Bayesian Model Selection and Bayesian Model Selection and Multi-target TrackingMulti-target Tracking

Presenters: Xingqiu Zhao and Nikki Hu

Joint work with M. A. Kouritzin, H. Long, J. McCrosky, W. Sun

University of Alberta

Supported by NSERC, MITACS, PIMS Lockheed Martin Naval Electronics and Surveillance System

Lockheed Martin Canada, APR. Inc

Outline Outline

• Introduction

• Simulation Studies

• Filtering Equations

• Markov Chain Approximations

• Model Selection

• Future Work

1. Introduction1. Introduction• Motivation: Submarine tracking and fish farming

• Model:

- Signal:

(1)

d

- Observation:

(2)

• Goal: to find the best estimation for the number of targets and the location of each target.

2. Simulation Studies2. Simulation Studies

3. filtering equations3. filtering equations

• Notations : the space of bounded continuous functions on ; : the set of all cadlag functions from into ; : the spaces of probability measures; : the spaces of positive finite measures on ; : state space of .

Let , , and .

Define

• The generator of Let

where .

For any ,

we define

where

and

• Conditions:

C1. and satisfy the Lipschitz conditions.

C2.

C3.

C4.

• Theorem 1. The equation (1) has a unique solution

a.s.,

which is an -valued Markov process.

• Bayes formula and filtering equations

Theorem 2. Suppose that C1-C3 hold. Then

(i)

(ii)

where

is the innovation process.

(iii)

• Uniqueness

Theorem 3. Suppose that C1-C4 hold. Let be an -

adapted cadlag process which is a solution of the Kushner-FKK equation

where

Then , for all a.s.

Theorem 4 Suppose that C1-C4 hold. If is an - adapted

-valued cadlag process satisfying

and

Then , for all a.s.

4. 4. Markov chain approximationsMarkov chain approximations

• Step 1: Constructing smooth approximation

of the observation process

• Step 2: Dividing D and

Let ,

For , let

For , let

Note that if is a rearrangement

of . Let

then . For , let .

For , with 1 in the i-th coordinate.

• Step 3: Constructing the Markov chain approximations

— Method 1:Method 1:

Let .

Set . One can find that

and

Define as

and for , define as

let

──Method 2Method 2:: Let and ,

Then

and

Define as for .

(μ ) (μ ) μNNA F L f k k

• Let as , take denote the integer part,

set

and let satisfy

Then, the Markov chain approximation is given by

Theorem 5.

in probability on

for almost every sample path of .

5. Model selection5. Model selection • Assume that the possible number of targets is , .

Model k: , . Which model is better?

• Bayesian FactorsBayesian Factors

Define the filter ratio processes as

• The Evolution of Bayesian Factors Let and be independent and Y be Brownian

motion on some probability space.

Theorem 3.

Let be the generator of , . Suppose that is continuous. Then is the unique measure-valued pair solution of the following system of SDEs,

(3)

for , and

(4)

for , where is the optimal filter for model k, and

• Markov chain approximations Applying the method in Section 3, one can construct

Markov chain approximations to equations (3) and (4).

6. Future work6. Future work

• Number of targets is a random variable

• Number of Targets is a random process