state estimation and kalman filtering cs b659 spring 2013 kris hauser

State Estimation and Kalman FilteringCS B659Spring 2013Kris Hauser

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

Markov Chains

• Sequence of probabilistic state variables X0,X1,X2,…• E.g., robot’s position, target’s position and velocity, …

X0 X1 X2 X3

Observe X1

X0 independent of X2, X3, …

P(Xt|Xt-1) known as transition model

Inference in MC• Prediction: the probability of future state?

• P(Xt) = Sx0,…,xt-1P (X0,…,Xt)

= Sx0,…,xt-1P (X0) Px1,…,xt P(Xi|Xi-1)

= Sxt-1P(Xt|Xt-1) P(Xt-1)

• “Blurs” over time, and approaches stationary distribution as t grows• Limited prediction power• Rate of blurring known as mixing time

[Incremental approach]

Modeling Partial Observability• Hidden Markov Model (HMM)

X0 X1 X2 X3

O1 O2 O3

Hidden state variables

Observed variables

P(Ot|Xt) called the observation model (or sensor model)

Filtering• Name comes from signal processing• Goal: Compute the probability distribution over current state

given observations up to this point

X0 X1 X2

O1 O2

Query variable

Known

Distribution given

Unknown

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

Hidden Markov Model for Robot Localization• Use observations + transition dynamics to get a better idea of

where the robot is at time t

X0 X1 X2 X3

z1 z2 z3


Observed variables

Predict – observe – predict – observe…

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 bt over time

• bt(x) = P(Xt=x|z1:t)

X0 X1 X2 X3

z1 z2 z3


Observed variables

Predict – observe – predict – observe…

Bayesian Filtering with Belief States• Compute bt, given zt and prior belief bt

• Recursive filtering equation

Update via the observation ztPredict P(Xt|z1:t-1) using dynamics alone

Bayesian Filtering with Belief States• Compute bt, given zt and prior belief bt

• Recursive filtering equation

In Continuous State Spaces…

• Compute bt, given zt and prior belief bt

• Continuous filtering equation

In Continuous State Spaces…

• Compute bt, given zt and prior belief bt

• Continuous filtering equation

• How to evaluate this integral?• How to calculate Z?• How to even represent a belief state?

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?

Gaussian Distribution• Mean m, standard deviation s• Distribution is denoted N(m,s)• If X ~ N(m,s), then

• With a normalization factor

Linear Gaussian Transition Model for Moving 1D Point

• Consider position and velocity xt, vt

• Time step h• Without noise

xt+1 = xt + h vt

vt+1 = vt

• With Gaussian noise of std s1

P(xt+1|xt) exp(-(xt+1 – (xt + h vt))2/(2s12)

i.e. Xt+1 ~ N(xt + h vt, s1)

Linear Gaussian Transition Model• If prior on position is Gaussian, then the posterior is also

Gaussian

vh s1

N(m,s) N(m+vh,s+s1)

Linear Gaussian Observation Model

• Position observation zt

• Gaussian noise of std s2

zt ~ N(xt,s2)

Linear Gaussian Observation Model• If prior on position is Gaussian, then the posterior is also

Gaussian

m (s2z+s22m)/(s2+s2

2)

s2 s2s22/(s2+s2

2)

Position prior

Posterior probability

Observation probability

Multivariate Gaussians• Multivariate analog in N-D space• Mean (vector) m, covariance (matrix) S

• With a normalization factor

X ~ N(m,S)

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 + e e ~ N(m,S)

Multivariate Computations• Linear transformations of gaussians

• If x ~ N(m,S), y = A x + b• Then y ~ N(Am+b, ASAT)

• Consequence• If x ~ N(mx,Sx), y ~ N(my,Sy), z=x+y• Then z ~ N(mx+my,Sx+Sy)

• Conditional of gaussian• If [x1,x2] ~ N([m1 m2],[S11,S12;S21,S22])• Then on observing x2=z, we have

x1 ~ N(m1-S12S22-1(z-m2), S11-S12S22

-1S21)

Presentation

Next time• Principles Ch. 9• Rekleitis (2004)

state estimation and kalman filtering cs b659 spring 2013 kris hauser

Documents