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 pp1, p2, pN
• piP(q1Si)
• Transition matrix A NxN
• aijP(qt1Sj qtSi)

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

• ? states observed through an observation x

• Model parameter ?p1pN, µ1...µ?, S1...S?

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 OO1, O2,

9
Hidden Markov ModelDefinition (2/2)
Similar to Markov Chain

• ?(A, B, p) Hidden Markov Model
• Aaij State transition probabilities
• aijP(qt1Sj qtSi)
• ppi initial state distribution
• piP(q1Si)

Similar to Mixture Model

• ?bi(v) Observation probability distribution
• bi(v)P(Otv qtSi)

Similar to Markov Chain
10
HMM Graph
Similar to Markov Chain
Similar to Mixture Model
11
The three basic problems

• Evaluation
• O, ? ? P(O?)
• Uncover the hidden part
• O, ? ? Q that P(QO, ?) is maximum
• Learning
• ? ? ? that P(O?) is maximum

12
Evaluation

• O, ? ? P(O?)
• 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, ? ? Q that P(QO, ?) 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

• ? ? ? that P(O?) 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?