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 Covariance Optimal Kalman Gain Updated State Estimate Updated 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 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

Particle Filter Example Raint+1 Raint+1 Raint Raint+1 (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.