80
Chapter 3
The solution to the minimization problem in Eq. (3.47) is given by
8 =
FR7P
‘e =
FR7P !R rqm
(3.48)
where the "hat" specifies that it is an estimate. The estimate in Eq. (3.48) coincides with the
maximum likelihood estimate, since \$ is normally distributed.2 Using Eq. (3.48) we find an
estimate of the measured rates to be given by
qm
= q „ - ^ = a - F R ^ P -1R r)q„
(3.49)
where I is a unity matrix. In Note 3.3 it is shown that the estimate qM
given by Eq. (3.49) has a
smaller standard deviation than the raw measurement q*, and the estimate is therefore likely to be
more reliable than the measured data* Application of Eq. (3.49) to a set of experimental data is
illustrated in Example 3.10.
Note 3.3
Variance covariance matrix of the rate estimates
The variance covariance matrix for the rate estimates is given by
F = e[(4 . - 0 ( 4 . - q .) 1 = E[(8-8)(8-8r)J
(
1
)
and
e |(5 - S)(8 -
à T)\ = E(bèT)
-
E(àèT)
- E(8ôr; + E(5S7)
(2)
The first term represents the variance-covariance matrix for 8 , which by using Eqs. (3.48) and (3.46) we
find to be given by
E(08r ) = FR7p -1E(££7)(FR7P~1)7 = FR 7(P"')r R rF r
(3)
For the second term in Eq. (2), we find ■
E(S87) = E[<?(FR7
P"‘e) 7 ]
= E[S(FR7 P*R,8) 7 ]
= E(8Sr)R7 (P"l)rR ,F7 = FR7 (P~l)rRrF7
2 When 5 is normally distributed, the function to be minimized is the same for the least-square minimization problem
and for the maximum-likelihood minimization problem. If the error vector is not normally distributed the estimate in
Eq. (3.48) remains valid for the least-squares minimization problem, whereas it will no longer be the
maximum-likelihood estimate (Wang and Stephanopouios, 1983).