(545f) Parameter Set Selection Via Clustering of Parameters into Pair-Wise Indistinguishable Groups of Parameters

Detailed
first-principles-based models often exhibit dynamics that is occurring at
multiple time and length scales. Additionally, these models tend to include a
large number of parameters, some of which may be estimated where selecting the
set of parameters to be estimated is an important task. Identifiability of a
set of parameters is determined by the effect that changes in the parameter
values have on the output. This effect is represented by the sensitivity
vectors. Several methods for parameter selection based on sensitivity vectors
have been proposed in the literature. These include, but are not limited to, a collinearity
index method,¹ a column pivoting method,² an extension of
the relative gain array,³ a Gram-Schmidt orthogonalization method,⁴
a recursive approach based upon principal component analysis,⁵ and a
combination of Hankel singular values and singular value decomposition.⁶
A systematic approach for parameter selection is based on optimality criteria
computed from the Fisher information matrix as the inverse of Fisher
information matrix provides a lower bound for the covariance matrix of
parameter estimators. A subset of identifiable parameters can be selected based
upon optimizing some experimental criteria such as the D-optimality or
the modified E-optimality criterion of the Fisher information matrix.^7,8

Most of the parameter selection approaches mentioned can be formulated as
a combinatorial optimization problem, however, solving these optimization
problems is nontrivial. For systems with only few parameters an exhaustive search⁷
can be used. However, the total number of possible combination of parameters is
too large to be enumerated even for systems with only a few dozen parameters.
Stochastic search techniques such as genetic algorithms⁹ can provide
a solution for larger systems, however, convergence of these algorithms is not
guaranteed. Another approach is to use a sequential approach where one
parameter is selected at a time, e.g., the orthogonalization method.^2,4
The main disadvantage of these types of approaches is that some combinations of
parameters may give better results, but could be excluded because of parameters
selected at earlier steps. Techniques used for the orthogonalization method
include the householder transformation² and the Gram-Schmidt
procedure⁴. Both of these procedures can be regarded as an approach
to maximize the D-optimality criterion one parameter at a time.⁹

The methods mentioned above focus on searching the space of possible
parameter sets, a procedure that is strongly affected by the number of
parameters of the model. This work presents a different approach for parameter
selection. The number of parameters to be considered is first reduced by
determining several groups of parameters where the parameters within a group
are pair-wise indistinguishable, i.e., they cannot be estimated together. It is
then possible to only consider one parameter per group for the parameter set
selection procedure. This technique significantly reduces the combinatorial
problem resulting from a large number of parameters and enables solution of the
parameter set selection problem using existing approaches.

The main contribution of this work consists of developing the analytic
and numerical methods for sorting parameters into groups for models consisting
of nonlinear differential equations. It will be shown that parameters in an
analytical pair-wise indistinguishable set can be re-parameterized by a new
parameter. A procedure for carrying out this step numerically will also be
presented.

References

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Brun, R.; Kuhni, M.; Siegrist, H.; Gujer, W.; Reichert, P. Practical
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