放射医学科学研究所台北荣总整合性脑功能研究室课件

1、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Introduction To Linear Discriminant Analysis吳育德Introduction To Linear Disc 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 maximum bet

2、ween-class scatter and minimum within-class scatter simultaneously. Linear Discr Linear Discriminant Analysis Linear Discriminant Analysis seeks a projection that best separate the data .Sb : between-class scatter matrixSw : within-class scatter matrix LinSol:LDA Fisher discriminant analysisSol:LDA

3、where , = k1+k2and let LDA Fisher discriminant analysiswhereLDA LDA Fisher discriminant analysisLDA Let M be a real symmetric matrix with largest eigenvalue thenand the maximum occurs when , i.e. the unit eigenvector associated with .Proof : LDA Generalized eigenvalue problem.Theorem 2Let M be a rea

4、l symmetric matrLDA Generalized eigenvalue of of Theorem 2LDA If M is a real symmetric matrix with largest eigenvalue .And the maximum is achieved whenever ,where is the unit eigenvector associated with .Cor: LDA Generalized eigenvalue of of Theorem 2If M is a real symmetric ma

5、triLDA Generalized eigenvalue problem. Theorem 1Let 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 1.n satisfy the generalized eiegenvalue equation : : generalized eigenvector : generalized eigenvalueLDA Genera

6、lized eigenvalue of of Theorem 1Let and be the unit eigenvectors and eigenvalues of Sw, i.eNow define then where Since ri 0 (Sw is positive definite) , exist LDA Generalized eigenvalue problemLDA Generalized eigenvalue of of Theorem 1LDA LDA We need to claim : (applying a unita

7、ry matrix to a whitening process doesnt 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) 0Generalized eigenvalue of of Theorem 1LDA Procedure for diagonalizing Sw (real symmetric and positive definite) and Sb (real symmetric) simultaneously is as follows :1. Find i by solving And then find nor