REPOSITORY > RESULTS

Abstract

The task of mathematical modeling of dynamic systems from observed system behavior,
widely known under the name of system identification, breaks down into two subtasks.
The first task, referred to as structure identification, is to specify the model structure,
i.e., the functional form of the model. In practice, the model structure is usually given by
a human domain expert and re
ects prior domain knowledge: this is called knowledgedriven
identification (as opposed to data-driven identification, which is based only on
data). Structure identification plays an important role in modeling as it defines the
choice available for the selection of the \best model".
The second task, referred to as parameter identification, aims to estimate the values of
the model parameters that define a best possible fit of the model to the measured data. It
assumes that the model structure is known and the observed system behavior is given in
the form of measured data. Accurate estimation of the model parameters is important for
describing and analyzing the behavior of the modeled system. Parameter identification
is therefore a crucial step in almost all approaches for reconstructing system dynamics
from measured data, including knowledge-driven and data-driven system identification as
well as traditional (human) and automated modeling, i.e., the automated discovery of
appropriate model structures and model parameter values by equation discovery tools.
In this dissertation, we address the task of parameter identification in dynamic models
of real-life systems. The models are represented by ordinary differential equations
(ODEs), as considered in the fields of systems biology and ecological modeling. The task
is approached as a least-squares estimation problem within the frequentist framework.
The latter means that the model parameters have fixed unique values and their optimal
values are the ones that minimize a quadratic cost function, i.e., the sum of squared errors
between the model prediction and the experimentally measured data. Least-squares estimation
is essentially an optimization task. However, it can turn into a difficult problem
for traditional (gradient-based) optimization methods when modeling complex system dynamics.
Therefore, it should be addressed by advanced meta-heuristic approaches, such
as evolutionary or swarm intelligence methods.
Typically, biological and ecosystem models are nonlinear and have many parameters,
the studied systems can often be only partially observed, and their measurements are
sparse and imperfect due to noise. All of these constraints can lead to identifiability
problems, i.e., the inability to uniquely identify the unknown model parameters, making
parameter estimation an even harder optimization task. Furthermore, the implicit definition
of the cost function requires expensive numerical ODE simulations that have to
be performed for every parameter solution investigated during the optimization process.
As a result, parameter identification is a challenging and computationally expensive step
in the process of reconstructing the structure and behavior of biological and ecological
systems.
This dissertation attempts to improve the quality of reconstructed system dynamics
by improving parameter identification. In this context, we perform a thorough empirical
evaluation of representative meta-heuristic methods on the task of estimating parameters
in two nonlinear ODE models. The considered models describe two practically relevant
and representative real-life systems, i.e., endosome maturation in endocytosis and a
food web of Lake Bled. The compared meta-heuristic methods are the differential antstigmergy
algorithm, the continuous differential ant-stigmergy algorithm, particle swarm
optimization, and differential evolution. As a baseline method for the experimental comparison,
we use Algorithm 717, a gradient-based local search method essentially designed
for nonlinear least-squares estimation. Different experimental scenarios are considered to
investigate the effect of limited observability of the system dynamics, the in
uence of the
ODE simulation method, and the impact of the noise in the data, on the complexity of
the parameter identification task, as well as the applicability and performance of different
optimization methods in this context.
The empirical evaluation shows that the meta-heuristic global optimization methods
for parameter identification are clearly superior and should be preferred over local optimization
methods. While the differences in performance between the different methods
within the class of meta-heuristics are not significant across all conditions, differential
evolution yields the best results in terms of the quality of the reconstructed system dynamics
as well as the speed of convergence. The observability of the system shows a
strong in
uence, where less complete observations make the optimization task much more
dicult. The results clearly indicate the importance of choosing a relevant cost function
when the modeled systems dynamics is only partially observed. While the use of a simple
one-step trapezoidal-based integrator for supervised prediction makes parameter identifi-
cation much faster, the use of a multistep variable-coecient integrator for unsupervised
prediction produces much better parameter estimates from real-measured data.
Furthermore, we consider the problem of parameter identification within the process
of automated modeling of dynamic systems, where a large number of model structures
is considered. One major drawback of existing automated modeling approaches is the
use of local search methods for parameter identification. In this context, we investigate
the in
uence of parameter identification (in terms of a global and a local optimization
method) on the outcome of the automated modeling process, i.e., on what models are
selected. We consider eight tasks of automated modeling of phytoplankton dynamics in
Lake Bled from single-year data measured in eight different years. The outcome of the
experiments empirically demonstrate the benefit of estimating model parameters by global
optimization methods for the model (structure) selection process, opening the opportunity
to model long term system dynamics.
Many challenges still remain concerning the use of optimization methods for parameter
identification in dynamic systems, especially in the context of automated modeling by
equation discovery methods. Besides the need to extend our study by including additional
dynamic systems from different domains, several lines for further improvement of existing
automated modeling methods can be followed. These include the use of more appropriate
and informative cost functions, as well as more robust and faster methods for parameter
identification. Finally, explicit integration of the feedback from identifiability analysis
within the process of model selection is highly desirable.