77 research outputs found
The Origins of Computational Mechanics: A Brief Intellectual History and Several Clarifications
The principle goal of computational mechanics is to define pattern and
structure so that the organization of complex systems can be detected and
quantified. Computational mechanics developed from efforts in the 1970s and
early 1980s to identify strange attractors as the mechanism driving weak fluid
turbulence via the method of reconstructing attractor geometry from measurement
time series and in the mid-1980s to estimate equations of motion directly from
complex time series. In providing a mathematical and operational definition of
structure it addressed weaknesses of these early approaches to discovering
patterns in natural systems.
Since then, computational mechanics has led to a range of results from
theoretical physics and nonlinear mathematics to diverse applications---from
closed-form analysis of Markov and non-Markov stochastic processes that are
ergodic or nonergodic and their measures of information and intrinsic
computation to complex materials and deterministic chaos and intelligence in
Maxwellian demons to quantum compression of classical processes and the
evolution of computation and language.
This brief review clarifies several misunderstandings and addresses concerns
recently raised regarding early works in the field (1980s). We show that
misguided evaluations of the contributions of computational mechanics are
groundless and stem from a lack of familiarity with its basic goals and from a
failure to consider its historical context. For all practical purposes, its
modern methods and results largely supersede the early works. This not only
renders recent criticism moot and shows the solid ground on which computational
mechanics stands but, most importantly, shows the significant progress achieved
over three decades and points to the many intriguing and outstanding challenges
in understanding the computational nature of complex dynamic systems.Comment: 11 pages, 123 citations;
http://csc.ucdavis.edu/~cmg/compmech/pubs/cmr.ht
Structure or Noise?
We show how rate-distortion theory provides a mechanism for automated theory
building by naturally distinguishing between regularity and randomness. We
start from the simple principle that model variables should, as much as
possible, render the future and past conditionally independent. From this, we
construct an objective function for model making whose extrema embody the
trade-off between a model's structural complexity and its predictive power. The
solutions correspond to a hierarchy of models that, at each level of
complexity, achieve optimal predictive power at minimal cost. In the limit of
maximal prediction the resulting optimal model identifies a process's intrinsic
organization by extracting the underlying causal states. In this limit, the
model's complexity is given by the statistical complexity, which is known to be
minimal for achieving maximum prediction. Examples show how theory building can
profit from analyzing a process's causal compressibility, which is reflected in
the optimal models' rate-distortion curve--the process's characteristic for
optimally balancing structure and noise at different levels of representation.Comment: 6 pages, 2 figures;
http://cse.ucdavis.edu/~cmg/compmech/pubs/son.htm
Efficient inference of parsimonious phenomenological models of cellular dynamics using S-systems and alternating regression
The nonlinearity of dynamics in systems biology makes it hard to infer them
from experimental data. Simple linear models are computationally efficient, but
cannot incorporate these important nonlinearities. An adaptive method based on
the S-system formalism, which is a sensible representation of nonlinear
mass-action kinetics typically found in cellular dynamics, maintains the
efficiency of linear regression. We combine this approach with adaptive model
selection to obtain efficient and parsimonious representations of cellular
dynamics. The approach is tested by inferring the dynamics of yeast glycolysis
from simulated data. With little computing time, it produces dynamical models
with high predictive power and with structural complexity adapted to the
difficulty of the inference problem.Comment: 14 pages, 2 figure
Parametric, nonparametric and parametric modelling of a chaotic circuit time series
The determination of a differential equation underlying a measured time
series is a frequently arising task in nonlinear time series analysis. In the
validation of a proposed model one often faces the dilemma that it is hard to
decide whether possible discrepancies between the time series and model output
are caused by an inappropriate model or by bad estimates of parameters in a
correct type of model, or both. We propose a combination of parametric
modelling based on Bock's multiple shooting algorithm and nonparametric
modelling based on optimal transformations as a strategy to test proposed
models and if rejected suggest and test new ones. We exemplify this strategy on
an experimental time series from a chaotic circuit where we obtain an extremely
accurate reconstruction of the observed attractor.Comment: 19 pages, 8 Fig
Identification of parameters in amplitude equations describing coupled wakes
We study the flow behind an array of equally spaced parallel cylinders. A
system of Stuart-Landau equations with complex parameters is used to model the
oscillating wakes. Our purpose is to identify the 6 scalar parameters which
most accurately reproduce the experimental data of Chauve and Le Gal [{Physica
D {\bf 58}}, pp 407--413, (1992)]. To do so, we perform a computational search
for the minimum of a distance \calj. We define \calj as the sum-square
difference of the data and amplitudes reconstructed using coupled equations.
The search algorithm is made more efficient through the use of a partially
analytical expression for the gradient . Indeed
can be obtained by the integration of a dynamical system propagating backwards
in time (a backpropagation equation for the Lagrange multipliers). Using the
parameters computed via the backpropagation method, the coupled Stuart-Landau
equations accurately predicted the experimental data from Chauve and Le Gal
over a correlation time of the system. Our method turns out to be quite robust
as evidenced by using noisy synthetic data obtained from integrations of the
coupled Stuart-Landau equations. However, a difficulty remains with
experimental data: in that case the several sets of identified parameters are
shown to yield equivalent predictions. This is due to a strong discretization
or ``round-off" error arising from the digitalization of the video images in
the experiment. This ambiguity in parameter identification has been reproduced
with synthetic data subjected to the same kind of discretization.Comment: 25 pages uuencoded compressed PostScript file (58K) with 13 figures
(155K in separated file) Submitted to Physica
An alternative solution to the model structure selection problem
An alternative solution to the model structure selection problem is introduced by conducting a forward search through the many possible candidate model terms initially and then performing an exhaustive all subset model selection on the resulting model. An example is included to demonstrate that this approach leads to dynamically valid nonlinear model
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