3/8/2016. Contents in latter part PATTERN RECOGNITION AND MACHINE LEARNING. Dynamical Systems. Dynamical Systems. Linear Dynamical Systems

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1 Cotets i latter part PATTERN RECOGNITION AND MACHINE LEARNING CHAPTER 13: SEQUENTIAL DATA Liear Dyamical Systems What is differet from HMM? Kalma filter Its stregth ad limitatio Particle Filter Its simple algorithm ad broad applicatio Dyamical Systems State space model of Dyamical systems Now latet variable is cotiuous rather tha discrete z F( z 1, w) State equatio x H( z, v) Observatio equatio Dyamical Systems State space model of Dyamical systems Now latet variable is cotiuous rather tha discrete z F( z 1, w) x H( z, v) or p( z z 1) p x z ) ( Liear Dyamical Systems Special case of dyamical systems Gaussia assumptio o distributio Liear Dyamical Systems Bayesia form of LDS Where z: Latet variable x: Observatio A: Dyamics matrix C: Emissio matrix w, v, u: Noise Sice LDS is liear Gaussia model, joit distributio over all latet ad observed variables is simply Gaussia. 1

2 Kalma Filter Kalma filter does exact iferece i LDS i which all latet ad observed variables are Gaussia (icl. multivariate Gaussia). Kalma filter hadles multiple dimesios i a sigle set of calculatios. Applicatio of Kalma Filter Trackig movig object Blue: True positio Gree: Measuremet Red: Post. estimate Kalma filter has two distict phases: Predict ad Two phases i Kalma Filter Predict Predictio of state estimate ad estimate covariace of state estimate ad estimate covariace with Kalma gai Estimatio of Parameter i LDS Distributio of Z-1 is used as a prior for estimatio of Z Predict Blue: Red: Red: Gree: Blue: Derivatio of Kalma Filter We use sum-product algorithm for efficiet iferece of latet variables. LDS is cotiuous case of HMM (sum -> iteger) Sum-product algorithm Solve (Kalma Gai Matrix) 2

3 What we have estimated? What we have estimated? Predict: : Predict: : Predictio error Predicted mea of Z Observed X Predicted mea of X Limitatio of Kalma Filter Due to assumptio of Gaussia distributio i KF, KF ca ot estimate well i oliear/o- Gaussia problem. Oe simple extesio is mixture of Gaussias I mixture of K Gaussias, is mixture of K Gaussias, ad will comprise mixture of K Gaussias. -> Computatioally itractable Limitatio of Kalma Filter To resolve oliear dyamical system problem, other methods are developed. Exteded Kalma filter: Gaussia approximatio by liearizig aroud the mea of predicted distributio Particle filter: Resamplig method, see later Switchig state-space model: cotiuous type of switchig HMM Particle Filter I oliear/o-gaussia dyamical systems, it is hard to estimate posterior distributio by KF. Apply the samplig-importace-resamplig (SIR) to obtai a sequetial Mote Carlo algorithm, particle filter. Advatages of Particle Filter Simple algorithm -> Easy to implemet Good estimatio i oliear/o-gaussia problem How to Represet Distributio Origial distributio (mixture of Gaussia) Gaussia approximatio Approximatio by PF (distributio of particle) Approximatio by PF (histogram) 3

4 Where s a ladmie? Use metal detector to fid a ladmie (orage star). Where s a ladmie? Radom survey i the field (red circle). The filter draws a umber of radomly distributed estimates, called particles. All particles are give the same likelihood Where s a ladmie? Get respose from each poit (stregth: size). Assig a likelihood to each particle such that the particular particle ca explai the measuremet. Where s a ladmie? Decide the place to survey i ext step. Scale the weight of particles to select the particle for resamplig Where s a ladmie? Itesive survey of possible place Draw radom particles based upo their likelihood (Resamplig). High likelihood -> more particle; low likelihood -> less particle All particle have equal likelihood agai. Operatio of Particle Filter 4

5 Algorithm of Particle Filter Sample represetatio of the posterior distributio p(z X) expressed as a samples {z (l) } with correspodig weights {w (l) }. Draw samples from mixture distributio Use ew observatio x+1 to evaluate the correspodig weights. Limitatio of Particle Filter I high dimesioal models, eormous particles are required to approximate posterior distributio. Repeated resamplig cause degeeracy of algorithm. All but oe of the importace weights are close to zero Avoided by proper choice of resamplig method 5