A Probability of a mode sequence(2/2) Challenge: Can we incorporate the latest observation? p(d,d, =ok.ok failed |@ -z-1) ok ok ok Sequence(i) 2. Update: Bayes rule Kalman Filter p(,=ok.ok failed, z.a, z,) p(d1…d,|a121…a-1z1-1a1z,) (z,|d1d1,a121…a1-1z1-)p(d1 …aa121…a,-1z1-1) const observation likelihood 0.5rS r given the mode sequence 2T S Prediction 1.p(x,|d1.d1,a11-a1-1z1)=N(x (i)-C 2p(z,|d1d2,a121a11-)=N(H(-,HCH+R Hybrid Mode Estimation and Gaussian Filtering with Hybrid HMM Models 16412/6.834 Lecture,15 March2004 Outline Applications: fault diagnosis, visual tracking Switching linear Gaussian models exact filtering Probabilistic Hybrid Automata filtering Approximate Gaussian filtering with hybrid HMM models Hybrid Mode Estimation and Gaussian Filtering with Hybrid HMM Models 6412/6.834 Lecture,15 March2004Hybrid Mode Estimation and Gaussian Filtering with Hybrid HMM Models 16.412 / 6.834 Lecture, 15 March 2004 21 Probability of a mode sequence (2/2) Challenge: Can we incorporate the latest observation? ok ok ok ok failed ( ... ... | ... ) 1 t 1 1 t t p d d ok ok failed a z a z 2. Update: Bayes rule + Kalman Filter ( ... ... | ... ) 1 t 1 1 t1 t1 p d d ok ok failed a z a z ( ... | ... ) 1 t 1 1 t 1 t 1 t t p d d a z a z a z   Observation likelihood given the mode sequence Prediction ( | ... , ... ) (ˆ , ) ( ) ( ) 1 1 1 1 1     i i 1. p xt d dt a z at zt N xt C Sequence (i) 2. ( | ... , ... ) ( ˆ , ) ( ) ( ) p z d1 d a1z1 a 1z 1 N Hx HC H R i i T t t t t t      S T rS r e S 1 0.5 1/ 2 | 2 | 1   S p z d d a z a z p d d a z a z const t t t t t t t ( | ... , ... ) ( ... | ... ) 1 1 1 1 1 1 1 1 1 1 Hybrid Mode Estimation and Gaussian Filtering with Hybrid HMM Models 16.412 / 6.834 Lecture, 15 March 2004 22 Outline z Applications: fault diagnosis, visual tracking z Switching linear Gaussian models + exact filtering z Probabilistic Hybrid Automata + filtering z Approximate Gaussian filtering with hybrid HMM models