GeMCA EXIN THEORY - Neural-based Orthogonal Data Fitting: The EXIN Neural Networks - page 213

σ 2

= σ 2

2 1 + ( 1 − ζ ) + o ( 1 − ζ )

i
2 ζ

i 2

(6.41)

which is a good approximation in a neighborhood of ζ = 1(DLS).For ζ = 1
these bounds have the lowest values with respect to ζ (i.e., σ 2
i

/ 2 ∀ i ). Considering
the change of variable µ = 1 − ζ , these bounds can be analyzed for increasing
µ (from zero on); that is, for problems with decreasing ζ but still close to DLS:

≤ σ 2

σ 2
i
2

i − 1

( 1 + µ) ≤ α i ( 1 + µ) ∀ i = 2, ... , n (6.42)

2

All bounds increase linearly with , which implies that all the eigenvalues of K

but the smallest and largest ones 1 increase linearly with

µ

. However, the increase
rate is not the same. Obviously, larger eigenvalues increase more than do smaller
eigenvalues.
About the TLS case

µ

ζ = 0 . 5, the bounds can be approximated as

σ 2
i
2 ζ

= σ i 1 − 2 (ζ − 0 . 5 ) + o (ζ − 0 . 5 )

2

(6.43)

By using the substitution µ = 0 . 5 − ζ , the approximation around the TLS case
becomes

σ 2 ≤ σ 2

( 1 + 2 µ) ≤ α i ( 1 + 2 µ) ∀ i = 2, ... , n (6.44)

i i − 1

and the same analysis as for DLS can be repeated here in a neighborhood of
ζ = 0 . 5 (i.e., around µ = 0).

6.3 ANALYSIS OF THE DERIVATIVE OF THE EIGENSYSTEM
OF GeTLS EXIN

6.3.1 Eigenvalue Derivative

From eq. (6.2), repeated here:

Ru = α Du (6.45)

it follows, by deriving both sides with respect to ζ :

R d u
d

d d d D

d

D d u
d

= Du + α u + α

ζ ζ ζ ζ

1 This Taylor approximation analysis cannot take these two extreme cases into account because only
one bound is given for each case.