# MATH 685/CSI 700 Lecture Notes

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

Example Example

Example Existence/Uniqueness

Normal Equations Orthogonality

Orthogonality Orthogonal Projector

Pseudoinverse Sensitivity and Conditioning

Sensitivity and Conditioning Solving normal equations

Example Example

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

Possible methods include

Householder transformations Givens rotations Gram-Schmidt orthogonalization

Householder Transformation Example

Householder QR factorization Householder QR factorization

Householder QR factorization For solving linear least squares problem, product Q of Householder transformations need not be formed explicitly

R can be stored in upper triangle of array initially containing A

Householder vectors v can be stored in (now zero) lower triangular portion of A (almost)

Householder transformations most easily applied in this form anyway

Example Example

Example Example

Givens Rotations Givens Rotations

Example Givens QR factorization

Givens QR factorization Straightforward implementation of Givens method requires about 50% more work than Householder method, and also

requires more storage, since each rotation requires two numbers, c and s, to define it

These disadvantages can be overcome, but requires more complicated implementation

Givens can be advantageous for computing QR factorization when many entries of matrix are already zero, since those annihilations can then be skipped

Gram-Schmidt orthogonalization Gram-Schmidt algorithm

Modified Gram-Schmidt Modified Gram-Schmidt

QR factorization Rank Deficiency If rank(A) < n, then QR factorization still exists, but yields

singular upper triangular factor R, and multiple vectors x give minimum residual norm

Common practice selects minimum residual solution x having smallest norm

Can be computed by QR factorization with column pivoting or by singular value decomposition (SVD)

Rank of matrix is often not clear cut in practice, so relative tolerance is used to determine rank

Near Rank Deficiency QR with Column Pivoting

QR with Column Pivoting Singular Value Decomposition

Example: SVD Applications of SVD

Pseudoinverse Orthogonal Bases

Lower-rank Matrix Approximation Total Least Squares

Ordinary least squares is applicable when right-hand side b is subject to random error but matrix A is known accurately

When all data, including A, are subject to error, then

total least squares is more appropriate

Total least squares minimizes orthogonal distances, rather than vertical distances, between model and data

Total least squares solution can be computed from SVD of [A, b]

Comparison of Methods Comparison of Methods

Comparison of Methods

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