IN THE NAME OF GOD

Title:

Particle Filter and Sampling Algorithms 

 By:Mohammad Reza Jabbari

Email: Mo.re.jabbari@gmail.com

January 2017

Outline1. Estimation Concepts2. Bayesian Estimation 3. Monte Carlo Integration Methods4. Particle Filter 5. Sampling Algorithms6. Application7. Summery8. End

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1.Estimation Concepts The purpose of estimation is to obtain an Approximate Value of an Unknown Parameter,

based on Noisy Observations made from measurements.

Estimation Theory

Classic (Estimation of a Parameter)

Bayesian (Estimation of a Random Variable)

Point Estimation

Interval Estimation

Method of Moment

Maximum Likelihood (ML)

Kalman Filter (KF)

Extended Kalman Filter (EKF)

Unscented Kalman Filter (UKF)

Particle Filter (PF)

Linearity / Gaussian

low nonlinearity / Gaussian

high nonlinearity / Gaussian Recursive

form

nonlinearity / non Gaussian

Definition And Classification

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Representation of Systems (Modeling) State-Space Model Many Process and systems can described with by state-space models

System Equation:

Dynamic Equation

Measurement Equation

𝒚 𝒕=𝒉𝒕 (𝒙 𝒕 ,𝒗𝒕 )

𝒙 𝒕= 𝒇 𝒕−𝟏 (𝒙 𝒕−𝟏 ,𝒘𝒕−𝟏 ) System State at time instant t State Transition Function Process Noise

Observation at time instant t Observation Function Obseravtion Noise

𝒑 (𝒙𝒕∨𝒙 𝒕−𝟏)

𝒑 (𝒚 𝒕|𝒙 𝒕 )

Probabilistic

form

Probabilistic

form

Likelihood Density

State Transition Density

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2. Bayesian Estimation The Goal

The goal of a Bayesian estimator is to approximate the unknown state base on base on previous measurements :

=

Aposterior Density By knowing posterior distribution, all kind of Estimation can be compute

The goals of Bayesian estimator

Find Aposterior Distribution 5/19

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Bayesian Estimator Recursive Equations Updating: Prediction:

𝒑 (𝒙 𝒕|𝒀 𝒕 )=𝒑 (𝒚 𝒕|𝒙 𝒕 )𝒑 (𝒙 𝒕|𝒀 𝒕−𝟏 )

𝒑 (𝒚 𝒕|𝒀 𝒕−𝟏 )𝒑 (𝒙 𝒕|𝒀 𝒕−𝟏 )=∫𝒑 (𝒙 𝒕|𝒙 𝒕−𝟏 )𝒑 (𝒙 𝒕−𝟏|𝒀 𝒕−𝟏 )𝒅𝒙𝒕−𝟏

State transition

density

Aprior at time t

Aposteriori at time t-1Aposteriori

at time tAprior at time tLikelihood

𝒑 (𝒙𝟎|𝒚𝟎 ) 𝒑 (𝒙𝟏|𝒚𝟎 ) 𝒑 (𝒙𝟏|𝒚𝟏 ) 𝒑 (𝒙𝟐|𝒚𝟏 )

Prediction

Prediction

Update Update Update

𝒑 (𝒙𝟎)

𝒚𝟎 𝒚𝟏 𝒚𝟐

Time instant t=0

Time instant t=1

Time instant t=2

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Problems The solution is conceptual because integral are not tractable

Close form solution are possible in a small number of situation

Solution Use Monte Carlo Integration Methods

For linear systems with Gaussian noise distribution

Optimal Estimation Using the Kalman Filter (KF)

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3. Monte Carlo Integration Methods Monte Carlo Integration is a Simple but Powerful technique for approximating complicated

integrals. Assume we are trying to estimate the integral of a function f over some domain D :

Assume that we have a PDF p defined over a domain D :

Its means that we generate samples according to p, computing f/p for each sample, and finding the average of these values.

This equality is true for any PDF on D, as long as p(x)0 whenever f(x)0

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Question What happens when we generate a random sample where the value of p is very

small?if p is very small for a given sample, f/p will be arbitrarily large. This large sample will greatly skew the sample mean away from the true mean, and the sample variance will also increase greatly.

Bad Samples

but one general rule of thumb to follow is that p should “look like” f (Importance Sampling)

Answer

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4. Particle Filter The particle filter is technique for implementing Recursive Bayesian Filter By Monte

Carlo Sampling Particles, with corresponding Weights are used to form an approximation of a probability density

function (PDF)

{𝑥𝑡 −1𝑖∗ }𝑖=1

𝑁 {𝑥𝑡𝑖 }𝑖=1𝑁 {𝑥𝑡

𝑖∗ }𝑖=1𝑁

Time instant t-1

Time instant t

𝒑 (𝒙 𝒕−𝟏|𝒀 𝒕−𝟏 ) 𝒑 (𝒙 𝒕|𝒀 𝒕−𝟏 )𝒑 (𝒙 𝒕|𝒀 𝒕 )Prediction

¿

Updat

e

𝑾 𝒕

𝒊=~𝑾 𝒕

𝒊

∑𝒋=𝟏

𝑵 ~𝑾 𝒕𝒋 Normalization 11/19

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Sample Representation of the posterior pdf The representation of the posterior pdf in the form of a set of samples is very convenientFor example: threat analysis, decision and control problems,

𝐸 {𝐶 (𝑥𝑡 )|𝑌 𝑡 }=∫𝐶 (𝑥𝑡 )𝑝 (𝑥𝑡|𝑌 𝑡 )𝑑𝑥𝑡≈∑𝑖=1

𝑁

𝑊 𝑡𝑖 𝐶 (𝑥¿¿ 𝑡𝑖)≈ 1

𝑁 ∑𝑖=1

𝑁

¿¿¿

In many cases, the requirement is find some particular function of the posterior, and the sample representation is often ideal for this.

Empirical Distribution

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Unfortunately, it’s usually impossible to sample efficiently from the posteriori distribution at any time t,

because being multivariate , non standard and only known up to a proportionality constant.

Importance Sampling Generate sample from another distribution (Proposal Distribution) Weight them according to how they fit the Posterior distribution

Notice: Free to choose proposal density but: It should be easy to sample from proposal density Proposal density should resemble the original density as closely as possible

Problem

Solution

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5. Sampling Algorithms Importance Sampling (IS) Importance

Weight

¿¿

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Sequential Importance Sampling (SIS) Importance sampling in its simplest form, it’s not adequate for recursive estimation.

Because, it needs to get all data before estimating . So the computational complexity increase with time.

If we can consider:~𝑊 𝑡

𝑖=𝑝 (𝑦𝑡|𝑥𝑡𝑖 ∗)𝑝 (𝑥𝑡

𝑖∗|𝑥𝑡− 1𝑖∗ )

π (𝑥𝑡𝑖∗|𝑥𝑡− 1

𝑖∗ ,𝑌 𝑡 )~𝑊 𝑡−1

𝑖

If Proposal distribution equal to

Apriori distribution

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Sequential Importance Resampling (SIR) Problem: The problem encountered by the SIS method is that, as t increase, the distribution of the importance weight becomes more and more skewed. And after a few time step, only one particle has a non-zero importance weight.

(Degeneracy) Solution: the key idea is to eliminate the particles having low importance weights and multiply particles having high importance weigh (Resampling)

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Problem of Resampling Impoverishment of the sample set particles with large weights may be selected many times so that the new set of samples may contain multiple copies of just a few distinct values.

Solution: Effective Sample Size

𝑁 𝑒𝑓𝑓=1

∑𝑗=1

𝑁

(𝑊 𝑡𝑗 )2𝑎𝑛𝑑1≤ �̂�𝑒𝑓𝑓 ≤𝑁

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6. Application Image Processing Sound Processing Tracking and Navigation Channel Estimation Biology ….

Base on image 1. Image Processing and Extract features

2. Estimation

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7. Summary Particle filter is very powerful framework for estimating parameter in nonlinear / non

Gaussian model Adapting with state-space model

Finding new application for particle filter Developing new implementation to reduce complexity Finding a mechanism to optimize number of particle

Advantages

Disadvantages High Computational Complexity It’s difficult to determine optimal Number of Particles Increasing particle with increasing model dimension Choice of importance density

Main Research Directory

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THE END