Tools from Nonlinear Dynamical Systems Theory Offer New Methods
Paul K. Newton; Signals, Spring 2002
In grade school we are all warned not to compare apples with oranges.
The size of two objects can't be compared unless we measure each
in some common unit. Comparing the speed of two runners requires
that each be measured in elapsed time over a set course. But
even quantities measured with the same units can be difficult
to compare if the measurements are taken at different times and
under different circumstances.
Is it really possible to compare Roger Bannister's 1954 world
record performance, running the mile in 3:59.4 minutes, with
Eamon Coghlan's 1983 time of 3:49.78? The two runs were completed
under very different circumstances (Bannister's was outdoor,
while Coghlan's was indoor), under different training conditions,
and with different expectations. How is it possible to compare
data that are recorded at different times, subject to different
environmental conditions?
Comparing the performance of consumer portfolios that evolve
over different time periods and that are subject to different
economic and management conditions offer many of the same challenges.
Classical equations offer one possible solution to some of these
comparison problems. Equations governing weather patterns, called
the Navier-Stokes equations of fluid turbulence, are widely agreed
upon, but the exact initial conditions that correspond to actual
weather patterns are not known, and small errors between the
model and reality are amplified exponentially. For instance,
it is widely acknowledged that the state-of-the-art in weather
prediction is a seven-day forecast. Experts using sophisticated
models and data from places like the National Center for Atmospheric
Research greatly outperform chance in their weather projections
out to one week. On the 8th day, they are better off flipping
a coin for prediction. Therefore, knowing the equations that
theoretically guide a process is an important first step, but
how do you measure and set initial conditions accurately so that
you end up with a good long-range forecast?
Some help to this question arrived late in the last century.
The modern era of dynamical systems theory began in 1890 with
the work of the French mathematician Henri Poincaré who
was primarily concerned with predicting the motion of planets.
Focusing on the so-called 3-body problem, his concern was the
analysis of the earth-moon-sun system under mutual gravitational
attraction. The equations for this system were well known at
the time and are relatively simple to write down. Their solutions,
Poincaré discovered, were highly sensitive to changes
in the initial conditions. Small changes in them, as in the equations
governing weather systems, are exponentially amplified, a phenomenon
now called the Butterfly Effect. Poincaré discovered chaos
theory, and a new era in scientific inquiry was opened. Novel
techniques for mining data and estimating initial conditions
are being developed from this new field driving exciting practical
applications.
The Genesis Discovery Mission, launched in August 2001, was
sent along a trajectory to Mars that was designed to take advantage
of the dynamical structure of the gravitational fields produced
by the planets, something that would not have been possible without
the use of nonlinear dynamics tools developed in the last 20
years.
Techniques from nonlinear dynamics have also had a profound
impact on the field of time-series analysis. They have given
us a way to understand these time-series and make meaningful
comparisons between them allowing us to create new forecasting
strategies. To return to our original question, how can one compare
financial data that is collected over different periods of time?
The problem of comparing consumer portfolio data is very much
like trying to compare data collected on two different graduating
classes from a university. Most alumni offices send out surveys
to their graduates every five years in order to collect data
such as salary information, and more subjective data, such as
the "level of personal satisfaction" achieved. Suppose
the Class of 1981 has an average annual salary of $100,000 while
the average annual salary from the more recent Class of 1991
is only $50,000. From this data, it is not at all easy to determine
which of these classes has a "more successful" student
body, even if you were willing to accept average salaries as
a measure of success. The Class of 1981 has been out longer,
has more experience; hence their salaries should be higher. To
compare more directly, we would need to compare the average salaries
of the two classes the same number of years after graduation.
But this too is difficult. Economic conditions were booming when
the Class of 1981 graduated and were not as favorable to the
Class of 1991. Shouldn't this be factored in?
One solution to this kind problem being pioneered by Strategic
Analytics is the use of sophisticated data discovery and decomposition
techniques that break down a signal into components between which
fair comparisons be made. In the context of consumer portfolio
performances, the key components are the maturation effects found
in a portfolio and the exogenous impacts coming primarily from
the outside environment. By measuring these effects at the component
level, we can begin to measure them and make meaningful comparisons
among them. Back to our alumni satisfaction example, using these
techniques we can adjust satisfaction responses for the economic
(exogenous) conditions at graduation, thereby arriving at a "pure" measure
of satisfaction normalized for the environment.
There are many new areas that the field of nonlinear dynamics
has influenced over the past ten years, with the analysis of
time-series being one of the most exciting and promising. When
coupled with solid business experience and in-depth domain knowledge,
we can expect new breakthroughs in modeling and prediction for
business processes. Techniques from nonlinear dynamics have also
had a profound impact on the field of time-series analysis and
are beginning to be used to analyze data both from the financial
markets and from complex weather patterns.
Paul Newton received his B.S. degree (cum
laude) from Harvard University majoring in applied mathematics
and physics, and his Ph.D. degree in applied mathematics from
Brown University. He has been a Professor of Mathematics at the
University of Illinois, Urbana-Champaign where he was also a
member of the Center for Complex Systems Research, and has held
visiting appointments at Stanford University, Brown University,
and the Institute for Theoretical Physics at UC Santa Barbara.
He is the author of the book "The N-Vortex Problem: Analytical
Techniques," published by Springer-Verlag, as well as over
50 journal articles on various topics in nonlinear and chaotic
dynamics, nonlinear partial differential equations, and turbulent
fluid flows. He serves on the editorial boards of the "Journal
of Nonlinear Science" and Springer-Verlag's "Texts in Applied
Mathematics."
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