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Lecture Notes 23 - Lecture23 Lecture 23 Applications of the...

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Lecture23 1 December 03, 2007 Outline: 1) Quick Review 2) Application 1: Total Least Squares A recipe Interpretation 3) Application 2: Pattern Recognition and Empirical Orthogonal Functions (EOF's/PCA etc.) Examples: Image reconstruction and compression Classification of 1-D Topography (EOF analysis) Lecture 23: Applications of the SVD SVD of a general mxn matrix A with rank r Quick Review = A V T U Σ Matlab: [U,S,V]=svd(A) SVD of a general mxn matrix A with rank r SVD: the economy sized version = A V T U Σ Matlab: [U,S,V]=svd(A,0) Least squares problems using the SVD if A is mxn overdetermined then the solution x that minimizes ||e ||=||b -Ax || is x =x + =A + b where A + =V Σ + U T is the pseudo-inverse Comments: 1) if A is invertible A + =______ and ||e ||=_______ 2) if A is full column rank x + is the unique solution to _____________________ 3) in general p =AA + b is the projection of b onto _______ ^ ^ ^
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Lecture23 2 December 03, 2007 Least squares problems using the SVD Total Least squares problems using the SVD Total Least squares problems using the SVD A recipe (in matlab): Total Least squares problems using the SVD
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