matrices . If Sw is positive definite, then there exists an n*n matrix V which achieves

The real numbers λ1….λn satisfy the generalized eiegenvalue equation :

: generalized eigenvector

: generalized eigenvalue

slide11
LDA

Generalized eigenvalue problem.....proof of Theorem 1

Let and be the unit eigenvectors and

eigenvalues of Sw, i.e

Now define then

where

Since ri ﹥0 (Sw is positive definite) , exist

slide12
LDA

Generalized eigenvalue problem.....proof of Theorem 1

slide13
LDA

Generalized eigenvalue problem.....proof of Theorem 1

We need to claim :

(applying a unitary matrix to a whitening process doesn’t affect it!)

(VT)-1 exists since det(VTSwV) = det (I )

→ det(VT) det(Sw) det(V) = det(I)

Because det(VT)= det(V)

→ [det(VT)]2 det(Sw) = 1 > 0

→ det(VT) 0

slide14
LDA

Generalized eigenvalue problem.....proof of Theorem 1

Procedure for diagonalizing Sw (real symmetric and positive definite) and Sb (real symmetric) simultaneously is as follows :

1. Find λi by solving

And then find normalized , i=1,2…..,n

2. normalized