Nonlinear Dynamics and
Complex Systems
Rick Gorvett, FCAS, MAAA, ARM, FRM, Ph.D.
Actuarial Science Professor
University of Illinois at Urbana-Champaign
ERM Symposium
Chicago, IL
April 2004
Agenda
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Purpose and framework
Historical background
Chaos and complexity
Nonlinear modeling techniques
Sample references
Contact information
Questions / ideas / suggestions
Purpose and Framework
• What this presentation is
– Description of historical evolution
– An overview of concepts
– Hopefully, inspirational
• What this presentation is not
– A cookbook of tried-and-true formulas
– An encyclopedia of applications
• This material is much more a way of thinking
than rote application of equations and techniques
Purpose and Framework (cont.)
A Personal Anecdote
• Some common student questions
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“Will this be on the exam?”
“Is the final cumulative?”
“What do I say at an interview?”
“How do I decide between casualty and life?”
• One particular (very good) student asked recently
– “How do I know I won’t get bored with being an
actuary,” which morphed into
– “Where is the beauty in actuarial science?”
Purpose and Framework (cont.)
The “Beauty” in Actuarial Science
• Virtually everything can be considered to be
relevant to actuarial science
– Economic and financial conditions
– Social, political, and religious conditions and trends
– Science, technology, medicine
• In a fast-changing, dynamic world, the profession
must also evolve and adapt to the underlying
factors
Historical Background
• Plato (427-347 BC)
– Forms, Ideas, Ideals
– Eternal, absolute, unchanging
– Outside the sense-world
• Pythagoras (570-490 BC)
– Leader of a religious sect
– Numbers are primary
– Relationship between plucked string length and the
resulting musical note
– Pythagorean theorem, etc.
Historical Background (cont.)
• Euclid (c. 300 BC)
– Compiled mathematical thought into his Elements
– Systematized theorem and methodology of proof,
as well as geometric reasoning (which held
primacy until quite recently)
• Ptolemy (c. 140 AD)
– Astronomer: wrote the Almagest (“The Greatest”)
– Geocentric universe
– Complicated system of circles (deferents,
epicycles, eccentrics, etc.)
Historical Background (cont.)
• Copernicus (1473-1543 AD)
– Heliocentric universe
– Planetary movements still circular
– Complicated: 48 cycles and epicycles
• Kepler (1571-1630 AD)
– Originally tried to place planetary orbits within a
framework of “nested solids”
– Ultimately, determined that planets orbit according
to ellipses
Historical Background (cont.)
• Nineteenth-century
– Non-Euclidean geometry: space need not be
“flat”
• Twentieth-century
– Relativity: space-time is warped by matter and
energy
– Quantum mechanics: probabilistic; breakdown
of causality principle
So….
• We naturally (and/or have been conditioned to)
love and accept
– Linearity
– Smoothness
– Stability
• This, in the face of a world that is largely
– Nonlinear
– Unsmooth
– Random
Chaos
• Random results from simple equations
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• Finding order in random results
• Sensitivity to initial conditions
– Butterfly effect
– Measurement issues (parameter uncertainty)
• Local randomness vs. global stability
• Deterministic – not total disorder
Chaos (cont.)
• Consider a non-linear time series
– E.g., it can converge, enter into periodic motion, or
enter into chaotic motion
• Example: the logistic function
xt+1 = a xt (1-xt)
– Stability depends upon the coefficient value
– Note: no noise or chaotic provision built into rule
Sante Fe Institute
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Founded in 1984
Private, non-profit
Multidisciplinary research and education
Primarily a “visiting” institution
Current research focus areas
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Cognitive neuroscience
Computation in physical and biological systems
Economic and social interactions
Evolutionary dynamics
Network dynamics
Robustness
Complexity
• A commonly heard definition: “the edge of
chaos”
– Between order and randomness
• Simple rules can lead to complex systems
• Related to “entropy”
– Entropy = disorder
– Second law of thermodynamics
Quotation
Nonlinear Dynamics and Chaos:
Where do we go from here? (Preface)
“This book was born out of the lingering
suspicion that the theory and practice of
dynamical systems had reached a plateau of
development…. The over-riding message is
clear: if dynamical systems theory is to make
a significant long-term impact, it needs to get
smart, because most systems are illdefined….”
Quotation
War and Peace, by Leo Tolstoy
Book Eleven, Chapter 1
“Only by taking infinitesimally small units for
observation (the differential of history, that is,
the individual tendencies of men) and attaining
to the art of integrating them (that is, finding
the sum of these infinitesimals) can we hope to
arrive at the laws of history.”
Quotation (cont.)
War and Peace, by Leo Tolstoy
Second Epilogue, Chapter 9
“All cases without exception in which our conception of
freedom and necessity is increased and diminished
depend on three considerations:
(1) The relation to the external world of the man who
commits the deeds.
(2) His relation to time.
(3) His relation to the causes leading to the action.”
Quotation (cont.)
War and Peace, by Leo Tolstoy
Second Epilogue, Chapter 11
“And if history has for its object the study of the
movement of the nations and of humanity
and not the narration of episodes in the lives
of individuals, it too, …, should seek the
laws common to all the inseparably
interconnected infinitesimal elements of free
will.”
Fractal Geometry and Analysis
• Think of a tree
– Picture from a distance, or
– A drawing or representation of a tree
• Move closer
– Individual branches and patterns are unique
• Quote from Peters (1994):
“Euclidean geometry cannot replicate a tree…. Euclidean
geometry recreates the perceived symmetry of the tree, but
not the variety that actually builds its structure. Underlying
this perceived symmetry is a controlled randomness, and
increasing complexity at finer levels of resolution.”
Finance and Economics
• Traditional (“classical”) paradigm
– Random walk
– Efficient markets hypothesis
– Rational behavior
• Emerging paradigm
– Behavioral and utility issues
– Possible path-dependence
– Learning from experience
Fractal Market Hypothesis
• Behavioral issues
– Importance of liquidity and investors’ horizons
• Investment horizons
– If there are a large number of traders with many
investment horizons in the aggregate, the longerhorizon traders can provide liquidity to the shorthorizon traders when the latter experience a
significant event (e.g., crash, discontinuity)
Nonlinear Modeling Techniques
• Neural networks
• Genetic algorithms
• Fuzzy logic
We will discuss
• Others (e.g., mentioned by Shapiro (2000))
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Statistical pattern recognition
Simulated annealing
Rule induction
Case-based reasoning
Neural Networks
• Artificial intelligence model
• Characteristics:
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Pattern recognition / reconstruction ability
Ability to “learn”
Adapts to changing environment
Resistance to input noise
• Bottom line:
– Data is input
– Behavior is output
Neural Networks (cont.)
• Process
– Data (neurons) input into the network
– Weights are assigned
– Weights are changed until output is “optimal”
• Brockett, et al (1994)
– Feed forward / back propagation
– Predictability of insurer insolvencies
Genetic Algorithms
• Inspired by biological evolutionary processes
• Iterative process
– Start with an initial population of “solutions”
(think: chromosomes)
– Evaluate fitness of solutions
– Allow for evolution of new (and potentially
better) solution populations
• E.g., via “crossover,” “mutation”
– Stop when “optimality” criteria are satisfied
Fuzzy Set Theory
Insurance Problems
• Risk classification
– Acceptance decision, pricing decision
– Few versus many class dimensions
– Many factors are “clear and crisp”
• Pricing
– Class-dependent
– Incorporating company philosophy / subjective
information
Fuzzy Set Theory (cont.)
A Possible Solution
• Provide a systematic, mathematical framework
to reflect vague, linguistic criteria
• Instead of a Boolean-type bifurcation, assigns a
membership function:
For fuzzy set A, mA(x): X ==> [0,1]
• Young (1996, 1997): pricing (WC, health)
• Cummins & Derrig (1997): pricing
• Horgby (1998): risk classification (life)
Some Other Techniques
• Agent-based modeling
– Simple agents + simple rules  societies
• Cellular automata
– Start with simple set of rules
– Can produce complex and interesting patterns
• Percolation theory
– Lattice
– Probability associated with “yes” or “no” in each
cell of the lattice
– Clustering and pathways
Sample References
• Casti, 2003, “Money is Funny, or Why Finance is Too Complex
for Physics,” Complexity, 8(2): 14-18
• Craighead, 1994, “Chaotic Analysis on U.S. Treasury Interest
Rates,” 4th AFIR International Colloquium, pp. 497-536
• Hogan, et al, eds., 2003, Nonlinear Dynamics and Chaos: Where
do we go from here?, Institute of Physics Publishing
• Horgan, 1995, “From Complexity to Perplexity,” Scientific
American, 272(6): p. 104
• Peters, 1994, Fractal Market Analysis: Applying Chaos Theory
to Investment and Economics, John Wiley & Sons
• Shapiro, 2000, “A Hitchhiker’s Guide to the Techniques of
Adaptive Nonlinear Models,” Insurance: Mathematics &
Economics, 26: 119-132
For a Copy of the Presentation
• E-mail: gorvett@uiuc.edu
• Web page: http://www.math.uiuc.edu/~gorvett/