Hidden Markov Models CS 4495 Computer Vision – A. Bobick CS 4495 Computer Vision Hidden Markov Models Aaron Bobick School of Interactive Computing
Hidden Markov Models CS 4495 Computer Vision – A. Bobick Administrivia • PS4 – going OK? • Please share your experiences on Piazza – e.g. discovered something that is subtle about using vl_sift. If you want to talk about what scales worked and why that’s ok too.
Hidden Markov Models CS 4495 Computer Vision – A. Bobick Outline • Time Series • Markov Models • Hidden Markov Models • 3 computational problems of HMMs • Applying HMMs in vision- Gesture Slides “borrowed” from UMd and elsewhere Material from: slides from Sebastian Thrun, and Yair Weiss
Hidden Markov Models CS 4495 Computer Vision – A. Bobick Audio Spectrum Audio Spectrum of the Song of the Prothonotary Warbler
Hidden Markov Models CS 4495 Computer Vision – A. Bobick Bird Sounds Prothonotary Warbler Chestnut-sided Warbler
Hidden Markov Models CS 4495 Computer Vision – A. Bobick Questions One Could Ask Time series classification What bird is this? • Time series prediction How will the song • continue? Is this bird sick? • Outlier detection What phases does this • Time series segmentation song have?
Hidden Markov Models CS 4495 Computer Vision – A. Bobick Other Sound Samples
Hidden Markov Models CS 4495 Computer Vision – A. Bobick Another Time Series Problem Cisco General Electric Intel Microsoft
Hidden Markov Models CS 4495 Computer Vision – A. Bobick Questions One Could Ask Time series prediction Will the stock go up or • down? What type stock is this (eg, • Time series classification risky)? Is the behavior abnormal? • Outlier detection
Hidden Markov Models CS 4495 Computer Vision – A. Bobick Music Analysis
Hidden Markov Models CS 4495 Computer Vision – A. Bobick Questions One Could Ask Time series classification Is this Beethoven or Bach? • Time series Can we compose more of • prediction/generation that? Time series segmentation Can we segment the piece • into themes?
Hidden Markov Models CS 4495 Computer Vision – A. Bobick For vision: Waving, pointing, controlling?
Hidden Markov Models CS 4495 Computer Vision – A. Bobick The Real Question How do we model these problems? • How do we formulate these questions as a • inference/learning problems?
Hidden Markov Models CS 4495 Computer Vision – A. Bobick Outline For Today • Time Series • Markov Models • Hidden Markov Models • 3 computational problems of HMMs • Applying HMMs in vision- Gesture • Summary
Hidden Markov Models CS 4495 Computer Vision – A. Bobick Weather: A Markov Model (maybe?) 80% 60% Sunny Rainy 15% 38% 5% 2% 75% 5% Snowy Probability of moving to a given state depends only on the current state: 1 st Order Markovian 20%
Hidden Markov Models CS 4495 Computer Vision – A. Bobick Ingredients of a Markov Model { , S S ,..., S } States: • 1 2 N State transition probabilities: • = = = a P q ( S | q S ) + ij t 1 i t j Initial state distribution: • π = = P q [ S ] i 1 i 80% Sunny 60% Rainy 15% 38% 2% 5% 5% 75% Snowy 20%
Hidden Markov Models CS 4495 Computer Vision – A. Bobick Ingredients of Our Markov Model { S , S , S } • States: sunny rainy snowy • State transition probabilities: .8 .15 .05 = A .38 .6 .02 .75 .05 .2 • Initial state distribution: 80% Sunny π = 60% Rainy (.7 .25 .05) 15% 38% 2% 5% 5% 75% Snowy 20%
Hidden Markov Models CS 4495 Computer Vision – A. Bobick Probability of a Time Series • Given: • What is the probability of this series? ⋅ ⋅ ⋅ P ( S ) P ( S | S ) P ( S | S ) P ( S | S ) sunny rainy sunny rainy rainy rainy rainy ⋅ ⋅ P ( S | S ) P ( S | S ) snowy rainy snowy snowy = ⋅ ⋅ ⋅ ⋅ ⋅ = 0 . 7 0 . 15 0 . 6 0 . 6 0 . 02 0 . 2 0 . 0001512 . 8 . 15 . 05 π = = ( . 7 . 25 . 05 ) A . 38 . 6 . 02 . 75 . 05 . 2
Hidden Markov Models CS 4495 Computer Vision – A. Bobick Outline For Today • Time Series • Markov Models • Hidden Markov Models • 3 computational problems of HMMs • Applying HMMs in vision- Gesture • Summary
Hidden Markov Models CS 4495 Computer Vision – A. Bobick Hidden Markov Models 80% 60% Sunny 60% 5% NOT OBSERVABLE Rainy 80% 30% 30% Sunny 15% 60% Rainy 15% 38% 38% 2% 5% 5% 75% 10% 5% 65% Snowy 2% 75% 5% Snowy 20% 50% 0% 50% OBSERVABLE 20%
Hidden Markov Models CS 4495 Computer Vision – A. Bobick Probability of a Time Series • Given: • What is the probability of this series? = P ( O ) P ( O , O , O ,..., O ) coat coat umbrella umbrella ∑ ∑ = = P ( O | Q ) P ( Q ) P ( O | q ,..., q ) P ( q ,..., q ) 1 7 1 7 all Q q ,..., q 1 7 = ⋅ ⋅ ⋅ ⋅ + 2 4 6 (0.3 0.1 0.6) (0.7 0.8 ) ... . 8 . 15 . 05 . 6 . 3 . 1 π = = = ( . 7 . 25 . 05 ) A . 38 . 6 . 02 B . 05 . 3 . 65 . 75 . 05 . 2 0 . 5 . 5
Hidden Markov Models CS 4495 Computer Vision – A. Bobick Specification of an HMM • N - number of states • Q = {q 1 ; q 2 ; : : : ;q T } – sequence of states • Some form of output symbols • Discrete – finite vocabulary of symbols of size M. One symbol is “emitted” each time a state is visited (or transition taken). • Continuous – an output density in some feature space associated with each state where a output is emitted with each visit • For a given sequence observation O • O = {o 1 ; o 2 ; : : : ;o T } – o i observed symbol or feature at time i
Hidden Markov Models CS 4495 Computer Vision – A. Bobick Specification of an HMM • A - the state transition probability matrix • a ij = P ( q t +1 = j|q t = i ) • B - observation probability distribution • Discrete: b j ( k ) = P ( o t = k |q t = j ) i ≤ k ≤ M • Continuous b j ( x ) = p( o t = x | q t = j ) • π - the initial state distribution S 2 S 3 S 1 • π (j) = P(q 1 = j) • Full HMM over a of states and output space is thus specified as a triplet: λ = (A,B, π )
Hidden Markov Models CS 4495 Computer Vision – A. Bobick What does this have to do with Vision? • Given some sequence of observations, what “model” generated those? • Using the previous example: given some observation sequence of clothing: • Is this Philadelphia, Boston or Newark? • Notice that if Boston vs Arizona would not need the sequence!
Hidden Markov Models CS 4495 Computer Vision – A. Bobick Outline For Today • Time Series • Markov Models • Hidden Markov Models • 3 computational problems of HMMs • Applying HMMs in vision- Gesture • Summary
Hidden Markov Models CS 4495 Computer Vision – A. Bobick The 3 great problems in HMM modelling 1. Evaluation: Given the model 𝜇 = ( 𝐵 , 𝐶 , 𝜌 ) what is the probability of occurrence of a particular observation sequence 𝑃 = { 𝑝 1 , … , 𝑝 𝑈 } = 𝑄 ( 𝑃 | 𝜇 ) • This is the heart of the classification/recognition problem: I have a trained model for each of a set of classes, which one would most likely generate what I saw. 2. Decoding: Optimal state sequence to produce an observation sequence 𝑃 = { 𝑝 1 , … , 𝑝 𝑈 } • Useful in recognition problems – helps give meaning to states – which is not exactly legal but often done anyway. 3. Learning: Determine model λ , given a training set of observations • Find λ , such that 𝑄 ( 𝑃 | 𝜇 ) is maximal
Hidden Markov Models CS 4495 Computer Vision – A. Bobick Problem 1: Naïve solution • State sequence 𝑅 = ( 𝑟 1 , … 𝑟 𝑈 ) • Assume independent observations: ) ∏ T λ = λ = P ( O | q , P ( o | q , ) b ( o ) b ( o )... b ( o ) t t q 1 1 q 2 2 qT T = i 1 NB: Observations are mutually independent, given the hidden states. That is, if I know the states then the previous observations don’t help me predict new observation. The states encode *all* the information. Usually only kind-of true – see CRFs.
Hidden Markov Models CS 4495 Computer Vision – A. Bobick Problem 1: Naïve solution • But we know the probability of any given sequence of states: λ = π P q ( | ) a a ... a − q 1 q q 1 2 q q 2 3 q T ( 1) qT
Hidden Markov Models CS 4495 Computer Vision – A. Bobick Problem 1: Naïve solution • Given: ) ∏ T λ = λ = P ( O | q , P ( o | q , ) b ( o ) b ( o )... b ( o ) t t q 1 1 q 2 2 qT T = i 1 λ = π P q ( | ) a a ... a − q 1 q q 1 2 q q 2 3 q T ( 1) qT • We get: ∑ λ = λ λ P ( O | ) P ( O | q , ) P ( q | ) q NB: -The above sum is over all state paths -There are 𝑂 𝑈 states paths, each ‘costing’ 𝑃 ( 𝑈 ) calculations, leading to 𝑃 ( 𝑈𝑂 𝑈 ) time complexity.
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