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### Kalman Filter Notes

Prateek Tandon

Generic Problem

- Imagine watching a small bird flying through a dense jungle.
- You glimpse intermittent flashes of motion.
- You want to guess where the bird is and where it may be in the next time step.
- Bird’s state might be 6-dimensional:

[x,y,z,x’,y’,z’] – three variables for position and three for velocity.

Kalman Filter

Xk = Fk xk-1 + Bk uk + wk (state update)

Zk = Hkxk + vk (measurement update)

Xk – current state

Xk-1 – last state

Uk – control input

Wk ~ N(0,Qk), represents process noise distributed via multivariate zero-mean normal distribution with covariance Qk

Vk ~ N(0,Rk), represents observation nose distributed via multivariate zero-mean normal distribution with covariance Rk

Fk – state transition model

Bk – control input model

Hk – observation model

Kalman Filter Algorithm

PREDICT:

Predicted State

Predicted Covariance

UPDATE:

Innovation and Measurement Residual

Innovation on CovarianceOptimal Kalman GainUpdated State EstimateUpdated Covariance Estimate

Applications

- Radar tracking of planes/missles/navigation
- Smoothing time series data
- Stock market
- People tracking / hand tracking / etc
- Sensor Data
- GPS Location Data smoothing application

Particle Filter Algorithm

Function PARTICLE-FILTERING(e,N,dbn) returns a set of samples for the next time step

Inputs: e, the new incoming evidence

N, the number of samples to be maintained

Dbn, a DBN with prior P(X0), transition model P(X1|X0), sensor model P(E1|X1)

Persistent: S, a vector of samples of size N, initially generated from P(X0)

Local variables: W, a vector of weights of size N

For i=1 to N do

S[i] sample from P(X1 | X0 = S[i])

W[i} P(E | X1 = S[i])

S WEIGHTED-SAMPLE-WITH-REPLACEMENT(N,S,W)

Return S

Particle Filter Example

Raint+1

Raint+1

Raint+1

Raint

(a) Propagate

(b) Weight,

[Not Umbrella observed.]

(c) Resample

References

- "Kalman Filter." . WIKIPEDIA, 13 APRIL 2013. Web. 13 Apr 2013. <http://en.wikipedia.org/wiki/Kalman_filter>.
- Russell, Stuart, and Peter Norvig. Artificial Intelligence: A Modern Approach. 3rd. New Jersey: Pearson Education Inc., 2010. Print.

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