(702b) Generalization of a Parameter Set Selection Procedure for Nonlinear Systems

Generalization of a Parameter
Set Selection Procedure for Nonlinear Systems

Daniel Howsmon, Wei
Dai, and Juergen Hahn

Systems of ordinary differential equations (ODEs) or
differential algebraic equations (DAEs) are often used to describe dynamic
systems, with applications in biochemical reaction networks [1], quantitative systems pharmacology [2], pharmaceutical processes [3], fermentation processes [4], and ecological systems [5] to name a few. The accuracy of these models depends on both the model structure and parameters. Whereas the structure is often chosen to best reflect the fundamental physics, chemistry, and/or biology of the system under study, the model parameters are sourced from available literature or estimated from experimental data.

To avoid over-fitting of the model to data, it is common
that only a subset of the model parameters are estimated from experimental data
and the remaining parameters are set to their nominal values. Parameter set
selection procedures using collinearity index methods [5], genetic algorithms [6], column-pivoting methods [7], Gram-Schmidt orthogonalization methods [8], and clustering methods [9] have been proposed. All of these methods rely on local sensitivity analysis to describe the relationship between the model parameters and model output. Therefore, selecting uncertain parameters for estimation is dependent on their uncertain nominal values. This may result in suboptimal parameter sets, especially as the confidence in nominal parameter values decreases [10]. Global sensitivity analysis can overcome some of the challenges associated with local sensitivity analysis by simultaneously varying model parameters over the range of interest, directly incorporating the uncertainty in parameters. However, results from the Morris method [11], sampling-based methods [12], variance-based methods [13], and other global sensitivity analyses are difficult to interpret for the parameter set selection problem since they do not use the concept of sensitivity vectors.

To overcome the shortcomings of both local and global
sensitivity analyses for parameter set selection, a procedure that uses local
sensitivity analysis and dynamic optimization to simultaneously vary a large
number of parameters was previously developed [14]. Briefly, a sensitivity cone containing all sensitivity vectors associated with a given parameter evaluated at different combinations of parameter values is constructed. The sensitivity cone is completely characterized by the nominal sensitivity vector and the angle between the cone surface and the nominal sensitivity vector. For a given parameter p_i1, this
angle ϕ_i1 can be found from the following dynamic optimization
problem:

Here, x ∈ Rⁿ  is the state vector, u ∈
R^m is
the input vector, p ∈ R^ℓ is the parameter vector, and s_ik is the sensitivity of state k with respect to parameter i. The ODEs are discretized
using collocation on finite elements to create a nonlinear programming problem.
After the ℓ optimization
problems are solved for a given uncertainty level, the angle between two
sensitivity cones is determined as

where  is the
angle between the nominal sensitivity vectors for parameters i₁ and i₂. The cosine
similarity scores  are then
used to cluster the parameters via a hierarchical clustering technique. However,
clustering based upon such sensitivity cones may be too restrictive in cases of
high nonlinearity or asymmetric uncertainty resulting in a very conservative
estimate of the number of parameters to be estimated.

To overcome this limitation, a procedure that directly
computes the minimum angle between groups of sensitivity vectors is developed. For
each set of parameters {p_i1,p_i2} this minimum angle
can be directly found from the following dynamic optimization problem:

The angles computed from the solution to these optimization
problems are then directly used in hierarchical clustering. This procedure is
more flexible and can more accurately determine the minimum distance between
groups of sensitivity vectors resulting in a more realistic number and
selection of parameters for estimation. The only drawback is that the procedure
is more computationally expensive as the sensitivity cone approach requires
solutions to ℓ optimization
problems and the presented technique requires solutions to ℓ(ℓ ? 1)/2 optimization
problems with 1.5x more optimization
variables. However, this downside is not significant for the number of
parameters that are typically found in problems under investigation. The work
includes a comparison with the parameter cone approach, highlighting scenarios
where the increased computational cost could be justified.

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