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
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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
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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
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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
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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)
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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
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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
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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
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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
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Introduction to CA Models
Time CA are dynamic models, time is one of the keys vectors of the concept
time
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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?
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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
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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
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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
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1
0,5
0,5
0 0
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-0,5
Ni,s|j,r
Ni,s|j,r
demand, public facility location, public space, …
0 0
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-0,5
Distance (km) -1
Distance (km) -1
(a) Attraction
(b) Repulsion
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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
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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
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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
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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
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SmallUrb|CA Model Structure
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Model Calibration [2]
SmallUrb|CA Model Structure
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Software – SmallUrb|CA
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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
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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
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10 11 12 13 14 15 16 17 18 19 20
Problem kMod
k
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SmallUrb|CA Model Performance
Test Instance #11 Reference Initial Land Use Map
Simulation Final Land Use Map
Final Land Use Map
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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
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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
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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)
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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
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Ongoing Work on CA
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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
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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
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Ongoing Work on CA
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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
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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
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Ongoing Work on CA
Metropolitan Area of Barcelona – Calibration [1] Reference Map 1996
Reference Map 2001
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Ongoing Work on CA
Metropolitan Area of Barcelona – Calibration [2] Reference Map 2001
Simulation Map 2001
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Ongoing Work on CA
Metropolitan Area of Barcelona – Prospective [1] Reference Map 2001
Simulation Map 2011
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Ongoing Work on CA
Metropolitan Area of Barcelona – Prospective [2] Reference Map 2011
Simulation Map 2021
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Ongoing Work on CA
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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
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Computational issues
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Parallel Research Projects
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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
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Model Comparison – JRC/UPC/UC Coll. Agreement MOLAND Architecture
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Parallel Research Projects
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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
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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
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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
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