DARPA ULTRALOG Final Report - Industrial and Manufacturing ...

• Classification of the signal: System identification in nonlinear chaotic systems means
establishing a set of invariants for each system of interest and then comparing
observations to that library of invariants. The invariants are properties of attractor and are
independent of any particular trajectory of the attractor. Invariants can be divided into
two classes: fractal dimensions (Farmer et. al. 1983) and Lyapunov exponents (Sano and
Sawada 1985). Fractal dimensions characterize geometrical complexity of dynamics i.e.
how the sample of points along a system orbit are distributed spatially. Lyapunov
exponents on the other hand describe the dynamical complexity i.e. “stretching and
folding” in the dynamical process.
• Making models and Prediction: This step involves determination of the parameters of
the assumed model of the dynamics:
y(
n)
→ y(
n + 1)
y(
n + 1) = F(
y(
n),
a , a
,..... a
1 2 p
which is consistent with invariant classifiers (Lyapunov exponents, dimensions). The functional
form F (⋅) often used, includes polynomials, radial basis functions etc. Local False Nearest
Neighbor (Abarbanel and Kennel 1993) test is used to determine how many dimensions are
locally required to describe the dynamics generating the time series, without knowing the
equations of motion and hence gives the dimension for the assumed model. The methods for
building nonlinear models can be classified as Global and Local (Farmer and Sidorowich 1987;
Casdalgi 1989). By definition Local methods vary from point to point in the phase space while
Global Models are constructed once and for all in the whole phase space. Models based on
Machine Learning techniques such as radial basis functions or Neural Networks (Powell 1987)
and Support Vector Machines (Mukherjee et al. 1997) carry features of both. They are usually
used as global functional forms, but they clearly demonstrate localized behavior too.
The techniques from nonlinear time series analysis are well suited for modeling the
nonlinearities in the supply chains. For an application of nonlinear time series analysis in supply
chains, the reader is referred to Lee et al., 2002. Using it one can deduce that the time series is
deterministic, so that it should be possible in principle to build predictive models. The invariants
can be used to effectively characterize the complex behavior. For e.g., the largest Lyapunov
exponent gives an indication of how far into the future, reliable predictions can be made while the
fractal dimensions gives an indication of how complex a model should be chosen to represent the
data. These models then provide the basis for systematically developing the control strategies. It
should be noted the functional forms used for modeling in the step (4) above, are continuous in
their argument. This approach builds models viewing a dynamical system as obeying laws of
physics. From another perspective a dynamical system can be considered as processing
information. So an alternative class of discrete “computational” models inspired from the theory
of automata and formal languages can also be used for modeling the dynamics (Marcus 1996).
“Computational Mechanics”, considers this viewpoint and describes the system behavior in terms
of its intrinsic computational architecture i.e. how it stores and processes information.
6.2 Computational Mechanics
Computational mechanics is a method for inferring the causal structure of stochastic processes
from empirical data or arbitrary probabilistic representations. It combines ideas and techniques
from nonlinear dynamics, information theory and automata theory, and is, as it were, an “inverse”
to statistical mechanics. Instead of starting with a microscopic description of particles and their
interactions, and deriving macroscopic phenomena, it starts with observed macroscopic data, and
infers the simplest causal structure: the “ε -machine” capable of generating the observations. The
ε -machine in turn describes the system's intrinsic computation, i.e., how it stores and processes
information. This is developed using the statistical mechanics of orbit ensembles, rather than
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