1. Kalman Filter Notes
Prateek Tandon

2. 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.

3. 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

4. Kalman Filter Algorithm
PREDICT:
Predicted State
Predicted Covariance
UPDATE:
Innovation and Measurement Residual
Innovation on Covariance Optimal Kalman Gain Updated State Estimate Updated Covariance Estimate

5. 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

6. 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

7. Particle Filter Example
P(R0)
0.7 R0
P(R1) T
0.7 F
0.3
Rain0
Rain1
Umbrella1 R1
P(U1) T
0.9 F
0.2

8. Particle Filter Example
Raint+1
Raint+1
Raint
Raint+1
(a) Propagate
(b) Weight,
[Not Umbrella observed.]
(c) Resample

9. References
"Kalman Filter." . WIKIPEDIA, 13 APRIL Web. 13 Apr <
Russell, Stuart, and Peter Norvig. Artificial Intelligence: A Modern Approach. 3rd. New Jersey: Pearson Education Inc., Print.