Nuno Norte Pinto University of Coimbra, Portugal
António Pais Antunes University of Coimbra, Portugal
Josep Roca Cladera Technical University of Catalonia, Spain
Cellular Automata Modeling
MIT
Boston, MA, USA
July 22nd, 2009
Introduction to CA Models
2
Definition and Historical Timeline
The concept of Cellular Automata (CA) has its origins in the work of von Neumann
and Ulam, two mathematicians that were facing (independently) the problem of
devising mathematical rules for biological systems and evolution
Automata comes from the consideration of theoretical mechanisms capable of
universally process any given code (defined by a set of states) – the Universal Turing
machine
Important dates
1940s – the work of von Neumann and Ulam
1970 – Conway’s Game of Life
1979 – “Cellular Geography”, Waldo Tobler
1980s – Stephen Wolfram’s work on CA (mathematical approach, wide set of
applications)
1985 – dissemination of Geographical CA, Helen Couclelis, Mike Batty, Roger White
1990s, 2000s – Intensive research on Geographical/Urban CA
1
3
Introduction to CA Models
CA and Urban Studies
Waldo Tobler introduced the concept of cellular models to geography
He stated the first law of geography – Everything is related with everything else
but near things are more related than distant things
Source:
Tobler,
1979
Waldo
Tobler,
“Cellular Geography”,
1979
4
Introduction to CA Models
Classic CA Structure
Five components
A
A
A
S
A
D
Cell and Cell Space
Neighborhood
Cell States
A
A
S
A
A
A
A
D
A
A
A
A
Transition Rules
A
A
B
A
A
A
Time
Classic (mathematical) approach
1D (vector), 2D (matrix) cell space
Predefined, continuous cell
neighborhood
Binary cell states
Probabilistic transition rules
2
5
Introduction to CA Models
Classic CA Structure
“…an automaton is a processing mechanism with characteristics that change over
time based on its internal characteristics, rules and external input…” (Benenson and
Torrens, 2004)
Mathematical formulation of a 2D CA
Each cell A (an automaton) is defined by a given state from a finite set of cell states
S and evolves in time according to a set of transition rules T, considering an
external input I
At ← ( S , T )
S = {S1 , S 2 ,..., S N }
T : (S t , I t ) → S t +1
At +1 ← ( S , T )
If we consider the neighborhood R of cell A and the cross influence of every cell
state of every cell in R in the state of A than we have the definition of CA
At ← ( S , T , R)
S = {S1 , S 2 ,..., S N } T : (S t , I t ) → S t +1
Introduction to CA Models
At +1 ← ( S , T , R)
6
Cells and Cell Space
CA models are based on regular cells
obtained from remote sense imagery (pixels)
Easy to get, easy to automatic classify land
use
Standard resolutions: 500×500 m2 up to
25×25 m2
Cells only contain land use information
Increasing image resolution improves land
use classification
Higher resolutions may produce a shift to
vector-based simulation
3
7
Introduction to CA Models
Neighborhood
Very important concept in CA
Implementation for Tobler’s First Law
of Geography
Traditional implementations of the
concept: Von Neumann, Moore
Von
Neumann
Moore
Combined
This relationship is valid for land use:
two land uses can attract (urbanecological) or repulse (urban-industrial)
their location
Spatial interaction occurs not only
because of direct and close by
neighboring but also because of more
distant effects – action-at-a-distance
Introduction to CA Models
8
Cell States
Mathematical approach
Binary: 1 (occupied), 0 (unoccupied)
Geographical approach
Cell State <=> Land Use
More or less disaggregated set of cell states
Dominant land uses (urban, commercial,
industrial, agriculture,…) are typical aggregated
cell states
Disaggregated subdivisions allow more detail:
residential low/medium/high density, public
facilities, retail, logistics, scrubs, forest,
permanent agriculture, seasonal crops,…
Issue with homogeneity
4
9
Introduction to CA Models
Transition Rules
CA’s engine
Can be purely probabilistic
Can derive from declared transition rules
Can derive from more complex measures of transition that integrate different
observed components
Take into account regulatory planning that can be considered as restrictions and
land suitabilities to (stochastic) growth
A
A
A
S
A
A
A
D
A
S
Survival
A
A
A
D
A
Death
A
A
A
A
B
A
A
A
A
Birth
Dead
10
Introduction to CA Models
Time
CA are dynamic models, time is one of the keys vectors of the concept
time
5
11
Introduction to CA Models
Relaxations
CA simplicity allows the generation
of complex patterns from simple rules
Applications to geography implied
a series of convenient relaxations
It is arguable if these relaxations
produce models classified as CA
Results on cellular based models
(not CA)
Can we really classify actual
models as CA?
12
SmallUrb|CA Model Structure
Main Components
Approach – Constrained CA model with land use demand based on population
density calibrated by an optimization procedure (Particle Swarm)
Five major CA components
Cell and Cell Structure
Neighborhood
Cell States
T : (St , It ) → St +1
Transition Rules
Time
time
6
13
SmallUrb|CA Model Structure
Cell and Cell Space
Irregular cells drawn from census blocks are (more) representative of urban form
They contain structured demographic and socioeconomic information
Easy to classify their land use
Conjugation of three important issues: information reliability, land use, and urban
form
14
SmallUrb|CA Model Structure
Neighborhood
It is a calibration parameter for the model
Neighborhood effect defined as a measure of the interaction between land uses at
two locations that decreases linearly until zero with the distance between them
Normalized measure: 1 if there is total attraction; 0 if there is no relationship; -1 if
there is total repulsion
Simplification of a very complex relationship of interdependent factors: housing
1
1
0,5
0,5
0
0
1
2
3
4
5
6
-0,5
Ni,s|j,r
Ni,s|j,r
demand, public facility location, public space, …
0
0
1
2
3
4
5
6
-0,5
Distance (km)
-1
Distance (km)
-1
(a) Attraction
(b) Repulsion
7
15
SmallUrb|CA Model Structure
Cell States
Aggregated set of cell states, close to simple definition of CA’s concept
Set of 6 cell states
Urban Low Density (UL)
Urban High Density (UH)
Industrial (I)
Non-urbanized urban areas (XU)
Non-urbanized industrial areas (XI)
Restricted areas (R)
State transition period of 5 years
Cells classified by their possible population density, close to urban regulations
Urban land uses integrate not only residential land but also network infrastructures,
public facilities, and public spaces
16
SmallUrb|CA Model Structure
Transition Rules
State transition occurs following the variation of the transition potential for each cell
at each time step, that takes into account three components
Accessibility
A *i = α A × Di ,C + β A × Di ,V + γ A × Di , I , ∀i ∈ C
Ai = 1 −
Ai*
∑
i∈C
, ∀i ∈ C
Ai*
Land Use Suitability – binary variable (admissible 1, non-admissible 0)
Neighborhood effect


d i, j
 N i ,s| j ,r = 1 − max

 d s ,r

0 ; otherwise
Ni,s =
∑N
j∈Vi
Transition Potential
i ,s j ,r

 × N max , ∀ i, j ∈ C , s, r ∈ S ; if d ≤ d max
s ,r
ij
s,r


, ∀i ∈ C, Vi = {j ∈ C : d ij ≤ δ }, s, r ∈ S
Pi ,s = (ν P × S i ,s + χ P × Ai + θ P × N i ,s )× ξ , ∀i ∈ C, s ∈ S
8
17
SmallUrb|CA Model Structure
Model Fitness
Approach – Using contingency matrixes for comparing two categorical maps –
reference LU map in final year and simulated LU map for final year
Measure – a modified kappa index (kmod), to reduce the distortion induced by nonchangeable cells: only cells that are able to change state (all but the state R) were
accounted
Modeled map
1
2
Reference map
1
2
m11
m12
m21
m22
kmod
…
j
…
…
…
…
…
s
m1s
m2s
…
…
…
…
…
…
…
…
…
i
…
…
…
mij
…
…
Agreement
< 0.00
poor
0.00 - 0.20
very week
0.21 - 0.40
week
0.41 - 0.60
moderate
…
…
…
…
…
…
…
0.61 - 0.80
substantial
s
ms1
ms2
…
…
…
mss
0.81 - 1.00
perfect
Contingency Matrix, s is the total number of cell states
k mod


n ∑ mii − ∑  ∑ mij × ∑ m ji 
i∈S*
i∈S*  j∈S*
j∈S*
 , S* = S /{R}
=


n 2 − ∑  ∑ mij × ∑ m ji 
i∈S*  j∈S*
j∈S*

SmallUrb|CA Model Structure
18
Model Calibration [1]
The high number of calibration parameters (48) and
The strong interdependence between the modeled phenomena suggested the
consideration of an efficient calibration procedure => Optimization
Particle Swarm (PS) technique
PS consists on having a swarm of n particles – each particle is a set of CA
calibration parameters, a point of the space of solutions – to fly during j iterations
Given the particle’s and the swarm leader’s records of position and velocity, the
algorithm converges to an optimal set of calibration parameters
Computational intensive
Objective function – to maximize the value of kmod
9
SmallUrb|CA Model Structure
19
Model Calibration [2]
SmallUrb|CA Model Structure
20
Software – SmallUrb|CA
10
21
SmallUrb|CA Model Performance
Test Instances
Set of 20 test instances
Initial Land Use Map
Final Land Use Map
20 test instances were randomly generated to
simulate plausible spatial structures
#8
Several features were taken into account:
problem size, number of cells, population
densities
1.000
#14
0.950
0.900
0.850
0.800
0.750
0.700
#19
0.650
0.600
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20
Problem
kMod
k
22
SmallUrb|CA Model Performance
Test Instance #11
Reference
Initial Land Use Map
Simulation
Final Land Use Map
Final Land Use Map
11
23
Ongoing Work on CA
Conceptual Issues [1.1]
Multi-scale approach
A regional scale of analysis with a CA model oriented for assessing aggregate
measures of land use demand considering population, employment, and flows data
A local scale of analysis with a CA model oriented for the allocation of land use
considering local problems separately
Issues regarding local scale problem identification
Traditional Approach
Proposed Approach
Local-scale CA, One regional problem
Macro-scale CA, One regional problem
Land use
demand
Local-scale CA, Local problems
24
Ongoing Work on CA
Conceptual Issues [1.2]
Multi-scale approach
MacroScale CA
t0
t0+5
Planning Scale
Time step t: ∆P;∆E;∆APla∆LU
t0+10
Dyn∆LU Input
Pl
a
∆L
U
Inp
ut
MultiScale CA
t
t
t0+1
t0
t0+5
MicroScale CA
t0+10
LU Dynamics
Time step t: Pla∆LULU Allocation
12
25
Ongoing Work on CA
Conceptual Issues [2]
Cells
A regional scale of analysis with a CA model oriented for assessing aggregate
measures of land use demand considering population, employment, and flows data
A local scale of analysis with a CA model oriented for the allocation of land use
considering local problems separately
The use of urban (or municipal) form and reliable data is a major strength of the
approach
Macro-scale cells – municipalities are representative of regional spatial interactions
Micro-scale cells – census blocks or smaller units closer to the urban form
Issues regarding homogeneity
Research on the possibility of using cell division (particularly at the local scale)
26
Ongoing Work on CA
Conceptual Issues [3.1]
Neighborhood
Variable neighborhoods are more representative of real world conditions
Regional neighborhood should be a mixture of local spatial interaction and long
range functional relationships
Importance of considering natural constraints
ti
ti+m
ti+p
13
Ongoing Work on CA
27
Conceptual Issues [3.2]
Neighborhood
Local scale neighborhood must be closer to the urban concept of neighborhood
Importance of considering natural and built constraints
Ongoing Work on CA
28
Conceptual Issues [4]
Transition rules and cell states
Use of aggregate indicators for population,
employment, and commuting flows at a regional
scale of analysis
Macro scale cell states are defined by an
aggregate or disaggregate degree of urbanization
Local scale transition rules must reflect not only
urban dynamics but also main planning
regulations
Local scale cell states can be classified at a
disaggregate level
Further research on the concept of urban
transition potential
Issues regarding cell homogeneity
14
Ongoing Work on CA
29
Conceptual Issues [5]
Land suitability
Regional scale uses general environmental and morphological conditions to assess
aggregate land suitability at a municipal scale
Local scale analysis implies the development of a robust set of land suitabilities
indicators, external to CA (working as an input)
Accessibility
Accessibility should be measured at a both scales considering an external multimodal accessibility model (working as an input)
Aggregate measures of accessibility at a regional level
Possible use of agent-based simulation at a local level
Ongoing Work on CA
30
Conceptual Issues [6]
Enhancing simulation capabilities
Models are criticized for being unable to understand anything that takes place
outside historical trends
Historical trends face ruptures that are the result of unique, time located, decisions
Examples: major urban renewal projects, Brownfield redevelopments, Olympic
Games, etc.
Critical issue to seduce planners into applying models
It also contributes to bringing models and their components (cell, neighborhood,
etc.) closer to urban reality
It is possible to introduce modeling parameters to try to understand and model
decision making stochasticity
15
31
Ongoing Work on CA
Metropolitan Area of Barcelona – Calibration [1]
Reference Map 1996
Reference Map 2001
32
Ongoing Work on CA
Metropolitan Area of Barcelona – Calibration [2]
Reference Map 2001
Simulation Map 2001
16
33
Ongoing Work on CA
Metropolitan Area of Barcelona – Prospective [1]
Reference Map 2001
Simulation Map 2011
34
Ongoing Work on CA
Metropolitan Area of Barcelona – Prospective [2]
Reference Map 2011
Simulation Map 2021
17
Ongoing Work on CA
35
Computational issues
Although processing speed is not determinant in this type of simulation, average
processing times (around 18 hours) must be significantly improved
Two solutions: parallel processing and throughput (distributed) computing
Parallel processing – more powerful solution
Possible use of Marenostrum (Barcelona Super Computing) at UPC
This solution is not suited for the problem at hand (at least at this stage)
Throughput computing – more economical solution
A pool of a maximum of 42 machines was assembled to work with the Condor
software in a Windows environment
Promising results for processing times and capacity with a low cost
Ongoing Work on CA
36
Computational issues
18
Parallel Research Projects
37
Model Comparison – JRC/UPC/UC Coll. Agreement
Sharing of historic and updated data on land use, demography and socioeconomic indicators
Comparative application of Urb|CA and MOLAND to a series of urban areas in
Europe
The Metropolitan Area of Barcelona, Spain
The Metropolitan Area of Porto, Portugal
The Spanish Mediterranean Coast
The Algarve, Portugal
Other areas of interest
Joint application to FP7+/National research projects
Innovative perspective of model comparison
Parallel Research Projects
38
Model Comparison – JRC/UPC/UC Coll. Agreement
MOLAND Architecture
19
Parallel Research Projects
39
Spanish Mediterranean Coast
Celebrating the 50th anniversary of the first Land Law in Spain (1956) and
following a major revision in 2006, we are assessing urban sprawl on the 50km offcoast area along the Mediterranean Coast
The project is using 4 datasets in time
The Marshall Plan aerial photography database (1950s)
The Corine Land Cover datasets for 1990 and 2000
SPOT 5 photography (2006)
First test area on Catalonia
Some difficulties on dealing with black and white photography from 1956, lack of
geographic reference marks
Problems regarding the size of the study area
Parallel Research Projects
40
COST TU0602 – Land Management for Urban Dynamics
Modular simulation package capable of simulate different contexts, considering
different datasets and different calibration procedures
Focus on policy testing (soil consumption, Brownfield regeneration, sprawl
assessment, LU/Transport interaction, urbanization costs)
Importance of having assessment tools to evaluate urban change, able of
producing reliable information for generating future plausible scenarios
Comparison of different urban growth contexts under different regulation
frameworks
20
41
Parallel Research Projects
MPP SOTUR
Possibility of loosely coupled modeling with other modeling approaches – MAS,
UrbanSim/OPUS
Interoperability for the modeling platform
Java programming for integration with MAS programming – Geographic
Automata Systems
GIS-based programming for producing geo-information in standard file systems
(shapefiles, metadata generation)
Simplified data access (through data infrastructure protocols)
Comparative case study application
M. C. Escher – segment of Metamorphosis III
Nuno Norte Pinto University of Coimbra, Portugal
António Pais Antunes University of Coimbra, Portugal
Josep Roca Cladera Technical University of Catalonia, Spain
Cellular Automata Modeling
MIT
Boston, MA, USA
July 22nd, 2009
21