Cellular Automata and Geographic Modeling

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Cellular Automata and Geographic Modeling

• Conways Game of Life1
• Mathematician John Horton Conway develops the
  Game of Life in 1967 based on principles
  developed by John von Neumann (see 1966 paper
  Theory of Self-Reproducing Automata by von
  Neumann)
• start with a simple set of organisms (cells)
  guided by a set of genetic laws for birth,
  deaths and survivals based on the immediate
  configuration of each organisms neighborhood
  region
• genetic laws
• No initial pattern for which there is a simple
  proof that a population can grow without limits
• No initial patterns that apparently do grow
  without limits
• There should be simple initial patterns that grow
  and change for a considerable period of time
  before coming to an end in three possible ways
  1) fading away completely (overcrowding or too
  sparse), 2) settling into a stable configuration
  that remains unchanged, 3) entering into an
  oscillating phase.

Martin Gardners Article in Scientific American
223 (October 1970) 120-123
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Conways Game of Life1

• Rules were such as to make behavior unpredictable
  and make the initial condition critical (at
  times)
• Each cell has eight neighbors
• Rules
• Survivals. Every cell with two or three
  neighboring cells survives for the next
  generation
• Deaths. Each cell with four or more neighbors
  dies (removed) from overpopulation. Every
  counter with one neighbor or none dies from
  isolation.
• Births. Each empty cell adjacent to exactly
  three neighbors no more, no feweris a birth
  cell (populated before the next iteration).
• All births and deaths occur simultaneously
  constituting a single generation (iteration).

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Conways Game of Life1

• Run it! game of life\GameOfLife.exe
• Start with a pattern consisting of black cells.
• Locate all cells that will die and delete after
  iteration
• Locate all empty cells where birth occurs and
  populate after iteration complete
• What youll find
• From simple initial conditions and a simple set
  of rules complex patterns emerge
• This game has prompted a whole new nature of
  inquire that involves the emergent behavior of
  complexity from simple localized rules
• The set of systems developed from these
  principles are called Cellular Automata

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Cellular Automata (CA) Applications in Geographic
Analysis

• Early work by Couclelis, 1985 White and Engelen
  1992) in the application of CA to urban modeling.
• We will focus on Clarkes et al. 1997 on
  development and calibration of a CA model for
  urban growth.
• Model premise
• Conversion of natural to artificial cover one of
  the most profound transformation to occur on
  earths surface
• Most models of urbanization focus on social and
  economic patterns and size relationships between
  cities (i.e. CPT)
• Few models have examined the rural to urban
  transition as a physical process
• The diffusion-limited aggregation model (DLA) is
  an exception (Batty and Longley, 1994)
• This model lends itself to cellular automata
  implementations.

Couclelis H., 1985 Cellular worldsa framework
for modeling micro-macro dynamics Environment
and Planning A 17 585-596. White, R. Engelen, G.
1992, Cellular automata and fractal urban form
a cellular modeling approach to the evolution of
urban land use patterns, WP-9264, RIKS,
Maastricht, The Netherlands. Clarke, K.C., L.
Gaydos, S. Hopen. 1997. A self-modifying
cellular automaton model of historical
urbanization in the San Francisco Bay area,
Environment and Planning B Planning and Design
1997, vol. 24, pg. 247-261
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Clarkes (et al.) model

• Rules for CA after White and Engelen (1992)
• reduction of space to a grid or tessellation of
  cells
• Establishment of an initial set of conditions,
  which does not have to be the origin of the
  entire system but can be any spatial arrangement
  of the phenomenon.
• Establishment of a set of transition rules
  between iterations and
• Recursive application of the rules in a sequence
  of iterations for the spatial pattern.
• Implementation
• Determine rules from an existing system or
  knowledge base
• Use historical data to calibrate the transition
  rules
• Predict the future by allowing the model to
  continue to iterate in time with the same rules

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Clarkes (et al.) model (cont.)

• Self-modifying CA
• Rules are allowed to change as system grows or
  changes (essentially a feedback mechanism which
  amplifies or attenuates some parameter)
• Example if all flat urban land is used by
  existing settlements the rules for penalizing
  building on steep slopes may soften (i.e. can
  build on steeper slopes)

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Clarkes (et al.) model (cont.)

• Study area San Francisco Bay area
• Available data
• Extensive growth
• Stresses on natural systems, especially water,
  intense
• Major policy issues
• Diverse landscape (sea level to 2,500 meters) and
  natural to wild conditions
• Data
• Seven raster images maps 1850, 1900, 1940, 1954,
  1962, 1974, 1990
• Temporal interpolation between dates to build
  time-series of change
• Maps before 1974, satellite images after
• Issue of generalization vs pixel(y) satellite data

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Clarkes (et al.) model (cont.)

• Grid size 300 meters
• Initial conditions set by seed cell determined
  by locating and dating founding of various
  settlements identified from historical maps etc.
• Input layers
• Slope
• Exempt areas (water bodies, parks etc.)
• Roads
• Seed layer

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Clarkes (et al.) model (cont.)

• Behavioral rules
• Selecting random locations
• Investigating spatial properties of neighboring
  cells (urban?,slope?distance to road?)
• Urbanize cell or not depending on stochastic
  process
• Factors
• Diffusion factor determines overall
  dispersiveness of distribution both from single
  cell and movement of new settlements outward
  through road systems.
• Breed coefficient determines how likely a newly
  genrated detached settlement is to begin its own
  growth cycle.
• Spread coefficient which controls how much normal
  outward organic expansion takes place within
  the system.
• Slope-resistance factor which influence
  likelihood of settlemtns extending up steeper
  slopes.
• Road-gravity factor which has the effect of
  attracting new settlements onto the existing road
  system if they fall within a given distance of
  the road.

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Clarkes (et al.) model (cont.)

• Growth rate is the sum of four different types of
  urban growth
• Spontaneous urban growth, which occurs when a
  randomly chosen cell falls close enough to an
  urbanized cell, simulating the influence of urban
  areas on their surroundings
• Diffusive growth urbanizes cells which are flat
  enough to be desirable locations for development
  even if no close to established urban areas.
• Organic growth spreads outward from existing
  urban centers, representing tendency of city to
  expand.
• Road-influence growth encourages urbanized cells
  to develop along road networks accessibility
  attracts development.

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Clarkes (et al.) model (cont.)

• Most growth is of the organic kind followed by
  spontaneous
• Statistics related to growth magnitude and type
  are recorded and shown to the user in real time
• Self-modification rules allow much control for
  feedback mechanisms for critical high growth
  rates and critical lo growth rates
• When growth exceeds critical value diffusion,
  spread and breed factors increased by multiplier
  greater than one.
• When growth rate drops to a critical low rate
  diffusion, spread and breed factors increased by
  multiplier less than one.
• Road-gravity factor is increase as size of road
  network increases
• Slope-resistance factor is increased allowing
  urbanization on steeper slopes
• Rules went through extensive calibration phase to
  establish stability

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Clarkes (et al.) model (cont.)

• Calibration
• Comparison made between historical data and model
  output
• Visual comparisons necessary for verifying model
  replicating historical patterns and played key
  role in first phase
• Area, edge and cluster analysis of urban areas
• Visual tools for comparing center of gravity of
  urban centers
• Statistical
• Pearsons r2 for three values urban area number
  of edge pixels number of pixel clusters for
  model and real distributions in key years.

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Clarkes (et al.) model (cont.)

• Calibration (cont)
• Four steps
• Validation
• Vary each parameter and fix others
• 101 separate runs per parameter
• Write GUI tools for visualization and make
  multiple runs
• Batch version of model which calculates
  correlations between the predicted and observed
  data
• Total area converted to urban
• Number of pixels defined as edge as definition of
  urban-rural interface
• Number of separate spreading centers
• Create Monte Carlo averages (100 iterations) to
  analyze mean and variance of outcomes

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Clarkes (et al.) model (cont.)

• Properties and features of model
• Step rules are relatively simple to explain and
  understand
• Model not dependent on generalized probability
  distributions derived from observed or
  hypothetical data but allows each cell to respond
  to its geographic context and condition. This is
  similar to the individual choices that are made
  in the urbanization process.
• Model is conducive to interaction with user
• Monte Carlo average enable by multiple initial
  conditions permitted
• Results can be linked to environmental models
  that require landuse (i.e. heat island analysis,
  run-off models, etc.)
• Spatial impact of model moves from local to
  global influence across space as number of
  iterations increase
• Results of simulation beyond prediction by an
  algorithm
• Self-modification increases range of possible
  outcomes and more closely simulates natural
  process