Tutorial on Hidden Markov Models

1. Tutorial onHidden Markov Models

2. Overview • Markov chains • Mixture Models • Hidden Markov Model • Definition • Three basic problems • Issues

3. Markov chain: an example Weather model: • 3 states {rainy, cloudy, sunny} Problem: • Forecast weather state, based on the current weather state

4. Markov chain – Model Definition • N States, {S1, S2,… SN} • Sequence of states Q ={q1, q2,…} • Initial probabilities π={π1, π2,… πN} • πi=P(q1=Si) • Transition matrix A NxN • aij=P(qt+1=Sj | qt=Si)

5. Mixture Models: an example Weather model: • 3 “hidden” states • {rainy, cloudy, sunny} • Measure weather-related variables (e.g. temperature, humidity, barometric pressure) Problem: • Given the values of the weather variables, what is the state?

6. Gaussian Mixture Model Definition • Ν statesobserved through an observation x • Model parameter θ={p1…pN, μ1...μΝ, Σ1...ΣΝ}

7. HMM: an example Weather model: • 3 “hidden” states • {rainy, cloudy, sunny} • Measure weather-related variables (e.g. temperature, humidity, barometric pressure) Problem: • Forecast the weather state, given the current weather variables

8. Hidden Markov ModelDefinition (1/2) • N “hidden” States, {S1, S2,… SN} • Sequence of states Q ={q1, q2,…} • Sequence of observations O={O1, O2, …}

9. Hidden Markov ModelDefinition (2/2) Similar to Markov Chain • λ=(A, B, π): Hidden Markov Model • A={aij}: State transition probabilities • aij=P(qt+1=Sj | qt=Si) • π={πi}: initial state distribution • πi=P(q1=Si) Similar to Mixture Model • Β={bi(v)}: Observation probability distribution • bi(v)=P(Ot=v | qt=Si) Similar to Markov Chain

10. HMM Graph Similar to Markov Chain Similar to Mixture Model

11. The three basic problems • Evaluation • O, λ → P(O|λ) • Uncover the hidden part • O, λ →Q that P(Q|O, λ) is maximum • Learning • {Ο} → λ that P(O|λ) is maximum

12. Evaluation • O, λ → P(O|λ) • Solved by using the forward-backward procedure • Applications • Evaluation of a sequence of observations • Find most suitable HMM • Used in the other two problems

13. Uncover the hidden part • O, λ →Q that P(Q|O, λ) is maximum • Solved by Viterbi algorithm • Applications • Find the real states • Learn about the structure of the model • Estimate statistics of the states • Used in the learning problem

14. Learning • {Ο} → λ that P(O|λ) is maximum • No analytic solution • Usually solved by Baum-Welch (EM variation) • Applications • Unsupervised Learning (single HMM) • Supervised Learning (multiple HMM)

15. Some issues • Limitations imposed by • Markov chain • Mixture model • Scalability • Learning • Initialisation • Model order

16. Questions?