按一下以編輯母片標題樣式,按一下以編輯母片,第二層,第三層,第四層,第五層,*,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,1,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,between-class scatter,and,minimum within-class scatter,simultaneously.,Linear Discr,2,Linear Discriminant Analysis,Linear Discriminant Analysis seeks a projection that best,separate,the data.,Sb:between-class scatter matrix,Sw:within-class scatter matrix,Lin,3,Sol:,LDA,Fisher discriminant analysis,Sol:LDA,4,where,=k,1,+k,2,and let,LDA,Fisher discriminant analysis,whereLDA,5,LDA,Fisher discriminant analysis,LDA,6,Let,M,be a real symmetric matrix with largest eigenvalue,then,and the maximum occurs when,i.e.the unit eigenvector associated with .,Proof:,LDA,Generalized eigenvalue problem,.,Theorem 2,Let M be a real symmetric matr,7,LDA,Generalized eigenvalue problem,.proof of Theorem 2,LDA,8,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 problem,.proof of Theorem 2,If M is a real symmetric matri,9,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,1,.,n,satisfy the generalized eiegenvalue equation:,:generalized eigenvector,:generalized eigenvalue,LDA,10,Generalized eigenvalue problem,.proof of Theorem 1,Let and,be the unit eigenvectors and,eigenvalues of S,w,i.e,Now define then,where,Since r,i,0(,S,w,is positive definite),exist,LDA,Generalized eigenvalue problem,11,LDA,Generalized eigenvalue problem,.proof of Theorem 1,LDA,12,LDA,We need to claim:,(applying a unitary matrix to a whitening process doesnt affect it!),(V,T,),-1,exists since det(V,T,S,w,V)=det(I),det(V,T,)det(,S,w,)det(V)=det(I),Because det(V,T,)=det(V),det(V,T,),2,det(S,w,)=1 0,det(V,T,)0,Generalized eigenvalue problem,.proof of Theorem 1,LDA,13,Procedure for diagonalizing S,w,(real symmetric and positive definite)and S,b,(real symmetric)simultaneously is as follows:,1.Find,i,by solving,And then find normalized,i=1,2.,n,2.normalized,LDA,Generalized eigenvalue problem,.proof of Theorem 1,Procedure for diagonalizing Sw,14,