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1.8 MIXED OLS-TLS PROBLEM
If n 1 columns of the m × n data matrix A are known exactly , the problem is called
mixed OLS-TLS [98]. It is natural to require that the TLS solution not perturb
the exact columns. After some column permutations in A such that A = [ A 1 ; A 2 ],
where A 1
n 2 , perform n 1
Householder transformations Q on the matrix [ A ; B ] (QR factorization) so that
m
×
n 1
m
×
is made of the exact n 1 columns and A 2
R 11 R 12 R 1 b
0
n 1
R 22 R 2 b
m
n 1
[ A 1 ; A 2 ; B ]
= Q
(1.46)
n 1
n n 1
d
where R 11 is a n 1
×
n 1 upper triangular matrix. Then compute the TLS solution
X 2
X 2
of R 22 X
R 2 b .
yields the last n
n 1
components of each solution
X = X 1 ;
X 2 T ,
X 1 of the solution matrix
vector x i .Tofindthefirst n 1 rows
R 12 X 2 . Thus, the entire method amounts to a
preprocessing step, a TLS problem, an OLS problem, and a postprocessing step
(inverse row permutations) [72].
(OLS) R 11 X 1
solve
=
R 1 b
Theorem 29 (Closed-Form Mixed OLS-TLS Solution) Let rank A 1 = n 1 ;
denote by
σ ( respectively,
σ ) the smallest [ respectively,
( n 2 + 1 ) th ] singular
value of R 22
( respectively,
[ R 22 ; R 2 b ]); assume that
the smallest singular
σ > σ , then the mixed OLS-TLS
values
σ = σ n 2 + 1 =···= σ n 2 + d coincide. If
solution is
A T A σ
2 00
0
1
X =
A T B
(1.47)
I n 2
Proof. It is a special case of [97, Th. 4].
1.9 ALGEBRAIC COMPARISONS BETWEEN TLS AND OLS
Comparing (1.29) with the LS solution
X = A T A 1 A T B
(1.48)
shows that σ n + 1 completely determines the difference between the solutions.
Assuming that A is of full rank, σ n + 1 = 0 means that both solutions coincide
( AX B compatible or underdetermined). As σ n + 1 deviates from zero, the set
AX B becomes more and more incompatible and the differences between the
TLS and OLS solutions become deeper and deeper.
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