Biochemical Reactions - A First Look
79
where R, is the reduced redundancy matrix containing only the independent rows of R If rank(R) =
0, the system is obviously not over-determined.
The case
L
=
N
+
M
+ 1
is obviously not of interest unless there are experimental errors
(“inaccuracies”) in the measured data. This case will be treated shortly for the general case of
L
measurements with
L
bounded as shown above. Usually some of the rates can, however, not be
found with satisfactory precision.
qw
is certainly one example, but
q%
can be difficult to measure if
the medium is strongly colored or contains solid particles as is often the case in industrial
fermentations. Consequently equation (3.37) can be used to calculate a set of rates qc where qc
contains less than 4 components when only C, H, O and N are considered.
We now assume that all the measured rates contain random errors. With qm
and q m
being the
vectors of, respectively, the true and the measured values, we have
q* =q*+&
(3.41)
where 5 is the vector of measurement errors. It is now assumed that the error vector is normally
distributed, with a mean value of zero and with a variance covariance matrix F, i.e.,
E(S) = 0
(3.42)
F = 4 f l . - ‘I « ) 6 . - <
l-)TJ= E(88'')
(3.43)
where E is a matrix with each element being represented by the expected value operator. If the
model is conect and there are no measurement errors (8 = 0), all / - rank(R) equations in Eq. (3.40)
will be satisfied, i.e., the elemental balances close exactly. In any real experimental investigation (5
* 0) there is a residual in each part of Eq. (3.41). The vector of residuals g is given by
e = R rd = R r(qm
—q
J = R rqm
(3.44)
since by Eq. (3.40) R rq ffl= 0. Since E(g) = 0, the residuals g defined in Eq. (3.44) also have a mean
of zero and the variance covariance matrix for g is given by Eq. (3.45):
E(e) = R rE(8) = 0
(3.45)
P = E(££r ) = R rE(S8r )R^ = R rFR^
(3.46)
The minimum variance estimate of the error vector 8 is obtained by minimizing the sum of squared
errors scaled according to the level of confidence placed on the individual measurements, i.e., to
compute,
Min(8TF l8)
(3.47)