hidden markov models - idiap research instituteodobez/teaching/ee613/ee613-hmm.pdf ·...
TRANSCRIPT
![Page 1: HIDDEN MARKOV MODELS - Idiap Research Instituteodobez/TEACHING/EE613/EE613-HMM.pdf · 2019-12-05 · Hidden Markov model (HMM) L. R. Rabiner. A tutorial on hidden Markov models and](https://reader034.vdocuments.net/reader034/viewer/2022042323/5f0dc8767e708231d43c1046/html5/thumbnails/1.jpg)
EE613Machine Learning for Engineers
HIDDEN MARKOV MODELS
Sylvain CalinonRobot Learning & Interaction Group
Idiap Research InstituteDec. 5, 2019
![Page 2: HIDDEN MARKOV MODELS - Idiap Research Instituteodobez/TEACHING/EE613/EE613-HMM.pdf · 2019-12-05 · Hidden Markov model (HMM) L. R. Rabiner. A tutorial on hidden Markov models and](https://reader034.vdocuments.net/reader034/viewer/2022042323/5f0dc8767e708231d43c1046/html5/thumbnails/2.jpg)
Outline
2
• Markov models
• Hidden Markov model (HMM)
• Forward-backward algorithm
• Viterbi decoding (dynamic programming)
• Hidden semi-Markov model (HSMM)
• HMM with dynamic features (Trajectory-HMM)
![Page 3: HIDDEN MARKOV MODELS - Idiap Research Instituteodobez/TEACHING/EE613/EE613-HMM.pdf · 2019-12-05 · Hidden Markov model (HMM) L. R. Rabiner. A tutorial on hidden Markov models and](https://reader034.vdocuments.net/reader034/viewer/2022042323/5f0dc8767e708231d43c1046/html5/thumbnails/3.jpg)
Markov models
3
![Page 4: HIDDEN MARKOV MODELS - Idiap Research Instituteodobez/TEACHING/EE613/EE613-HMM.pdf · 2019-12-05 · Hidden Markov model (HMM) L. R. Rabiner. A tutorial on hidden Markov models and](https://reader034.vdocuments.net/reader034/viewer/2022042323/5f0dc8767e708231d43c1046/html5/thumbnails/4.jpg)
Markov models - Parameters
4
K possible states
![Page 5: HIDDEN MARKOV MODELS - Idiap Research Instituteodobez/TEACHING/EE613/EE613-HMM.pdf · 2019-12-05 · Hidden Markov model (HMM) L. R. Rabiner. A tutorial on hidden Markov models and](https://reader034.vdocuments.net/reader034/viewer/2022042323/5f0dc8767e708231d43c1046/html5/thumbnails/5.jpg)
Markov models in language modeling
5
![Page 6: HIDDEN MARKOV MODELS - Idiap Research Instituteodobez/TEACHING/EE613/EE613-HMM.pdf · 2019-12-05 · Hidden Markov model (HMM) L. R. Rabiner. A tutorial on hidden Markov models and](https://reader034.vdocuments.net/reader034/viewer/2022042323/5f0dc8767e708231d43c1046/html5/thumbnails/6.jpg)
Markov models in language modeling
6
![Page 7: HIDDEN MARKOV MODELS - Idiap Research Instituteodobez/TEACHING/EE613/EE613-HMM.pdf · 2019-12-05 · Hidden Markov model (HMM) L. R. Rabiner. A tutorial on hidden Markov models and](https://reader034.vdocuments.net/reader034/viewer/2022042323/5f0dc8767e708231d43c1046/html5/thumbnails/7.jpg)
Markov models in language modeling
7
Example of text generated from a 4-gram model, trained on a corpus of 400 million words.
The first 4 words are specified by hand, the model generates the 5th word, and then the results are fed back into the model.
Source: http://www.fit.vutbr.cz/~imikolov/rnnlm/gen-4gram.txt
SAYS IT’S NOT IN THE CARDS LEGENDARY RECONNAISSANCE BY ROLLIE DEMOCRACIES UNSUSTAINABLE COULD STRIKE REDLINING VISITS TO PROFIT BOOKING WAIT HERE AT MADISON SQUARE GARDEN COUNTY COURTHOUSE WHERE HE HAD BEEN DONE IN THREE ALREADY IN ANY WAY IN WHICH A TEACHER …
![Page 8: HIDDEN MARKOV MODELS - Idiap Research Instituteodobez/TEACHING/EE613/EE613-HMM.pdf · 2019-12-05 · Hidden Markov model (HMM) L. R. Rabiner. A tutorial on hidden Markov models and](https://reader034.vdocuments.net/reader034/viewer/2022042323/5f0dc8767e708231d43c1046/html5/thumbnails/8.jpg)
MLE of transition matrix in Markov models
8
![Page 9: HIDDEN MARKOV MODELS - Idiap Research Instituteodobez/TEACHING/EE613/EE613-HMM.pdf · 2019-12-05 · Hidden Markov model (HMM) L. R. Rabiner. A tutorial on hidden Markov models and](https://reader034.vdocuments.net/reader034/viewer/2022042323/5f0dc8767e708231d43c1046/html5/thumbnails/9.jpg)
Hidden Markov model (HMM)
Python notebook: demo_HMM.ipynb
Matlab code: demo_HMM01.m
9
![Page 10: HIDDEN MARKOV MODELS - Idiap Research Instituteodobez/TEACHING/EE613/EE613-HMM.pdf · 2019-12-05 · Hidden Markov model (HMM) L. R. Rabiner. A tutorial on hidden Markov models and](https://reader034.vdocuments.net/reader034/viewer/2022042323/5f0dc8767e708231d43c1046/html5/thumbnails/10.jpg)
In a Markov chain, the state is directly visible to the observer → the transition probabilities are the only parameters.
In an HMM, the state is not directly visible, but an output dependent on the state is visible.
Hidden Markov model (HMM)
10
Hiddenstates
Observedoutput
(emissionprobability)
Image adapted from Wikipedia
Initial state: All we know is that 60% of the days are rainy on average.
The transition probability represents the change of the weather in the underlying Markov chain. Here, there is a 30% chance that tomorrow will be sunny if today is rainy.
![Page 11: HIDDEN MARKOV MODELS - Idiap Research Instituteodobez/TEACHING/EE613/EE613-HMM.pdf · 2019-12-05 · Hidden Markov model (HMM) L. R. Rabiner. A tutorial on hidden Markov models and](https://reader034.vdocuments.net/reader034/viewer/2022042323/5f0dc8767e708231d43c1046/html5/thumbnails/11.jpg)
Hidden Markov model (HMM)
11
Hiddenstates
Observedoutput
(emissionprobability)
Image adapted from Wikipedia
You can think of an HMM either as:
• a Markov chain with stochastic measurements
• a GMM with latent variables changing over time
The emission probability represents how likely Bob performs a certain activity on each day. if it is sunny, there is a 60% chance that he is outside for a walk. If it is rainy, there is a 50% chance that he cleans his apartment, etc.
![Page 12: HIDDEN MARKOV MODELS - Idiap Research Instituteodobez/TEACHING/EE613/EE613-HMM.pdf · 2019-12-05 · Hidden Markov model (HMM) L. R. Rabiner. A tutorial on hidden Markov models and](https://reader034.vdocuments.net/reader034/viewer/2022042323/5f0dc8767e708231d43c1046/html5/thumbnails/12.jpg)
Inference problems associated with HMMs
12
![Page 13: HIDDEN MARKOV MODELS - Idiap Research Instituteodobez/TEACHING/EE613/EE613-HMM.pdf · 2019-12-05 · Hidden Markov model (HMM) L. R. Rabiner. A tutorial on hidden Markov models and](https://reader034.vdocuments.net/reader034/viewer/2022042323/5f0dc8767e708231d43c1046/html5/thumbnails/13.jpg)
Emission/output distributions in HMM
Discrete tables
V1
V2
V3
V1
V2
V3
Gaussian distribution Mixture of Gaussians
13
![Page 14: HIDDEN MARKOV MODELS - Idiap Research Instituteodobez/TEACHING/EE613/EE613-HMM.pdf · 2019-12-05 · Hidden Markov model (HMM) L. R. Rabiner. A tutorial on hidden Markov models and](https://reader034.vdocuments.net/reader034/viewer/2022042323/5f0dc8767e708231d43c1046/html5/thumbnails/14.jpg)
Transition matrix structures in HMM
14
![Page 15: HIDDEN MARKOV MODELS - Idiap Research Instituteodobez/TEACHING/EE613/EE613-HMM.pdf · 2019-12-05 · Hidden Markov model (HMM) L. R. Rabiner. A tutorial on hidden Markov models and](https://reader034.vdocuments.net/reader034/viewer/2022042323/5f0dc8767e708231d43c1046/html5/thumbnails/15.jpg)
HMM - Examples of application
15
HMM is used in many fields as a tool for time series or sequences analysis, and in fields where the goal is to recover a data sequence that is not immediately observable:
Speech recognitionSpeech synthesisPart-of-speech taggingNatural language modelingMachine translationGene predictionMolecule kinetic analysisDNA motif discoveryAlignment of bio-sequences (e.g., proteins)Metamorphic virus detectionDocument separation in scanning solutions
CryptoanalysisActivity recognitionProtein foldingHuman motion scienceOnline handwriting recognitionRobotics
![Page 16: HIDDEN MARKOV MODELS - Idiap Research Instituteodobez/TEACHING/EE613/EE613-HMM.pdf · 2019-12-05 · Hidden Markov model (HMM) L. R. Rabiner. A tutorial on hidden Markov models and](https://reader034.vdocuments.net/reader034/viewer/2022042323/5f0dc8767e708231d43c1046/html5/thumbnails/16.jpg)
Automatic speech recognition
ξt can represent features extracted from the speech signal, and st can represent
the word being spoken. The transition model P(st|st-1) represents the language
model, and the observation model P(ξt|st) represents the acoustic model.
Part of speech tagging
ξt can represent a word, and st represents its part of speech (noun, verb,
adjective, etc.)
Activity recognition
ξt can represent features extracted from a video frame, and st is the class of
activity the person is engaged in (e.g., running, walking, sitting, etc.).
Gene finding
ξt can represent the DNA nucleotides (A,T,G,C), and st can represent whether we
are inside a gene-coding region or not.
16
HMM - Examples of application ξt Observationst Hidden state
![Page 17: HIDDEN MARKOV MODELS - Idiap Research Instituteodobez/TEACHING/EE613/EE613-HMM.pdf · 2019-12-05 · Hidden Markov model (HMM) L. R. Rabiner. A tutorial on hidden Markov models and](https://reader034.vdocuments.net/reader034/viewer/2022042323/5f0dc8767e708231d43c1046/html5/thumbnails/17.jpg)
GMM
HMM
HMM parameters
From now on, we will consider a single Gaussian as state output
17
![Page 18: HIDDEN MARKOV MODELS - Idiap Research Instituteodobez/TEACHING/EE613/EE613-HMM.pdf · 2019-12-05 · Hidden Markov model (HMM) L. R. Rabiner. A tutorial on hidden Markov models and](https://reader034.vdocuments.net/reader034/viewer/2022042323/5f0dc8767e708231d43c1046/html5/thumbnails/18.jpg)
Useful intermediary variables in HMM
18
Forward variable
Backward variable
Smoothed node marginals
Smoothed edge marginals
![Page 19: HIDDEN MARKOV MODELS - Idiap Research Instituteodobez/TEACHING/EE613/EE613-HMM.pdf · 2019-12-05 · Hidden Markov model (HMM) L. R. Rabiner. A tutorial on hidden Markov models and](https://reader034.vdocuments.net/reader034/viewer/2022042323/5f0dc8767e708231d43c1046/html5/thumbnails/19.jpg)
KK
K
Forward algorithm
19
![Page 20: HIDDEN MARKOV MODELS - Idiap Research Instituteodobez/TEACHING/EE613/EE613-HMM.pdf · 2019-12-05 · Hidden Markov model (HMM) L. R. Rabiner. A tutorial on hidden Markov models and](https://reader034.vdocuments.net/reader034/viewer/2022042323/5f0dc8767e708231d43c1046/html5/thumbnails/20.jpg)
Forward algorithm
20
![Page 21: HIDDEN MARKOV MODELS - Idiap Research Instituteodobez/TEACHING/EE613/EE613-HMM.pdf · 2019-12-05 · Hidden Markov model (HMM) L. R. Rabiner. A tutorial on hidden Markov models and](https://reader034.vdocuments.net/reader034/viewer/2022042323/5f0dc8767e708231d43c1046/html5/thumbnails/21.jpg)
Forward algorithm
21
ξ1
s1
ξ2
s2
ξ3
s3
ξ4
s4
![Page 22: HIDDEN MARKOV MODELS - Idiap Research Instituteodobez/TEACHING/EE613/EE613-HMM.pdf · 2019-12-05 · Hidden Markov model (HMM) L. R. Rabiner. A tutorial on hidden Markov models and](https://reader034.vdocuments.net/reader034/viewer/2022042323/5f0dc8767e708231d43c1046/html5/thumbnails/22.jpg)
Forward algorithm
22
ξ1
s1
ξ2
s2
ξ3
s3
ξ4
s4
![Page 23: HIDDEN MARKOV MODELS - Idiap Research Instituteodobez/TEACHING/EE613/EE613-HMM.pdf · 2019-12-05 · Hidden Markov model (HMM) L. R. Rabiner. A tutorial on hidden Markov models and](https://reader034.vdocuments.net/reader034/viewer/2022042323/5f0dc8767e708231d43c1046/html5/thumbnails/23.jpg)
KK
K
Forward algorithm
23
![Page 24: HIDDEN MARKOV MODELS - Idiap Research Instituteodobez/TEACHING/EE613/EE613-HMM.pdf · 2019-12-05 · Hidden Markov model (HMM) L. R. Rabiner. A tutorial on hidden Markov models and](https://reader034.vdocuments.net/reader034/viewer/2022042323/5f0dc8767e708231d43c1046/html5/thumbnails/24.jpg)
.94
.06
.94 .94 .94 .94 1
1.06 .06 .06 .06
Low influence of transition probabilities w.r.t. emission probabilities in HMM
.06
.94
.06 .06 .06 .06 1
1.94 .94 .94 .94
24
Learned transition probabilities Transition probabilities manually set
Direction of motion
The color of each datapointcorresponds to the value of the forward variable 𝛼
![Page 25: HIDDEN MARKOV MODELS - Idiap Research Instituteodobez/TEACHING/EE613/EE613-HMM.pdf · 2019-12-05 · Hidden Markov model (HMM) L. R. Rabiner. A tutorial on hidden Markov models and](https://reader034.vdocuments.net/reader034/viewer/2022042323/5f0dc8767e708231d43c1046/html5/thumbnails/25.jpg)
Useful intermediary variables in HMM
25
Forward variable
Backward variable
Smoothed node marginals
Smoothed edge marginals
![Page 26: HIDDEN MARKOV MODELS - Idiap Research Instituteodobez/TEACHING/EE613/EE613-HMM.pdf · 2019-12-05 · Hidden Markov model (HMM) L. R. Rabiner. A tutorial on hidden Markov models and](https://reader034.vdocuments.net/reader034/viewer/2022042323/5f0dc8767e708231d43c1046/html5/thumbnails/26.jpg)
26
Backward algorithm
![Page 27: HIDDEN MARKOV MODELS - Idiap Research Instituteodobez/TEACHING/EE613/EE613-HMM.pdf · 2019-12-05 · Hidden Markov model (HMM) L. R. Rabiner. A tutorial on hidden Markov models and](https://reader034.vdocuments.net/reader034/viewer/2022042323/5f0dc8767e708231d43c1046/html5/thumbnails/27.jpg)
K
K
K
Backward algorithm
27
![Page 28: HIDDEN MARKOV MODELS - Idiap Research Instituteodobez/TEACHING/EE613/EE613-HMM.pdf · 2019-12-05 · Hidden Markov model (HMM) L. R. Rabiner. A tutorial on hidden Markov models and](https://reader034.vdocuments.net/reader034/viewer/2022042323/5f0dc8767e708231d43c1046/html5/thumbnails/28.jpg)
Useful intermediary variables in HMM
28
Forward variable
Backward variable
Smoothed node marginals
Smoothed edge marginals
These variable are sometimes called "smoothed values" as they combine forward and backward probabilities in the computation.
You can think of their roles as passing "messages" from left to right, and from right to left, and then combining the information at each node.
![Page 29: HIDDEN MARKOV MODELS - Idiap Research Instituteodobez/TEACHING/EE613/EE613-HMM.pdf · 2019-12-05 · Hidden Markov model (HMM) L. R. Rabiner. A tutorial on hidden Markov models and](https://reader034.vdocuments.net/reader034/viewer/2022042323/5f0dc8767e708231d43c1046/html5/thumbnails/29.jpg)
Smoothed node marginals
29
![Page 30: HIDDEN MARKOV MODELS - Idiap Research Instituteodobez/TEACHING/EE613/EE613-HMM.pdf · 2019-12-05 · Hidden Markov model (HMM) L. R. Rabiner. A tutorial on hidden Markov models and](https://reader034.vdocuments.net/reader034/viewer/2022042323/5f0dc8767e708231d43c1046/html5/thumbnails/30.jpg)
Smoothed node marginals
30
Conditional independenceproperty
ξ1
s1
ξ2
s2
ξ3
s3
ξ4
s4
![Page 31: HIDDEN MARKOV MODELS - Idiap Research Instituteodobez/TEACHING/EE613/EE613-HMM.pdf · 2019-12-05 · Hidden Markov model (HMM) L. R. Rabiner. A tutorial on hidden Markov models and](https://reader034.vdocuments.net/reader034/viewer/2022042323/5f0dc8767e708231d43c1046/html5/thumbnails/31.jpg)
Smoothed node marginals
31
![Page 32: HIDDEN MARKOV MODELS - Idiap Research Instituteodobez/TEACHING/EE613/EE613-HMM.pdf · 2019-12-05 · Hidden Markov model (HMM) L. R. Rabiner. A tutorial on hidden Markov models and](https://reader034.vdocuments.net/reader034/viewer/2022042323/5f0dc8767e708231d43c1046/html5/thumbnails/32.jpg)
These variable are sometimes called "smoothed values" as they combine forward and backward probabilities in the computation.
You can think of their roles as passing "messages" from left to right, and from right to left, and then combining the information at each node.
Useful intermediary variables in HMM
32
Forward variable
Backward variable
Smoothed node marginals
Smoothed edge marginals
![Page 33: HIDDEN MARKOV MODELS - Idiap Research Instituteodobez/TEACHING/EE613/EE613-HMM.pdf · 2019-12-05 · Hidden Markov model (HMM) L. R. Rabiner. A tutorial on hidden Markov models and](https://reader034.vdocuments.net/reader034/viewer/2022042323/5f0dc8767e708231d43c1046/html5/thumbnails/33.jpg)
Smoothed edge marginals
K
![Page 34: HIDDEN MARKOV MODELS - Idiap Research Instituteodobez/TEACHING/EE613/EE613-HMM.pdf · 2019-12-05 · Hidden Markov model (HMM) L. R. Rabiner. A tutorial on hidden Markov models and](https://reader034.vdocuments.net/reader034/viewer/2022042323/5f0dc8767e708231d43c1046/html5/thumbnails/34.jpg)
Smoothed edge marginals
34
![Page 35: HIDDEN MARKOV MODELS - Idiap Research Instituteodobez/TEACHING/EE613/EE613-HMM.pdf · 2019-12-05 · Hidden Markov model (HMM) L. R. Rabiner. A tutorial on hidden Markov models and](https://reader034.vdocuments.net/reader034/viewer/2022042323/5f0dc8767e708231d43c1046/html5/thumbnails/35.jpg)
Smoothed edge marginals
35
![Page 36: HIDDEN MARKOV MODELS - Idiap Research Instituteodobez/TEACHING/EE613/EE613-HMM.pdf · 2019-12-05 · Hidden Markov model (HMM) L. R. Rabiner. A tutorial on hidden Markov models and](https://reader034.vdocuments.net/reader034/viewer/2022042323/5f0dc8767e708231d43c1046/html5/thumbnails/36.jpg)
EM for HMM
Similar to Markov models
Similarto GMM
K GaussiansM trajectoriesTm points per traj.
36
![Page 37: HIDDEN MARKOV MODELS - Idiap Research Instituteodobez/TEACHING/EE613/EE613-HMM.pdf · 2019-12-05 · Hidden Markov model (HMM) L. R. Rabiner. A tutorial on hidden Markov models and](https://reader034.vdocuments.net/reader034/viewer/2022042323/5f0dc8767e708231d43c1046/html5/thumbnails/37.jpg)
EM for HMM
37
K GaussiansM trajectoriesTm points per traj.
![Page 38: HIDDEN MARKOV MODELS - Idiap Research Instituteodobez/TEACHING/EE613/EE613-HMM.pdf · 2019-12-05 · Hidden Markov model (HMM) L. R. Rabiner. A tutorial on hidden Markov models and](https://reader034.vdocuments.net/reader034/viewer/2022042323/5f0dc8767e708231d43c1046/html5/thumbnails/38.jpg)
EM for HMM - Summary
38
K GaussiansM trajectoriesTm points per traj.
![Page 39: HIDDEN MARKOV MODELS - Idiap Research Instituteodobez/TEACHING/EE613/EE613-HMM.pdf · 2019-12-05 · Hidden Markov model (HMM) L. R. Rabiner. A tutorial on hidden Markov models and](https://reader034.vdocuments.net/reader034/viewer/2022042323/5f0dc8767e708231d43c1046/html5/thumbnails/39.jpg)
EM for HMM - Summary
39
These results can be formally retrieved with EM (also called Baum-Welch algorithm in the context of HMM).
The update rules can be interpreted as normalized counts, with several types of weighted averages required in the computation.
K GaussiansM trajectoriesTm points per traj.
![Page 40: HIDDEN MARKOV MODELS - Idiap Research Instituteodobez/TEACHING/EE613/EE613-HMM.pdf · 2019-12-05 · Hidden Markov model (HMM) L. R. Rabiner. A tutorial on hidden Markov models and](https://reader034.vdocuments.net/reader034/viewer/2022042323/5f0dc8767e708231d43c1046/html5/thumbnails/40.jpg)
Numerical underflow issue in HMM
40
![Page 41: HIDDEN MARKOV MODELS - Idiap Research Instituteodobez/TEACHING/EE613/EE613-HMM.pdf · 2019-12-05 · Hidden Markov model (HMM) L. R. Rabiner. A tutorial on hidden Markov models and](https://reader034.vdocuments.net/reader034/viewer/2022042323/5f0dc8767e708231d43c1046/html5/thumbnails/41.jpg)
Numerical underflow issue in HMM
41
This issue is sometimes not covered in textbooks, although it remains very important for practical implementation of HMM!
![Page 42: HIDDEN MARKOV MODELS - Idiap Research Instituteodobez/TEACHING/EE613/EE613-HMM.pdf · 2019-12-05 · Hidden Markov model (HMM) L. R. Rabiner. A tutorial on hidden Markov models and](https://reader034.vdocuments.net/reader034/viewer/2022042323/5f0dc8767e708231d43c1046/html5/thumbnails/42.jpg)
Summary - Why did we introduce these four intermediary variables in HMM?
Forward variable
Backward variable
Smoothed node marginals
Smoothed edge marginals
42
![Page 43: HIDDEN MARKOV MODELS - Idiap Research Instituteodobez/TEACHING/EE613/EE613-HMM.pdf · 2019-12-05 · Hidden Markov model (HMM) L. R. Rabiner. A tutorial on hidden Markov models and](https://reader034.vdocuments.net/reader034/viewer/2022042323/5f0dc8767e708231d43c1046/html5/thumbnails/43.jpg)
Summary - Why did we introduce these four intermediary variables in HMM?
43
![Page 44: HIDDEN MARKOV MODELS - Idiap Research Instituteodobez/TEACHING/EE613/EE613-HMM.pdf · 2019-12-05 · Hidden Markov model (HMM) L. R. Rabiner. A tutorial on hidden Markov models and](https://reader034.vdocuments.net/reader034/viewer/2022042323/5f0dc8767e708231d43c1046/html5/thumbnails/44.jpg)
Viterbi decoding (MAP vs MPE estimates)
Python notebook: demo_HMM.ipynb
Matlab code: demo_HMM_Viterbi01.m
44
Maximum a posteriori Most probable explanation
![Page 45: HIDDEN MARKOV MODELS - Idiap Research Instituteodobez/TEACHING/EE613/EE613-HMM.pdf · 2019-12-05 · Hidden Markov model (HMM) L. R. Rabiner. A tutorial on hidden Markov models and](https://reader034.vdocuments.net/reader034/viewer/2022042323/5f0dc8767e708231d43c1046/html5/thumbnails/45.jpg)
Viterbi decoding (MAP vs MPE estimates)
45
Maximum a posteriori Most probable explanation
![Page 46: HIDDEN MARKOV MODELS - Idiap Research Instituteodobez/TEACHING/EE613/EE613-HMM.pdf · 2019-12-05 · Hidden Markov model (HMM) L. R. Rabiner. A tutorial on hidden Markov models and](https://reader034.vdocuments.net/reader034/viewer/2022042323/5f0dc8767e708231d43c1046/html5/thumbnails/46.jpg)
Viterbi decoding - Trellis representation
46
![Page 47: HIDDEN MARKOV MODELS - Idiap Research Instituteodobez/TEACHING/EE613/EE613-HMM.pdf · 2019-12-05 · Hidden Markov model (HMM) L. R. Rabiner. A tutorial on hidden Markov models and](https://reader034.vdocuments.net/reader034/viewer/2022042323/5f0dc8767e708231d43c1046/html5/thumbnails/47.jpg)
Viterbi decoding - Algorithm
This is the probability of ending up in state i at time step t by taking the most probable path
It tells us the most likely previous state on the most probable path to st = i
47
![Page 48: HIDDEN MARKOV MODELS - Idiap Research Instituteodobez/TEACHING/EE613/EE613-HMM.pdf · 2019-12-05 · Hidden Markov model (HMM) L. R. Rabiner. A tutorial on hidden Markov models and](https://reader034.vdocuments.net/reader034/viewer/2022042323/5f0dc8767e708231d43c1046/html5/thumbnails/48.jpg)
Viterbi decoding - Trellis representation
48
![Page 49: HIDDEN MARKOV MODELS - Idiap Research Instituteodobez/TEACHING/EE613/EE613-HMM.pdf · 2019-12-05 · Hidden Markov model (HMM) L. R. Rabiner. A tutorial on hidden Markov models and](https://reader034.vdocuments.net/reader034/viewer/2022042323/5f0dc8767e708231d43c1046/html5/thumbnails/49.jpg)
Viterbi decoding - Example
Image adapted from Kevin P. Murphy (2012), Machine Learning: A Probabilistic Perspective49
![Page 50: HIDDEN MARKOV MODELS - Idiap Research Instituteodobez/TEACHING/EE613/EE613-HMM.pdf · 2019-12-05 · Hidden Markov model (HMM) L. R. Rabiner. A tutorial on hidden Markov models and](https://reader034.vdocuments.net/reader034/viewer/2022042323/5f0dc8767e708231d43c1046/html5/thumbnails/50.jpg)
Numerical underflow issue in Viterbi
50
![Page 51: HIDDEN MARKOV MODELS - Idiap Research Instituteodobez/TEACHING/EE613/EE613-HMM.pdf · 2019-12-05 · Hidden Markov model (HMM) L. R. Rabiner. A tutorial on hidden Markov models and](https://reader034.vdocuments.net/reader034/viewer/2022042323/5f0dc8767e708231d43c1046/html5/thumbnails/51.jpg)
Numerical underflow issue in Viterbi
51
![Page 52: HIDDEN MARKOV MODELS - Idiap Research Instituteodobez/TEACHING/EE613/EE613-HMM.pdf · 2019-12-05 · Hidden Markov model (HMM) L. R. Rabiner. A tutorial on hidden Markov models and](https://reader034.vdocuments.net/reader034/viewer/2022042323/5f0dc8767e708231d43c1046/html5/thumbnails/52.jpg)
Hidden semi-Markov model (HSMM)
Python notebook: demo_HSMM.ipynb
Matlab code: demo_HSMM01.m
52
![Page 53: HIDDEN MARKOV MODELS - Idiap Research Instituteodobez/TEACHING/EE613/EE613-HMM.pdf · 2019-12-05 · Hidden Markov model (HMM) L. R. Rabiner. A tutorial on hidden Markov models and](https://reader034.vdocuments.net/reader034/viewer/2022042323/5f0dc8767e708231d43c1046/html5/thumbnails/53.jpg)
By artificially duplicating the number of states while keeping the same emission
distribution, other state duration distributions can be modeled
State duration probability in standard HMM
53
The state duration follows a geometric distribution
![Page 54: HIDDEN MARKOV MODELS - Idiap Research Instituteodobez/TEACHING/EE613/EE613-HMM.pdf · 2019-12-05 · Hidden Markov model (HMM) L. R. Rabiner. A tutorial on hidden Markov models and](https://reader034.vdocuments.net/reader034/viewer/2022042323/5f0dc8767e708231d43c1046/html5/thumbnails/54.jpg)
54
Another approach is to provide an explicit model of the state duration instead of relying on
self-transition probabilities
Hidden semi-Markov model (HSMM)
![Page 55: HIDDEN MARKOV MODELS - Idiap Research Instituteodobez/TEACHING/EE613/EE613-HMM.pdf · 2019-12-05 · Hidden Markov model (HMM) L. R. Rabiner. A tutorial on hidden Markov models and](https://reader034.vdocuments.net/reader034/viewer/2022042323/5f0dc8767e708231d43c1046/html5/thumbnails/55.jpg)
GMM
HMM
HSMM
Hidden semi-Markov model (HSMM)
Parametric duration distribution
![Page 56: HIDDEN MARKOV MODELS - Idiap Research Instituteodobez/TEACHING/EE613/EE613-HMM.pdf · 2019-12-05 · Hidden Markov model (HMM) L. R. Rabiner. A tutorial on hidden Markov models and](https://reader034.vdocuments.net/reader034/viewer/2022042323/5f0dc8767e708231d43c1046/html5/thumbnails/56.jpg)
Hidden semi-Markov model (HSMM)
56
![Page 57: HIDDEN MARKOV MODELS - Idiap Research Instituteodobez/TEACHING/EE613/EE613-HMM.pdf · 2019-12-05 · Hidden Markov model (HMM) L. R. Rabiner. A tutorial on hidden Markov models and](https://reader034.vdocuments.net/reader034/viewer/2022042323/5f0dc8767e708231d43c1046/html5/thumbnails/57.jpg)
Hidden semi-Markov model (HSMM)
57
![Page 58: HIDDEN MARKOV MODELS - Idiap Research Instituteodobez/TEACHING/EE613/EE613-HMM.pdf · 2019-12-05 · Hidden Markov model (HMM) L. R. Rabiner. A tutorial on hidden Markov models and](https://reader034.vdocuments.net/reader034/viewer/2022042323/5f0dc8767e708231d43c1046/html5/thumbnails/58.jpg)
Hidden semi-Markov model (HSMM)
58
![Page 59: HIDDEN MARKOV MODELS - Idiap Research Instituteodobez/TEACHING/EE613/EE613-HMM.pdf · 2019-12-05 · Hidden Markov model (HMM) L. R. Rabiner. A tutorial on hidden Markov models and](https://reader034.vdocuments.net/reader034/viewer/2022042323/5f0dc8767e708231d43c1046/html5/thumbnails/59.jpg)
HMM with dynamic features
(Trajectory-HMM)
Matlab code: demo_trajHSMM01.m
59
![Page 60: HIDDEN MARKOV MODELS - Idiap Research Instituteodobez/TEACHING/EE613/EE613-HMM.pdf · 2019-12-05 · Hidden Markov model (HMM) L. R. Rabiner. A tutorial on hidden Markov models and](https://reader034.vdocuments.net/reader034/viewer/2022042323/5f0dc8767e708231d43c1046/html5/thumbnails/60.jpg)
HMM with dynamic features
60
![Page 61: HIDDEN MARKOV MODELS - Idiap Research Instituteodobez/TEACHING/EE613/EE613-HMM.pdf · 2019-12-05 · Hidden Markov model (HMM) L. R. Rabiner. A tutorial on hidden Markov models and](https://reader034.vdocuments.net/reader034/viewer/2022042323/5f0dc8767e708231d43c1046/html5/thumbnails/61.jpg)
61
HMM with dynamic features
![Page 62: HIDDEN MARKOV MODELS - Idiap Research Instituteodobez/TEACHING/EE613/EE613-HMM.pdf · 2019-12-05 · Hidden Markov model (HMM) L. R. Rabiner. A tutorial on hidden Markov models and](https://reader034.vdocuments.net/reader034/viewer/2022042323/5f0dc8767e708231d43c1046/html5/thumbnails/62.jpg)
62
HMM with dynamic features
![Page 63: HIDDEN MARKOV MODELS - Idiap Research Instituteodobez/TEACHING/EE613/EE613-HMM.pdf · 2019-12-05 · Hidden Markov model (HMM) L. R. Rabiner. A tutorial on hidden Markov models and](https://reader034.vdocuments.net/reader034/viewer/2022042323/5f0dc8767e708231d43c1046/html5/thumbnails/63.jpg)
63
HMM with dynamic features
![Page 64: HIDDEN MARKOV MODELS - Idiap Research Instituteodobez/TEACHING/EE613/EE613-HMM.pdf · 2019-12-05 · Hidden Markov model (HMM) L. R. Rabiner. A tutorial on hidden Markov models and](https://reader034.vdocuments.net/reader034/viewer/2022042323/5f0dc8767e708231d43c1046/html5/thumbnails/64.jpg)
64
(C=3 here)
D dimensions C derivativesT time steps
HMM with dynamic features
Large sparse matrix
![Page 65: HIDDEN MARKOV MODELS - Idiap Research Instituteodobez/TEACHING/EE613/EE613-HMM.pdf · 2019-12-05 · Hidden Markov model (HMM) L. R. Rabiner. A tutorial on hidden Markov models and](https://reader034.vdocuments.net/reader034/viewer/2022042323/5f0dc8767e708231d43c1046/html5/thumbnails/65.jpg)
65
HMM with dynamic features
![Page 66: HIDDEN MARKOV MODELS - Idiap Research Instituteodobez/TEACHING/EE613/EE613-HMM.pdf · 2019-12-05 · Hidden Markov model (HMM) L. R. Rabiner. A tutorial on hidden Markov models and](https://reader034.vdocuments.net/reader034/viewer/2022042323/5f0dc8767e708231d43c1046/html5/thumbnails/66.jpg)
66
HMM with dynamic features
![Page 67: HIDDEN MARKOV MODELS - Idiap Research Instituteodobez/TEACHING/EE613/EE613-HMM.pdf · 2019-12-05 · Hidden Markov model (HMM) L. R. Rabiner. A tutorial on hidden Markov models and](https://reader034.vdocuments.net/reader034/viewer/2022042323/5f0dc8767e708231d43c1046/html5/thumbnails/67.jpg)
67
Weighted Least Squares!
HMM with dynamic features
![Page 68: HIDDEN MARKOV MODELS - Idiap Research Instituteodobez/TEACHING/EE613/EE613-HMM.pdf · 2019-12-05 · Hidden Markov model (HMM) L. R. Rabiner. A tutorial on hidden Markov models and](https://reader034.vdocuments.net/reader034/viewer/2022042323/5f0dc8767e708231d43c1046/html5/thumbnails/68.jpg)
HMM with dynamic features
68
![Page 69: HIDDEN MARKOV MODELS - Idiap Research Instituteodobez/TEACHING/EE613/EE613-HMM.pdf · 2019-12-05 · Hidden Markov model (HMM) L. R. Rabiner. A tutorial on hidden Markov models and](https://reader034.vdocuments.net/reader034/viewer/2022042323/5f0dc8767e708231d43c1046/html5/thumbnails/69.jpg)
HMM with dynamic features - Summary
69
![Page 70: HIDDEN MARKOV MODELS - Idiap Research Instituteodobez/TEACHING/EE613/EE613-HMM.pdf · 2019-12-05 · Hidden Markov model (HMM) L. R. Rabiner. A tutorial on hidden Markov models and](https://reader034.vdocuments.net/reader034/viewer/2022042323/5f0dc8767e708231d43c1046/html5/thumbnails/70.jpg)
References
70
Hidden Markov model (HMM)
L. R. Rabiner. A tutorial on hidden Markov models and selected applications in speech recognition. Proc. IEEE, 77:2:257–285, February 1989
Hidden semi-Markov model (HSMM)
S.-Z. Yu. Hidden semi-Markov models. Artificial Intelligence, 174:215–243, 2010
S. E. Levinson. Continuously variable duration hidden Markov models for automatic speech recognition. Computer Speech & Language, 1(1):29–45, 1986
HMM with dynamic features (Trajectory HMM)
S. Furui. Speaker-independent isolated word recognition using dynamic features of speech spectrum. IEEE Trans. on Acoustics, Speech, and Signal Processing, 34(1):52–59, 1986
H. Zen, K. Tokuda, and T. Kitamura. Reformulating the HMM as a trajectory model by imposing explicit relationships between static and dynamic feature vector sequences. Computer Speech and Language, 21(1):153–173, 2007
![Page 71: HIDDEN MARKOV MODELS - Idiap Research Instituteodobez/TEACHING/EE613/EE613-HMM.pdf · 2019-12-05 · Hidden Markov model (HMM) L. R. Rabiner. A tutorial on hidden Markov models and](https://reader034.vdocuments.net/reader034/viewer/2022042323/5f0dc8767e708231d43c1046/html5/thumbnails/71.jpg)
Appendix
71
![Page 72: HIDDEN MARKOV MODELS - Idiap Research Instituteodobez/TEACHING/EE613/EE613-HMM.pdf · 2019-12-05 · Hidden Markov model (HMM) L. R. Rabiner. A tutorial on hidden Markov models and](https://reader034.vdocuments.net/reader034/viewer/2022042323/5f0dc8767e708231d43c1046/html5/thumbnails/72.jpg)
72
Markov models - Transition matrix
![Page 73: HIDDEN MARKOV MODELS - Idiap Research Instituteodobez/TEACHING/EE613/EE613-HMM.pdf · 2019-12-05 · Hidden Markov model (HMM) L. R. Rabiner. A tutorial on hidden Markov models and](https://reader034.vdocuments.net/reader034/viewer/2022042323/5f0dc8767e708231d43c1046/html5/thumbnails/73.jpg)
MLE of transition matrix in Markov models
73
![Page 74: HIDDEN MARKOV MODELS - Idiap Research Instituteodobez/TEACHING/EE613/EE613-HMM.pdf · 2019-12-05 · Hidden Markov model (HMM) L. R. Rabiner. A tutorial on hidden Markov models and](https://reader034.vdocuments.net/reader034/viewer/2022042323/5f0dc8767e708231d43c1046/html5/thumbnails/74.jpg)
HMM: Smoothed edge marginals
74
Conditional independence property
ξ1
s1
ξ2
s2
ξ3
s3
ξ4
s4
![Page 75: HIDDEN MARKOV MODELS - Idiap Research Instituteodobez/TEACHING/EE613/EE613-HMM.pdf · 2019-12-05 · Hidden Markov model (HMM) L. R. Rabiner. A tutorial on hidden Markov models and](https://reader034.vdocuments.net/reader034/viewer/2022042323/5f0dc8767e708231d43c1046/html5/thumbnails/75.jpg)
HSMM: Initialization of forward variable
75