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Cellular Automata and Geographic Modeling

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Title: Cellular Automata and Geographic Modeling


1
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
2
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

6
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
7
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

8
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)

9
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

10
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

11
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.

12
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.

13
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

14
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.

15
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

16
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
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