Schmucker-Weidelt Lecture Notes, Aarhus, 1975 - MTNet

It remains to show a way to minimize Q, subject to (6.15). The way,
however, is well-known. One simply intrsduces as (N+1 )-st unknown
a Lagrangian parameter X and minimizes the quantity
a
Q' = Q + h (Cai~li - 1 ) , ui = / Gi (r)dr (6.16)
0
The differentiation of (6.16) with respect to the (N4-1) unknowns
yields the (N+1) linear equations
where Qik = W Sik + (l-W) c Eik.
This system of equations is easily solved. The meaning of X is
revealed by multiplying (6.17a) by ak, adding and using (6.17b)
and (6.15) . It results
In the case W = 1, we have X = -2s. With a knowledge of a (roll
i
is obtained from (6.8) with gi = y. and E' from (6.9).
3-
When is inserted in (6.1) instead of m(r) , it w ill in
general not exactly reproduce the data.
The minimization (6.15)
N
subject to the constraint C aiui = 1 admits for two data (N:=2)
i=1
a simple geometrical interpretation: For constant s and E' these
positive definite quantities are represented by ellipses, the con-
straint is a line in the (al,a2)-plane. For uncorrelated errors,
the principal axes of E~ are the al and a2 axis.
hT1a W varies from 0 to 1 the combinatiomof (al ,a2) on the fat
line are obtained. s and. E' are determined from the ellipses
throughthese points,Sj-nce all s-(~~)-elli~ses are similar, s and^
2