Sanjay Patil1 and Ryan Irwin2

Graduate research assistant1,REU undergrad2

Human and Systems Engineering

URL: www.isip.msstate.edu/publications/seminars/msstate/2005/particle/

HUMAN AND SYSTEMS ENGINEERING:Gentle Introduction to Particle Filtering

Page 2 of 20Particle Filtering – Gentle Introduction and Implementation Demo

Abstract

Particle Filtering:

• Most conventional techniques for speech analysis are based on modeling signals as Gaussian Mixture Models in Hidden Markov Model based systems.

• To overcome the mismatched channel conditions, and/or significantly reduce

the complexity of the models, Nonlinear approaches are expected to perform better than the conventional techniques.

• Particle filters, based on sequential Monte Carlo methods, is one such nonlinear methods.

• Particle filtering allows complete presentation of the posterior distribution of the states. Statistical estimates can be computed easily even in the presence of nonlinearities.

Page 3 of 20Particle Filtering – Gentle Introduction and Implementation Demo

Outline of Presentation

• Nonlinear Methods – necessity

• Drawing Samples from a Probability distribution. (introduce ‘Particle’)

• Sequential Monte Carlo Methods – necessity, different names – bootstrap, condensation algorithm, survival of the fittest.

• Steps in particle filtering (explaining the algorithm – block schematic)

• Actual example – (along with all the steps)

• Brief review and applications for tracking

• As can be applied to Speaker Verification

• Demo

Page 4 of 20Particle Filtering – Gentle Introduction and Implementation Demo

0 2 4 6 8 10 12 14 16 18 200

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 2 4 6 8 10 12 14 16 18 200

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0 5 10 15 20 250

0.05

0.1

0.15

0.2

0.25

5000 samples500 samples

200 samplesTake p(x)=Gamma(4,1)

Generate some random samples

Plot basic approximation to pdf

Each sample is called as ‘Particle’

Drawing samples from a probability distribution function

• Concept of samples and its weights

Page 5 of 20Particle Filtering – Gentle Introduction and Implementation Demo

Particle filtering -

Different Names –

• Sequential Monte Carlo filters

• Bootstrap filters

• Condensation Algorithm

• Survival of the fittest

General Problem Statement – Filtering – estimation of the states

• Tracking the state (parameters or hidden variables) as it evolves over time

• Sequentially arriving (noisy and non-Gaussian) observations

• Idea is to have best possible estimate of hidden variables

Page 6 of 20Particle Filtering – Gentle Introduction and Implementation Demo

Assume that pdf p(xk-1 | y1:k-1) is available at time k -1.

• Prediction stage:

This is the prior of the state at time k ( without the information on measurement). Thus, it is the probability of the the state given only the previous measurements

• Update stage:

This is posterior pdf from predicted prior pdf and newly available measurement.

Particle filtering algorithm continue…

)|(

)|()|()|(

1:1

1:1:1

kk

kkkkkk yyp

yxpyxpyxp

General two-stage Framework

(Prediction-Update stages)

11:111:1 )|()|()|( kkkkkkk dxyxpxxpyxp

Page 7 of 20Particle Filtering – Gentle Introduction and Implementation Demo

Particle filtering algorithm step-by-step (1)

Page 8 of 20Particle Filtering – Gentle Introduction and Implementation Demo

Particle filtering step-by-step (2)

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Particle filtering step-by-step (3)

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Particle filtering step-by-step (4)

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Particle filtering step-by-step (5)

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Particle filtering step-by-step (6)

Page 13 of 20Particle Filtering – Gentle Introduction and Implementation Demo

Particle filtering - visualization

Drawing samples

Predicting next state

Updating this state

What is THIS STEP???

Resampling….

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Sampling Importance Resample algorithm (necessity)

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Applications:

All the applications are mostly tracking applications in different forms….

Visual Tracking – e.g. human motion (body parts)

Prediction of (financial) time series – e.g. mapping gold price, stocks

Quality control in semiconductor industry

Military Applications

Target recognition from single or multiple images

Guidance of missiles

What is the application for IES NSF funded project –

Time series estimation for speech signal (Java demo)

Speaker Verification (details on next slide)

Page 16 of 20Particle Filtering – Gentle Introduction and Implementation Demo

Pattern Recognition Applet

• Java applet that gives a visual of algorithms implemented at IES

• Classification of Signals: • PCA - Principle Component Analysis

• LDA - Linear Discrimination Analysis

• SVM - Support Vector Machines

• RVM - Relevance Vector Machines

• Tracking of Signals • LP - Linear Prediction

• KF - Kalman Filtering

• PF – Particle Filtering

Page 17 of 20Particle Filtering – Gentle Introduction and Implementation Demo

Pattern Classification

• Different data sets need to be differentiated without looking at all the data samples

• Classifications distinguishes between sets of data without the samples

• Algorithms separate data sets with a line of discrimination

• To have zero error the line of discrimination should completely separate the classes

• These patterns are easy to classify

Page 18 of 20Particle Filtering – Gentle Introduction and Implementation Demo

Pattern Classification

• Toroidals are not classified very successfully with a straight line

• Error should be around 50% because half of each class is separated

• A proper line of discrimination of a toroidal would be a circle enclosing only the inside set

Page 19 of 20Particle Filtering – Gentle Introduction and Implementation Demo

Signal Tracking

• The input signals are now time based with the x-axis representing time

• All the signal tracking algorithms are implemented with interpolated data

• The interpolation ensures that the input samples are at regular intervals

• Sampling is always done on regular intervals

• The linear prediction algorithm is a linear way to predict signals with no noise

Page 20 of 20Particle Filtering – Gentle Introduction and Implementation Demo

Signal Tracking

• The Kalman filter and particle filter are based on prediction of the states of the signal

• States are related to the observations through the state equation

• The particle filtering algorithm introduces process and measurement noise

• At each iteration possible states are given by the black points

• The average of the black points is where the overall state is predicted to be

Page 21 of 20Particle Filtering – Gentle Introduction and Implementation Demo

References:

• S. Haykin and E. Moulines, "From Kalman to Particle Filters," IEEE International Conference on Acoustics, Speech, and Signal Processing, Philadelphia, Pennsylvania, USA, March 2005.

• M.W. Andrews, "Learning And Inference In Nonlinear State-Space Models," Gatsby Unit for Computational Neuroscience, University College, London, U.K., December 2004.

• P.M. Djuric, J.H. Kotecha, J. Zhang, Y. Huang, T. Ghirmai, M. Bugallo, and J. Miguez, "Particle Filtering," IEEE Magazine on Signal Processing, vol 20, no 5, pp. 19-38, September 2003.

• N. Arulampalam, S. Maskell, N. Gordan, and T. Clapp, "Tutorial On Particle Filters For Online Nonlinear/ Non-Gaussian Bayesian Tracking," IEEE Transactions on Signal Processing, vol. 50, no. 2, pp. 174-188, February 2002.

• R. van der Merve, N. de Freitas, A. Doucet, and E. Wan, "The Unscented Particle Filter," Technical Report CUED/F-INFENG/TR 380, Cambridge University Engineering Department, Cambridge University, U.K., August 2000.

• S. Gannot, and M. Moonen, "On The Application Of The Unscented Kalman Filter To Speech Processing," International Workshop on Acoustic Echo and Noise, Kyoto, Japan, pp 27-30, September 2003.

• J.P. Norton, and G.V. Veres, "Improvement Of The Particle Filter By Better Choice Of The Predicted Sample Set," 15th IFAC Triennial World Congress, Barcelona, Spain, July 2002.

• J. Vermaak, C. Andrieu, A. Doucet, and S.J. Godsill, "Particle Methods For Bayesian Modeling And Enhancement Of Speech Signals," IEEE Transaction on Speech and Audio Processing, vol 10, no. 3, pp 173-185, March 2002.