Title: Catastrophe Theory Real Time Strategy
1Catastrophe TheoryReal Time Strategy Decision
SupportTuesday 4 December 2001, DSTL
- J Q Smith
- University of Warwick
- Coventry, CV4 7AL
2Contents 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
3Catastrophe 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)
4The Cusp Catastrophe
5The Cusp Catastrophe
6Some Essential Features of Catastrophe Models
- Dynamic
- Smooth underlying potential
- Real time
- Discontinuous behaviour arising from conflicting
objectives/information
7But 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
8Rational 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?
9Problems 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
-
10Smith, (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
11New 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
12New 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
13New 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
14The 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
15Reconciliation 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)
16Three 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
17Three 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)
18Three 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
19Why 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.
20Some 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