Bayesian Model Selection and Multi-target Tracking Presenters: Xingqiu Zhao and Nikki Hu Joint work...
-
date post
18-Dec-2015 -
Category
Documents
-
view
213 -
download
0
Transcript of Bayesian Model Selection and Multi-target Tracking Presenters: Xingqiu Zhao and Nikki Hu Joint work...
![Page 1: Bayesian Model Selection and Multi-target Tracking Presenters: Xingqiu Zhao and Nikki Hu Joint work with M. A. Kouritzin, H. Long, J. McCrosky, W. Sun.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649d235503460f949f92bc/html5/thumbnails/1.jpg)
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
![Page 2: Bayesian Model Selection and Multi-target Tracking Presenters: Xingqiu Zhao and Nikki Hu Joint work with M. A. Kouritzin, H. Long, J. McCrosky, W. Sun.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649d235503460f949f92bc/html5/thumbnails/2.jpg)
Outline Outline
• Introduction
• Simulation Studies
• Filtering Equations
• Markov Chain Approximations
• Model Selection
• Future Work
![Page 3: Bayesian Model Selection and Multi-target Tracking Presenters: Xingqiu Zhao and Nikki Hu Joint work with M. A. Kouritzin, H. Long, J. McCrosky, W. Sun.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649d235503460f949f92bc/html5/thumbnails/3.jpg)
1. Introduction1. Introduction• Motivation: Submarine tracking and fish farming
• Model:
- Signal:
(1)
d
![Page 4: Bayesian Model Selection and Multi-target Tracking Presenters: Xingqiu Zhao and Nikki Hu Joint work with M. A. Kouritzin, H. Long, J. McCrosky, W. Sun.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649d235503460f949f92bc/html5/thumbnails/4.jpg)
- Observation:
(2)
• Goal: to find the best estimation for the number of targets and the location of each target.
![Page 5: Bayesian Model Selection and Multi-target Tracking Presenters: Xingqiu Zhao and Nikki Hu Joint work with M. A. Kouritzin, H. Long, J. McCrosky, W. Sun.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649d235503460f949f92bc/html5/thumbnails/5.jpg)
2. Simulation Studies2. Simulation Studies
![Page 6: Bayesian Model Selection and Multi-target Tracking Presenters: Xingqiu Zhao and Nikki Hu Joint work with M. A. Kouritzin, H. Long, J. McCrosky, W. Sun.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649d235503460f949f92bc/html5/thumbnails/6.jpg)
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 .
![Page 7: Bayesian Model Selection and Multi-target Tracking Presenters: Xingqiu Zhao and Nikki Hu Joint work with M. A. Kouritzin, H. Long, J. McCrosky, W. Sun.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649d235503460f949f92bc/html5/thumbnails/7.jpg)
Let , , and .
Define
![Page 8: Bayesian Model Selection and Multi-target Tracking Presenters: Xingqiu Zhao and Nikki Hu Joint work with M. A. Kouritzin, H. Long, J. McCrosky, W. Sun.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649d235503460f949f92bc/html5/thumbnails/8.jpg)
• The generator of Let
where .
![Page 9: Bayesian Model Selection and Multi-target Tracking Presenters: Xingqiu Zhao and Nikki Hu Joint work with M. A. Kouritzin, H. Long, J. McCrosky, W. Sun.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649d235503460f949f92bc/html5/thumbnails/9.jpg)
For any ,
we define
where
and
![Page 10: Bayesian Model Selection and Multi-target Tracking Presenters: Xingqiu Zhao and Nikki Hu Joint work with M. A. Kouritzin, H. Long, J. McCrosky, W. Sun.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649d235503460f949f92bc/html5/thumbnails/10.jpg)
• Conditions:
C1. and satisfy the Lipschitz conditions.
C2.
C3.
C4.
![Page 11: Bayesian Model Selection and Multi-target Tracking Presenters: Xingqiu Zhao and Nikki Hu Joint work with M. A. Kouritzin, H. Long, J. McCrosky, W. Sun.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649d235503460f949f92bc/html5/thumbnails/11.jpg)
• Theorem 1. The equation (1) has a unique solution
a.s.,
which is an -valued Markov process.
![Page 12: Bayesian Model Selection and Multi-target Tracking Presenters: Xingqiu Zhao and Nikki Hu Joint work with M. A. Kouritzin, H. Long, J. McCrosky, W. Sun.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649d235503460f949f92bc/html5/thumbnails/12.jpg)
• Bayes formula and filtering equations
Theorem 2. Suppose that C1-C3 hold. Then
(i)
(ii)
where
is the innovation process.
(iii)
![Page 13: Bayesian Model Selection and Multi-target Tracking Presenters: Xingqiu Zhao and Nikki Hu Joint work with M. A. Kouritzin, H. Long, J. McCrosky, W. Sun.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649d235503460f949f92bc/html5/thumbnails/13.jpg)
• 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.
![Page 14: Bayesian Model Selection and Multi-target Tracking Presenters: Xingqiu Zhao and Nikki Hu Joint work with M. A. Kouritzin, H. Long, J. McCrosky, W. Sun.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649d235503460f949f92bc/html5/thumbnails/14.jpg)
Theorem 4 Suppose that C1-C4 hold. If is an - adapted
-valued cadlag process satisfying
and
Then , for all a.s.
![Page 15: Bayesian Model Selection and Multi-target Tracking Presenters: Xingqiu Zhao and Nikki Hu Joint work with M. A. Kouritzin, H. Long, J. McCrosky, W. Sun.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649d235503460f949f92bc/html5/thumbnails/15.jpg)
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
![Page 16: Bayesian Model Selection and Multi-target Tracking Presenters: Xingqiu Zhao and Nikki Hu Joint work with M. A. Kouritzin, H. Long, J. McCrosky, W. Sun.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649d235503460f949f92bc/html5/thumbnails/16.jpg)
Note that if is a rearrangement
of . Let
then . For , let .
For , with 1 in the i-th coordinate.
![Page 17: Bayesian Model Selection and Multi-target Tracking Presenters: Xingqiu Zhao and Nikki Hu Joint work with M. A. Kouritzin, H. Long, J. McCrosky, W. Sun.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649d235503460f949f92bc/html5/thumbnails/17.jpg)
• Step 3: Constructing the Markov chain approximations
— Method 1:Method 1:
Let .
Set . One can find that
and
![Page 18: Bayesian Model Selection and Multi-target Tracking Presenters: Xingqiu Zhao and Nikki Hu Joint work with M. A. Kouritzin, H. Long, J. McCrosky, W. Sun.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649d235503460f949f92bc/html5/thumbnails/18.jpg)
Define as
and for , define as
let
![Page 19: Bayesian Model Selection and Multi-target Tracking Presenters: Xingqiu Zhao and Nikki Hu Joint work with M. A. Kouritzin, H. Long, J. McCrosky, W. Sun.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649d235503460f949f92bc/html5/thumbnails/19.jpg)
──Method 2Method 2:: Let and ,
Then
and
Define as for .
(μ ) (μ ) μNNA F L f k k
![Page 20: Bayesian Model Selection and Multi-target Tracking Presenters: Xingqiu Zhao and Nikki Hu Joint work with M. A. Kouritzin, H. Long, J. McCrosky, W. Sun.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649d235503460f949f92bc/html5/thumbnails/20.jpg)
• Let as , take denote the integer part,
set
and let satisfy
![Page 21: Bayesian Model Selection and Multi-target Tracking Presenters: Xingqiu Zhao and Nikki Hu Joint work with M. A. Kouritzin, H. Long, J. McCrosky, W. Sun.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649d235503460f949f92bc/html5/thumbnails/21.jpg)
Then, the Markov chain approximation is given by
Theorem 5.
in probability on
for almost every sample path of .
![Page 22: Bayesian Model Selection and Multi-target Tracking Presenters: Xingqiu Zhao and Nikki Hu Joint work with M. A. Kouritzin, H. Long, J. McCrosky, W. Sun.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649d235503460f949f92bc/html5/thumbnails/22.jpg)
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
![Page 23: Bayesian Model Selection and Multi-target Tracking Presenters: Xingqiu Zhao and Nikki Hu Joint work with M. A. Kouritzin, H. Long, J. McCrosky, W. Sun.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649d235503460f949f92bc/html5/thumbnails/23.jpg)
• 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)
![Page 24: Bayesian Model Selection and Multi-target Tracking Presenters: Xingqiu Zhao and Nikki Hu Joint work with M. A. Kouritzin, H. Long, J. McCrosky, W. Sun.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649d235503460f949f92bc/html5/thumbnails/24.jpg)
for , and
(4)
for , where is the optimal filter for model k, and
![Page 25: Bayesian Model Selection and Multi-target Tracking Presenters: Xingqiu Zhao and Nikki Hu Joint work with M. A. Kouritzin, H. Long, J. McCrosky, W. Sun.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649d235503460f949f92bc/html5/thumbnails/25.jpg)
• Markov chain approximations Applying the method in Section 3, one can construct
Markov chain approximations to equations (3) and (4).
![Page 26: Bayesian Model Selection and Multi-target Tracking Presenters: Xingqiu Zhao and Nikki Hu Joint work with M. A. Kouritzin, H. Long, J. McCrosky, W. Sun.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649d235503460f949f92bc/html5/thumbnails/26.jpg)
6. Future work6. Future work
• Number of targets is a random variable
• Number of Targets is a random process