Catastrophe Theory Real Time Strategy

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Catastrophe TheoryReal Time Strategy Decision
SupportTuesday 4 December 2001, DSTL

• J Q Smith
• University of Warwick
• Coventry, CV4 7AL

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Contents of Talk

• The Rise Fall of Catastrophe Theory
• The Development of Bayesian Decision Theory
• Theoretical links between Catastrophe Theory
  Decision Theory
• M.U.I.A.
• Non-Linear Forecasting
• An automated Decision Support System
• Links with Game Theory
• Roles for Catastrophe Theory in Real Time
• Strategy Formation

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Catastrophe Theory

• Classification Theorem (1972) Maths
• Elegant local, generic classification of smooth
  families of
• potential functions in high dimensions
  parametered by
• finite low dimensional parameters
• Explanation of Morphogenesis (1972-1980)
  Applications
• Descriptions of dynamic processes
• Examples from natural phenomena (breaking waves)
• biology (heart function)
• finance (stock market)
• psychology (behaviour of drivers)
• sociology (censorship)
• decision-making (prison riots)

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The Cusp Catastrophe
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The Cusp Catastrophe
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Some Essential Features of Catastrophe Models

• Dynamic
• Smooth underlying potential
• Real time
• Discontinuous behaviour arising from conflicting
  objectives/information

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But What Rôle Catastrophe Theory?

• Certainly appears to describe common phenomena
• Appears purely descriptive
• Questions asked in the late 70s early 80s?
• Where is the potential? Why is it smooth?
• Why should we expect a local classification to be
  global?
• Often simple games Li Yorke (1975), Rand (1977)
  do not exhibit catastrophes, but chaos
• Without a background theory it was difficult to
  justify

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Rational Behaviour Bayesian Decision Theory

• Savage (1950), Lindley (1998), De Groot (1970),
  De Finetti
• (1972)
• To be rational choose an act maximising
  expected
• utility
• is a rational potential determining best
  actions .
• Q1. Do standard dynamic decision problems exhibit
• catastrophes?
• Q2. Can we assert that the local classification
  is valid
• globally for wide classes of these
  problems?

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Problems Then

• 1) Utilities usually 1 dimensional and nearly
  always
• concave
• (so objects are e.g. expectations)
• No
  conflict of objectives
• 
  single local global maximum
• 2) Usable forecasting/control models e.g.
  K.Fs were
• second order or Gaussian
• No
  conflict of information
• 
  encoded
• Either of these model descriptions
  Catastrophe
• Theory does not apply
• 
  

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Smith, (1978) Theoretical Thesis

• Subsequent Papers with Harrison Zeeman
• If was bounded and if
• (i) was a mixture or was a mixture
  (Smith, 1979)
• (ii) the prior or likelihood had thick tails
  (Smith, 1979)
• (iii) had a mean/variance link (Harrison
  Smith, 1979)
• then the classification of Catastrophe Theory was
  valid and
• global - often exhibiting cusp or butterfly
  catastrophes
• Problems
• 1) These types of models were hardly every used
• 2) When appropriate, few suitable calculation
  techniques
• were available to make these dynamic and
  real time

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New Developments 1) M.U.I.A.

• Keeney Reiffa (1976), von Winterfeldt Edwards
  (1986),
• French (1989), Clemens (1990), Keeney (1992)
• Developed a practical and operational Bayesian
  Decision
• Analysis
• Utilities need several attributes conflicting
  objectives
• Attributes often utility independent so
• where are bounded in 0,1
  attributes
• Note In most of these problems was not
  dynamic

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New Developments 2) Dynamic MCMC Particle
Filters Control

• Gordon (1993), Shepherd Pitt (1997), Aquila
  West
• (2000), Tanner (1994), Doucet et al (2000),
  Santos Smith
• (2001)
• Military problems (bearings only problem
  Gordon
• missile hit/loss
  trade off Tanner)
• Financial problems (dynamic portfolio choice,
  Aquila
• West 2000,
  Santos Smith 2001)
• Features
• Applications typically have thick tailed
  likelihoods or priors
• Exhibiting catastrophes globally
• Quick new algorithms, coping with intrinsic
  conflict of information

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New Developments 3) Decision Support Systems

• RODOS-DAONEM (1990-2005)
• Decision making through M.U.I.A.
• Dynamic processes of complex non-linear spatial
  time series
• On line accommodation of (sometimes conflicting)
  disparate sources of information
• Real time support

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The Stress between Statistical Models Game
Theory Smith, 1996

• Statistics Game
  Theory
• Use opponents past acts Appeal to
  rationality of
• to predict future opponents
  to predict their
• 
  future acts
• Problems with Statistical Models
• Why should opponent be consistent with past?
  Particularly if we change our own strategy?
• Problems with Game Theory Models
• How can we take account of the rationality of
  opponent when we dont know their objectives,
  information, what they believe?
• Why should we believe our opponent rational?
  Kadane Larkley, 1983

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Reconciliation of Statistics Game Theory,
(Smith 1996)

• Choose to maximise
• where is chosen so as not to contradict
  conclusions
• from rationality
• i.e. make sure is not obviously wrong!
• Link with Catastrophe Models
• Away from current , opponents reaction less
  certain
• increased variance on our prediction of
  opponents response
• larger spread in (see Moffat
  Larkley, 2000)

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Three Basic Models of Conflict

• 1) Conflicting Objectives
• Attributes cannot be simultaneously satisfied
• Cost to self, cost to enemy, public
  response.
• With two attributes
• Normal Factor attribute weights
• Splitting Factor distance between good
  strategies for
• individual
  attributes
• e.g. Either attack or retreat DONT attack in
  limited way

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Three Basic Models of Conflict (Contd)

• 2) Conflict of Information
• Science predicts what should be happening
• Early data observations is not consistent with
  this
• Normal Factor relative belief in outliers
  proneness
• reliability of model
  -v- observations
• (modelled through tail
  distributions of
• likelihood and prior)
• Splitting Factor measure of discrepancy between
• predictions of model
  and data indication
• Note Also mixture models for different
  explanations,
• Draper (1997)

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Three Basic Models of Conflict (Contd)

• 3) Rationality -v- Past Action
• Some, (e.g. current) strategies provoke a
  predictable
• response from adversary
• Other strategies will produce a response which is
• unpredictable
• Normal Factor relative gain of speculative
  strategies/
• increased uncertainty
• Splitting Factor potential gain, potential
  uncertainty

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Why its timely to reconsider Catastrophe Theory

• 1) Its constructive unlike Chaos, evocative
  and easy to appreciate from a technical view
  point
• 2) It is now properly justified e.g. through
  recent development in M.U.I.A. and Bayesian
  non-linear dynamic models
• 3) It is possible to use models which exhibit the
  non-linear dynamics it classifies.
• 4) It is feasible to operationalise within a
  decision support system.

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Some Potential Uses of Catastrophe Theory

• Help to construct interesting scenarios for
  emergency training, that standard models avoid
• Produce evocative diagnostics normal/splitting
  factors for feedback support to real time
  decision-makers
• Classify types of strategies contrast efficacy
  of different types/focus on, classes of
  decision worth checking