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•Inevery
z
i
z
j
plane, the positions of the asymptotes are inversely proportional
to the differences between the corresponding
A
T
A
eigenvalues
λ
i
and
λ
j
[eq. (5.106) . Hence, the positions of the barriers depend on the clustering
of these eigenvalues (a big gap has a negative effect on the domain). In
particular, in the
z
n
z
1
plane, the position of the barrier given by the first
saddle (the highest) is inversely proportional to the distance between
1
and
λ
n
, and therefore an ill-conditioned data matrix
A
worsens the domain.
λ
5.3.10 TLS Domain of Convergence
The TLS domain of convergence is affected by the action of the following bor-
ders:
1.
The barriers (asymptotes)
. This was shown in Section 5.3.7. Furthermore,
in every plane
z
i
z
j
with
j
v
i
direc-
<
i
there is a unique barrier for the
v
j
, generated by the first saddle having parameter
t
> λ
i
.
Note that the solution locus stays on the hyperbola branch approaching the
horizontal asymptote for
t
tion, parallel to
→
λ
i
, and the saddle locus giving rise to the
barrier begins from the opposite infinite point on the asymptote. Hence, the
origin is always between the minimum and the barrier.
2.
The saddle cone projections in the TLS hyperplane
. These repulsive cone
projections border that part of the frontier between the saddle and the
corresponding barrier.
3.
The maximum locus tangent
. If the maximum is near the vertical asymp-
tote, it corresponds closely to this asymptote. With respect to a vertical
asymptote corresponding to a far eigenvalue
λ
i
(
λ
i
λ
1
), its effect is not
negligible.
4.
The saddle-maximum hypercrest projection
. In the TLS hyperplane, the
hyperplane representing the maxima in the minimum direction (see Section
2.5.1) projects to an
(
n
−
1
)
-dimensional plane which corresponds to a
straight line in every plane
z
i
z
j
.
The equation of the hypercrest in the
(
n
+
1
)
-dimensional space
(MCA
space, see Figure 4.1 for the three-dimensional case) is given by
n
+
1
i
=
1
v
i
,
n
+
1
i
T
n
v
=
=
0
(5.115)
+
1
where
v
n
+
1
is the “smallest” right singular vector of [
A
;
b
]. In the TLS hyper-
plane, the minimum is given by the intersection with the straight line parallel to
v
n
+
1
:
1
v
1,
n
+
1
,
...
,
v
n
,
n
+
1
T
−
1
x
min
=
x
=
(5.116)
v
n
+
1,
n
+
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