1 / 30

CPSC 7373: Artificial Intelligence Lecture 12: Hidden Markov Models and Filters

CPSC 7373: Artificial Intelligence Lecture 12: Hidden Markov Models and Filters. Jiang Bian, Fall 2012 University of Arkansas at Little Rock. Hidden Markov Models. Hidden Markov Models (HMMs) t o analyze, or t o predict t ime series Applications: Robotics Medical Finance Speech and

ozzy
Download Presentation

CPSC 7373: Artificial Intelligence Lecture 12: Hidden Markov Models and Filters

An Image/Link below is provided (as is) to download presentation 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. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. CPSC 7373: Artificial IntelligenceLecture 12: Hidden Markov Models and Filters Jiang Bian, Fall 2012 University of Arkansas at Little Rock

  2. Hidden Markov Models • Hidden Markov Models (HMMs) • to analyze, or • to predict • time series • Applications: • Robotics • Medical • Finance • Speech and • Language Technologies • etc.

  3. Bayes Network of HMMs • HMMs • A sequence of states that evolves over time; • Each state only depends on the previous state; and • Each state emits a measurement • Filters • Kalman Filters • Particle Filters S1 S2 S3 Sn . . . Zn Z1 Z2 Z3

  4. Markov Chain The initial state: P(R0) = 1 P(S0) = 0 0.4 R S 0.8 0.6 0.2 P(R1) = ??? P(R2) = ??? P(R3) = ???

  5. Markov Chain The initial state: P(R0) = 1 P(S0) = 0 0.4 R S 0.8 0.6 0.2 P(R1) = 0.6 P(R1|R0) * P(R0) = 0.6 P(R2) = 0.44 P(R2) = P(R2|R1) * P(R1) + P(R2|S1)*P(S1) = 0.44 P(R3) = 0.376 P(R3) = P(R3|R2) * P(R2) + P(R3|S2)*P(S2) = 0.376

  6. Markov Chain 0.5 • P(A0) = 1 • P(A1) = ??? • P(A2) = ??? • P(A3) = ??? A B 0.5 1

  7. Markov Chain 0.5 • P(A0) = 1 • P(A1) = 0.5 • P(A2) = 0.75 • P(A3) = 0.625 A B 0.5 1

  8. Stationary Distribution 0.5 • P(A1000) = ??? • Stationary Distribution: • P(At) = P(At-1) • P(At|At-1)*P(At-1) +P(At|Bt-1)*P(Bt-1) • P(At) = X • X = 0.5 * X + 1 * (1- X) • X = 2/3 = P(A) • P(B) = 1 – P(A) = 1/3 A B 0.5 1

  9. Stationary Distribution • ??? 0.4 R S 0.8 0.6 0.2

  10. Stationary Distribution • 1/3 • X = P(Rt) = P(Rt-1) • = 0.6X + 0.2(1-x) • 0.6x = 0.2 • X = 1/3 0.4 R S 0.8 0.6 0.2

  11. Transition Probabilities ? • Finding the transition probabilities from observation. • e.g., R, S, S, S, R, S, R (7 days) • Maximum Likelihood • P(R0) = 1 • P(S|R) = 1, P(R|R) = 0 • P(S|S) = 0.5, P(R|S) = 0.5 R S ? ? ?

  12. Transition Probabilities - Quiz ? • Finding the transition probabilities from observation. • e.g., S, S, S, S, S, R, S, S, S, R, R • Maximum Likelihood • P(R0) = ??; P(S0) = ?? • P(S|R) = ??, P(R|R) = ?? • P(S|S) = ??, P(R|S) = ?? R S ? ? ?

  13. Transition Probabilities - Quiz ? • Finding the transition probabilities from observation. • e.g., S, S, S, S, S, R, S, S, S, R, R • Maximum Likelihood • P(R0) = 0; P(S0) = 1 • P(S|R) = 0.5, P(R|R) = 0.5 • P(S|S) = 0.75, P(R|S) = 0.25 R S ? ? ?

  14. Laplacian Smoothing ? • Laplacian smoothing; k = 1 • P(R0) = ??; P(S0) = ?? • P(S|R) = ??, P(R|R) = ?? • P(S|S) = ??, P(R|S) = ?? R S ? ? ?

  15. Laplacian Smoothing ? • R, S, S, S, S • Laplacian smoothing; k = 1 • P(R0) = 2/3; P(S0) = 1/3 • P(S|R) = 4/5, P(R|R) = 1/5 • P(S|S) = 2/3, P(R|S) = 1/3 R S ? ? ?

  16. Hidden Markov Models • Suppose that I can’t observe the weather (e.g., I am grounded with no windows); • But I can make a guess based on whether my wife carries a umbrella; and • P(U|R) = 0.9; P(-U|R) = 0.1 • P(U|S) = 0.2; P(-U|S) = 0.8 • P(R1|U1) = P(U1|R1) * P(R1) / P(U1) = 0.9 * 0.4 / (0.4*0.9 + 0.2 *0.6) = 0.75 • P(R1) = P(R1|R0)P(R0) + P(R1|S0) P(S0) = 0.4 • P(R0) = ½ • P(S0) = ½ 0.4 R S 0.8 0.6 0.2 U U

  17. Specification of an HMM • N - number of states • Q = {q1; q2; : : : ;qT} - set of states • M - the number of symbols (observables) • O = {o1; o2; : : : ;oT} - set of symbols • A - the state transition probability matrix • aij= P(qt+1 = j|qt= i) • B- observation probability distribution • bj(k) = P(ot= k|qt= j) i≤ k ≤ M • π - the initial state distribution

  18. Specification of an HMM • Full HMM is thus specified as a triplet: • λ = (A,B,π)

  19. Central problems in HMM modelling • Problem 1 Evaluation: • Probability of occurrence of a particular observation sequence, O = {o1,…,ok}, given the model • P(O|λ) • Complicated – hidden states • Useful in sequence classification

  20. Central problems in HMM modelling • Problem 2 Decoding: • Optimal state sequence to produce given observations, O = {o1,…,ok}, given model • Optimality criterion • Useful in recognition problems

  21. Central problems in HMM modelling • Problem 3 Learning: • Determine optimum model, given a training set of observations • Find λ, such that P(O|λ) is maximal

  22. Particle Filter

  23. Particle Filters

  24. Particle Filters

  25. Sensor Information: Importance Sampling

  26. Robot Motion

  27. Sensor Information: Importance Sampling

  28. Robot Motion

  29. Particle Filter Algorithm • Algorithm particle_filter( St-1, ut-1zt): • For Generate new samples • Sample index j(i) from the discrete distribution given by wt-1 • Sample from using and • Compute importance weight • Update normalization factor • Insert • For • Normalize weights

  30. Particle Filters • Pros: • easy to implement • Work well in many applications • Cons: • Don’t work in high dimension spaces • Problems with degenerate conditions. • Very few particles, and • Not much noise in either measurements or controls.

More Related