Lecture Notes 12

Lecture Notes 12 - Lecture12 Lecture 12: Applications of...

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Lecture12 1 Outline: 1) Quick Review: A) projections and least-squares problems B) properties of projection matrices 2) Applications: Putting it to work A) fitting a straight line to noisy data B) fitting polynomials to data C) Using Matlab for least-squares problems D) General linear least Squares f(x )= c i Φ i (x ) 3) Caveats and Cautions Lecture 12: Applications of Projections: Linear Least Squares problems Orthogonal Projection onto a Subspace Properties of projection matrices P=A(A T A) -1 A T 1) Projection matrices are always symmetric 2) Projection matrices are usually singular (if N(A T ) Z) 3) if A is invertible P=_____. Why? 4) if p =Pb , then Pp =_____? therefore ___________ Fitting of a straight-line as a least-squares problem problem: find the best fit straight line through 3 points (-3,-2),(0,1),(3,1) (x 1 ,y 1 ) (x 2 ,y 2 ) (x 3 ,y 3 )
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Lecture12 2 Fitting of a straight-line as a least-squares problem problem: find the best fit straight line through 3 points (-3,-2),(0,1),(3,1)
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Lecture Notes 12 - Lecture12 Lecture 12: Applications of...

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