CS325 Artificial Intelligence Ch. 15,20 Hidden Markov Models and Particle Filtering
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1 CS325 Artificial Intelligence Ch. 15,20 Hidden Markov Models and Particle Filtering Cengiz Günay, Emory Univ. Günay Ch. 15,20 Hidden Markov Models and Particle FilteringSpring / 21
2 Get Rich Fast! Günay Ch. 15,20 Hidden Markov Models and Particle FilteringSpring / 21
3 Get Rich Fast! Or go bankrupt? Günay Ch. 15,20 Hidden Markov Models and Particle FilteringSpring / 21
4 Get Rich Fast! Or go bankrupt? So, how can we predict time-series data? Günay Ch. 15,20 Hidden Markov Models and Particle FilteringSpring / 21
5 Hidden Markov Models Günay Ch. 15,20 Hidden Markov Models and Particle FilteringSpring / 21
6 Hidden Markov Models Günay Ch. 15,20 Hidden Markov Models and Particle FilteringSpring / 21
7 Entry/Exit Surveys Exit survey: Reinforcement Learning What s the difference between MDPs and Reinforcement Learning? What is the dilemma between exploration and exploitation? Entry survey: Hidden Markov Models (0.25 points of final grade) What previous algorithm would you use for time series prediction? What time series do you wish you could predict? Günay Ch. 15,20 Hidden Markov Models and Particle FilteringSpring / 21
8 Time Series Prediction? Have we done this before? Günay Ch. 15,20 Hidden Markov Models and Particle FilteringSpring / 21
9 Time Series Prediction? Have we done this before? Belief states with action schemas? Günay Ch. 15,20 Hidden Markov Models and Particle FilteringSpring / 21
10 Time Series Prediction? Have we done this before? Belief states with action schemas? Not for continuous variables Goal-based Günay Ch. 15,20 Hidden Markov Models and Particle FilteringSpring / 21
11 Time Series Prediction? Have we done this before? Belief states with action schemas? Not for continuous variables Goal-based MDPs and RL? Günay Ch. 15,20 Hidden Markov Models and Particle FilteringSpring / 21
12 Time Series Prediction? Have we done this before? Belief states with action schemas? Not for continuous variables Goal-based MDPs and RL? Goal-based No time sequence Günay Ch. 15,20 Hidden Markov Models and Particle FilteringSpring / 21
13 Time Series Prediction with Hidden Markov Models (HMMs) Dr. Thrun is very happy HMMs are his specialty. HMMs: analyze & predict time series data can deal with noisy sensors Günay Ch. 15,20 Hidden Markov Models and Particle FilteringSpring / 21
14 Time Series Prediction with Hidden Markov Models (HMMs) Dr. Thrun is very happy HMMs are his specialty. HMMs: Example domains: analyze & predict time series data can deal with noisy sensors finance (get rich fast!) robotics medical speech and language Alternatives: Recurrent neural networks (not probabilistic) Günay Ch. 15,20 Hidden Markov Models and Particle FilteringSpring / 21
15 What are HMMs? Markov chain: Hidden states : S 1 S 2 S n Measurements : Z 1 Z n Günay Ch. 15,20 Hidden Markov Models and Particle FilteringSpring / 21
16 What are HMMs? Markov chain: Hidden states : S 1 S 2 S n Measurements : Z 1 Z n It s essentially a Bayes Net! Günay Ch. 15,20 Hidden Markov Models and Particle FilteringSpring / 21
17 What are HMMs? Markov chain: Hidden states : S 1 S 2 S n Measurements : Z 1 Z n It s essentially a Bayes Net! Implementations: Kalman Filter (see Ch. 15) Particle Filter Günay Ch. 15,20 Hidden Markov Models and Particle FilteringSpring / 21
18 Video: Lost Robots, Speech Recognition
19 Future Prediction with Markov Chains Is tomorrow going to be Rainy or Sunny? R S Günay Ch. 15,20 Hidden Markov Models and Particle FilteringSpring / 21
20 Future Prediction with Markov Chains Is tomorrow going to be Rainy or Sunny? R S Start with today is rainy : P(R 0 ) = 1, then P(S 0 ) = 0 Günay Ch. 15,20 Hidden Markov Models and Particle FilteringSpring / 21
21 Future Prediction with Markov Chains Is tomorrow going to be Rainy or Sunny? R S Start with today is rainy : P(R 0 ) = 1, then P(S 0 ) = 0 What s P(S 1 ) =? P(S 2 ) =? P(S 3 ) =? Günay Ch. 15,20 Hidden Markov Models and Particle FilteringSpring / 21
22 Future Prediction with Markov Chains Is tomorrow going to be Rainy or Sunny? R S Start with today is rainy : P(R 0 ) = 1, then P(S 0 ) = 0 What s P(S 1 ) = 0.4 P(S 2 ) =? P(S 3 ) =? Günay Ch. 15,20 Hidden Markov Models and Particle FilteringSpring / 21
23 Future Prediction with Markov Chains Is tomorrow going to be Rainy or Sunny? R S Start with today is rainy : P(R 0 ) = 1, then P(S 0 ) = 0 What s P(S 1 ) = 0.4 P(S 2 ) = 0.56 P(S 3 ) =? Günay Ch. 15,20 Hidden Markov Models and Particle FilteringSpring / 21
24 Future Prediction with Markov Chains Is tomorrow going to be Rainy or Sunny? R S Start with today is rainy : P(R 0 ) = 1, then P(S 0 ) = 0 What s P(S 1 ) = 0.4 P(S 2 ) = 0.56 P(S 3 ) = Günay Ch. 15,20 Hidden Markov Models and Particle FilteringSpring / 21
25 Future Prediction with Markov Chains Is tomorrow going to be Rainy or Sunny? R S Start with today is rainy : P(R 0 ) = 1, then P(S 0 ) = 0 What s P(S 1 ) = 0.4 P(S 2 ) = 0.56 P(S 3 ) = P(S t+1 ) = 0.4 P(R t ) P(S t ) Günay Ch. 15,20 Hidden Markov Models and Particle FilteringSpring / 21
26 Back to the Future? How far can we see into the future? P(A ) =? Until it reaches a stationary state (or limit cycle) Günay Ch. 15,20 Hidden Markov Models and Particle FilteringSpring / 21
27 Back to the Future? How far can we see into the future? P(A ) =? Until it reaches a stationary state (or limit cycle) Use calculus: lim t P(A t+1) = P(A t ) Günay Ch. 15,20 Hidden Markov Models and Particle FilteringSpring / 21
28 Back to the Future? How far can we see into the future? P(A ) =? Until it reaches a stationary state (or limit cycle) Use calculus: lim t P(A t+1) = P(A t ) R S P(S ) =? Günay Ch. 15,20 Hidden Markov Models and Particle FilteringSpring / 21
29 Back to the Future? How far can we see into the future? P(A ) =? Until it reaches a stationary state (or limit cycle) Use calculus: lim t P(A t+1) = P(A t ) R S P(S ) = 2/3 Günay Ch. 15,20 Hidden Markov Models and Particle FilteringSpring / 21
30 Back to the Future? How far can we see into the future? P(A ) =? Until it reaches a stationary state (or limit cycle) Use calculus: lim t P(A t+1) = P(A t ) R S P(S ) = 2/3 lim P(S t+1) = 0.4 P(R t ) P(S t ), t subst. x = P(S t+1 ) = P(S t ) = 1 P(R t ) = Günay Ch. 15,20 Hidden Markov Models and Particle FilteringSpring / 21
31 And How Do We Get The Transition Probabilities??? R S?? Observed sequence in Atlanta : RRSRRRSR Use Maximum Likelihood Günay Ch. 15,20 Hidden Markov Models and Particle FilteringSpring / 21
32 And How Do We Get The Transition Probabilities??? R S?? Observed sequence in Atlanta : RRSRRRSR Use Maximum Likelihood P(S S) =? Günay Ch. 15,20 Hidden Markov Models and Particle FilteringSpring / 21
33 And How Do We Get The Transition Probabilities??? R S?? Observed sequence in Atlanta : RRSRRRSR Use Maximum Likelihood P(S S) = observed transitions total transitions from S = 0 2 Günay Ch. 15,20 Hidden Markov Models and Particle FilteringSpring / 21
34 And How Do We Get The Transition Probabilities??? R S?? Observed sequence in Atlanta : RRSRRRSR Use Maximum Likelihood P(S S) = observed transitions total transitions from S = 0 2 P(R S) = 2/2, P(S R) = 2/5, P(R R) = 3/5 Günay Ch. 15,20 Hidden Markov Models and Particle FilteringSpring / 21
35 And How Do We Get The Transition Probabilities??? R S?? Observed sequence in Atlanta : RRSRRRSR Use Maximum Likelihood P(S S) = Edge effects? P(S S) = 0? observed transitions total transitions from S = 0 2 P(R S) = 2/2, P(S R) = 2/5, P(R R) = 3/5 Günay Ch. 15,20 Hidden Markov Models and Particle FilteringSpring / 21
36 Overcoming Overfitting: Remember Laplacian Smoothing? Observed sequence in Atlanta : RRSRRRSR Laplacian smoothing K = 1 P(S S) = observed transitions total transitions from S Günay Ch. 15,20 Hidden Markov Models and Particle FilteringSpring / 21
37 Overcoming Overfitting: Remember Laplacian Smoothing? Observed sequence in Atlanta : RRSRRRSR Laplacian smoothing K = 1 P(S S) = observed transitions + K total transitions from S + N Günay Ch. 15,20 Hidden Markov Models and Particle FilteringSpring / 21
38 Overcoming Overfitting: Remember Laplacian Smoothing? Observed sequence in Atlanta : RRSRRRSR Laplacian smoothing K = 1 P(S S) = K, N selected such that 0 P 1. observed transitions + K total transitions from S + N = Günay Ch. 15,20 Hidden Markov Models and Particle FilteringSpring / 21
39 Where is Markov Hidden? R S H G H G Hidden: rainy or sunny Observe: happy or grumpy Günay Ch. 15,20 Hidden Markov Models and Particle FilteringSpring / 21
40 Where is Markov Hidden? R S H G H G Hidden: rainy or sunny Observe: happy or grumpy Initial conditions P(R 0 ) = 1/2, P(S 0 ) = 1/2 P(S 1 H 1 ) =? Günay Ch. 15,20 Hidden Markov Models and Particle FilteringSpring / 21
41 Where is Markov Hidden? R S H G H G Hidden: Observe: rainy or sunny happy or grumpy Initial conditions P(R 0 ) = 1/2, P(S 0 ) = 1/2 P(S 1 H 1 ) = P(H 1 S 1 )P(S 1 ) P(H 1 ) Bayes rule! Günay Ch. 15,20 Hidden Markov Models and Particle FilteringSpring / 21
42 Congrats, Done with Prediction and State Estimation What else can we do with HMMs? Localization of the lost robot Blindfolded person Günay Ch. 15,20 Hidden Markov Models and Particle FilteringSpring / 21
43 Congrats, Done with Prediction and State Estimation What else can we do with HMMs? Localization of the lost robot Blindfolded person Video: Robot localization Günay Ch. 15,20 Hidden Markov Models and Particle FilteringSpring / 21
44 HMMs, Formally Hidden states : S 1 S 2 S n Measurements : Z 1 Z n Question: P(S 1 S 2 ) P(S n S 2 )? Günay Ch. 15,20 Hidden Markov Models and Particle FilteringSpring / 21
45 HMMs, Formally Hidden states : S 1 S 2 S n Measurements : Z 1 Z n Question: P(S 1 S 2 ) P(S n S 2 ) Yes! Past and future are independent. Günay Ch. 15,20 Hidden Markov Models and Particle FilteringSpring / 21
46 HMMs, Formally Hidden states : S 1 S 2 S n Measurements : Z 1 Z n Question: P(S 1 S 2 ) P(S n S 2 ) Yes! Past and future are independent. HMM equations: State estimation: P(S 1 Z 1 ) = αp(z 1 S 1 )P(S 1 ) Günay Ch. 15,20 Hidden Markov Models and Particle FilteringSpring / 21
47 HMMs, Formally Hidden states : S 1 S 2 S n Measurements : Z 1 Z n Question: P(S 1 S 2 ) P(S n S 2 ) Yes! Past and future are independent. HMM equations: State estimation: P(S 1 Z 1 ) = αp(z 1 S 1 )P(S 1 ) Prediction: P(S 2 ) = S 1 P(S 2 S 1 )P(S 1 ) Günay Ch. 15,20 Hidden Markov Models and Particle FilteringSpring / 21
48 HMMs for Localization Example Robot knows map, but not location: use multiplication and convolution Günay Ch. 15,20 Hidden Markov Models and Particle FilteringSpring / 21
49 HMMs for Localization Example Robot knows map, but not location: use multiplication and convolution Günay Ch. 15,20 Hidden Markov Models and Particle FilteringSpring / 21
50 HMMs for Localization Example Robot knows map, but not location: use multiplication and convolution Günay Ch. 15,20 Hidden Markov Models and Particle FilteringSpring / 21
51 HMMs for Localization Example Robot knows map, but not location: use multiplication and convolution Günay Ch. 15,20 Hidden Markov Models and Particle FilteringSpring / 21
52 HMMs for Localization Example Robot knows map, but not location: use multiplication and convolution Günay Ch. 15,20 Hidden Markov Models and Particle FilteringSpring / 21
53 HMMs for Localization Example Robot knows map, but not location: use multiplication and convolution Günay Ch. 15,20 Hidden Markov Models and Particle FilteringSpring / 21
54 Particle Filters: For Clean Water? Günay Ch. 15,20 Hidden Markov Models and Particle FilteringSpring / 21
55 Particle Filters: For Clean Water? Nope, but same idea. Video: Robot localization with particle filters Günay Ch. 15,20 Hidden Markov Models and Particle FilteringSpring / 21
56 Particle Filters: For Clean Water? Nope, but same idea. Video: Robot localization with particle filters Belief representation Points are hypotheses Particles survive if consistent with measurements Easy implementation! Günay Ch. 15,20 Hidden Markov Models and Particle FilteringSpring / 21
57 Localization with Particle Filters Particle filtering: weights show likelihood; pick particles, shift, and repeat Günay Ch. 15,20 Hidden Markov Models and Particle FilteringSpring / 21
58 Localization with Particle Filters Particle filtering: weights show likelihood; pick particles, shift, and repeat Günay Ch. 15,20 Hidden Markov Models and Particle FilteringSpring / 21
59 Localization with Particle Filters Particle filtering: weights show likelihood; pick particles, shift, and repeat Günay Ch. 15,20 Hidden Markov Models and Particle FilteringSpring / 21
60 Localization with Particle Filters Particle filtering: weights show likelihood; pick particles, shift, and repeat Günay Ch. 15,20 Hidden Markov Models and Particle FilteringSpring / 21
61 Localization with Particle Filters Particle filtering: weights show likelihood; pick particles, shift, and repeat Günay Ch. 15,20 Hidden Markov Models and Particle FilteringSpring / 21
62 Localization with Particle Filters Particle filtering: weights show likelihood; pick particles, shift, and repeat Continuous space! Computational resources used efficiently! Günay Ch. 15,20 Hidden Markov Models and Particle FilteringSpring / 21
63 Particle Filter Algorithm S: Particle set {< x, w >,...}, U: Control vector (e.g., map), Z: Measure vector S = Ø, η = 0 For i=1... n sample j {w} w/ replacement x P(x U, S j ) w = P(Z x ) η = η + w S = S {< x, w >} End For i=1... n // Normalization step w i = 1 η w i End Günay Ch. 15,20 Hidden Markov Models and Particle FilteringSpring / 21
64 Particle Filter Pros & Cons In general works well! Stanley uses it for navigation. Pros: Easy to implement Efficient Complex and changing environments in robotics Cons: Dimensionality problem: need many particles Problems with degenerate conditions (adding noise may help) Günay Ch. 15,20 Hidden Markov Models and Particle FilteringSpring / 21
65 Time Series Prediction Conclusion Particle filtering: Most widely used algorithm! Can handle time series and uncertainty Other application areas: Financial prediction Weather Alternative methods: Kalman filters Recurrent neural nets Günay Ch. 15,20 Hidden Markov Models and Particle FilteringSpring / 21
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