Brief Introduction to the Alberta Group Brief Introduction to the Alberta Group of MITACS-PINTS Centerof MITACS-PINTS Center

I. Groups: (Project Leader : Prof. Mike Kouritzin) University of Alberta (base), H.E.C. (led by Prof. Bruno Remillard), University of Waterloo (led by Prof. Andrew Heunis).

II. Sponsors and their interests (1) Lockheed Martin Naval Electronics and Surveillance System: Surveillance and tracking, search and rescue, anti-narcotic smuggling, air traffic management, and global positioning.

II. Sponsors and their interests (cont.)(2) Lockheed Martin Canada, Montreal: same interests as above.

(3) Acoustic Positioning Research Inc., Edmonton: Track stage performers using acoustic techniques and adjust lighting, sound effects. Create performer-movement- controlled product.

(4) VisionSmart, Edmonton: Quality control of industrial processes such as real time analysis of oriented strand board (OSB) density variations using thermography techniques and pattern recognition of naturally occurring substances etc.

(5) Stantec (future) : environmental monitoring, pollution tracking.

III. Postdoctoral Fellow and Graduate StudentsDr. Hongwei Long (MITACS Industrial PDF)Dr. Wei Sun (PIms-MITACS Industrial PDF)

David Ballantyne (Graduate Student)Calvin Chan (Graduate Student)Christina Popescu (Graduate Student)Cody Hyndman (Graduate Student, Waterloo)Nabel Saimi (Graduate Student, Trois Rivieres)Abderhamane Ait-Simmou (Graduate Student , Trois Rivieres)Paul Wiebe (Graduate Student)

Xinjian Ma (Undergraduate Student)Surrey Kim (Undergraduate Student)

IV. Sample from Simulation Front(1) Particle filters for combined state and parameter

estimation: application to tracking of a dinghy lost at sea. (By Hubert Chan and Michael Kouritzin).

(2) Branching particle nonlinear filtering: application to highly directed signal dynamics, highly definitive observation or ``clipped’’ observation data. (By David Ballantyne, John Hoffman and Michael Kouritzin).

(3) Convolutional filters: application to financial engineering and tracking by combining signal prediction and parameter estimation. (By Michael Kouritzin, Bruno Remillard and Calvin P. Chan).

(4) Interacting particle filters: application to tracking of multiple targets. (By David Ballantyne, Hubert Chan and Michael Kouritzin).

Simulation Front Cont.

(5) Markov chain approximations: application to a filtering

model for reflecting diffusions (e.g. fish in a tank).

(By David Ballantyne, Michael Kouritzin, Hongwei Long

and Wei Sun).

(6) Branching particle filtering: application to acoustic

positioning system in theater. (By Michael Kouritzin,

Surrey Kim and Victor Ma)

(V) Sample from Theoretical Front

(1) Markov chain approximation to nonlinear filtering equations for reflecting diffusion processes. (By Michael Kouritzin, Hongwei Long and Wei Sun).

(2) On the generality of the classical filtering equations. (By Michael Kouritzin and Hongwei Long).

(3) Empirical processes based on pseudo-observations: the multivariate case. (By K. Ghoudi and Bruno Remillard)

(4) Nonparameter weighted-symmetry tests. (By B. Abdous, K. Ghoudi and Bruno Remillard).

(5) Testing for randomness against serial dependence. (By C. Genest, J.F. Quessy and B. Remillard).

Theoretical Front (continued)

(6) Functional central limit theorems for interacting particle systems. (By Pierre Del Moral and Michael Kouritzin).

(7) Explicit solutions for vector Ito’s equations. (By Michael Kouritzin and Bruno Remillard).

(8) On Uniqueness of Solutions for the Stochastic Differnectial Equations of Nonlinear Filtering. (By Vladimir Lucic and Andrew Heunis)

(VI.) Ideas for the future

(1) Implement chaos method and try out with random environments.

(2) Filtering when observations in random environments

(3) Tracking and estimation of bacteria and other species.

OverviewOverview

1. Surrey Kim - Overview and Review of Filters and Filtering Procedure. (8-10 mins).

2. Hong Wei Long. - Markov Chain Approximation to Non-Linear Filters. (13-15 mins).

3. Q & A.

Signal SimulationSignal Simulation

A vessel with at sea transporting narcotics or a dinghy lost at sea.

Can You Find The Signal ?Can You Find The Signal ?(Mild Observation Noise)(Mild Observation Noise)

Can You Find The Signal ?Can You Find The Signal ?(Medium Observation Noise)(Medium Observation Noise)

Can You Find The Signal ?Can You Find The Signal ?(Realistic Observation Noise)(Realistic Observation Noise)

Filtering ProcedureFiltering Procedure

1. Modeling an unobserved signal.

2. Modeling the partial/noisy/distorted observation.

3. Filter outputs a conditional distribution estimate of the signal’s past/present/future state, based on observations.

+ =

Signal & Observation Model

Noisy Observations Filter’s Estimate

Mathematical Formulation of FilteringMathematical Formulation of Filtering To do filtering we require a predictive model for (signal

observations). The most classical predictive model is:

a) Signal is a measurable Markov process

b) The observation are :

where is a measurable function and B is a Brownian

motion independent of . Is there a stochastic evolution equation for

Yes !

Kushner (1967) guessed right. Fujisaki, Kallianpur, and Kunita (1972) proved it rigorously

under Kurtz and Ocone (1988) wondered if this condition

could be weakened. Kouritzin and Long (2001) proved that if

then

where is the innovation process. is the weak generator for . The new condition is much more general, allowing for

example and with - stable distributions with .

(old)

(new)

Filter Simulation 1Filter Simulation 1Single Target Tracking: Dinghy Lost At SeaSingle Target Tracking: Dinghy Lost At Sea

SignaSignall

Noisy Noisy ObservationObservation Branching Particle Branching Particle

Filter’s estimateFilter’s estimate

Filter Simulation 2Filter Simulation 2Multiple Target Tracking: 3 DinghiesMultiple Target Tracking: 3 Dinghies

Filter ApplicationsFilter Applications

1. Tracking: current real-time state of a signal.

2. Smoothing: past state of a signal.

3. Prediction: future state of a signal.

4. Historical Path: includes all estimates from 1, 2, 3.

Filter Simulation 3Filter Simulation 3Historical Filter: Dinghy Lost At SeaHistorical Filter: Dinghy Lost At Sea

Green – past path & estimate of the confidence.White - current state & estimate of the confidence.Magenta – future path & estimate of the confidence.

Markov Chain Approximations to Markov Chain Approximations to Nonlinear FiltersNonlinear Filters

The Fish Farming ProblemThe Fish Farming Problem

Unknown number of fish Swimming randomly (assumption: fish don’t have

dinner plans) . Very distorted and noisy pictures. Very dirty water which add to observation

corruption (fish eat => fish s!*#) . Blind spots due to hiding places & other fishes.

I. Formulation of Filtering ProblemI. Formulation of Filtering Problem

Signal: reflecting diffusions in rectangular region D (e.g. fish in a tank).

Mathematical model described by Skorohod SDE:

Formulation of Filtering Formulation of Filtering (cont.)(cont.)

The associated diffusion generator

Observation: distorted, corrupted, partial:

Optimal filter: to evaluate

Formulation of Filtering Formulation of Filtering (cont.)(cont.)

Reference probability measure:

Under : and are independent,

is a standard Brownian motion.

Kallianpur-Striebel formula (Bayes formula):

Formulation of Filtering Formulation of Filtering (cont.)(cont.)

has a density under our assumptions,

which solves the Zakai equation:

Kushner-Huang’s wide-band observation noise approximation

converges to in distribution

Formulation of Filtering Formulation of Filtering (cont.)(cont.)

Find numerical solutions to the above random PDE by replacing with ,

Kushner or Bhatt-Karandikar’s robustness

result can handle this part: the approximate

filter converges to optimal filter.

II. Construction of Markov ChainsII. Construction of Markov Chains Use stochastic particle method developed by Kurtz,

Arnold, Kotelenez, Blount, Kouritzin and Long.We have the following Dirichlet form :

Divide the region D into cells and Construct discretized operator via (discretized) Dirichlet form.

Construction of Markov Chains Construction of Markov Chains (cont.)(cont.)

Dirichlet form: a coercive closed bilinear formassociated with the diffusion generator.

(Signal observation) defined on a probability space

Use another probability space to construct independent Poisson processes and a sequence of Bernoulli trials.

Define a product probability space

Construction of Markov ChainsConstruction of Markov Chains (cont.)(cont.)

: number of particle in cell k at time t

Markov Chain Particle Based Filter SimulationMarkov Chain Particle Based Filter Simulation

Construction of Markov Chains Construction of Markov Chains (cont.)(cont.)

{ } is modelled as an inhomogeneous Markov chain via random time changes.

Use much slower transition rates which makes the implementation much more efficient.

Particles evolve in each cell according to birth and death from reaction (involving observation data), random walks from diffusion and drift.

Construction of Markov Chains Construction of Markov Chains (cont.)(cont.)

denotes mass of each particle The approximate Markov process is given by

From , we can construct a unique probability measure defined on the cadlag path space for each .

Using martingale theory and Dirichlet form theory to analyze the mathematical structure of our Markov chains

P~

III. Laws of Large NumbersIII. Laws of Large Numbers

We have both the quenched and annealed laws of large numbers :

Quenched approach: fixing the sample path of observation process

Annealed approach: considering the observation process as a random medium for Markov chains

IV. Concluding RemarksIV. Concluding Remarks

Find implementable approximate solutions to filtering equations.

Our method differs from previous ones

such as Monte Carlo method (using Markov

chains to approximate signals, Kushner 1977), interacting particle method (Del Moral, 1997), weighted particle method (Kurtz and Xiong, 1999, analyze), and branching particle method (Kouritzin, 2000)

Our algorithm is far more efficient for the class

of reflecting signals considered in this work.