84
Chapter З
variance-covariance matrix F. Madron
et al
(1977) describe a method by which the variance-
covariance matrix can be found from knowledge of the errors in the primary variables (see Note
3.4), but even if we know that there is often a certain coupling between measurement errors we will
use a diagonal F matrix since we can rarely do anything better. The diagonal F matrix is preferable,
partly because the calculations are simplified but also because we usually know little beyond the
order of magnitude of the measurement errors. Thus in some cases F is expressed as
which is the classical least-squares estimate. Equation (3.51) may be useful in many situations, and
its application to the data analyzed in Example 3.10 is illustrated in Example 3.11.
Note 3.4 Calculation of the variance-covariance matrix from the errors in the primary variables
Normally the measured rates are determined from several measurements of so-called primary variables,
e.g., the volumetric glucose utilization rate in a chemostat is measured from the glucose concentration in the
exit stream and the feed flow rate. Specification of the variance-covariance matrix is therefore not
straightforward, but Madron
et al.
(1977) describe a simple approach to find F. The measured rates are
specified as functions of the primary variables, which are collected in the vector v (dimension
L
):
Generally the functions/ are nonlinear, but in order to obtain an approximate estimate of the variance and
the co-variances the functions are linearized. Thus the errors
in the
L
measured rates are expressed as
F = cl
(3.50)
where I is the unity matrix and, inserting Eq. (3.50) in Eq. (3.49), we obtain
qffl= ( I - R r (R
X ) - lR ryqm
(3.51)
(
1
)
£**
i*
linear combinations of the errors
Oj
of the
L
primary variables:
(
2
)
The sensitivities
gji
are collected in the matrix G (of dimension
L*. L * )t
and the variance-covariance
matrix F is calculated from
F = GF*Gr
(3)
where F* is the diagonal matrix containing the
L*
variances of the primary variables. The accuracy of the
computed variances is limited by the accuracy of the linear approximation in Eq. (2) involved in the
computation of the sensitivities._______________________________________________________________
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