Geospatial Simulation

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Geospatial Simulation

• Maa 123.3570
• March 17th April 28th

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"Give me space and motionand I will give you the
world"

• R. Descartes

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Cellular Automata - A Discrete Universe

• Sini Ooperi
• Maa 123.3570
• Geospatial Simulation

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Definition

• "A cellular automaton is a collection of
  "colored" cells on a grid of specified shape that
  evolves through a number of discrete time steps
  according to a set of rules based on the states
  of neighboring cells. The rules are then applied
  iteratively for as many time steps as desired.
• Cellular Automaton -- from Wolfram MathWorld

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CA consists of four components

• cs cellar space, i.e. the cells
• n neighborhood template
• ss finite set of automaton states
• tr transition rules

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Cellular Environments

• Regular tesselations
• raster, matrix, grid
• consists of pixels
• resolution
• shape of cells square, triangle, hexagon etc.
• 3-d pixelvoxel (volumetric pixel)

Square http//www.alife.pl/portal/ca/e/index.html
Triangle and hexagon http//www.cse.sc.edu/7Eba
ys/CAhomePage
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Benefits of Regular Tesselations

• use of map algebra (square cells)
• weighting of neighbors is basically
  straight-forward
• 1) von Neumann 4 edge neighbors of equal
  edge length, each weighted 1/4 0.25
• 2) Moore 4 edge neighbors, 4 corner neighbors,
• weighting scheme, for example
• edge neighbors 1.00
• corner neighbors 0.75
• 3) Honeycomb 6 edge neighbors of equal edge
  lengths, each weighted 1/6 0.166
• honeycomb a wax structure consisting of rows of
  six-sided cells where bees store their honey

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Cellular Neighborhoods 1. von Neumann and Moore

• von Neumann
• a cell has four neighbors
• Moore
• a cell has eight neighbors

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Cellular Environments irregular grids

• Voronoi diagrams
• Triangulations
• Cadastral maps (boundaries and ownership of land
  parcels)
• in some cases more realistic
• simulation of animal movement where boundary
  lengths have effect on movement behavior
• Source of Upper Picture Peuquet D.J. 2002.
  Representation of Space and Time p.250
• Source of Lower Picture McCullagh M.J Ross
  G.C. 1980. Delaunay Triangulation of a Random
  Data Set for Isarithmic Mapping p.95, The
  Cartographic Journal 17(2), 93-99.

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Challenges with Irregular Grid

• more care in design of workable data structure
• operations between layers are more complicated to
  perform than in a regular grid environment
• design of a realistic weighting scheme for the
  neighbors

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More Neighborhood Templates Extended
neighborhoods

• all neighborhoods that are larger than the
  classical 5 or 9 cells

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Hierarchical Neighborhoods

• Two neighborhood categories
• 1)
• 2)

5
6
3
4
1
2
7
8
9
10 12
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Asymmetric Neighborhoods

• each polygon is related to varying number of
  adjacent polygons

Source Benenson I. and Torrens P.M. 2004.
Geosimulation p.193
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Asymmetric Neighborhoods

• weighting schemes
• common boundary with the home cell
• common boundary with the home cell and the
  perimeter or area of the neighboring cell
• common boundary with the home cell and the area
  of the neighbor inside some specific spatial
  window

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Transition Rules

• define how the next state (value) of the central
  cell is calculated from the previous states,
• state(previous)-gtstate(next)
• two most common options
• 1. Additive rule
• 1.1. Totalistic
• States of the neighbors and the central cell
  are all added
• 1.2. Outer-Totalistic
• Only states of the neighboring cells are
  added
• 2. Multiplicative rule
• 2.1. Totalistic
• States of the neighbors and the central cell
• are all multiplied
• 2.2. Outer-Totalistic
• Only states of the neighboring cells are
  multiplied

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Totalistic Rule (T)

• the states (values) of the central cell and its
  neighbors are included into the sum or product
  when calculating the next state.

northern state
northern state
nw state
ne state
home state
eastern state
western state
home state
western state
eastern state
southern state
se state
sw state
southern state
vonNeumann Neighborhood
Moore neighborhood
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Outer-Totalistic (OT)

• Only the states of the neighbors are included
  into the sum or product when calculating the next
  state. (home cell is left out)

northern state
NW state
NE state
northern state
western state
western state
eastern state
eastern state
southern state
SW state
SE state
southern state
vonNeumann Neighborhood
Moore Neighborhood
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Summing-up table
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Analog to FocalSum or FocalProduct

• FocalProduct specifies that each location's
  NEWLAYER value should indicate the multiplicative
  product of the FIRSTLAYER values at all locations
  within its neighborhood.
• FocalSum specifies that each location's NEWLAYER
  value should indicate the sum of the FIRSTLAYER
  values at all locations within its neighborhood.
• Source Tomlin C.D. 1990. Geographic Information
  Systems and Cartographic Modeling.p.230-231

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Combinations within Rules

• threshold rules
• exact rules
• threshold exact rules

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Example of Transition Rule Outer Totalistic
Rule Game of Life

• public byte output(Neighbor neighbor)
• int sum (int)(neighbor.northState
  neighbor.eastState
  neighbor.southState neighbor.westState
  neighbor.neState neighbor.seState
  neighbor.swState neighbor.nwState)
  
• if (neighbor.homeState 0 )
• if (sum3) return (byte) 1
  else return (byte) 0 else
  if (neighbor.homeState 1)
  if (sum2 or sum3) return (byte) 1
  else if (sum1 or sumgt4) return (byte) 0
  return (byte) 0

Eight neighbor states are summed up.
State of the home cell plays a vital role at this
step when the result of the sum is included into
the algorithm
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Example of Transition Rule Totalistic -Rule

• public byte output(Neighbor neighbor)
• int sum (int)(neighbor.homeState
  neighbor.northState
• neighbor.eastState neighbor.southState
• neighbor.westState neighbor.neState
• neighbor.seState neighbor.swState
• neighbor.nwState)
• if (sumgt4 and sumlt8) return (byte) 1
  else return (byte) 0

All nine cells states are summed up.
State of the central cell has been taken into
account in sum total. It doesn't play any further
role. Compare with outer-totalistic rule.
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So all you have to remember is that

• if the home cell is included into the sum or
  product with the neighbors we have
• Totalistic Rule
• in all other cases we have
• Outer-Totalistic Rule

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Mathematical Notation-Additive Rule

• VonNeumann Neighforhood
• 1.Totalistic Rule
• 2.Outer-Totalistic Rule

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Mathematical Notation-Additive Rule

• Moore Neighborhood
• 1.Totalistic Rule
• 2.Outer-Totalistic Rule

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General Form of Transitional Rule

• defines the dynamics of the output of a discrete
  time autonomous cell, defined by a weighted
  summation of all neighborhood cell outputs at the
  previous time step.
• The neighborhood is represented by the set N and
  a unique index k is chosen to identify the
  neighboring cell in a particular neighborhood
• (http//cnn.com.au/)

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Game of Life - rule

• Rule1 Survival a live cell with exactly two or
  three neighbors stays alive.
• Rule2 Birth a dead cell with exactly three
  live neighbors becomes alive.
• Rule3 Death- a cell dies due to 'loneliness' if
  it has only one neighbor or due to 'crowding' if
  it has more than four neighbors.

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Rules can be global or partitioned
Spatial Partitioning (two sub areas)
step 1 step 1
A
B
Rule One Rule Two

step 1 step 2
Temporal partitioning (two phases)
A
B
Rule One Rule One
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Update can be synchronized or unsynchronized

• Cells can be updated
• all at the same time synchronously
• in sequential order asynchronously
• Block of cells can be updated
• all at the same time synchronously
• in sequential order asynchronously

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Torus can be open or closed

• Close torus, finite space
• in geographic automata

• Open torus, infinite space
• when one goes off the top, one comes in at the
  corresponding position on the bottom, and when
  one goes off the left, one comes in on the right

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Basic Elements in All Cellular Automata Systems

• initial configuration (user-defined or random)
• declaration of neighborhood (von Neumann, Moore,
  own)
• transition rules (additive or multiplicative)
• update method (synchronized or unsynchronized)
• potential partitioning options
• torus (closed or open)
• in user interfaces of CA software
• step, advance the simulation by one time step
• start, start the simulation, update the pattern
  at specified interval (milliseconds)
• stop, stop the simulation from running

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Where to focus our attention when simulating?

• 1. Existential changes of individual cells or
  patterns
• appearing (birth)
• disappearing (death)
• 2. Changes of spatial properties of individual
  patterns
• location
• size
• shape
• 3. Structure emerging from the patterns
• level of saturation/percolation of the area
• 4. Temporal behaviors of cell attribute values

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Temporal Behaviors

• 1. Temporal variation of attribute values
• in a particular place (cell, block of cells)
• in one area compared to the other areas
• 2. General dynamics of attribute values over the
  selected area within a specified time period
• statistical descriptors spatial averages,
  standard deviations etc.
• 3. Locations with behaviors having specific
  features like
• 1. exceptionally high, low values
• 2. exceptionally wide fluctuations of values
• 3. continuous increase or decrease of values
  during
• a given time period
• 4. Identification of spatial clusters of similar
  behaviors

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Literature

• Benenson I, Torrens P.M. 2004. GeoSimulation-autom
  ata-based
• modeling of urban phenomena (287pp.)
• DeMers M.N. 2002. GIS Modeling in Raster.(203pp.)
• Ilachinski A. 2002.Cellular Automata-A Discrete
  Universe.p.117(808pp.)
• MacEachren A.M. How Maps Work- representation,
  visualisation, and Design chapters 8 and
  9(513pp.)
• New Constructions in Cellular Automata. 2003.
  Griffeath D., Moore C. (editors).(340pp.)
• Peuquet D.J. 2002. Representation of Space and
  Time (380pp.)
• Tomlin D.C. 1990. Geographic Information Systems
  and Cartographic Modeling.(249pp.)
• Toffoli T., Margolus N. 1985. Cellular Automata
  Machines. (259pp.)
• Wolfram S. 2002. A New Kind of Science chapter
  5 p.169-221.(1196pp.)

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Next time geographic automata (GA)

• heterogeneous environment
• constrained space obstacles
• advanced rules (capacity, potentiality of cells,
  probabilistic transition rules, memory-based
  rules etc.)
• urban automata applications
• Hands-on geographic automata