ENVIRONMENTAL MODELLING PROBABILISTIC MODELS

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ENVIRONMENTAL MODELLINGPROBABILISTIC MODELS

• Dr Claire H. Jarvis, chj2_at_le.ac.uk

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The next three lectures

• Introduction Categories of models
• Focusing on bottom-up models
• Types of bottom up models
• For cellular automata in particular
• Theoretical concerns
• Practical applications
• Advantages disadvantages of cellular automata
  for environmental modelling
• A glimpse beyond cellular automata

• Practical work
• Cellular automata model for land use change
  in Bolivia

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Categories of environmental model
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Categories of environmental model

• Environmental models may be categorised in a
  variety of ways (e.g. Burrough et al 1996,
  Heuvelink 1998, Steyaert 1993).

Complexity of form
Time
Dependent Independent
Physical Complex
conceptual Simple conceptual
Resolution
Cell Block Global
Important axes include stochastic/ deterministic,
inductive/ deductive, bottom-up/ top-down
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Model types
Bottom-up models are particularly, but not
exclusively or necessarily, used for stochastic
modelling. We shall indeed approach
stochasticity, but for now consider the following
categorisation
Top-down models (equations) Bottom-up models
(A-life models)
Cellular automata, a primitive class of A-Life
(artificial life) model, are always developed
from the bottom-up
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Why consider model taxonomies?

• To provide a framework within which you can place
  the different techniques
• Useful to weigh up the strengths and weaknesses
  of the various model types.
• No one model type will be best for all
  applications
• Transfer knowledge between different application
  areas of the environmental sciences (Skidmore
  2002, p8)
• Reminder of the many uncertainties surrounding
  the modelling process
• You might build a model with which you are well
  pleased, but others might approach the subject
  from a different but equally valid perspective

Skidmore, A. (2002) Taxonomy of environmental
models in the spatial sciences, In Environmental
Modelling with GIS and Remote Sensing, Taylor
Francis, London, p8-25.
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Cellular automataEquations vs CA models

• This module deliberately introduces both top-down
  environmental models rooted in equations and
  bottom-up cellular automata models
• This part of the module focuses on the
  possibilities of cellular automata. You need to
  consider what these models are, when these models
  might be most appropriately used and how. Neither
  equations nor artificial life models are a
  universal modellers panacea!
• Viewing geographical problems and models from
  different perspectives is healthy and
  cross-comparisons provide further stimulation of
  research and debate

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Models based on the principles of artificial life
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ARTIFICIAL LIFE MODELS

• Artificial life is the study of how to create
  man-made systems that behave as if they are
  alive.
• Rucker (1993 p5, In Openshaw Openshaw 1997
  p239)
• We can describe artificial life as the study of
  artificially created systems that embody at least
  some of the behaviours of real life
• Prata (1993 p1, In Openshaw Openshaw 1997 p239)

Environmental Modelling GY7202 PROBABILISTIC
MODELS (1)
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Artificial life models

• Examples of this type of model, rooted in
  biological principles, include
• Cellular automata (Behaviour of simple cells)
• Genetic algorithms (Complex swapping of
  alleles)
• Neural networks (Inspired by brain cell
  activity)

These three lectures focus on the simplest
category of these a-life models the cellular
automata
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Potential significance of A-Life technologies

• They may lead to a better understanding of
  natures self-organising laws and the nature of
  life, with enormous philosophical and practical
  consequences
• They may lead to self-replicating factories
• They may carry life to the next level of
  evolution
• A rich source of inspiration for new types of
  research on artificial intelligence
• Creations of artificial life may be useful in
  their own right.

(Prata 1993, In Openshaw Openshaw 1997 p240)
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Cellular automata
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Cellular AutomataHistory (1)

• The THEORY of cellular automata was introduced in
  the late 1940s by John von Neumann and
  Stanislaw Ulam, working at the Los Alamos
  National Laboratory in the United States (von
  Neumann, 1966 Toffoli, 1987)
• PRACTICAL aspects of CAs were developed in the
  late 1960s when John Horton Conway developed the
  Game of Life (Gardner, 1970) and with the
  increased popularity of the computer

Environmental Modelling GY7202 PROBABILISTIC
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Cellular AutomataHistory (2)

• Much interest in CAs has been sparked by the work
  of Wolfram during the 1980s
• The use of artificial life models in the
  geographical sciences more generally can be
  attributed to the work of Chris Langton at the
  Santa Fe Institute

Environmental Modelling GY7202 PROBABILISTIC
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Cellular automataWhat are they? (1)
Informally

• The cellular automaton consists of a line of
  cells, each colored either black or white. At
  every step there is then a definite rule that
  determines the color of a given cell from the
  color of that cell and its immediate left and
  right neighbors on the step before.
• Stephen Wolfram (A New Kind of Science, 2002)

Environmental Modelling GY7202 PROBABILISTIC
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Using various simple rules, Stephen Wolfram
generates a variety of complex images generated
from a simple base
Rule 30
Rule 2
Rule 18

See Wolfram(2002), pp51-114
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Lets work through a simple two-dimensional
example ourselves
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Game of life (Mathematician John Conway)

• A cell becomes live if it is surrounded by three
  life neighbours
• Cell becomes dead if surrounded by more than
  three (overcrowding) or less than two (isolation)
  
• The number of life sites fluctuates over time
  according to a power law until steady state is
  reached

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Starting point
After 10 iterations
Graphics from Engelen, http// )
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Game of life Potential patterns

• The cell patterns resulting from the initial
  configuration of 0/1 have been given all sorts of
  names
• Blinkers, ponds, gliders, shuttling bees, puffer
  trains
• Wolframs (2002) book discusses many of these
  phenomena in more detail
• There are a number of Internet sites where you
  can try creating your own universes
• The purpose of this modelling course is to
  explore build custom-build universes that
  demonstrate geographical purposes

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Cellular automataWhat are they? (2)
More formally
Consider a space divided into cells, where each
cell is repeatedly "updated" to a new state in an
evolving sequence. A program of this nature is
specifically called a cellular automaton when it
is 1) parallel, 2) local, and 3) homogeneous.
Environmental Modelling GY7202 PROBABILISTIC
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Cellular automataWhat are they? (3)

• Parallelism - the individual cell updates are
  performed independently of each other, i.e. all
  of the updates are done at once
• Locality - when a cell is updated, its new value
  is based solely on the old values of the cell and
  of its nearest neighbours.
• (3) Homogeneity - each cell is updated according
  to the same rules.

Environmental Modelling GY7202 PROBABILISTIC
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Cellular automataWhat are they? (4)
Importantly, using a CA, a complex system is not
described using complex equations, but rather
complexity emerges through the interaction of
simple individuals (cells) following simple
rules. A localised rather than centralised
mindset is adopted. There is no global
blueprint but one emerges bottom-up (Openshaw
Openshaw 1997)
Environmental Modelling GY7202 PROBABILISTIC
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Major elements of cellular automata
Cell space The space is composed of individual
cells. Theoretically, these cells may be in any
geometric shape. Cell states The states of
each cell may represent any spatial variable,
e.g. the various types of land use. Time steps
A CA will evolve at a sequence of discrete time
steps. At each step, the cells will be updated
simultaneously based on transition rules.
Transition rules A transition rule normally
specifies the states of cell before and after
updating based on its neighbourhood conditions.
White, R., and G. Engelen, 2000,
High-resolution integrated modeling of the
spatial dynamics of urban and regional systems,
Computer, Environment and Urban Systems
24383-400.
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Major elements of cellular automata (1)Cell space

• Theoretically, cells may be in any geometric
  shape.
• Most CA adopt regular grids to represent such
  space, which make CA very similar to a raster
  GIS.
• Recent work in GIScience by Wu challenges this
  notion, and looks at x spaces

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Major elements of cellular automata (2) Cell
states

• The states of each cell may represent any
  spatial variable.
• In your practical exercises, as for much of the
  geographical literature, the states that you will
  consider will be various types of land use.
• Land use is a discrete state.

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Major elements of cellular automata (3) Time
steps

• As you saw in the Game of Life example, a CA
  will evolve at a sequence of discrete time steps.
  
• At each step, the cells will be updated
  simultaneously based on transition rules.
• These transition rules may alter across time.
• In the special case of the reversible cellular
  automata, at any time-step of the development the
  rules may to go forwards or backwards in time
  without losing any information.

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Major elements of cellular automata (4)
Transition rules

• Establishing transition rules is a key issue
  when building cellular automata
• Transition parameters include
• The neighbourhood
• The nature of the transition

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Transition rules The neighbourhood (1)

• What configuration of neighbourhood?
• The neighbourhood need not be square, but may be
  circular or hexagonal or even triangular
• The Margolus Neighbourhood takes a different
  approach considers 2x2 cells of a lattice at
  once.

Neumann Moore
(Figures from A. Schalten, http//www.ifs.tuwien.a
c.at/aschatt/info/ca/Behav_Univers, Accessed 21
October 2003)
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Transition rules The neighbourhood (2)

• How broad is the neighbourhood?
• Should this extend only to the neighbouring
  cells, or beyond them?
• Should the neighbourhood be symmetrical?

Moore Extended Moore
(Figures from A. Schalten, http//www.ifs.tuwien.a
c.at/aschatt/info/ca/Behav_Univers, Accessed 21
October 2003)
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Transition rules Nature of transition

• Standard rules
• Every group of states of the neighbourhood cells
  is related a state of the core cell
• Totalistic Rules
• The state of the next state core cell is only
  dependent upon the sum of the states of the
  neighbourhood cells and not their individual
  configurations.
• Legal Rules
• Related to totalistic rules, these cover the
  special case where all cells start with the same
  state. Essentially, these rules are the subset of
  totalistic rules that enforce a change of state
  where applied

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Classes of CA
"...many (perhaps all) cellular automata fall
into four basic behaviour classes.", Stephen
Wolfram (Wolfram, 1984) Class 1 Reaches a
unique state from limited points Class 2
Creates repeating patterns Class 3 - Creates
chaotic, a-periodic patterns which re often
self-similar (fractal) in nature Class 4 - More
complex behaviour, irreversible, die after
initially stable period (After A.
Schalten, http//www.ifs.tuwien.ac.at/aschatt/inf
o/ca/Behav_Univers, Accessed 21 October 2003)
Real problems need to hit here
On the edge of chaos (Langton)
Too simple
Too chaotic
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Cellular automataWhat relevance to geography?

• CAs are inherently spatial, incorporating the
  intrinsically spatial concept of the
  neighbourhood.
• Relationships can be seen between CAs and earlier
  work such as Tobler's (1979) cellular geography,
  raster GIS models and Hagerstrand's (1968)
  diffusion models.
• The relevance of CAs to geography depends on the
  extent to which simple rules underlie the
  apparent complexity of geographical systems
  particularly human systems
• CAs may be seen as a simple way of integrating
  time into GIS/spatial data structures.

Environmental Modelling GY7202 PROBABILISTIC
MODELS (1)
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Cellular automataWhy not continue with
equations? (1)

• In many models it is common to refer to the
  derivative of a variable with respect to
  population size N. It assumes that N is large,
  not good when modelling small population effects
• Equations are generally poor at dealing with
  highly non-linear effects such as thresholding or
  if-then-else conditionals, which arise very
  frequently in the description of animal and human
  behaviour
• It would take tens to hundreds of lines of
  equations to express even a simple model of many
  geographical processes as a function of the many
  genetic, memory and environmental variables that
  affect their behaviour

(After Taylor Jefferson, 1996, p5-6)
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Cellular automataWhy not continue with
equations? (2)

• Most differential equations, especially of the
  real world, have no closed form solution
• One of the well known CA-scientists Tommaso
  Toffoli contributed to this problematic (Toffoli,
  1984)
• "... in modeling physics with the traditional
  approach, we start for historical reasons ...
  with mathematical machinery that probably has
  much more than we need, and we have to spend much
  effort disabling these 'advanced features' so
  that we can get our job done in spite of them."

(After A. Schalten, http//www.ifs.tuwien.ac.at/a
schatt/info/ca/Behav_Univers, Accessed 21
October 2003)
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NEXT WEEK

• Practical steps to consider when building a
  cellular automata model
• Applications of cellular automata, with a focus
  on land use change

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READING

• Wolfram, S. (2002) A new kind of science
• Two dimensions and beyond, Chapter 5, pp169-221
• Implications for everyday systems, Chapter 8,
  pp 363-432.
• Resnick, M. (1997) Turtles, Termites, and
  Traffic Jams, MIT Press, pp163.
• Openshaw, S., Openshaw, C. (1997) Artificial
  intelligence in Geography
• Artificial Life, Chapter 8, p236-266