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σ
n
> σ
n
+
1
=···=
σ
n
+
d
,then
X
F
≥
X
F
If
(1.49)
σ
n
+
1
also influences the difference in condition between the TLS and OLS prob-
lems and thus the difference in numerical accuracy of their respective solutions
in the presence of worst-case perturbations.
σ
n
2
n
+
1
is a measure of how close
AX
≈
B
is to the class of nongeneric TLS problems. Assuming that
A
0
X
≈
B
0
is the corresponding unperturbed set, rank
A
0
=
n
and the perturbations in
A
and
B
have approximately the same size, the TLS solution is more accurate than
the OLS solution, provided that the ratio
(σ
n
−
σ
−
σ
0
n
)/σ
n
>
1, where
σ
0
n
1
is the
(
n
+
1
)
th singular value of [
A
0
;
B
0
]. The advantage of TLS is more remarkable
with increasing ratio: for example, when
+
1
+
σ
n
≈
B
0
F
is large, or when
0, when
v
n
of
A
0
associated with its smallest
X
0
becomes close to the singular vector
singular value.
1.9.1 About the Residuals
Proposition 30
Define the LS residua R
as B
−
AX
and the TLS residual
ˆ
Ras
B
−
A X; then
1
A
A
T
A
−
σ
1
I
−
1
X
R
−
R
=
σ
2
n
2
n
(1.50)
+
+
R
F
≥
R
F
(1.51)
The TLS and LS residuals approach each other if:
1.
1
is small (slightly incompatible set).
2.
B
F
is small (i.e., the TLS solution becomes close to the LS solution).
3.
σ
n
σ
n
+
1
[i.e.,
A
may not be (nearly) rank deficient].
4.
B
is close to the largest singular vectors of
A
.
σ
n
+
If
σ
n
+
1
=
0
,then R
=
R
=
0
.
Proposition 31
1.10 STATISTICAL PROPERTIES AND VALIDITY
Under the assumption that
all
errors in the augmented matrix [
A
;
B
] are row-
wise independently and identically distributed (i.i.d.) with
zero mean
and
common
covariance matrix
of the form
σ
2
ν
0
,
0
known and positive definite (e.g., the
identity matrix), the TLS method offers the
best
estimate and is more accurate
than the LS solution in estimating the parameters of a model. The most suitable
model for the TLS concept is the
errors-in-variables
(EIV) model, which assumes
an unknown but
exact
linear relation (
zero residual
problems) among the true
variables that can only be observed with errors.
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