Cellular Automata for Spatial Data Analysis

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

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 • eachcellcanbewhite (alive) orblack (dead) • for eachcell, theirneighbors are the 8 closercells • ateach time step, thestateofeachcellobeythefollowingrules (executedsimultaneously): • thecellsurvivesifthere are 2 or 3 aliveneighborcells, otherwisethecelldies • a diedcellcanchange to analivecellif it hasexatly 3 aliveneighbors, otherwise it remainsdead

Game ofLife 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/

CA = (G, N, S, IC, R, BC, UC) G: Geometry N: Neighborhood S: States IC: Initialcondition R: Rules BC: Boundaryconditions UC: Updatingcriteria The CA Structure Source: Adapted from Leonardo Santos’ slide

The Grid

The GeometryExample: Two-DimensionalGrids 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

CA = (G, N, S, IC, R, BC, UC) G: Geometry N: Neighborhood S: States IC: Initialcondition R: Rules BC: Boundaryconditions UC: Updatingcriteria The CA Structure Adapted from Leonardo Santos’ slide

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

Moore Neighborhood First neighbors Second neighbors Adaptedfrom 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

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

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

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 time Extrinsic Incubation Period Mosquito infectshumans Momentof infection 8 to 12 days 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.

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

Cellular Automata for Spatial Data Analysis