RIKEN-UMD Data Assimilation Conference 2015 7-9 October 2015@UMD Data assimilation of chaotic Cellular Automata using a particle filter with localization Marimo Ohhigashi (University of Hyogo), RIKEN Shin-ichiro Shima (University of Hyogo, RIKEN, Osaka University) Travis Sluka (University of Maryland) Takemasa Miyoshi (RIKEN)
Self Introduction ● Marimo Ohhigashi – Ph.D student @ University of HYOGO, Japan – Research assistant @RIKEN DA-team ● My interests – Human modeling ● Image Processing – Extracting human behavior from image sequences ● Crowd Simulation / Agent Simulation – Modeling human movements in the city
Outlines ● Problem of DA on Discrete System ● What is CA ● DA methods ● Experiments & Results
Problem of DA on Discrete System ● It's not straightforward to apply data assimilation methods based on the Gaussian error distribution to discrete-state systems ● We explore data assimilation with Cellular Automata which exhibit chaotic dynamics for discrete state variables. – A particle filter approach – To reduce the errors with a limited number of particles, we apply localization for a particle filter
What are Cellular Automata (CA) ● System discrete in time, space, and values ● World that consists of a regular grid of cells ● Each cell has a finite number of state, such as “on” or “off” ● The cells are updated synchronously and the rules for updating are the same throughout all of space and time ● The value of a cell at the next time step depends only on the value of it and its neighbor cells at the current time step Von Neumann neighborhood, Moore neighborhood, Von Neumann neighborhood, radius = 1 radius = 1 radius = 2 We used Moore neighborhood of radius = 1 Figures by Travis Sluka “Data assimilation of discrete-valued chaotic non-linear systems, simple Bayesian approach”
CA : Game of Life ● One of the most well known 2D CA is Conway’s game of life. ● 2 states, on and off, simple rule: – OFF cell: turns on if 3 neighbors on – ON cell: stays on only if it has 2 or 3 neighbors on ● Chaotic behavior and complex patterns emerges ● Game of life is difficult to use for data assimilation, because tends to end up at steady or cyclic state fairly quickly Conway’s Game of Life A “glider gun” In the game of life (Black is on, white is ofg)
CA : Sheep ● A similar CA to those which have been used to simulate predator/prey relationships and the spread of infectious diseases through a population. ● Can be thought of as a heard of sheep that reproduce, move, and eat grass that grows and spreads. ● Results in a chaotic nonlinear dynamic system ● Patterns grow and last longer than Game of Life, suitable to use with developing discrete data assimilation methods. – 3 states ● Empty(black) ● Grass(green) ● Sheep(white)
CA : Sheep Rules ● Sheep CA – 3-state Cellular Automata （ Empty, Grass, Sheep ） – World is discrete in time, space, and values – Value of a cell depends only on Moore neighborhood – Periodic boundary condition Empty Grass Sheep >> Grass >> Empty >> Empty ● no Sheep ● more then 3 sheep ● no Grass ● more than 3 Grass >> Sheep ● more then 2 Sheep Moore neighborhood, ● more then 3 Grass radius = 1 Image from https://dreamsinvitro.wordpress.com/2009/05/24/cellular-automata/
DA methods ● Particle Filter approach ● Create multiple randomized CA state as initial particles ● Calculate posterior probability density function by Bayes' rule Prior prob. Posterior prob. Forecast Likelihood Analysis ● The posterior and prior probabilities are maintained by an ensemble of models of the CA. ● The likelihood values are a function of the observation and forecast values for a given cell as well as the anticipated error rate in the observation data mode of each cell particles + Data Assimilation truth noise observation analysis
Observation model True Obs. noise Obs. mask Obs. value mod(3) + + → on each cells : 0 (empty) : observed : +1 : 1 (grass) : masked : +2 : 2 (sheep)
Resampling methods ● We tested 3 types of resampling methods pf0 Number of pf1 Number of Particles: N Particles: N Resample all cells Resample cells at once one by one ... (straightforward PF) (perfectly localized) ... Number of pf2 Particles: N Resample cells one by one with changing localization range
Resampling method: pf0 Straightforward Particle Filter Particles Number of Weight of the CA state at time t Particles: N can be calculated as ... Calculate likelihood on each CA (particle) Resampling all cells at once with the weight Probability by Roulette Selection Particle
Resampling method: pf1 Originally developed by Travis Sluka @ UMD Perfectly localized PF Particles Number of Particles: N Weight of the state s of cell( i , j ) at time t can be calculated as ... Calculate model probability (modelProb) on each cells Resampling cells one by one with the weight Probability by Roulette Selection Empty Grass Sheep ModelProb STATE (prior prob. mass function)
Resampling method: pf2 proposed new method Easy to change the range of the Number of Particles localization comparable to pf1 Particles: N Weight of the state s of cell( i , j ) at time t can be calculated as Range = 0 Resampling cells one by one with Range = 1 changing localization range by Range = 1 Roulette Selection Range = ２ If Range=0 then, it equivalent with pf1
CellSize 50x50 NumParticles 20 Results : pf0 Spinup 499 ObsErr 0.1 ObsMask none ● assim_cycle_1 (DA@every 1 step) assimCycle 1 & 100 ● assim_cycle_100 (DA@every 100 step) – Both DA doesn't work well because of the number of particle is small Spinup DA Free Run DA@every 100 step DA@every 1 step Observation
CellSize 50x50 NumParticles 20 Results : pf1 Spinup 499 ObsErr 0.1 ObsMask none ● assim_cycle_1 (DA@every 1 step) assimCycle 1 & 100 – DA Works well with full spacial resolution (No obs. mask) ● assim_cycle_100 (DA@every 100 step) – Small errors grow rapidly because of the system is chaotic Spinup DA Free Run DA@every 100 step Observation DA@every 1 step
CellSize 50x50 NumParticles 20 Spinup 499 Results : pf2 ObsErr 0.1 ObsMask none ● assim_cycle_1 (DA@every 1 step) assimCycle 1 & 100 locRange 0 – DA Works well with full spacial resolution (No obs. mask) ● assim_cycle_100 (DA@every 100 step) – Small errors grow rapidly because of the system is chaotic Spinup DA Free Run DA@every 100 step Observation DA@every 1 step
Result : pf2 DA cycle: 50 True Analysis CellSize 50x50 NumParticles 20 Spinup 499 ObsErr 0.1 ObsMask none assimCycle 50 locRange 0 Obs. Difference
Effect of the number of particles NumParticles 2,5,10,20,50,100, The number of particles (N) efgects 200, 500, 1000 Spinup 499 the performance of the DA ObsErr 0.01-0.5 assimCycle 1 locRange 0
Effect of the spacial resolution NumParticles 1000 ● Observation mask decrease Spinup 499 ObsErr 0.01-0.5 the performance of the DA AssimCycle 1 LocRange 0 – Large mask, Decrease the accuracy No obs. Obs all cells Mask rate (FULL MASK) (NO MASK)
Effect of changing localization range NumParticles 5000 ● Changing of the localization range contributes to Spinup 499 reduce the analysis errors, especially in coarse obsErr 0.1 AssimCycle 1, 5, 10 , 50 temporal resolution Mask all cells Obs. all cells DA Every 5 step DA every 1step DA Every 50step DA every 10step
Effect of changing localization range NumParticles 2 - 1000 ● Changing of the localization range contributes to Spinup 499 reduce the analysis errors, especially in coarse obsErr 0.1 AssimCycle 10 temporal resolution Mask 0.2 Mask 0.0 (NO MASK) numParticle Mask 0.6 Mask 0.4
Summary ● We explore data assimilation with Cellular Automata – New method: Resampling on each cells with neighborhood (Localized Partical Filter) – Evaluated DA performance by changing localization range ● Changing of the localization range contributes to reduce the analysis errors with a limited number of particles – Especially in coarse temporal resolution in the observation
Future work: ● Blocked resampling Number of Particles Particles: N Assimilate multiple cells (block) at once Evaluate the performance of DA By changing block size & localization range Execute more detailed statistical analysis ● Estimate the rules of the CA – Currently each ensemble run use perfect model ● Apply to a real problem
Thank you
System model and True value CA state at time t : Time development of the system: ● We used the result of specific single run as a true value – This result shows high activity rate continuously until about 5000 step – Activity rate show how many cell changed in a single time development – Figure on the left side show the activity rate of the true result ● Execute DA from 500step to 5000step
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