T EMPORAL P ROBABILISTIC M ODELS P T 2

A GENDA Kalman filtering Dynamic Bayesian Networks Particle filtering

K ALMAN F ILTERING In a nutshell Efficient filtering in continuous state spaces Gaussian transition and observation models Ubiquitous for tracking with noisy sensors, e.g. radar, GPS, cameras

H IDDEN M ARKOV M ODEL FOR R OBOT L OCALIZATION 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…

H IDDEN M ARKOV M ODEL FOR R OBOT L OCALIZATION 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…

B AYESIAN F ILTERING WITH B ELIEF S TATES

Update via the observation z t Predict P(X t |z 1:t-1 ) using dynamics alone B AYESIAN F ILTERING WITH B ELIEF S TATES

I N C ONTINUOUS S TATE S PACES …

G ENERAL B AYESIAN F ILTERING IN C ONTINUOUS S TATE S PACES

K EY R EPRESENTATIONAL D ECISIONS 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?

G AUSSIAN D ISTRIBUTION

L INEAR G AUSSIAN T RANSITION M ODEL FOR M OVING 1D P OINT 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,   )

L INEAR G AUSSIAN T RANSITION M ODEL If prior on position is Gaussian, then the posterior is also Gaussian vh 11 N( ,  )  N(  +vh,  +  1 )

L INEAR G AUSSIAN O BSERVATION M ODEL Position observation z t Gaussian noise of std  2 z t ~ N (x t,   )

L INEAR G AUSSIAN O BSERVATION M ODEL 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

M ULTIVARIATE G AUSSIANS X ~ N( ,  )

M ULTIVARIATE L INEAR G AUSSIAN P ROCESS 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( ,  )

M ULTIVARIATE C OMPUTATIONS 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  (z-  2 ),  11 -  12   21 )

K ALMAN F ILTER A SSUMPTIONS x t ~ N (  x,  x ) x t+1 = F x t + g + v z t+1 = H x t+1 + w v ~ N ( ,  v ), w ~ N ( ,  w ) Dynamics noise Observation noise

T WO S TEPS Maintain  t,  t the parameters of the gaussian distribution over state x t Predict Compute distribution of x t+1 using dynamics model alone Update (observe z t+1 ) Compute P(x t+1 |z t+1 ) with Bayes rule

T WO S TEPS Maintain  t,  t the parameters of the gaussian distribution over state x t Predict Compute distribution of x t+1 using dynamics model alone x t+1 ~ N (F  t + g, F  t F T +  v ) Let these be N(  ’,  ’) Update Compute P(x t+1 |z t+1 ) with Bayes rule

T WO S TEPS Maintain  t,  t the parameters of the gaussian distribution over state x t Predict Compute distribution of x t+1 using dynamics model alone x t+1 ~ N (F  t + g, F  t F T +  v ) Let these be N(  ’,  ’) Update Compute P(x t+1 |z t+1 ) with Bayes rule Parameters of final distribution  t+1 and  t+1 derived using the conditional distribution formulas

D ERIVING THE U PDATE R ULE xtztxtzt ’a’a = N (, )  ’ B B T C x t ~ N(  ’,  ’) (1) Unknowns a,B,C (3) Assumption (7) Conditioning (1) x t | z t ~ N(  ’-BC -1 (z t -a),  ’-BC -1 B T ) (2) Assumption z t | x t ~ N(H x t,  W ) C-B T  ’ -1 B =  W => C = H  ’ H T +  W H x t = a-B T  ’ -1 (x t -  ’) => a=H  ’, B T =H  ’ (5) Set mean (4)=(3) (6) Set cov. (4)=(3) (8,9) Kalman filter  t =  ’ -  ’H T C -1 (z t -H  ’) (4) Conditioning (1) z t | x t ~ N(a-B T  ’ -1 x t, C-B T  ’ -1 B)  t =  ’ -  ’H T C -1 H  ’

P UTTING IT TOGETHER Transition matrix F, covariance  x Observation matrix H, covariance  z  t+1 = F  t + K t+1 (z t+1 – HF  t )  t+1 = (I - K t+1 )(F  t F T +  x ) Where K t+1 = (F  t F T +  x )H T (H(F  t F T +  x )H T +  z ) -1 Got that memorized?

P ROPERTIES OF K ALMAN F ILTER Optimal Bayesian estimate for linear Gaussian transition/observation models Need estimates of covariance… model identification necessary Extensions to nonlinear transition/observation models work as long as they aren’t too nonlinear Extended Kalman Filter Unscented Kalman Filter

Tracking the velocity of a braking obstacle Learning that the road is slick Actual max deceleration Braking begins Estimated max deceleration Velocity initially uninformed More distance measurements arrive Obstacle slows Stopping distance (95% confidence interval) Braking initiatedGradual stop

N ON -G AUSSIAN DISTRIBUTIONS Gaussian distributions are a “lump” Kalman filter estimate

N ON -G AUSSIAN DISTRIBUTIONS Integrating continuous and discrete states Splitting with a binary choice “up” “down”

E XAMPLE : F AILURE DETECTION Consider a battery meter sensor Battery = true level of battery BMeter = sensor reading Transient failures: send garbage at time t Persistent failures: send garbage forever

E XAMPLE : F AILURE DETECTION Consider a battery meter sensor Battery = true level of battery BMeter = sensor reading Transient failures: send garbage at time t … Persistent failures: sensor is broken …

D YNAMIC B AYESIAN N ETWORK BMeter t Battery t Battery t-1 BMeter t ~ N(Battery t,  ) (Think of this structure “unrolled” forever…)

D YNAMIC B AYESIAN N ETWORK BMeter t Battery t Battery t-1 BMeter t ~ N(Battery t,  ) P(BMeter t =0 | Battery t =5) = 0.03 Transient failure model

R ESULTS ON T RANSIENT F AILURE E(Battery t ) Transient failure occurs Without model With model Meter reads …

R ESULTS ON P ERSISTENT F AILURE E(Battery t ) Persistent failure occurs With transient model Meter reads …

P ERSISTENT F AILURE M ODEL BMeter t Battery t Battery t-1 BMeter t ~ N(Battery t,  ) P(BMeter t =0 | Battery t =5) = 0.03 Broken t-1 Broken t P(BMeter t =0 | Broken t ) = 1 Example of a Dynamic Bayesian Network (DBN)

R ESULTS ON P ERSISTENT F AILURE E(Battery t ) Persistent failure occurs With transient model Meter reads … With persistent failure model

H OW TO PERFORM INFERENCE ON DBN? Exact inference on “unrolled” BN Variable Elimination – eliminate old time steps After a few time steps, all variables in the state space become dependent! Lost sparsity structure Approximate inference Particle Filtering

P ARTICLE F ILTERING ( AKA S EQUENTIAL M ONTE C ARLO ) Represent distributions as a set of particles Applicable to non- gaussian high-D distributions Convenient implementations Widely used in vision, robotics

P ARTICLE R EPRESENTATION Bel(x t ) = {(w k,x k )} w k are weights, x k are state hypotheses Weights sum to 1 Approximates the underlying distribution

Weighted resampling step P ARTICLE F ILTERING Represent a distribution at time t as a set of N “particles” S t 1,…,S t N Repeat for t=0,1,2,… Sample S[i] from P(X t+1 |X t =S t i ) for all i Compute weight w[i] = P(e|X t+1 =S[i]) for all i Sample S t+1 i from S[.] according to weights w[.]

B ATTERY E XAMPLE BMeter t Battery t Battery t-1 Broken t-1 Broken t Sampling step

B ATTERY E XAMPLE BMeter t Battery t Battery t-1 Broken t-1 Broken t Suppose we now observe BMeter=0 P(BMeter=0|sample) = ?

B ATTERY E XAMPLE BMeter t Battery t Battery t-1 Broken t-1 Broken t Compute weights (drawn as particle size) P(BMeter=0|sample) = ?

B ATTERY E XAMPLE BMeter t Battery t Battery t-1 Broken t-1 Broken t Weighted resampling P(BMeter=0|sample) = ?

B ATTERY E XAMPLE BMeter t Battery t Battery t-1 Broken t-1 Broken t Sampling Step

B ATTERY E XAMPLE BMeter t Battery t Battery t-1 Broken t-1 Broken t Now observe BMeter t = 5

B ATTERY E XAMPLE BMeter t Battery t Battery t-1 Broken t-1 Broken t Compute weights 1 0

B ATTERY E XAMPLE BMeter t Battery t Battery t-1 Broken t-1 Broken t Weighted resample

A PPLICATIONS OF P ARTICLE F ILTERING IN R OBOTICS Simultaneous Localization and Mapping (SLAM) Observations: laser rangefinder State variables: position, walls

S IMULTANEOUS L OCALIZATION AND M APPING (SLAM) Mobile robots Odometry Locally accurate Drifts significantly over time Vision/ladar/sonar Inaccurate locally Global reference frame Combine the two State: (robot pose, map) Observations: (sensor input)

G ENERAL PROBLEM x t ~ Bel (x t ) (arbitrary p.d.f.) x t+1 = f(x t,u,  p ) z t+1 = g(x t+1,  o )  p ~ arbitrary p.d.f.,  o ~ arbitrary p.d.f. Process noise Observation noise

S AMPLING I MPORTANCE R ESAMPLING (SIR) VARIANT Predict Update Resample

A DVANCED F ILTERING T OPICS Mixing exact and approximate representations (e.g., mixture models) Multiple hypothesis tracking (assignment problem) Model calibration Scaling up (e.g., 3D SLAM, huge maps)

N EXT T IME Putting it together: intelligent agents Read R&N 2