Download itm03 - University of Alberta

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
University of Alberta
Supported by NSERC, MITACS, PIMS
Lockheed Martin Naval Electronics and Surveillance System
Lockheed Martin Canada, APR. Inc
Outline
•
•
•
•
•
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Introduction
Simulation Studies
Filtering Equations
Markov Chain Approximations
Model Selection
Future Work
1. Introduction
• Motivation: Submarine tracking and fish farming
• Model:
- Signal:
d
(1)
- Observation:
(2)
• Goal: to find the best estimation for the number
of targets and the location of each target.
2. Simulation Studies
3. 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
Define
,
, and
.
• The generator of
Let
where
.
For any
we define
,
where
and
• Conditions:
C1.
C2.
C3.
C4.
and
satisfy the Lipschitz conditions.
• 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. Markov chain approximations
• Step 1: Constructing smooth approximation
of the observation process
• Step 2: Dividing D and
Let
For
For
,
, let
, let
Note that
of
if
. Let
then
For
is a rearrangement
. For
,
, let
.
with 1 in the i-th coordinate.
• Step 3: Constructing the Markov chain approximations
— Method 1:
Let
Set
that
and
.
. One can find
Define
and for
let
as
, define
as
AN Fk (μ )   L(μ ) fk μ  N
─Method 2
:
Let
and
,
Then
and
Define
for
as
.
• Let
set
and let
as
, take
satisfy
denote the integer part,
Then, the Markov chain approximation is given by
Theorem 5.
in probability on
for almost every sample path of
.
5. Model selection
• Assume that the possible number of targets is ,
Model k:
,
. Which model is better?
• Bayesian 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 measurevalued pair solution of the following system of SDEs,
(3)
for
, and
(4)
for
and
, where
is the optimal filter for model k,
• Markov chain approximations
Applying the method in Section 3, one can construct
Markov chain approximations to equations (3) and (4).
6. Future work
• Number of targets is a random variable
• Number of Targets is a random process