Análise Espacial de Dados Geográficos Autômatos Celulares Disciplina SER 301 Análise Espacial de Dados Geográficos Líliam C. Castro Medeiros lccastro@dpi.inpe.br

Cellular Automata Dynamic and self-reproducing sistems Discrete space and time The basic elements: cells The nth iteration Neumann JV, Burks AW (1966). The Theory of Self-Reproducing Automata, University of Illinois Press, Urbana

Cellular Automata Dynamic and self-reproducing sistems Discrete space and time The basic elements: cells The nth iteration Neumann JV, Burks AW (1966). The Theory of Self-Reproducing Automata, University of Illinois Press, Urbana

Cellular Automata Dynamic and self-reproducing sistems Discrete space and time The basic elements: cells The nth iteration Neumann JV, Burks AW (1966). The Theory of Self-Reproducing Automata, University of Illinois Press, Urbana

Dynamic and self-reproducing sistems Discrete space and time The basic elements: cells The nth iteration Neumann JV, Burks AW (1966). The Theory of Self-Reproducing Automata, University of Illinois Press, Urbana

Dynamic and self-reproducing sistems Discrete space and time The basic elements: cells The nth iteration Neumann JV, Burks AW (1966). The Theory of Self-Reproducing Automata, University of Illinois Press, Urbana

Dynamic and self-reproducing sistems Discrete space and time The basic elements: cells The nth iteration Neumann JV, Burks AW (1966). The Theory of Self-Reproducing Automata, University of Illinois Press, Urbana

Dynamic and self-reproducing sistems Discrete space and time The basic elements: cells The nth iteration Neumann JV, Burks AW (1966). The Theory of Self-Reproducing Automata, University of Illinois Press, Urbana

Each cell contains: A finite set of predeterminated states A set of transition rules (to change the states) which depend on the cell’s neighborhood The nth iteration Neumann JV, Burks AW (1966). The Theory of Self-Reproducing Automata, University of Illinois Press, Urbana

Source: Rita Zorzenon’s slide

The Cellular Automata Desenvolvido pelo matemático húngaro John von Neumann, que na década de 40, propôs um modelo baseado na ideia de sistemas lógicos que fossem auto-reprodutores e que imitassem a própria vida. Cooper NG (1983). From Turing and von Neumann to the present. Los Alamos Science.

An Example: John Conway’s Game of Life a regular grid with square cells

An Example: John Conway’s Game of Life each cell can be white (alive) or black (dead)

An Example: John Conway’s Game of Life each cell can be white (alive) or black (dead) for each cell, their neighbors are the 8 closer cells Figure: Leonardo Santos et al. (2011). A susceptible-infected model for exploring the effects of neighborhood structures on epidemic processes – a segregation analysis. Proceedings XII GEOINFO, November 27-29, 2011, Campos do Jordão, Brazil. p 85-96.

An Example: John Conway’s Game of Life each cell can be white (alive) or black (dead) for each cell, their neighbors are the 8 closer cells at each time step, the state of each cell obey the following rules (executed simultaneously): the cell survives if there are 2 or 3 alive neighbor cells, otherwise the cell dies a died cell can change to an alive cell if it has exatly 3 alive neighbors, otherwise it remains dead

Possible states: alive or dead Game of Life John Conway (1970) Possible states: alive or dead Death: by loneliness - one or zero neighbors by overpopulation – more than 4 neighbors Birth: cells with exactly 3 alive neighbors Survival: exactly 2 or exactly 3 alive neighbors Adapted from Adriana Racco’s slide

Rita Zorzenon’s slide

Game of Life Some sites to see the Game of Life simulation: http://www.math.com/students/wonders/life/life.html or http://www.bitstorm.org/gameoflife/

Source: Adapted from Leonardo Santos’ slide CA = (G, N, S, IC, R, BC, UC) G: Geometry N: Neighborhood S: States IC: Initial condition R: Rules BC: Boundary conditions UC: Updating criteria The CA Structure Source: Adapted from Leonardo Santos’ slide

The Grid

Source: Adapted from Leonardo Santos’ slide CA = (G, N, S, IC, R, BC, UC) G: Geometry N: Neighborhood S: States IC: Initial condition R: Rules BC: Boundary conditions UC: Updating criteria The CA Structure Source: Adapted from Leonardo Santos’ slide

The Geometry Example: Two-Dimensional Grids Cells that have a common edge with the involved are named as “main neighbors” of the cell (are showed with hatching) The set of actual neighbors of the cell a, which can be found according to N, is denoted as N(a) Source: Lev Naumov’ slide

Adapted from Leonardo Santos’ slide CA = (G, N, S, IC, R, BC, UC) G: Geometry N: Neighborhood S: States IC: Initial condition R: Rules BC: Boundary conditions UC: Updating criteria The CA Structure Adapted from Leonardo Santos’ slide

Von Neumann Neighborhood First neighbors Second neighbors Adapted from Adriana Racco’s slide

Adapted from Adriana Racco’s slide Moore Neighborhood First neighbors Second neighbors Adapted from Adriana Racco’s slide

Adapted from Adriana Racco’s slide Random Neighborhood Adapted from Adriana Racco’s slide

Other Neighborhoods The arbitrary neighborhood is determined by the model Examples: Based on people activity-space (Santos et al, 2011) First neighbors Second neighbors Based on data (Aguiar et al, 2003) Adapted from Adriana Racco’s slide

Neighborhoods in Time They can be static: the same neighbors all the time (classical CA) dynamic: the neighbors can change at each time step

when: December, 12th, at 2 p.m.! where: IAI auditorium

Source: Adapted from Leonardo Santos’ slide CA = (G, N, S, IC, R, BC, UC) G: Geometry N: Neighborhood S: States IC: Initial condition R: Rules BC: Boundary conditions UC: Updating criteria The CA Structure Source: Adapted from Leonardo Santos’ slide

Source: Adapted from Leonardo Santos’ slide CA = (G, N, S, IC, R, BC, UC) G: Geometry N: Neighborhood S: States IC: Initial condition R: Rules BC: Boundary conditions UC: Updating criteria The CA Structure Source: Adapted from Leonardo Santos’ slide

Source: Adapted from Leonardo Santos’ slide CA = (G, N, S, IC, R, BC, UC) G: Geometry N: Neighborhood S: States IC: Initial condition R: Rules BC: Boundary conditions UC: Updating criteria The CA Structure Source: Adapted from Leonardo Santos’ slide

Adapted from Adriana Racco’s slide Rules The rules may depend on the state of the own cell neighbor’s cells The rules may be based on influence fields of the geography of the system They may be deterministic or stochastic They can depend only on the actual state of the cells Adapted from Adriana Racco’s slide

Source: Adapted from Leonardo Santos’ slide CA = (G, N, S, IC, R, BC, UC) G: Geometry N: Neighborhood S: States IC: Initial condition R: Rules BC: Boundary conditions UC: Updating criteria The CA Structure Source: Adapted from Leonardo Santos’ slide

Boundary Conditions Periodic (1D - ring or 2D – torus)

Boundary Conditions Periodic (1D - ring or 2D – torus)

Boundary Conditions Periodic (1D - ring or 2D – torus)

Boundary Conditions Periodic (1D - ring or 2D – torus)

Boundary Conditions Periodic (1D - ring or 2D – torus) Reflexive

Boundary Conditions Periodic (1D - ring or 2D – torus) Reflexive Fixed

Boundary Conditions Periodic (1D - ring or 2D – torus) Reflexive Fixed Null (the cells located on the borders have as neighbors only those cells immediately adjacent to them into the grid) Others

Source: Adapted from Leonardo Santos’ slide CA = (G, N, S, IC, R, BC, UC) G: Geometry N: Neighborhood S: States IC: Initial condition R: Rules BC: Boundary conditions UC: Updating criteria The CA Structure Source: Adapted from Leonardo Santos’ slide

Examples of Bidimensional Cellular Automata Models

You can also see this in sites.google.com/site/amazonida/drops/forestfire

Other Example of Cellular Automata Model

Dengue Fever It is a viral disease trasmitted in Brazil mainly by Aedes aegypti mosquito

Stages of Infection In Mosquitoes Susceptible Infected 8 to 12 days Susceptible Infected Extrinsic Incubation Period time Mosquito infects humans Moment of infection Figure: Whitehead SS, Blaney JE, Durbin AP, Murphy BR (2007). Prospects for a dengue virus vaccine. Nature Reviews Microbiology, 5: 518-528.

Dengue Stages In Humans Susceptible Infected Recovered time Intrinsic Incubation Period Contagious Human infects mosquitoes Moment of infection 3 to 14 days Average between 4 and 5 days Average between 4 and 7 days Figure: Whitehead SS, Blaney JE, Durbin AP, Murphy BR (2007). Prospects for a dengue virus vaccine. Nature Reviews Microbiology, 5: 518-528.

There are four distinct serotypes of the virus: Dengue Virus There are four distinct serotypes of the virus: DENV1, DENV2, DENV3 e DENV4

The Model

A multi-level stochastic cellular automata Humans Mosquitoes A multi-level stochastic cellular automata

The Model Humans Mosquitoes

The Model Humans Mosquitoes

The Model Time of infection (days) State Humans Mosquitoes

The Model Humans Mosquitoes Time of infection (days) Age days

Patterns

Model Considerations Human mobility Asymptomatic people Human renewal House infestation Vector density per household Each iteration corresponds to a day Periodic boundary conditions

Simulation in Human Lattice Inicialmente:

Simulation in Mosquito Lattice Inicialmente: Um único humano infectado

Parameters of the Model Human occupation rate Number of humans at each residence Human/vector population radio House infestation rate Daily bite frequency Incubation periods Contagious period Mosquito daily survival probability Contamination probabilities

Other Example of Cellular Automata Model

Source: Leonardo Santos and Suani Pinho

Source: Leonardo Santos and Suani Pinho

Source: Leonardo Santos et al (2009)

Patterns Generated by Cellular Automata Models Rita Zorzenon’s slide

Patterns Generated by Cellular Automata Models Rita Zorzenon’s slide

TerraME www.terrame.org Is a programming environment for spatial dynamical modeling. It supports cellular automata, agent-based models and network models running in 2D cell spaces. It provides an interface to TerraLib geographical database, allowing models direct access to geospatial data. www.terrame.org

www.terrame.org

Other Example chuva chuva chuva N Pico do Itacolomi do Itambé Serra do Lobo N Fonte: (Carneiro, 2006) 72

Cellular Automata WET DRY (soilWater > infCap) ? Fonte: (Carneiro, 2006)

Simulation outcome fonte: Carneiro (2006)