A Graphical models revisited Actions Transition distribution P(x, ai,x,) Observation distribution Beliefs p(zx) p(x, ai,x Observations (Z1 Z2 p(21|x) observable Hidden States Hybrid Mode Estimation and Gaussian Filtering with Hybrid HMM Models 16412/6.834 Lecture,15 March2004 11 Review: Hidden markov models Actions Beliefs Observations(Z) Ave States Discrete states actions and observations Transition observation p. written as tables Belief update P,(x)=p(,|x∑px|a,x) Hybrid Mode Estimation and Gaussian Filtering with Hybrid HMM Models 6. 412/6.834 Lecture, 15 March 2004
Hybrid Mode Estimation and Gaussian Filtering with Hybrid HMM Models 16.412 / 6.834 Lecture, 15 March 2004 11 Graphical models revisited a1 b1 Z1 x1 b2 Z2 x2 ( | , ) j i i p x a x Actions Beliefs Observations States Observable Hidden ( | ) i i p z x Model: Transition distribution Observation distribution ( | ) i i p z x ( | , ) j i i p x a x Hybrid Mode Estimation and Gaussian Filtering with Hybrid HMM Models 16.412 / 6.834 Lecture, 15 March 2004 12 Review: Hidden Markov models z Discrete states, actions, and observations z Transition & observation p. written as tables a1 b1 Z1 x1 b2 Z2 x2 ( | , ) j i i p x a x Actions Beliefs Observations States Observable Hidden 7 11 5 9 77 Mass Ave Belief update:
A Review Linear models Kalman filter Actions States Acti Continuous states, actions and observations Linear(linearized) process and measurement model A x,+ Bu,+ qi z,=Hx, +r K,=C H(HC, H+R) 元=A1+B,元=x+K,(z,-H1) C,=ACHA+O C,=(I-K,)C Hybrid Mode Estimation and Gaussian Filtering with Hybrid HMM Models 16412/6.834 Lecture,15 March2004 Switching Linear Systems(SLDS) Discrete and continuous state Also known as jump Markov linear Gaussian model Actions Discrete states(modes) Beliefs Pr(d d r(d)=丌o P(r la, x,) Observations (Z, Continuous state p(zlx) Observable Hidden xu 1=A(duD),+b(dusu, Continuous (X1 +qn(d1+1) states y,=Hx, +r Discrete states d,bd( d2)
Hybrid Mode Estimation and Gaussian Filtering with Hybrid HMM Models 16.412 / 6.834 Lecture, 15 March 2004 13 Review: Linear models, Kalman Filter z Continuous states, actions, and observations z Linear (linearized) process and measurement model a1 b1 Z1 x1 b2 Z2 x2 ( | , ) j i i p x a x Actions Beliefs Observations States Observable Hidden t t1 t1 t1 x Ax Bu q t t t z Hx r Belief update: Hybrid Mode Estimation and Gaussian Filtering with Hybrid HMM Models 16.412 / 6.834 Lecture, 15 March 2004 14 Switching Linear Systems (SLDS) z Discrete and continuous state z Also known as jump Markov linear Gaussian model a1 b1 Z1 x1 b2 Z2 x2 ( | , ) t 1 t p x a x Actions Beliefs Observations Continuous states Observable Hidden ( | ) t t p z x d1 d2 Discrete states Discrete states (modes): 0 0 1 Pr( ) Pr( | ) S d dt dt Continuous state: ( ) ( ) ( ) ( ) 0 0 0 1 1 1 1 1 x v d y Hx r q d x A d x B d u t t t t t t t t t t ( | ) p dt1 dt