1. Scientific Computing Chapter 3 - Linear Least squares
Guanghua tan
School of computer and communication, HNU

2. Outline Least Squares Data Fitting Least Squares Data Fitting
Existence, Uniqueness, and Conditioning
Solving Linear Least Squares Problems

3. 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 over-determined, so usually there is no exact solution

4. Example: Fitting a plane
A plane can be determined by 3 points
Scan points may contain many points
Plane equation

5. Linear Least Squares
For linear problems, we obtain over- determined linear system Ax = b, with m×n matrix A, m > n
System is better written , since equality is not exactly satisfiable when m>n
Least squares solution x minimizes squared Euclidean norm of residual vector

6. Data Fitting
Given m data points (ti, yi), find n-vector x of parameters that gives “best fit” to model function f(t, x)
Problem is linear if function f is linear in components of x,
where functions depend only on t

7. Data Fitting Polynomial fitting
is linear, since polynomial linear in coefficients, though nonlinear in independent variable t
Fitting sum of exponentials
is example of nonlinear problem
For now, we will consider only linear least squares problems

8. Example: Data Fitting
Fitting quadratic polynomial to five data points gives linear squares problem

9. Example: Data Fitting For data over-determined 5×3 linear system is
Solution, we will see how to compute later, is
So approximating polynomial is

10. Outline Least Squares Data Fitting
Existence, Uniqueness, and Conditioning
Solving Linear Least Squares Problems

11. Existence and Uniqueness
Linear least squares problem always has solution
Solution is unique, if and only if, columns of A are linearly independent, i.e., rank(A) = n, where A is m × n
If rank(A) < n, then A is rank-deficient, and solution of linear least squares problem is not unique
For now we assume A has full column rank n

12. Necessary condition for a minimum:
Normal Equations
Necessary condition for a minimum:
To minimize squared Euclidean norm of residual vector
take derivative with respect to x and set it to 0
which reduces to n×n linear system of normal equations

13. Geometric Interpretation
Vector v1 and v2 are orthogonal if their product is zero,
Space spanned by columns of m×n matrix A, , is of dimension at most n
if m>n, b generally does not lie in span(A), so there is no exact solution to Ax = b

14. Geometric Interpretation
Vector y = Ax in span(A) closest to b in 2- norm occurs when residual r = b – Ax is orthogonal to span(A)
again giving system of normal equations
The solution to the least squares problem is

15. Pseudo-inverse and Condition Number
Non-square m×n matrix A has no inverse in usual sense
If rank(A) = n, pseudo-inverse is defined by
and condition number by
By convention,

16. Condition Number
Just as condition number of square matrix measures closeness to singularity, condition number of rectangular matrix measures closeness to rank deficiency
Least squares solution of is given by

17. Sensitivity and Conditioning
Sensitivity of least squares solution to depends on b as well as A
Define angle θ between b and y = Ax by
Bound on perturbation ∆x in solution x due to perturbation ∆b in b is given by

18. Sensitivity and Conditioning
Similarly, for perturbation E in matrix A,
Condition number of least squares solution is about cond(A) if residual is small, but can be squared or arbitrarily worse for large residual

19. Outline Least Squares Data Fitting
Existence, Uniqueness, and Conditioning
Solving Linear Least Squares Problems

20. Normal Equations Method
If m×n matrix A has rank n, then symmetric n×n matrix ATA is positive definite, so its Cholesky factorization
can be used to obtain solution x to system of normal equations
which has same solution as linear squares problem
Normal equations method involves transformations
rectangular
square
triangular

21. Example: Normal Equations Method
For polynomial data-fitting example given previously, normal equations method gives

22. Example, continued
Cholesky factorization of symmetric positive definite matrix gives
Solving lower triangular system Lz = ATb by forward-substitution gives
Solving upper triangular system LTx = z by back- substitution gives

23. Shortcomings of Normal Equations
Information can be lost in forming ATA and ATb
For example, take
where is positive number smaller than
Then in floating-point arithmetic
which is singular

24. Shortcomings of Normal Equations
Sensitivity of solution is also worsened, since

25. Augmented System Method
Definition of residual together with orthogonality requirement give (m+n)×(m+n) augmented system
Augmented system is symmetric but not positive definite, is larger than original system, and requires storing two copies of A
But it allows greater freedom in choosing pivots in computing LU factorization
Another way to transform a least squares problem into a square linear system is to embed it in a larger system.
The definition of the residual vector r, together with the requirement that the residual to the columns of A, gives the system of two equatiions

26. Augmented System Introducing scaling parameter gives system
which allows control over relative weights of two subsystems in choosing pivots
Reasonable rule of thumb is to take
Augmented system is sometimes useful, but is far from ideal in work and storage required

27. Orthogonal Transformation
We seek alternative method that avoids numerical difficulties of normal equations
We need numerically robust transformation that produces easier problem without changing solution
What kind of transformation leaves least squares solution unchanged?
In view f the potential numerical difficulties with the normal equations approach, we need an alternative that does not require formation of the cross-product matrix and right-hand-side vector.
Thus, we seek a more numerically robust type of transformation that will yield a simpler problem whose solution is the same as that of the original least square problem but is more easily computed
As with square linear system

28. Orthogonal Transformations
Square matrix Q is orthogonal if
Multiplication of vector by orthogonal matrix preserves Euclidean norm
Thus, multiplying both sides of least squares problem by orthogonal matrix does not change its solution

29. Triangular Least Squares
As with square linear systems, suitable target in simplifying least squares problems is triangular form
Upper triangular over-determined (m>n) least squares problem has form
where R is nxn upper triangular and b is partitioned similarly
Residual is

30. Triangular Least Squares
We have no control over second term, , but first term becomes zero if x satisfies n×n triangular system
which can be solved by back-substitution
Resulting x is least squares solution, and minimum sum of squares is

31. Triangular Least Squares
So our strategy is to transform general least squares problem to triangular form using orthogonal transformation so that least squares solution is preserved

32. QR Factorization
Given m×n matrix A, with m>n, we seek m×m orthogonal matrix Q such that
where R is n×n and upper triangular
Linear least squares problem is then transformed into triangular least squares problem
which has same solution, since

33. Orthogonal Bases
If we partition m×m orthogonal matrix Q = [Q1 Q2], where Q1 is m×n, then
is called reduced QR factorization of A
Columns of Q1 are orthonomal basis for span(A)
Solution to least squares problem is given by solution to square system

34. 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

35. Computing QR Factorization
Possible methods include
Householder transformations
Givens rotations
Gram-Schmidt orthogonalization
We will focus mainly on the use of Householder transformations, which is the most popular and generally the most effective approach in this context, but we will sketch the other two methods as well

36. Householder Transformations
Householder transformation has form
for nonzero vector v
H is orthogonal and symmetric:
Given vector a, we want to choose v so that

37. Householder transformation

38. Householder Transformation
Substituting into formula for H, we can take
And , with sign chosen to avoid cancellation, i.e.

39. Example: Householder Transformation
If , then we take
where
Since a1 is positive, we choose negative sign for to avoid cancellation, so

40. Example: Householder Transformation
To confirm that transformation works,

41. Householder QR Factorization
To compute QR factorization of A, use householder transformation to annihilate sub-diagonal entries of each successive column
Each Householder transformation is applied to entire matrix, but does not affect prior columns, so zeros are preserved
Process just described produces factorization
where R is n×n and upper triangular

42. Householder QR Factorization
If , then
In applying Householder transformation H to arbitrary vector u
which is much cheaper than general matrix-vector multiplication and requires only vector v, not full matrix H

43. Implementation of Householder QR Factorization
Pseudo description
For k = 1 to n {loop over columns}
{ compute Householder
vector for current col}
if βk = 0 then { skip current column if
continue with next k it’s already zero }
for j = k to n { apply transformation
to remaining
sub-matrix }
end

44. Householder QR Factorization
To preserve solution of linear least squares problem, right-hand side b is transformed by same sequence of Householder transformations
Then solve triangular least squares problem
For solving linear least squares problem, product Q of Householder transformations need not be formed explicitly

45. Householder QR Factorization
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

46. Example: Householder QR Factorization
For polynomial data fitting example given previously, with
Householder vector v1 for annihilating subdiagonal entries of first column of A is

47. Example, continued
Applying resulting Householder transformation H1 yields transformed matrix and right-hand side
Householder vector v2 for annihilating subdiagonal entries of second column of H1A is

48. Example, continued
Applying resulting Householder transformation H2 yields
Householder vector v3 for annihilating subdiagonal entries of third column of H2H1A

49. Example, continued
Applying resulting Householder transformation H3 yields
Now solve upper triangular system by back-substitution to obtain

50. Givens rotations Given rotations introduce zeros one at a time
Given vector , choose scalars c and s so that
Previous equation can be rewritten

51. Givens Rotations Gaussian elimination yields triangular system
Back-substitution then gives
Finally, , implies

52. Example: Givens Rotation
Let , to annihilate second entry we compute cosine and sine
Rotation is then given by
To confirm that rotation works,

53. Givens QR Factorization
More generally, to annihilate selected component of vector in n dimensions, rotate target component with another component
By systematically annihilating successive entries, we can reduce matrix to upper triangular form using sequence of Given rotations

54. Givens QR Factorization
Each rotation is orthogonal, so their product is orthogonal, producing 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

55. Givens QR Factorization
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 be skipped

56. Gram-Schmidt Orthogonalization
Given vectors a1 and a2, we seek orthonormal vectors q1 and q2 having same span
This can be accomplished by subtracting from second vector its projection onto first vector and normalizing both resulting vectors, as shown in diagram

57. Gram-Schmidt Orthogonalization
Process can be extended to any number of vectors , orthogonalizing each successive vector against all preceding ones, giving classical Gram-Schmidt
Resulting qk and rjk form reduced QR factorization of A

58. Modified Gram-Schmidt
Classical Gram-Schmidt procedure often suffers loss of orthogonality in finite-precision
Also, separate storage is required for A, Q, and R, since original ak are need in inner loop, so qk cannot overwrite columns of A
Both deficiencies are improved by modified Gram-Schmidt procedure, with each vector orthogonalized in turn against all subsequent vectors, so qk can overwrite ak

59. Modified Gram-Schmidt Factorization
Modified Gram-Schmidt algorithm

60. 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 solutions 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
So far we have assumed that A is of full column ran, i.e. rank(A)=n. If this is not the case, i.e., if A has linearly dependent columns, then the QR factorization still exists, but the upper triangular factor R is singular
This situation usually arises from a poorly designed experiment, insufficient data, or inadequare or redundamt model

61. Example: Near Rank Deficiency
Consider 3x2 matrix
Computing QR factorization
R is extremely close to singular (exactly singular to 3-digit accuracy of problem statement)

62. Example: Near Rank Deficiency
If R is used to solve linear least squares problem, result is highly sensitive to perturbations in right-hand side
For practical purpose, rank(A)=1 rather than 2, because columns are nearly linearly dependent

63. QR with Column Pivoting
Instead of processing columns in natural order, select for reduction at each stage column of remaining unreduced submatrix having maximum Euclidean norm
If rank(A)=k<n, then after k steps, norms of remaining unreduced columns will be zero (or “negligible” in finite-precision arithmetic) below row k
Yields orthogonal factorization of form
Where R is kxk, upper triangular, and nonsingular, and permutation matrix that performs column interchanges

64. QR with Column Pivoting
Basic Solution to least squares problem can now be computed by solving triangular system , where c1 contains first k components of QTb, and then taking
Minimum-norm solution can be computed, if desired, at expense of additional processing to annihilate S
Basic solution, i.e, a solution having at most k nonzero components

65. QR with Column Pivoting
Rank(A) is usually unknown, so rank is determined by monitoring norms of remaining unreduced columns and terminating factorization when maximum value falls below chosen tolerance

66. Singular Value Decomposition
Singular value decomposition (SVD) of m×n matrix A has form
where U is m×m orthogonal matrix, V is n×n orthogonal matrix, and ∑ is m×n diagonal matrix, with

67. Singular Value Decomposition
Diagonal entries , called singular values of A, are usually ordered so that
Columns ui of U and vi of V are called left and right singular vectors
Computation of SVD will be discussed in next chapter, here only covers the application of SVD

68. Example: SVD
SVD of is given by

69. Applications of SVD Minimum norm solution to is given by
For ill-conditioned or rank deficient problems, “small” singular values can be omitted from summation to stabilize solution
Euclidean matrix norm:
Euclidean condition number of matrix:
Rank of matrix: number of nonzero singular values

70. Pseudo-inverse
Define pseudo-inverse of ( possibly rectangular ) diagonal matrix by transposing and taking scalar pseudo-inverse of each entry
Then pseudo-inverse of general real m×n matrix A is given by
Pseudo-inverse always exists whether or not matrix is square or has full rank
If A is square and nonsingular, then
In all cases, minimum-norm solution to is given by

71. Lower-Rank Matrix Approximation
Another way to write SVD is
with
Ei has rank 1 and can be stored using only m+n storage locations
Condensed approximation to A is obtained by omitting from summation terms corresponding to small singular values
Approximation using k largest singular values is closest matrix of rank k to A
Approximation is useful in image processing, data compression , information retrieval, cryptography, etc

72. 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

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

74. Comparison of Methods
Forming normal equations matrix ATA requires about n2m/2 multiplications, and solving resulting symmetric linear system requires about n3/6 multiplications
Solving least squares problem using Householder QR factorization requires about mn2-n3/3 multiplications
If m ≈ n, both methods require about same amount of work
If m>>n, Householder QR requires about twice as much as normal equations
The choice among them depends on the particular problem being solved and involves tradeoffs among efficiency, accuracy and reliability

75. Comparison of Methods
Cost of SVD is proportional to mn2+n3, with proportionality constant ranging from 4 to 10, depending on algorithms used
Normal equations method produces solution whose relative error is proportional to [cond(A)]2
Required Cholesky factorization can be expected to break down if or worse

76. Comparison of Methods
Householder method produces solution whose relative error is proportional to
which is best possible, since is inherent sensitivity of solution to least squares problem
Householder method can be expected to break down (in back-substitution phase) only if cond(A) or worse

77. Comparison Methods
Householder is more accurate and more broadly applicable than normal equations
These advantages may not be worth additional cost, however, when problem is sufficiently well conditioned that normal equations provide sufficient accuracy
For rank-deficient or nearly rank-deficient problems, Householder with column pivoting can produce useful solution when normal equations method fails outright
SVD is even more robust and reliable than Householder, but substantially more expensive

78. Time Adjustment
I want to adjust the class next Thursday to next Tuesday
Is there any time available?
Morning or afternoon?