Time series, HMMs, Kalman Filters
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1 Classic HMM tutorial see class website: *L. R. Rabiner, "A Tutorial on Hidden Markov Models and Selected Applications in Speech Recognition," Proc. of the IEEE, Vol.77, No.2, pp , Time series, HMMs, Kalman Filters Machine Learning 10701/15781 Carlos Guestrin Carnegie Mellon University March 28 th, 2005
2 Adventures of our BN hero Compact representation for probability distributions Fast inference Fast learning 1. Naïve Bayes But Who are the most popular kids? 2 and 3. Hidden Markov models (HMMs) Kalman Filters
3 Handwriting recognition Character recognition, e.g., kernel SVMs r r r a c r rr z c c b
4 Example of a hidden Markov model (HMM)
5 Understanding the HMM Semantics X 1 = {a, z} X 2 = {a, z} X 3 = {a, z} X 4 = {a, z} X 5 = {a, z} O 1 = O 2 = O 3 = O 4 = O 5 =
6 HMMs semantics: Details X 1 = {a, z} X 2 = {a, z} X 3 = {a, z} X 4 = {a, z} X 5 = {a, z} O 1 = O 2 = O 3 = O 4 = O 5 = Just 3 distributions:
7 HMMs semantics: Joint distribution X 1 = {a, z} X 2 = {a, z} X 3 = {a, z} X 4 = {a, z} X 5 = {a, z} O 1 = O 2 = O 3 = O 4 = O 5 =
8 Learning HMMs from fully observable data is easy X 1 = {a, z} X 2 = {a, z} X 3 = {a, z} X 4 = {a, z} X 5 = {a, z} O 1 = O 2 = O 3 = O 4 = O 5 = Learn 3 distributions:
9 Possible inference tasks in an HMM X 1 = {a, z} X 2 = {a, z} X 3 = {a, z} X 4 = {a, z} X 5 = {a, z} O 1 = O 2 = O 3 = O 4 = O 5 = Marginal probability of a hidden variable: Viterbi decoding most likely trajectory for hidden vars:
10 Using variable elimination to compute P(X i o 1:n ) X 1 = {a, z} X 2 = {a, z} X 3 = {a, z} X 4 = {a, z} X 5 = {a, z} Compute: O 1 = O 2 = O 3 = O 4 = O 5 = Variable elimination order? Example:
11 What if I want to compute P(X i o 1:n ) for each i? X 1 = {a, z} X 2 = {a, z} X 3 = {a, z} X 4 = {a, z} X 5 = {a, z} Compute: O 1 = O 2 = O 3 = O 4 = O 5 = Variable elimination for each i? Variable elimination for each i, what s the complexity?
12 Reusing computation X 1 = {a, z} X 2 = {a, z} X 3 = {a, z} X 4 = {a, z} X 5 = {a, z} Compute: O 1 = O 2 = O 3 = O 4 = O 5 =
13 The forwards-backwards algorithm X 1 = {a, z} X 2 = {a, z} X 3 = {a, z} X 4 = {a, z} X 5 = {a, z} O 1 = O 2 = O 3 = O 4 = O 5 = Initialization: For i = 2 to n Generate a forwards factor by eliminating X i-1 Initialization: For i = n-1 to 1 Generate a backwards factor by eliminating X i+1 i, probability is:
14 Most likely explanation X 1 = {a, z} X 2 = {a, z} X 3 = {a, z} X 4 = {a, z} X 5 = {a, z} O 1 = O 2 = O 3 = O 4 = O 5 = Compute: Variable elimination order? Example:
15 The Viterbi algorithm X 1 = {a, z} X 2 = {a, z} X 3 = {a, z} X 4 = {a, z} X 5 = {a, z} O 1 = O 2 = O 3 = O 4 = O 5 = Initialization: For i = 2 to n Generate a forwards factor by eliminating X i-1 Computing best explanation: For i = n-1 to 1 Use argmax to get explanation:
16 What about continuous variables? In general, very hard! Must represent complex distributions A special case is very doable When everything is Gaussian Called a Kalman filter One of the most used algorithms in the history of probabilities!
17 Time series data example: Temperatures from sensor network OFFICE OFFICE CONFERENCE STORAGE QUIET PHONE ELEC COPY 5 6 LAB KITCHEN SERVER
18 Operations in Kalman filter X 1 X 2 X 3 X 4 X 5 O 1 = O 2 = O 3 = O 4 = O 5 = Compute Start with At each time step t: Condition on observation Roll-up (marginalize previous time step)
19 Detour: Understanding Multivariate Gaussians Observe attributes Example: Observe X 1 =18 P(X 2 X 1 =18)
20 Characterizing a multivariate Gaussian Mean vector: Covariance matrix:
21 Conditional Gaussians Conditional probabilities P(Y X)
22 Kalman filter with Gaussians X 1 X 2 X 3 X 4 X 5 O 1 = O 2 = O 3 = O 4 = O 5 = Equivalent to a linear system
23 Detour2: Canonical form Standard form and canonical forms are related: Conditioning is easy in canonical form Marginalization easy in standard form
24 Conditioning in canonical form First multiply: Then, condition on value B = y
25 Operations in Kalman filter X 1 X 2 X 3 X 4 X 5 O 1 = O 2 = O 3 = O 4 = O 5 = Compute Start with At each time step t: Condition on observation Roll-up (marginalize previous time step)
26 Roll-up in canonical form First multiply: Then, marginalize X t :
27 Operations in Kalman filter X 1 X 2 X 3 X 4 X 5 O 1 = O 2 = O 3 = O 4 = O 5 = Compute Start with At each time step t: Condition on observation Roll-up (marginalize previous time step)
28 Learning a Kalman filter Must learn: Learn joint, and use division rule:
29 Maximum likelihood learning of a multivariate Gaussian Data: Means are just empirical means: Empirical covariances:
30 What you need to know Hidden Markov models (HMMs) Very useful, very powerful! Speech, OCR, Parameter sharing, only learn 3 distributions Trick reduces inference from O(n 2 ) to O(n) Special case of BN Kalman filter Continuous vars version of HMMs Assumes Gaussian distributions Equivalent to linear system Simple matrix operations for computations
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