Hidden Markov Model - York University
Transcript of Hidden Markov Model - York University
![Page 1: Hidden Markov Model - York University](https://reader030.fdocuments.in/reader030/viewer/2022013018/61d14d607961420d3e1a7cf5/html5/thumbnails/1.jpg)
Markov ModelHidden
![Page 2: Hidden Markov Model - York University](https://reader030.fdocuments.in/reader030/viewer/2022013018/61d14d607961420d3e1a7cf5/html5/thumbnails/2.jpg)
Weather Forecast Person’s position over time
Sequential data often arise from measurements of time series
Motivation
Goal Predict new values given old observations
![Page 3: Hidden Markov Model - York University](https://reader030.fdocuments.in/reader030/viewer/2022013018/61d14d607961420d3e1a7cf5/html5/thumbnails/3.jpg)
Markov Model
IdeaRestrict the connectivity of future states to
previous states
![Page 4: Hidden Markov Model - York University](https://reader030.fdocuments.in/reader030/viewer/2022013018/61d14d607961420d3e1a7cf5/html5/thumbnails/4.jpg)
Outline
● Markov Model● Hidden Markov Model● Inference using HMM● Case Study: Speech Recognition● Conclusion
![Page 5: Hidden Markov Model - York University](https://reader030.fdocuments.in/reader030/viewer/2022013018/61d14d607961420d3e1a7cf5/html5/thumbnails/5.jpg)
Markov Model
• Markov Models: Only assume a certain number of previous states to be relevant
• First-order Markov Models: Only the last state is relevant for the current state
![Page 6: Hidden Markov Model - York University](https://reader030.fdocuments.in/reader030/viewer/2022013018/61d14d607961420d3e1a7cf5/html5/thumbnails/6.jpg)
• First-order Markov Models simplify this to
by only taking the last state into account
Markov Model
• Recall: Chain Rule
![Page 7: Hidden Markov Model - York University](https://reader030.fdocuments.in/reader030/viewer/2022013018/61d14d607961420d3e1a7cf5/html5/thumbnails/7.jpg)
Example
![Page 8: Hidden Markov Model - York University](https://reader030.fdocuments.in/reader030/viewer/2022013018/61d14d607961420d3e1a7cf5/html5/thumbnails/8.jpg)
State Space Model
Input: Human trajectory in 2d
Output: Classification• Walking straight• Right turn• Left turn
Problem:State space is not observation space
![Page 9: Hidden Markov Model - York University](https://reader030.fdocuments.in/reader030/viewer/2022013018/61d14d607961420d3e1a7cf5/html5/thumbnails/9.jpg)
Markov ModelHidden
![Page 10: Hidden Markov Model - York University](https://reader030.fdocuments.in/reader030/viewer/2022013018/61d14d607961420d3e1a7cf5/html5/thumbnails/10.jpg)
Is it raining outside??
![Page 11: Hidden Markov Model - York University](https://reader030.fdocuments.in/reader030/viewer/2022013018/61d14d607961420d3e1a7cf5/html5/thumbnails/11.jpg)
Weather (Rain/ No rain) → Hidden
Umbrella (Yes/ No) → Observation
The rain causes the umbrella to appear !
![Page 12: Hidden Markov Model - York University](https://reader030.fdocuments.in/reader030/viewer/2022013018/61d14d607961420d3e1a7cf5/html5/thumbnails/12.jpg)
Hidden Markov Model
Weather
Umbrella
Latent (Hidden) State Variable
Observation Variable Discrete or Continuous
Discrete
![Page 13: Hidden Markov Model - York University](https://reader030.fdocuments.in/reader030/viewer/2022013018/61d14d607961420d3e1a7cf5/html5/thumbnails/13.jpg)
State Space ModelAn observation at time k is only dependent on the state of time k and independent of all states from the beginning to k-1
![Page 14: Hidden Markov Model - York University](https://reader030.fdocuments.in/reader030/viewer/2022013018/61d14d607961420d3e1a7cf5/html5/thumbnails/14.jpg)
Hidden Markov Model
![Page 15: Hidden Markov Model - York University](https://reader030.fdocuments.in/reader030/viewer/2022013018/61d14d607961420d3e1a7cf5/html5/thumbnails/15.jpg)
Hidden Markov Model
![Page 16: Hidden Markov Model - York University](https://reader030.fdocuments.in/reader030/viewer/2022013018/61d14d607961420d3e1a7cf5/html5/thumbnails/16.jpg)
Transition Model
![Page 17: Hidden Markov Model - York University](https://reader030.fdocuments.in/reader030/viewer/2022013018/61d14d607961420d3e1a7cf5/html5/thumbnails/17.jpg)
Transition Model
![Page 18: Hidden Markov Model - York University](https://reader030.fdocuments.in/reader030/viewer/2022013018/61d14d607961420d3e1a7cf5/html5/thumbnails/18.jpg)
Observation Model
![Page 19: Hidden Markov Model - York University](https://reader030.fdocuments.in/reader030/viewer/2022013018/61d14d607961420d3e1a7cf5/html5/thumbnails/19.jpg)
Prior Probability
![Page 20: Hidden Markov Model - York University](https://reader030.fdocuments.in/reader030/viewer/2022013018/61d14d607961420d3e1a7cf5/html5/thumbnails/20.jpg)
Hidden Markov Model
![Page 21: Hidden Markov Model - York University](https://reader030.fdocuments.in/reader030/viewer/2022013018/61d14d607961420d3e1a7cf5/html5/thumbnails/21.jpg)
Hidden Markov Model
![Page 22: Hidden Markov Model - York University](https://reader030.fdocuments.in/reader030/viewer/2022013018/61d14d607961420d3e1a7cf5/html5/thumbnails/22.jpg)
Important Note
• First-order Markov Models are often inaccurate, since you throw away any information about the past
• Increasing the accuracy by more general models– Increasing the order of the Markov model– Adding more state variables
![Page 23: Hidden Markov Model - York University](https://reader030.fdocuments.in/reader030/viewer/2022013018/61d14d607961420d3e1a7cf5/html5/thumbnails/23.jpg)
Inference
![Page 24: Hidden Markov Model - York University](https://reader030.fdocuments.in/reader030/viewer/2022013018/61d14d607961420d3e1a7cf5/html5/thumbnails/24.jpg)
Hidden Markov Model
Inference
![Page 25: Hidden Markov Model - York University](https://reader030.fdocuments.in/reader030/viewer/2022013018/61d14d607961420d3e1a7cf5/html5/thumbnails/25.jpg)
Inference Tasks
1. Filtering
Present
![Page 26: Hidden Markov Model - York University](https://reader030.fdocuments.in/reader030/viewer/2022013018/61d14d607961420d3e1a7cf5/html5/thumbnails/26.jpg)
Inference Tasks
1. Filtering
2. Smoothing
Past
![Page 27: Hidden Markov Model - York University](https://reader030.fdocuments.in/reader030/viewer/2022013018/61d14d607961420d3e1a7cf5/html5/thumbnails/27.jpg)
Inference Tasks
1. Filtering
2. Smoothing
3. Prediction
Future
![Page 28: Hidden Markov Model - York University](https://reader030.fdocuments.in/reader030/viewer/2022013018/61d14d607961420d3e1a7cf5/html5/thumbnails/28.jpg)
Inference Tasks
1. Filtering
2. Smoothing
3. Prediction
4. Most Likely Sequence
![Page 29: Hidden Markov Model - York University](https://reader030.fdocuments.in/reader030/viewer/2022013018/61d14d607961420d3e1a7cf5/html5/thumbnails/29.jpg)
Inference Example
Inference??
Prediction????
![Page 30: Hidden Markov Model - York University](https://reader030.fdocuments.in/reader030/viewer/2022013018/61d14d607961420d3e1a7cf5/html5/thumbnails/30.jpg)
Observation:
Weather:
Day 1 Day 2 Day 3 Day 4
What is the most likely weather sequence??
?
![Page 31: Hidden Markov Model - York University](https://reader030.fdocuments.in/reader030/viewer/2022013018/61d14d607961420d3e1a7cf5/html5/thumbnails/31.jpg)
Most Likely Sequence
![Page 32: Hidden Markov Model - York University](https://reader030.fdocuments.in/reader030/viewer/2022013018/61d14d607961420d3e1a7cf5/html5/thumbnails/32.jpg)
● Maximizing path for k-1
Viterbi Algorithm
![Page 33: Hidden Markov Model - York University](https://reader030.fdocuments.in/reader030/viewer/2022013018/61d14d607961420d3e1a7cf5/html5/thumbnails/33.jpg)
Viterbi Algorithm
![Page 34: Hidden Markov Model - York University](https://reader030.fdocuments.in/reader030/viewer/2022013018/61d14d607961420d3e1a7cf5/html5/thumbnails/34.jpg)
Viterbi Algorithm
![Page 35: Hidden Markov Model - York University](https://reader030.fdocuments.in/reader030/viewer/2022013018/61d14d607961420d3e1a7cf5/html5/thumbnails/35.jpg)
Speech Recognition
Hello
States?Observations? “Hello”
Transition Model?
Observation Model? Prior?
![Page 36: Hidden Markov Model - York University](https://reader030.fdocuments.in/reader030/viewer/2022013018/61d14d607961420d3e1a7cf5/html5/thumbnails/36.jpg)
Strengths & Weaknesses of HMM
HMM provides better results than MM
Principle of HMM can be adapted to many problems, e.g. finding alignment
Computationally expensive, both in memory and time
Need more training than classical Markov Models
![Page 37: Hidden Markov Model - York University](https://reader030.fdocuments.in/reader030/viewer/2022013018/61d14d607961420d3e1a7cf5/html5/thumbnails/37.jpg)
Summary
Markov Model
Hidden MM
Transition Model Observation Model Prior
![Page 38: Hidden Markov Model - York University](https://reader030.fdocuments.in/reader030/viewer/2022013018/61d14d607961420d3e1a7cf5/html5/thumbnails/38.jpg)
Summary
Inference using HMM● Filtering● Smoothing● Prediction● Most likely Sequence
○ Viterbi Algorithm
![Page 39: Hidden Markov Model - York University](https://reader030.fdocuments.in/reader030/viewer/2022013018/61d14d607961420d3e1a7cf5/html5/thumbnails/39.jpg)
Thank you
![Page 40: Hidden Markov Model - York University](https://reader030.fdocuments.in/reader030/viewer/2022013018/61d14d607961420d3e1a7cf5/html5/thumbnails/40.jpg)
Sources
Most important graphics from this presentation are taken from http://srl.informatik.uni-freiburg.de/teachingdir/ws13/slides/09-TemporalReasoning-1.pdf