Title: Particle Filters : Monte Carlo approach to signal processing
1Particle Filters Monte Carlo approach to signal
processing
2Stochastic System and Signal Theory Group (SST)
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- Project Particle Filters and Their Applications
to - Target Tracking and Detection
- Financed by THALES Nederland B.V.
3Outline
- Historic development of filtering problems
- Basic algorithm of a generic PF
- Advantages of PF
- Some successful applications
- Ongoing research
4What is a filter ?
- X(t) true signal
- Y(t) measurement of X(t) corrupted by noise
N(t) - Objective
- Estimate X(t) based on the knowledge of Y( ) up
to time t
5Solved independently by Kolmogorov and Wiener in
early 40s
- Kolmogorovs Approach
- Based on
- Projection Theorem
- on Hilbert Space
6Wieners approach Based on (cross) correlation
of signal observation
- Key assumptions
- Scalar processes
- Joint stationarity of signal
noise processes - Leads to Wiener Hopf equation
- Wiener Hopf equation now has numerous
applications to diverse fields of Applied
Mathematics
7Extensions
- Non stationary not obvious in the Wiener set
up - Vector case analysis highly complicated
-
8Breakthrough Kalman Filter ( 1960 )
- Model for signal a dramatic paradigm shift
!!! - Dynamic State Space formulation
- Naturally multi dimensional
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-
9- Immediate sensational success in space missions
- Today, there is not a single branch in science
or engineering where KF is not used !!!!!! - Also routinely used in economics and finance
10Kalman Filter Discrete Version
In MMSE sense
11Note
filter density / posterior density
12Remarks
- KF is restricted to linear-Gaussian model
- Optimal filter density is Gaussian
- characterized by conditional mean and
- conditional covariance
13Nonlinear Filter
- filter density non Gaussian
- - work directly with filter density
- Question How to (recursively) update the filter
- density ?
14Emergence of Particle Filter (Early 90s)
- Represent the filter density by a large set of
(weighted) particles - For example
Each with weight 1/N, N Nr. Of particles
15Importance Sampling
- Ideally generate particles from
filter density
Not known !!!
- Sample from Proposal Density ( ?() )
- Known as Importance sampling density in Monte
Carlo - literature
16Principle of a PF
- Propose new particles
- Weight update
17One Cycleof the basic algorithm
18One Cycleof the basic algorithm
19Working principle of a PF Propose, weight,
resample
- Propose new particles
- Weight update
- Resample
20Advantages
- Linear-Gaussian assumption not required
- Leads to an (approximate) estimate of the
complete probability distribution - Approximation of pdf rather than approximating
state space model as in ad-hoc extensions of
Kalman filter - Basic version is very easy to code
- - simple core algorithms, modularity
21Some Applications
- (Radar) Target Tracking and Detection
- Financial Engineering
- Robotics and Computer Visions
- Mobile Communications, Image, Audio Signals
- Rare Event Simulation
- Electricity Load Forecasting
22PF in action ( Example 2 ).Courtsey Yvo Boers
( THALES Nederland )
23Ongoing research
Active field of research with a lot of open
problems (both theoretical practical) including
- Number of particles
- Choice of proposal
- Optimal
- Sampling in large dimensions
- Number of particles
- Choice of proposal
- Optimal
- Sampling in large dimensions
24Question ??....