Data assimilation of chaotic Cellular Automata using a particle - - PowerPoint PPT Presentation

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) RIKEN-UMD Data Assimilation Conference 2015 7-9 October 2015@UMD

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

Moore neighborhood, radius = 1 Von Neumann neighborhood, radius = 1 Von Neumann neighborhood, 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 (Black is on, white is ofg) A “glider gun” In the game of life

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

• no Sheep
• more than 3 Grass

>> Sheep

• more then 2 Sheep
• more then 3 Grass

>> Empty

• more then 3 sheep

>> Empty

• no Grass

Image from https://dreamsinvitro.wordpress.com/2009/05/24/cellular-automata/

Moore neighborhood, radius = 1

DA methods

• Particle Filter approach
• Create multiple randomized CA state as initial particles
• Calculate posterior probability density function by Bayes' rule
• 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

Posterior prob. Analysis Prior prob. Likelihood Forecast + truth noise

• bservation

particles analysis Data Assimilation

mode of each cell

Observation model

True

• Obs. noise

+ →

: +1 : +2 : 0 (empty) : 1 (grass) : 2 (sheep)

• Obs. value
• Obs. mask

+

: observed : masked mod(3)

• n each cells

Resampling methods

• We tested 3 types of resampling methods

Number of Particles: N

pf0 ...

Number of Particles: N

... pf1 pf2

Number of Particles: N Resample all cells at once (straightforward PF) Resample cells

• ne by one

(perfectly localized) Resample cells one by one with changing localization range

Resampling method: pf0

Number of Particles: N

Straightforward Particle Filter Weight of the CA state at time t can be calculated as Resampling all cells at once with the weight by Roulette Selection

Particles ...

Particle Probability

Calculate likelihood on each CA (particle)

Resampling method: pf1

Number of Particles: N

Empty Grass Sheep

STATE Probability

...

Calculate model probability (modelProb) on each cells

Weight of the state s of cell(i,j) at time t can be calculated as Resampling cells one by one with the weight by Roulette Selection

Particles

ModelProb (prior prob. mass function) Originally developed by Travis Sluka @ UMD

Perfectly localized PF

Resampling method: pf2

Weight of the state s of cell(i,j) at time t can be calculated as Resampling cells one by one with changing localization range by Roulette Selection

Particles

Number of Particles: N Range = 1 Range = 1 Range = ２ Range = 0

Easy to change the range of the localization comparable to pf1 If Range=0 then, it equivalent with pf1

proposed new method

Results : pf0

• assim_cycle_1 (DA@every 1 step)
• assim_cycle_100 (DA@every 100 step)

– Both DA doesn't work well because of the number of

particle is small

Spinup DA DA@every 1 step Observation DA@every 100 step Free Run CellSize 50x50 NumParticles 20 Spinup 499 ObsErr 0.1 ObsMask none assimCycle 1 & 100

Results : pf1

• assim_cycle_1 (DA@every 1 step)

– 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 DA@every 1 step Observation DA@every 100 step Free Run CellSize 50x50 NumParticles 20 Spinup 499 ObsErr 0.1 ObsMask none assimCycle 1 & 100

Results : pf2

• assim_cycle_1 (DA@every 1 step)

– 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

DA@every 1 step Observation DA@every 100 step Free Run Spinup DA CellSize 50x50 NumParticles 20 Spinup 499 ObsErr 0.1 ObsMask none assimCycle 1 & 100 locRange 0

DA cycle: 50

Result : pf2

True Analysis Difference Obs.

CellSize 50x50 NumParticles 20 Spinup 499 ObsErr 0.1 ObsMask none assimCycle 50 locRange 0

Effect of the number of particles

The number of particles (N) efgects the performance of the DA

NumParticles 2,5,10,20,50,100, 200, 500, 1000 Spinup 499 ObsErr 0.01-0.5 assimCycle 1 locRange 0

Effect of the spacial resolution

• Observation mask decrease

the performance of the DA

– Large mask, Decrease the accuracy

NumParticles 1000 Spinup 499 ObsErr 0.01-0.5 AssimCycle 1 LocRange 0 Obs all cells (NO MASK) No obs. (FULL MASK) Mask rate

Effect of changing localization range

• Changing of the localization range contributes to

reduce the analysis errors, especially in coarse temporal resolution

NumParticles 5000 Spinup 499

• bsErr 0.1

AssimCycle 1, 5, 10 , 50 DA Every 50step DA every 10step DA every 1step DA Every 5 step

• Obs. all cells

Mask all cells

Mask 0.0 (NO MASK) Mask 0.2 Mask 0.4 Mask 0.6

numParticle

• Changing of the localization range contributes to

reduce the analysis errors, especially in coarse temporal resolution

Effect of changing localization range

NumParticles 2 - 1000 Spinup 499

• bsErr 0.1

AssimCycle 10

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
• Estimate the rules of the CA

– Currently each ensemble run use perfect model

• Apply to a real problem

Particles

Number of Particles: N

Assimilate multiple cells (block) at once Evaluate the performance of DA By changing block size & localization range Execute more detailed statistical analysis

Thank you

System model and True value

• 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

CA state at time t : Time development of the system: