Transcript Markov Chain Approximations to Nonlinear Filters

Brief Introduction to the Alberta Group
of 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 Students
Dr. 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.
Overview
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 Simulation
A vessel with at sea
transporting narcotics or
a dinghy lost at sea.
Can You Find The Signal ?
(Mild Observation Noise)
Can You Find The Signal ?
(Medium Observation Noise)
Can You Find The Signal ?
(Realistic Observation Noise)
Filtering Procedure
1.
2.
3.
Modeling an unobserved signal.
Modeling the partial/noisy/distorted observation.
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 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
(old)
 Kurtz and Ocone (1988) wondered if this condition
could be weakened.
 Kouritzin and Long (2001) proved that if
then
(new)
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
.
Filter Simulation 1
Single Target Tracking: Dinghy Lost At Sea
Signal
Noisy Observation
Branching Particle Filter’s
estimate
Filter Simulation 2
Multiple Target Tracking: 3 Dinghies
Filter Applications
1.
2.
3.
4.
Tracking: current real-time state of a signal.
Smoothing: past state of a signal.
Prediction: future state of a signal.
Historical Path: includes all estimates from 1, 2, 3.
Filter Simulation 3
Historical 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
Nonlinear Filters
The 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 Problem


Signal: reflecting diffusions in rectangular region D
(e.g. fish in a tank).
Mathematical model described by Skorohod SDE:
Formulation of Filtering (cont.)

The associated diffusion generator

Observation: distorted, corrupted, partial:

Optimal filter: to evaluate
Formulation of Filtering (cont.)

Reference probability measure:

Under

:
and
are independent,
is a standard Brownian motion.
Kallianpur-Striebel formula (Bayes formula):
Formulation of Filtering (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 (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 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
Construct discretized operator
Dirichlet form.
cells and
via (discretized)
Construction of Markov Chains (cont.)


Dirichlet form: a coercive closed bilinear form
associated 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 Chains (cont.)
: number of particle in cell
k at time t
Markov Chain Particle Based Filter Simulation
Construction of Markov Chains (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 (cont.)


denotes mass of each particle
The approximate Markov process is given by
~
P

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
III. 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 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.