Sanjay Patil and Ryan Irwin Intelligent Electronics Systems, Human and Systems Engineering Center for Advanced Vehicular Systems URL: HUMAN AND SYSTEMS ENGINEERING: Introduction to Particle Filtering

Page 1 of 22 Introduction to Particle Filtering Abstract Most conventional techniques used for speech analysis are based on modeling the speech signal as Gaussian mixture models. Nonlinear approaches are expected to outperform the conventional techniques because of their abilities to compensate for the mismatched channel conditions and to significantly reduce the complexity of the models. Particle filtering is one such nonlinear method based on sequential Monte Carlo technique. Particle filtering works by approximating the target probability distribution. Thus, it greatly reduces the complexities associated with the models.

Page 2 of 22 Introduction to Particle Filtering 5000 samples 500 samples200 samples Consider a some pdf p(x) Generate some random samples Conclusion More the number of samples better is the distribution function represented. The number of samples drawn at a particular probability represent the weight (contribution) by those samples towards the distribution function The contribution is called as the weight of the sample. Each sample is called as ‘Particle’ Drawing samples to represent a probability distribution function Concept of particles and their weights weight

Page 3 of 22 Introduction to Particle Filtering Particle filtering algorithm Different Names Sequential Monte Carlo filters Bootstrap filters Condensation Algorithm Survival of the fittest Problem Statement 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 4 of 22 Introduction to Particle Filtering Assume that pdf p(x k-1 | y 1:k-1 ) is available at time k -1. Prediction stage: This is a priori of the state at time k ( without the information on measurement). Thus, it is the probability of the state given only the previous measurements Update stage: This is posterior pdf from predicted prior pdf and newly available measurement. Particle filtering algorithm continued General two-stage framework (Prediction-Update stages)

Page 5 of 20 Introduction to Particle Filtering Particle filtering algorithm step-by-step (1) time Measurements / Observations States (unknown / hidden) cannot be measured Initial set-up: No observations available Known parameters – x 0, p(x 0 ), p(x k |x k-1 ), p(y k |x k ), noise statistics Draw samples to represent x0 by its distribution p(x0)

Page 6 of 20 Introduction to Particle Filtering Particle filtering algorithm step-by-step (2) time Measurements / Observations States (unknown / hidden) cannot be measured Known parameters – x 0, p(x 0 ), p(x k |x k-1 ), p(y k |x k ), noise statistics Still no observations or measurements are available. Predict x1 using equation

Page 7 of 20 Introduction to Particle Filtering Particle filtering algorithm step-by-step (3) time Measurements / Observations States (unknown / hidden) cannot be measured Known parameters – x 0, p(x 0 ), p(x k |x k-1 ), p(y k |x k ), noise statistics First observation / measurement is available. Update x1 using equation

Page 8 of 20 Introduction to Particle Filtering Particle filtering algorithm step-by-step (4) time Measurements / Observations States (unknown / hidden) cannot be measured Known parameters – x 0, p(x 0 ), p(x k |x k-1 ), p(y k |x k ), noise statistics Second observation / measurement is NOT available. Predict x2 using equation

Page 9 of 20 Introduction to Particle Filtering Particle filtering algorithm step-by-step (5) time Measurements / Observations States (unknown / hidden) cannot be measured Known parameters – x 0, p(x 0 ), p(x k |x k-1 ), p(y k |x k ), noise statistics Second observation / measurement is available. Update x2 using equation

Page 10 of 22 Introduction to Particle Filtering Particle filtering - visualization Drawing samples Predicting next state Updating this state What is THIS STEP??? Resampling….

Page 11 of 22 Introduction to Particle Filtering Sampling Importance Resample algorithm (necessity)

Page 12 of 22 Introduction to Particle Filtering Applications Most of the applications involve tracking 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 For IES NSF funded project, particle filtering has been used for: Time series estimation for speech signal (Java demo) Speaker Verification (details on next slide)

Page 13 of 22 Introduction to Particle Filtering Speaker Verification Time series estimation of speech signal Speaker Verification: Hypothesis: particle filters approximate the probability distribution of a signal. If large number of particles are used, it approximates the pdf better. Only needed is the initial guess of the distribution. ! How are we going to achieve this..

Page 14 of 22 Introduction to Particle Filtering 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 URL:

Page 15 of 22 Introduction to Particle Filtering Classification – 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

Page 16 of 22 Introduction to Particle Filtering Classification – 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

Page 17 of 22 Introduction to Particle Filtering Classification – 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

Page 18 of 22 Introduction to Particle Filtering Signal Tracking – Kalman Filter The input signals are now time based with the x-axis representing time Signal tracking algorithms interpolate data Interpolation ensures that the input samples are at regular intervals Sampling is always done on regular intervals Kalman filter is shown here

Page 19 of 22 Introduction to Particle Filtering Signal Tracking – Particle Filter Algorithm 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 Average of the black particles is where the overall state is predicted

Page 20 of 22 Introduction to Particle Filtering Summary Particle filtering promises to be one of the nonlinear techniques. More points to follow

Page 21 of 22 Introduction to Particle Filtering References S. Haykin and E. Moulines, "From Kalman to Particle Filters," IEEE International Conference on Acoustics, Speech, and Signal Processing, Philadelphia, Pennsylvania, USA, March M.W. Andrews, "Learning And Inference In Nonlinear State-Space Models," Gatsby Unit for Computational Neuroscience, University College, London, U.K., December 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 , September 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 , February 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 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 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 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 , March 2002.