Corynebacterium glutamicum - JUWEL - Forschungszentrum Jülich

2.5. Experimental Design
This entropy is maximal when all models have an equal probability and decreases when
larger differences between the models are detected. The expected change in this entropy
should thus be maximized and this change R canbeformulatedas
R = � Pi,n−1
�
pi ln
pi
� dyn
Pi,n−1pi
(2.59)
where pi is the probability density function of the expected new measurement yn accordingtomodeli.
Box and Hill have used the maximization of a maximal expected value for R, asthe
criterion for a model discriminating design. They formulated their criterion for the
univariate static case with constant measurement variance σ 2 as:
1
arg max
ξ 2
�
m�
m�
i=1 j=i+1
Pi,n−1Pj,n−1
(σ∗2 i − σ∗2 j )2
(σ2 + σ∗2 i )(σ2 + σ∗2 j ) +[ˆyi − ˆyj] 2 ∗
�
1
σ2 + σ∗2 1
+
i σ2 + σ∗2 ��
j
(2.60)
The peculiar use of the upper bound on the expected value for R has been criticized
by several statisticians (Meeter et al., 1970). Reilly suggested a method to estimate the
expected value for R instead (Reilly, 1970).
The use of this maximal value is also the cause for the use of the differences in expected
model variances in the numerator of the first part of the criterion. Especially in dynamic
experiments and in models with poor parameter estimates, the model variances can take
extreme values which can change very fast. This happens, for instance, in situations of
sudden changes such as a depletion of substrate. Therefore it is very likely that, in these
cases, the differences in the expected measured values play no significant role any more,
but the criterion reduces to the maximization of the differences in the model variances,
which is intuitively strange. Several examples where basically only the differences in
the model variances determine the criterion, were also found in the simulative study in
chapter 4.
The calculation of the model variances with the requirement that the model
can be approximated by a linearized form, provides another source of problems
(Bajramovic and Reilly, 1977). Hsiang and Reilly have suggested a method to avoid
this, however requiring the use of a series of discrete parameter sets (Hsiang and Reilly,
1971).
The original criterion from Box and Hill was formulated for one variable and a constant
measurement variance. One option to enhance this criterion for several measured
variables and several measurements in the dynamic experiments, is the summation over
all measured values, assuming all measurements to be independent:
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