GENERALIZATION OF LINEAR REGRESSION PROBLEMS - Neural-based Orthogonal Data Fitting: The EXIN Neural Networks - page 167

bTb

min

s 2

ζ

s 1

Σ i q i 2 / λ i

t

λ 3 λ 2 λ 1

max

Figure 5.5 Graphical method for the determination of critical point parameters for a three-
dimensional case; h ( t ) is given by the nonvertical dashed straight line for two different GTLS
problems (the horizontal line is the DLS); the solid curve is l ( t ).

t min t min

B > 1 B < 1

A/(B − 1)

ζ ζ

− 1/(B − 1)

1/(1 − B)

− A/(1 − B)

Figure 5.6 Plots of the solution locus parameter as a function of the type of GeTLS problem
for all possible A and B .

1. The intersection for the maximum holds at t =+∞ ; it agrees with the
observation that the maximum for the DLS problem is at the TLS hyper-
plane origin.
2. The distance of a saddle to the asymptote at which its locus tends depends
on b T b .

The positions of the branches of the hyperbolas in the plane z i z j (see
Figures 5.7 and 5.8 for the two-dimensional case) depend on the signs of the
corresponding q i and q j [i.e., the signs of the coordinates of the OLS solution z
(see Theorem 101) because λ i ≥ 0 ∀ λ i ]. Hence, knowledge of the OLS solution
indicates the quadrant containing the solution locus for every plane z i z j .