1. An introduction to Particle filtering
Paul Sundvall
Presentation in course “Optimal filtering”
Signals, Sensors and Systems, KTH
November 11th 2004

2. Outline Introduction Comparison with the Kalman filter
Description of the algorithm
Implementation
Example
Paul Sundvall

3. Introduction Particle filtering is a method for state estimation
is a Monte Carlo method
handles nonlinear models with non- Gaussian noise
Paul Sundvall

4. Comparison to the discrete Kalman filter
Slow
Fast
Computational speed
Approximate
Exact, optimal
Solution
Output
Noise type
State equation
Particle filter
Kalman filter
Any distribution, uni- or multimodal
Gaussian, unimodal
Paul Sundvall

5. Significant property
The particle filter gives an approximate solution to an exact model, rather than the optimal solution to an approximate model.
Paul Sundvall

6. Algorithm Initializew
The propability density function is approximated using point weights
Each point is called a particle
Each particle has a positive weight
Basic algorithm:
Initialize
Time update (move particles)
Measurement update (change weights)
Resample (if needed)
Goto 2 when new measurement arrives
Each point is called a particle
Each particle has a positive weight
Initializew

7. Time Update One-step prediction of each particle
Note that a realization of the
process noise is used for every
particle.

8. Measurement update The weights are adjusted using the measurement
All weights are normalized
Particles that can explain the measurement gain weight
Particles far off the true state lose weight.
The density of the cloud changes

9. Resampling It can be shown that the algorithm degenerates
Allt particles but one become very light
Solved by resampling so that all weights become equal

10. Implementation
Calculation demand is proportional to the number of particles
The approximation error decreases as the number of particles grow
N can easily be changed during runtime
One needs to know what to do with p(x)
is not a good choice for multimodal distributions!

11. Example A boat travels on a one-dimensional sea
Noisy depth measurements are given
Given a perfect sea-chart d(x), estimate the position!
Matlab code for the example is available on

12. Final comments
Better the more multimodal, non-linear and non-gaussian the system is
The most basic variants are simple to implement
It is easy to add model knowledge (saturation, limit checking, nonnegativeness...)
Variants of the particle filters exist
To reduce the number of particles needed, by combining Kalman filters and particle filters
To ensure that states with low propability but high risk are tracked despite few particles (fault detection)
To use discrete states
To use both discrete and continuous states (hybrid)

13. References
A good introduction is given in ”A tutorial on Particle Filters for Online Nonlinear/Non-Gaussian Bayesian Tracking” M. Sanjeev Arulampalam et. al.
Very good application examples can be found in ”Particle Filters for Positioning, Navigation and Tracking” Fredrik Gustafsson et. al.
A description of how particle filters can be used for fault detection is found in ”Particle Filters for Rover Fault Diagnosis” Vandi Verma et. al.