MATH 685/ CSI 700/ OR 682 Lecture Notes Lecture 4. Least squares
Method of least squares Measurement errors are inevitable in observational and experimental sciences
Errors can be smoothed out by averaging over many cases, i.e., taking more measurements than are strictly necessary to determine parameters of system Resulting system is overdetermined, so usually there is no exact
solution In effect, higher dimensional data are projected into lower dimensional space to suppress irrelevant detail Such projection is most conveniently accomplished by
method of least squares Linear least squares
Data fitting Data fitting
Normal Equations Orthogonality
Orthogonality Orthogonal Projector
Pseudoinverse Sensitivity and Conditioning
Sensitivity and Conditioning Solving normal equations
Shortcomings Augmented system method
Augmented system method Orthogonal Transformations
Triangular Least Squares Triangular Least Squares
QR Factorization Orthogonal Bases
Computing QR factorization To compute QR factorization of m n matrix A, with m > n, we
annihilate subdiagonal entries of successive columns of A, eventually reaching upper triangular form
Similar to LU factorization by Gaussian elimination, but use orthogonal transformations instead of elementary elimination matrices
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