# Kalman Filter Notes - PowerPoint PPT Presentation

1 / 9

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:

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

Kalman Filter Notes

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

## 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

• 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

Rain0

Rain1

Umbrella1

### 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.