MOFT Tutoriials, Sequential Monte Carlo (SMC) and Particle Filters

1. MOFT Tutorials
Multi Object Filtering Multi Target Tracking
Sequential Monte Carlo (SMC) Methods
and Particle Filters

2. SMC Methods and Particle Filters
Bayesian Inference and Filtering
Classical Dynamic System
▪ system state evolves in the state space
▪ states are hidden & only partially observed in the observation space
▪ fundamental (dynamical) system problems, filtering, control, system
identification

3. SMC Methods and Particle Filters
State Space Models
▪ many process and system can be described by state-space models

4. SMC Methods and Particle Filters
Bayesian Estimation
▪ posterior density based on previous measurements

5. SMC Methods and Particle Filters
Bayesian
Filtering

6. SMC Methods and Particle Filters
Bayesian Filtering
▪ closed form solutions are possible only for some specific cases,
otherwise intractable
▪ one specific case is linear systems with Gaussian noise, optimal
solution available in closed form is Kalman filter
▪ for other cases MC-Monte Carlo integration methods are powerful
and provide simple, efficient approximations

7. SMC Methods and Particle Filters
Sequential Monte Carlo (SMC) – Particle Filter
▪ a class of approximate numerical solutions to the Bayes recursion
▪ applicable to nonlinear non-Gaussian dynamic and observation
models.
▪ use of random samples (particles) to approximate probability
distributions of interest
▪ 𝑁 independently and identically distributed (i.i.d.) samples 𝐱(𝒊)
𝑖=1
𝑁
from an arbitrary probability density 𝑝 of 𝐱.

8. SMC Methods and Particle Filters
▪ samples 𝐱(𝒊)
𝑖=1
𝑁
as a point mass approximation of 𝑝, i.e., 𝑝 𝐱 ∝ ෤𝑝 𝐱
𝑝 𝐱 ≈
1
𝑁
෍
𝑖=1
𝑁
𝛿(𝐱 − 𝐱(𝑖)
)
▪ usually density is only known up to a normalizing constant, i.e. 𝑝 𝐱 ∝
෤𝑝 𝐱 ,

9. SMC Methods and Particle Filters
Sequential Monte Carlo (SMC) – Particle Filter
▪ is approximation to Bayes filter
▪ particles with corresponding weights are used to form an
approximation to posterior density

10. SMC Methods and Particle Filters
Sequential Monte Carlo (SMC) – Particle Filter
▪ in Bayes recursion the normalizing constant is difficult to compute.
▪ because being multivariate , posterior distribution is not standard and
only known up to a proportionality constant.
▪ it’s usually impossible to sample efficiently from the posterior
distribution at any time t,
▪ draw 𝑁 i.i.d. samples 𝐱(𝒊)
𝑖=1
𝑁
from a known density 𝑞, proposal or
importance density,
▪ weight these samples accordingly to obtain a weighted point mass
approximation to 𝑝.

11. SMC Methods and Particle Filters
Sequential Monte Carlo (SMC) – Particle Filter
▪ ”good” proposal is one such that the weights 𝑤(𝒊)
𝑖=1
𝑁
all have
roughly the same value
▪ weighted samples 𝑤(𝒊), 𝐱(𝑖)
𝑖=1
𝑁
as a weighted point mass
approximation of 𝑝, i.e.,
𝑝 𝐱 ≈ ෍
𝑖=1
𝑁
𝑤(𝒊)
𝛿(𝐱 − 𝐱(𝑖)
)

12. SMC Methods and Particle Filters
Particle Filter – Importance Sampling
▪ generate sample from another distribution – proposal density
▪ weight them according to how they fit the posterior distribution
▪ should be easy to sample from proposal density
▪ proposal should resemble the original density closely as possible

13. SMC Methods and Particle Filters
Particle Filter – Sequential Importance Sampling
▪ in importance sampling as t increase, the distribution of the
importance weight becomes more and more skewed
▪ after a few time step, only one particle has a non-zero importance
weight – particle depletion/degeneracy
▪ variance of the importance weights increases over time, thereby
degrading the quality of the particle approximation

14. SMC Methods and Particle Filters
▪ particle depletion is generally mitigated by resampling the weighted
particles 𝑤 𝑘
𝑖
, 𝐱0:𝑘
𝑖
𝑖=1
𝑁
▪ to generate more replicas of particles with high weights and eliminate
those with low weights.
▪ many resampling schemes available,
▪ multinomial_resampling
▪ systematic_resampling
▪ stratified_sampling
▪ residual_sampling
▪ choice of resampling scheme affects computational load as well as
the quality of the particle approximation

15. SMC Methods and Particle Filters
Particle Filter – Sequential Importance Sampling
▪ resampling as solution to degeneracy, eliminate the particles having
low importance weights and multiply particles having high importance
weight

16. SMC Methods and Particle Filters
▪ for any function ℎ, the (finite) expectation of ℎ can be approximated
by the empirical expectation, i.e.
න ℎ(𝐱)𝑝 𝐱 𝑑𝐱 ≈
1
𝑁
෍
𝑖=1
𝑁
𝑤(𝑖)ℎ(𝐱(𝑖))
where
𝑤(𝑖) =
෥𝑤 𝐱(𝑖)
σ 𝑗=1
𝑁
෥𝑤 𝐱(𝑗)
෥𝑤 𝐱(𝑖) =
𝑝 𝐱(𝑖)
𝑞 𝐱(𝑖)
▪ are normalized importance weights and importance weights
respectively.

17. SMC Methods and Particle Filters
▪ posterior density 𝑝0:𝑘−1, at time 𝑘 − 1, is represented as a set of
weighted particles 𝑤 𝑘−1
𝑖
, 𝐱0:𝑘−1
𝑖
𝑖=1
𝑁
, i.e
𝑝0:𝑘−1 𝐱0:𝑘−1 𝐳 𝑘−1 ≈ ෍
𝑖=1
𝑁
𝑤 𝑘−1
𝑖
𝛿(𝐱0:𝑘−1 − 𝐱0:𝑘−1
𝑖
)
▪ a proposal density 𝑞 𝑘 . |𝐱 𝑘−1
𝑖
, 𝐳 𝑘 that we can easily sample from.

18. SMC Methods and Particle Filters
▪ posterior density 𝑝0:𝑘, at time 𝑘, is represented as a new set of
weighted particles 𝑤 𝑘
𝑖
, 𝐱0:𝑘
𝑖
𝑖=1
𝑁
𝑝0:𝑘 𝐱0:𝑘 𝐳 𝑘 ≈ ෍
𝑖=1
𝑁
𝑤 𝑘
𝑖
𝛿(𝐱0:𝑘 − 𝐱0:𝑘
𝑖
)
where
𝐱0:𝑘
𝑖
= (𝐱0:𝑘−1
𝑖
− 𝐱 𝑘
𝑖
)
𝐱 𝑘
𝑖
= 𝑞 𝑘 . |𝐱 𝑘−1
𝑖
, 𝐳 𝑘
𝑤 𝑘
𝑖
=
෥𝑤 𝑘
𝑖
σ𝑖=1
𝑁
෥𝑤 𝑘
𝑖
෥𝑤 𝑘
𝑖
= 𝑤 𝑘
𝑖
𝑔 𝑘 𝐳 𝑘|𝐱 𝑘
𝑖
𝑓𝑘|𝑘−1 𝐱 𝑘
𝑖
|𝐱 𝑘−1
𝑖
𝑞 𝑘 𝐱 𝑘
𝑖
|𝐱 𝑘−1
𝑖
, 𝐳 𝑘

19. References
[1] Olivier Cappe´, Simon J. Godsill, and Eric Moulines, An Overview of Existing
Methods and Recent Advances in Sequential Monte Carlo, No. 5, May 2007,
Proceedings of the IEEE
[2] Particle Filters and Their Applications, Kaijen Hsiao Henry de Plinval-Salgues
Jason Miller Cognitive Robotics April 11, 2005

20. Other MOFT Tutorials – Lists and Links
Introduction to Multi Target Tracking
Bayesian Inference and Filtering
Kalman Filtering
Sequential Monte Carlo (SMC) Methods and Particle Filtering
Single Object Filtering Single Target Tracking
Nearest Neighbor(NN) and Probabilistic Data Association Filter(PDAF)
Multi Object Filtering Multi Target Tracking
Global Nearest Neighbor and Joint Probabilistic Data Association Filter
Data Association in Multi Target Tracking
Multiple Hypothesis Tracking, MHT

21. Other MOFT Tutorials – Lists and Links
Random Finite Sets, RFS
Random Finite Set Based RFS Filters
RFS Filters, Probability Hypothesis Density, PHD
RFS Filters, Cardinalized Probability Hypothesis Density, CPHD Filter
RFS Filters, Multi Bernoulli MemBer and Cardinality Balanced MeMBer, CBMemBer Filter
RFS Labeled Filters, Generalized Labeled Multi Bernoulli, GLMB and Labeled Multi Bernoulli, LMB Filters
Multiple Model Methods in Multi Target Tracking
Multi Target Tracking Implementation
Multi Target Tracking Performance and Metrics

22. http://www.egniya.com/EN/MOFT/Tutorials/
moft@egniya.com