Introduction To Particle Filtering:

Sanjay Patil and Ryan IrwinIntelligent Electronics Systems

Human and Systems EngineeringCenter for Advanced Vehicular Systems

URL: www.cavs.msstate.edu/hse/ies/publications/seminars/msstate/2005/particle_filtering/

Introduction To Particle Filtering: Integrating Bayesian Models and State Space Representations

http://www.cavs.msstate.edu/hse/ies/publications/seminars/msstate/2005/particle_filtering/

of 22Introduction to Particle Filtering

Abstract

• Conventional approaches to speech recognition use:

Gaussian mixture models to model spectral variation; Hidden Markov models to model temporal variation.

• Particle filtering:

is based on a nonlinear state space representation; does not require Gaussian noise models; can be used for prediction or filtering of a signal;

Approximates the target probability distribution (e.g. amplitude of speech signal);

also known as survival of the fittest, the condensation algorithm, and sequential Monte Carlo filters.


State-Space Representation

),,(

),,( 1

ttttt

ttttt

VNXgY

UNXfX

State equation

Observation (measurement) equation

Where,

• Xt state vector

• Yt noisy observation vector

• Ut, Vt noise vectors

• Nt input (usually, not considered)

• ft(.) system transition function (state transition matrix)

• gt(.) measurement function (observation matrix)

General discrete-time nonlinear, non-Gaussian dynamic system


Nonlinear State Space Model – Phase Lock Loop“A device which continuously tries to track the phase of the incoming signal…”

1 2 1 2sin[ ( ) ( )] sin[2 ( ) ( )]

2 2

d di o i o

k A A k A At t wt t t

Low-pass filter, h(t)

Voltage ControlledOscillator

Si(t)

So(t)

Phase detector

Se(t)

1 2sin[ ( ) ( )]

2

di o

k A At t

1sin[ ( )]iA wt t

2cos[ ( )]oA wt t

•Nonlinear feedback system

•Consider a first order PLL, :

sin ( )

sin ( )

oe

e ie

AK tt

AK tt t

AKsin( )

0

t

-

( )i t ( )e t

( )o t

0

( )sin ( )t

o eAK h t x x dx .

( ) ( )h t t

tt

ttt

XSinKAY

XSinKAUX

1

1

kkk

kkkk

YXSinKAY

XXSinKAUX

State-space equations:


Hidden Markov Model and Nonlinear State-Space ModelNonlinear State-Space Model:

),(

),( 1

kkk

kkk

EXgY

VXfX

Hidden Markov Model:

• Models observations and states with non-linear and non-Gaussian functions g(.) and f(.) along with V and E being noise terms.

• .Generalization of HMM

• Models observations and states with transition probabilities (A), observation probabilities (B), and initial probabilities ().

• Usual HMMs are modeled by first-order Markov chain. The observation probabilities can be modeled as continuous or discrete densities. The model performance improves for continuous densities as compared to discrete densities.

• The calculation is based on forward-backward algorithm for evaluation (scoring)

),,( BA


Particle filter (NSSM) and CD Hidden Markov Model Nonlinear State-Space Model:

),()(1

11,

jmjm

M

mjmk

kkkk

CYbj

XaX

),(

),( 1

kkk

kkk

EXgY

VXfX

CD Hidden Markov Model:

• .Generalization of HMM

• The calculation from X1, to X2 to Xn goes through a prediction and update stage with observation used to update the (predicted value) states

• Use of particles to approximate the target distribution (if particle filtering is implemented).

• Output – depends on prob. formulation

• Can involve variable length of observations

• Models Gaussian Mixtures.

• The calculation is based on forward-backward algorithm for evaluation (scoring)

• Finite number of means and covariance used to model the target distribution.

• Output – depends on prob. formulation

• Most of the times, HMMs work on a uniform length of frame (data)

Both involve Bayes rules for state computation


Particle Filtering

What is a particle?

X

p(X) p(X) continuous probability distribution of

interest (blue)

M

m

mm xxwxp1

)()( )()(

probability distribution of interest (blue))(xp

)(mw weights of the particles

the particles)(mx

the Dirac delta function(.)

where,

Mmmm wxxp 1)()( ,)(

where, approximating random measure

M number of particles


11111 )|()|()|( kkkkkkk dXYXpXXpYXp

Nonlinear State-Space Model – particle filter

• Two stages:

1. Predict stage (using prior equation, transition matrix)

Nonlinear State-Space Model:

)|(

)|( 1

kkk

kxk

XYpY

XXpX

Transition matrix

Observation matrix

2. Update stage (using filtering equation, prior, observation matrix)

)|(

)|()|()|(

1

1

kk

kkkkkk YYp

YXpXYpYXp

)()|()()|(,1 01111 XpYXpandXpYXpkfor kkkk

cumbersome, intractable integrals

solution:

• approximate representation particle filter


Nonlinear State-Space Model – particle filter

Particle filter is

• A method to approximate the continuous pdf

• A method to sample the pdf to help compute the (intractable) integrals

• Generalization of HMM.

Steps in particle filtering algorithm: (similar to Viterbi algorithm)

1. Generate samples to represent the initial probability

2. Using the prior equation, predict the next state

3. Using the observation, get the weights for the states computed. Predicted states (from step 2) along with the weights collectively represent the state distribution

4. Resample it so as to have the uniformly distributed current state omitting the least-significant representation

5. Continue steps 2 through 4, till all the observations are exhausted


Applications

Most of the applications involve tracking

• Ice Hockey Game – tracking the players demo*Ref.* Kenji Okuma, Ali Taleghani, Nando de Freitas, Jim Little and David Lowe. A Boosted Particle Filter: Multitarget Detection and Tracking. 8th European Conference on Compute Vision, ECCV 2004, Prague, Czech Republic. http://www.cs.ubc.ca/~nando/publications.html

Ref.^M. Gabrea, “Robust adaptive Kalman Filtering-based speech enhancement algorithm,” ICASSP 2004, vol 1, pp I-301-4, May 2004.

K. Paliwal, “Estimation of noise variance from the noisy AR signal and its application in speech enhancement,” IEEE transaction on Acoustics, Speech, and Signal Processing, vol 36, no 2, pp 292-294, Feb 1988.

At IES – NSF funded project, particle filtering has been used for: • Time series estimation for speech signal^

• Speaker VerificationSpeech verification algorithm based on HMM and Particle Filtering algorithm.


Time Series Prediction

Implementation :

Problem statement : in presence of noise, estimate the clean speech signal.

Order defines the number of previous samples used for prediction.

Noise calculation is based on Modified Yule-Walker equations.

yt – speech amplitude in presence of noise, xt – cleaned speech signal.

Feature

Extraction

Order of Prediction

Model Estimation

State Predicts State Updates

tttt

ttttt

exHy

vGxFx

1

Number of particles

part of the figure (ref): www.bioid.com/sdk/docs/About_Preprocessing.htm


Particle filter – Detailed step by step analysis

Observation data

Set-up

• Speech signal is sampled at regular intervals – Observations

• Idea – to filter the speech signal by particle filters

• For every frame of signal, LP coefficients and noise covariance for calculated

• After this is – particle filtering algorithm : Assume: order = 4, particles = 5

Five Gaussian particles

samples process noise predicted state

X(k) = A * X(k-1) + V(k)

New Observation data

weights

Y(k)* = B * X(k)

Filtered Obsn data

update states

resampling


Speaker Verification

Hypothesis

Particle filters approximate the probability distribution of a signal

If large number of particles are used, it approximates the pdf better

Attempt will be made to use more Gaussian mixtures as compared to the existing system

Trade-off between number of passes and number of particles

Feature

Extraction

Claimed ID

Classifier Decision

Accept

Reject

Speaker Model Imposter Model

Changes will be made here…


Pattern Recognition Applet

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

• Classification of Signals • PCA - Principal 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

URL: http://www.cavs.msstate.edu/hse/ies/projects/speech/software/demonstrations/applets/util/pattern_recognition/current/index.html


Classification Algorithms – Best Case

• Data sets need to be differentiated

• Classifying 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


Classification Algorithms – Worst Case

• Toroidals are not classified easily 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

• The toroidal is not common in speech patterns


Classification Algorithms – Realistic Case

• A more realistic case of two mixed distributions using RVM

• This algorithm gives a more complex line of discrimination

• More involved computation for RVM yields better results than LDA and PCA

• Again, LDA, PCA, SVM, and RVM are pattern classification algorithms

• More information given online in tutorials about algorithms


Signal Tracking Algorithms – Kalman Filter

• Predicts the next state of the signal given prior information

• Signals must be time based or drawn from left to right

• X-axis represents time axis

• Algorithms interpolate data ensuring periodic sampling

• Kalman filter is shown here


Signal Tracking Algorithms – Particle Filter

• The model has realistic noise

• Gaussian noise is actually generated at each step

• Noise variances and number of particles can be customized

• Algorithm runs as previously described

1. State prediction stage

2. State update stage

• Each step gives a collection of possible next states of signal

• The collection is represented in the black particles

• Mean value of particles becomes the predicted state


Summary

• Particle filtering promises to be one of the nonlinear techniques.

• More points to follow


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.

• M. Gabrea, “Robust Adaptive Kalman Filtering-based Speech Enhancement Algorithm,” ICASSP 2004, vol 1, pp. I-301-I-304, May 2004.

• K. Paliwal, :Estiamtion og noise variance from the noisy AR signal and its application in speech enhancement,” IEEE transaction on Acoustics, Speech, and Signal Processing, vol 36, no 2, pp 292-294, Feb 1988.


References (for PLL):

• Modern Digital and Analog Communication Systems B.P. Lathi, Oxford University Press, Second Edition.• Andrew J. Viterbi, “Phase-Locked Loop Dynamics in the presence of noise by Fokker-Planck Techniques”, Proceedings of the IEEE, 1963.


References (HMM and particle):

• M. Andrews, “Learning and Inference in Nonlinear State-Space Models,” (in preparation).• V. Digalakis, J. Rohlicek, and M. Ostendorf, “,” IEEE transactions on Speech and Audio Processing, vol 1, no 4, October 1993, pp 431-434.


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Proof for the predict stage equation:

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baaaaapbAp

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Observations independent given the states


State-Space Equation and State-Variable Equation

State-Space Equation:

kkk

kkk

uJXHY

uGXFX

1

kkk

kkk

EXHY

VGXFX

1

State-Variable Equation:

• Parameters required are:

F, H, G, X0, p(X0), noise statistics, covariance terms,

Vk and Ek are noise terms

• The calculation from X1, to X2 to Xn goes through a prediction and update stage with observation used to update the (predicted value) states.

• Output term: Xk (hidden / unknown)

• Parameters required are:

F, H, G, X0, U0(input)

• The calculation from X1, to X2 to Xn goes through only one stage. Idea is to find observations Yk.

• Output term: Yk (output / not hidden)

Both involve matrix algebra, carry same names, similar meanings

Introduction To Particle Filtering:

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