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State Estimation and Kalman Filtering CS B659 Spring 2013 Kris Hauser

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Motivation Observing a stream of data Monitoring (of people, computer systems, etc) Surveillance, tracking Finance & economics Science Questions: Modeling & forecasting Handling partial and noisy observations

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Markov Chains Sequence of probabilistic state variables X 0,X 1,X 2,… E.g., robot’s position, target’s position and velocity, … X0X0 X1X1 X2X2 X3X3 Observe X 1 X 0 independent of X 2, X 3, … P(X t |X t-1 ) known as transition model

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Inference in MC Prediction: the probability of future state? P(X t ) = x0,…,xt-1 P (X 0,…,X t ) = x0,…,xt-1 P (X 0 ) x1,…,xt P(X i |X i-1 ) = xt-1 P(X t |X t-1 ) P(X t-1 ) “Blurs” over time, and approaches stationary distribution as t grows Limited prediction power Rate of blurring known as mixing time [Incremental approach]

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Modeling Partial Observability Hidden Markov Model (HMM) X0X0 X1X1 X2X2 X3X3 O1O1 O2O2 O3O3 Hidden state variables Observed variables P(O t |X t ) called the observation model (or sensor model)

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Filtering Name comes from signal processing Goal: Compute the probability distribution over current state given observations up to this point X0X0 X1X1 X2X2 O1O1 O2O2 Query variable Known Distribution given Unknown

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Filtering Name comes from signal processing Goal: Compute the probability distribution over current state given observations up to this point P(X t |o 1:t ) = x t-1 P(x t-1 |o 1:t-1 ) P(X t |x t-1,o t ) P(X t |X t-1,o t ) = P(o t |X t-1,X t )P(X t |X t-1 )/P(o t |X t-1 ) = P(o t |X t )P(X t |X t-1 ) X0X0 X1X1 X2X2 O1O1 O2O2 Query variable Known Distribution given Unknown

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Kalman Filtering In a nutshell Efficient probabilistic filtering in continuous state spaces Linear Gaussian transition and observation models Ubiquitous for state tracking with noisy sensors, e.g. radar, GPS, cameras

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Hidden Markov Model for Robot Localization Use observations + transition dynamics to get a better idea of where the robot is at time t X0X0 X1X1 X2X2 X3X3 z1z1 z2z2 z3z3 Hidden state variables Observed variables Predict – observe – predict – observe…

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Hidden Markov Model for Robot Localization Use observations + transition dynamics to get a better idea of where the robot is at time t Maintain a belief state b t over time b t (x) = P(X t =x|z 1:t ) X0X0 X1X1 X2X2 X3X3 z1z1 z2z2 z3z3 Hidden state variables Observed variables Predict – observe – predict – observe…

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Bayesian Filtering with Belief States

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Update via the observation z t Predict P(X t |z 1:t-1 ) using dynamics alone Bayesian Filtering with Belief States

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In Continuous State Spaces…

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Key Representational Decisions Pick a method for representing distributions Discrete: tables Continuous: fixed parameterized classes vs. particle-based techniques Devise methods to perform key calculations (marginalization, conditioning) on the representation Exact or approximate?

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Gaussian Distribution

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Linear Gaussian Transition Model for Moving 1D Point Consider position and velocity x t, v t Time step h Without noise x t+1 = x t + h v t v t+1 = v t With Gaussian noise of std P(x t+1 |x t ) exp(-(x t+1 – (x t + h v t )) 2 /(2 2 i.e. X t+1 ~ N(x t + h v t, )

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Linear Gaussian Transition Model If prior on position is Gaussian, then the posterior is also Gaussian vh 11 N( , ) N( +vh, + 1 )

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Linear Gaussian Observation Model Position observation z t Gaussian noise of std 2 z t ~ N(x t, )

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Linear Gaussian Observation Model If prior on position is Gaussian, then the posterior is also Gaussian ( 2 z+ 2 2 )/( 2 + 2 2 ) 2 2 2 2 /( 2 + 2 2 ) Position prior Posterior probability Observation probability

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Multivariate Gaussians X ~ N( , )

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Multivariate Linear Gaussian Process A linear transformation + multivariate Gaussian noise If prior state distribution is Gaussian, then posterior state distribution is Gaussian If we observe one component of a Gaussian, then its posterior is also Gaussian y = A x + ~ N( , )

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Multivariate Computations Linear transformations of gaussians If x ~ N( , ), y = A x + b Then y ~ N(A +b, A A T ) Consequence If x ~ N( x, x ), y ~ N( y, y ), z=x+y Then z ~ N( x + y, x + y ) Conditional of gaussian If [x 1,x 2 ] ~ N([ 1 2 ],[ 11, 12 ; 21, 22 ]) Then on observing x 2 =z, we have x 1 ~ N( 1 - 12 22 -1 (z- 2 ), 11 - 12 22 -1 21 )

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Presentation

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Next time Principles Ch. 9 Rekleitis (2004)

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