Geographic Cellular Automata

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Geographic Cellular Automata

• Maa 123.3570 Geospatial Simulation
• 3rd Session (April 7th)
• Sini Ooperi

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Today - advanced GCA

• probabilistic transition rules
• stochastic input variables in transition rules
  (random component)
• constrained automata
• rules with constrains road network usually grows
  from current nodes onwards, not from single
  fractions
• landscape with constrains obstacles and
  barriers (deep slopes, rivers and lakes)
• domain specific limiting constrains
• memory-based transition rules (current--,
  current-, current, next)
• neural network-based probabilities
• fuzzy input variables in transition rules
  (uncertainty in parameters)

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classic von Neumann or Moore
extended /hierarcical
Output layer / lattice at time step t1
neighborhood
transition rules
Initial configuration, states of the cells at
time step t0

• probability based
• memory-based
• constrained

Model (GCA)
heterogenous space
GIS
Intermediate layer(s), new input variable(s) to
be used in transfer rules

• Input variables can
• change between time steps (dynamic)
• be stochastic (random, Monte Carlo)
• fuzzy (uncertainty in the value)

Input variables (source layers)
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Spatial and Temporal Variation

• Variation can be
• Only spatial, i.e. stable cell values in input
  layers between the time steps
• vegetation zones
• annual average of daily temperatures
• a fixed aggregate index which describes the cell,
  computed from a set of different cell-specific
  indices
• Also temporal, i.e. an array of cell values in
  one or more input layers between the time steps
• daily maximum temperature values
• monthly precipitation values
• analyze the original values to form discrete
  statistical classes of them equal intervals,
  quantiles, mean-standard deviation, maximum
  breaks etc.

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Spatial variation of an input variable (site
attribute)

• Spatial heterogeneity means that the attribute
  values, for example, vegetation type of the cells
  are different but do not change in time
• Input layer (vegetation type) does not change
  between time steps

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Temporal variation of an input variable (site
attribute)

• attribute values of the cells change between time
  steps, for example, the rainfall
• Input for rainfall contains an array of three
  raster layers for the different type of years
• rainfall raster dryYear, averageYear,
  wetYear

dry year
3 input layer for annual rainfall
average year
wet year
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Why does temporal variation matter?

• If the phenomenon is modeled with aggregate or
  typical properties, values of parameters,
• it would smooth out
• the details -gt
• the variation -gt
• dynamics
• we would loose much of the richness actually at
  work within the system and get distorted results
  and inadequate predictions

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How to handle temporal variation in the attribute
values of the input variable?

• Original values Statistical distribution Statist
  ical categories
• discrete view, continuous view,
  categorized view,
• 25 different values values follow N
  min,max 5 different categories

median
Values on the number line Normal Distribution
Quintiles
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When simulating we randomly choose...

• Original values Statistical distribution Statist
  ical categories
• discrete view, continuous view,
  categorized view,
• 25 different values values follow N
  min,max 5 different categories

1.00
0.60-0.70
0.70
0.65
median
0.00
...which value to use ...any value from the
range ...which category to use
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Example Spread of Insect Species

• temporal variation in temperature conditions have
  a direct effect on the population growth and
  active flight take-off of invasive insects
• a set of temperature-driven classes to be used in
  different time steps
• for a summer step 5 classes of equal frequency
• for winter step 3 classes of equal frequency
• initialization load different kind of summers
  and winters to be used during the time interval
  to be simulated

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Temperature-based Classification of 30 Summers

• 5 categories quintiles

Hot
Cold
Cool
Moderate
Warm
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Temperature-based Classification of 30 Winters

• 3 categories

Cold
Moderate
Mild
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Running Resource-Constrained Cellular Automa Model

• Building the time sequence for simulation run
• Summer Options

Hot
Cold
Cool
Moderate
Warm
Winter Options
Cold
Moderate
Mild
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Time Sequence Example
Winters
t1 t2 t3 t4
t5 t6 t7

Summers
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Spatially constrained environment barrier zones

• Spatial barrier zones of different severity
• Unfeasible areas, for example, a city in a valley
  between two mountains cannot spread to the
  slopes if they are too steep due to building
  costs
• Unfavorable areas, an animal species can spread
  through it in some extent, but speed of spread
  changes at the border, it can either increase or
  decrease depending on the species
• Unsuitable areas, for example,
• a city at the coastline cannot spread into the
  sea
• a plant or animal species cannot spread to areas
  where there is no habitat (food, nesting sites,
  suitable climate etc.)
• Landscape barriers are merely fixed and staple
  than temporal and moving -gt they constrain spread
  and movement in every time step
• in general cannot be crossed ( exclusion layer)
  or if can be crossed then with higher costs

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Spatially constrained GCA - SLEUTH model

• spread of urban spatial pattern as a function of
• Slope
• Landcover
• Exclusion layer
• Urban
• Transportation network
• Hillshade
• constrains are implemented by two layers
• exclusion layer (for example, water, swamps,
  etc.)
• slope, slope above 21 cannot be urbanized. Given
  that the local slope (slope (i,j)) is below 22,
  the slope_coefficient determines the weight of
  the probability that the location (i,j) may be
  built upon
• permanent landscape-induced constrains which
  cannot be removed

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How do spatial barriers affect the results of
spread simulations?

• in spread dynamics we'll notice
• asymmetric spread due to barriers, figure b
• spread directed down towards the favorable area,
  into the spread corridor, figure c

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Temporal barriers temporally constrained spread

• time periods when the boundary conditions for the
  dynamics of the phenomenon are not met
• the flow and spread ceases, for example,
• water flow in the watershed due to severe period
  of drought
• spread of an invasive species in a novel
  ecosystem due to
• unfavorable weather conditions for spread
• spread of the urbanization due to strict land
  use policy, for instance, concerning
  neighboring wild life sanctuaries
• are not permanent but happen stochastically or
  with a certain probability -gt
• they constrain spread and movement only
  occasionally
• can last either one or several time steps in the
  simulation

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How do spatio-temporal barriers or gradients
affect the results?

• in spread dynamics we'll notice zonal differences
  in spread
• at primary progression zones the speed of spread
  is faster than in the neighborhood areas
• at secondary progression zones the speed of
  spread is lower compared to primary progression
  zones
• the excluded areas are not available for spread
  -gt
• spatial pattern can contain both permanent
  holes (landscape barriers) or temporal holes
  (unfavorable conditions during the time step)

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• Observed distribution of fire ant, together with
  locations where it is expected to invade (after
  Pimm and Bartell, 1980)

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Spread of muskrat (minkki in Finnisch) from
Prague in five sectors (after Andow et al., 1990)
Range distance as a function of time in five
sectors.
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Probabilistic rules dynamic probabilities

• probabilities are not constant, but depend on the
  states of neighbors -gt probability layer is
  updated between every time step
• for the cell C in state

where denotes C's neighbors
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Transition rules - categories of neighbors

• Spatial neighbors
• vonNeuman
• Moore etc.
• Time neighbors

t-1
t1
t
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Rules with memory component

• the transition rules contains a memory component
  so that the past states and the current states of
  the neighboring cells have an effect on the new
  state of the cell
• the values of the neighbors can be, for example,
  averaged prior to the use in transition rules
• the past and the new state of the central cell
  can be also averaged

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Implementing time neighbors
Source Griffeath Moore (2003) New
Constructions in Cellular Automata p.300
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Applications of memory-based rules,
continuous-valued CA

• to produce smooth wave patterns
• cell values are continuous (float, double) not
  integers like in traditional discrete CA

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Simulation output is a continuous valued grid map
but if you want you can extract isolines and
produce an isopleth map.
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Neural network based transition rules

• capacity to recognize and classify pattern
  through training and learning
• training is done by back-propagation procedure
  which is able to generate optimal weights from a
  set of training data
• thus, the flow of the phenomenon is taught to the
  GCA model with the historical empirical data
• future dynamics is simulated either with the
  original empirical data or the data can be
  modified to take into account some criteria, for
  example, planning objectives
• the system learns, for example, the spread of
  urbanization of New York between 1850 -2000
• the urbanization model is then run to predict the
  future extent of New York in year 2030s with a
  selected planning objectives
• at each iteration the neural networks determine
  the urbanization probability which is subject to
  input of site attributes and weights.

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Urban simulation using neural network
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Fuzzy set based transition rules

• fuzzy logic is used in modeling imprecision,
  vagueness, and uncertainty
• variables consist of partially overlapping fuzzy
  sets, which form qualitative groups of values
  within given ranges of values
• In order to convert crisp numerical variables
  into fuzzy,
• fuzzy sets are fully defined by membership
  functions, which return a membership value (µ)
  within 0,1 for a given crisp object in the
  fuzzy set.

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Membership function

• represents the degree of truth Not
  probability
• For any set X, a membership function on X is any
  function from X to the real unit interval 0, 1.
  
• The value 0 means that x is not a member of the
  fuzzy set the value 1 means that x is fully a
  member of the fuzzy set. The values between 0 and
  1 characterize fuzzy members, which belong to the
  fuzzy set only partially.

almost Normal distribution
trianglular
trapezoidal
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Fuzzy-constrained CA model of mountain pine
beetle infestations
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Part 1. Susceptability sub model

• Four indices indicating susceptability of a tree
  to mountain pine beetle (MPB) attack
• Proportion of logdepole pine trees in the stand
  in which the tree is located
• Distance to the nearest large deciduous stand
• Distance to the nearest tree attacked in the
  previous year
• Size of the tree

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Fuzzy membership functions of the four
susceptibility components
Fuzzy transition zone as a function of the
inverse distance to the nearest neighbor cell in
an adjacent stand Value of the cell in the
transition zone between stands of different
species is defined by
Memberships a) µ(LP) Lodgepole pine b) µ(DS)
Distance to deciduous distance c) µ(AT) Distance
to attack power function d) µ(LT) Large
trees
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Susceptibility of a tree to attack by MPB, µ(Su)

• The four variables for each cell are combined to
  a single value representing the susceptability of
  a tree
• Map for each site indicating the susceptibility
  values µ(Su) between 0 and 1 of each tree to
  MPB attack

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Part 2. CA sub model

• 3 sub-components
• MPB Winter Mortality Component
• Cold winter scenario
• Mild winter scenario
• Output a layer representing the location of MPB
  who has reached adulthood
• MPB Dispersal Component
• Transition rules a successful attack of cell at
  coordinates x,y will be initiated if number of
  adult beetles of cell xy is equal to or greater
  than the function for a given value of µ(Su)

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CA sub model continues

• MPB Attack Component
• Neighborhood 25 x 25 cells represents the area
  of potential local infestations called spot
  growth
• Transition rules describe the function between
  the percentage of trees currently under attack by
  MPB in the neighborhood and the susceptibility
  µ(Su)
• CALIBRATION SIMULATION

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Spatial metrics for analyzing pattern dynamics in
urbanization
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Assignment 2. Model Structure
Source Kirkby M. Limits to modelling in the
Earth and environmental science. In
GeoComputation. 2000. Openshaw S., R. Abrahart
(editors) p.373