Nonlinear Dynamics and Complex Systems

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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

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Agenda

• Purpose and framework
• Historical background
• Chaos and complexity
• Nonlinear modeling techniques
• Sample references
• Contact information
• Questions / ideas / suggestions

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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

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Purpose and Framework (cont.)A Personal Anecdote

• Some common student questions
• 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 wont get bored with being an
  actuary, which morphed into
• Where is the beauty in actuarial science?

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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

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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.

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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.)

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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

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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

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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

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Chaos

• Random results from simple equations
• OR
• Finding order in random results
• Sensitivity to initial conditions
• Butterfly effect
• Measurement issues (parameter uncertainty)
• Local randomness vs. global stability
• Deterministic not total disorder

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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
• xt1 a xt (1-xt)
• Stability depends upon the coefficient value
• Note no noise or chaotic provision built into
  rule

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Sante Fe Institute

• Founded in 1984
• Private, non-profit
• Multidisciplinary research and education
• Primarily a visiting institution
• Current research focus areas
• Cognitive neuroscience
• Computation in physical and biological systems
• Economic and social interactions
• Evolutionary dynamics
• Network dynamics
• Robustness

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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

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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 ill-defined.

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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.

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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
• The relation to the external world of the man who
  commits the deeds.
• His relation to time.
• His relation to the causes leading to the action.

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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.

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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.

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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

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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
  longer-horizon traders can provide liquidity to
  the short-horizon traders when the latter
  experience a significant event (e.g., crash,
  discontinuity)

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Nonlinear Modeling Techniques

• Neural networks
• Genetic algorithms
• Fuzzy logic
• Others (e.g., mentioned by Shapiro (2000))
• Statistical pattern recognition
• Simulated annealing
• Rule induction
• Case-based reasoning

We will discuss
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Neural Networks

• Artificial intelligence model
• Characteristics
• Pattern recognition / reconstruction ability
• Ability to learn
• Adapts to changing environment
• Resistance to input noise
• Bottom line
• Data is input
• Behavior is output

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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

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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

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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

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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 gt 0,1
• Young (1996, 1997) pricing (WC, health)
• Cummins Derrig (1997) pricing
• Horgby (1998) risk classification (life)

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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

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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 Hitchhikers Guide to the
  Techniques of Adaptive Nonlinear Models,
  Insurance Mathematics Economics, 26 119-132

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For a Copy of the Presentation

• E-mail gorvett_at_uiuc.edu
• Web page http//www.math.uiuc.edu/gorvett/