WML2007 Hidden Markov Model - PowerPoint PPT Presentation

Title: WML2007 Hidden Markov Model


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WML2007 Hidden Markov Model

• archer

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References

• 1 Hidden Markov model, From Wikipedia, the free
  encyclopedia
• http//en.wikipedia.org/wiki/Hidden_Mar
  kov_model
• 2 Layered hidden Markov model, From Wikipedia,
  the free encyclopedia
• http//en.wikipedia.org/wiki/Layered_hi
  dden_Markov_model
• 3 N. Oliver, A. Garg and E. Horvitz, "Layered
  Representations for Learning and
• Inferring Office Activity from
  Multiple Sensory Channels", Computer Vision and
• Image Understanding, vol. 96, p.
  163-180, 2004.
• 4 Kalman filter, From Wikipedia, the free
  encyclopedia
• http//en.wikipedia.org/wiki/Kalman_fil
  ter
• 5 B. Ristic and S. Arulampalam, Beyond the
  Kalman filter particle filters for
• tracking. Boston, MA Artech House,
  2004.
• 6 X. Huang, A. Acero, H.-W. Hon, Spoken
  Language Processing, Prentice Hall,
• 2001,
• 7 Mobility and Handover Prediction mechanism a
  performance comparison
• exploiting several filters
  http//lia.deis.unibo.it/Research/SOMA/MobilityPre
  diction/htmlDocs/filters.html

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Outline

• Hidden Markov Model
• Hierarchical HMM
• Layered HMM
• Continuous HMM
• Semi-continuous HMM
• Solving HMM problems
• Forward and backward algorithm
• Baum-Welch algorithm
• Viterbi Algorithm
• Particle Filter
• Conclusion and Comments

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Hidden Markov Model

• Dynamic system
• The systems state is function of time
• Regular Markov Model
• State is directly visible.
• Hidden Markov Model
• State is invisible, but
• variables influenced
• by the state are visible.
• Hidden x(t) X1,X2,X3
• Observable y(t) Y1, Y2, Y3, Y4

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Hidden Markov Model

• Markov property
• hidden variable x(t) (at time t) only depends on
  the value of the hidden variable x(t - 1) (at
  time t - 1). Governed by State transition
  function
• the value of the observed variable y(t) only
  depends on the value of the hidden variable x(t)
  (both at time t). Governed by likelyhood function

P(x(t)x(t-1))
P(y(t)x(t))
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Solving HMM HMM Decoding Problem

• Determine the hidden state sequence (x(0),
  x(1),..x(t)) from the observation sequence
  (y(0), y(1).y(t))

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Example Player Pose Estimation
Hidden States
Visible Observations
Feature Extraction ( Segmentation,Skeleton Body
part Localization )
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Some Applications of Hidden Markov Model

• Especially well-known for their application in
  temporal pattern recognition
• Speech recognition
• Handwriting recognition
• Gesture recognition
• bioinformatics

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Layered Hidden Markov Model

• Layered Hidden Markov Model (LHMM)
• Statistical model consists of N levels of HMMs
• Can be transfer to
• a more complex
• single HMM

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Layered Hidden Markov Model

• Application Example 3

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Layered Hidden Markov Model

• LHMM VS single HMM
• Smaller amount of data is required to achieve
  performance comparable to the HMM
• Any layer of the LHMM can be retrained separately
  without altering the other layers of the LHMM
• For example, the lower layers which are more
  sensitive to changes in the environment such as
  the type of sensors, sampling rate etc, can be
  retrained separately

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Continuous HMM

• Discrete HMM
• Likelyhood function p(ytxt) are discrete
• Continuous HMM
• Likelyhood function p(ytxt) are continuous
• Usually using mixture of Gaussians to approximate
  the continuous likelyhood function p(ytxt)

p(ytxt) continuous
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Semi-continuous HMM

• Modeling the discrete HMM with mixture of
  Gaussians model
• Usually select the most significant Gaussians
  only to model discrete likelyhood function
  p(ytxt)

p(ytxt)
yt
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Three Basic Problems

• The Evaluation Problem Given an HMM F and a
  sequence of observation, what is the probability
  that the model gnerates the observations ?
• The Learning Problem Given a HMM F and a set of
  observations, how can we adjust the HMM parameter
  F to maximize the joint probability (likehood)
  ?P(YF)
• The Decoding Problem (decode the hidden states,
  state estimation) Given a HMM F and a sequence
  of observations, what is the most likely state
  sequence that produces the observation ?

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Existing Algorithms

• HMM Evaluation
• Forward algorithm
• Backward algorithm
• HMM Learning
• Baum-Welch Algorithm
• HMM Decoding (state estimation)
• Viterbi algorithm
• Particle Filter

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HMM Evaluation Forward Algorithm

• The evaluation problem
• P(Y1, YT F) ?
• A problem of O(NT)
• Forward algorithm
• Solve the evaluation problem in recursive style
• Forward probability
• the probability of producing Yi,t-1 while ending
  up in state si
• At each time iteration, the only probability to
  be calculated is the forward probability

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HMM Evaluation Forward Algorithm
Initial state probabilities State transition
prob Aaij. Symbol emission prob Bbijk
Initialization
Induction
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Calculating Observation probability
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HMM Evaluation Backward Algorithm

• Backward probability
• The probability of producing the sequence Yt,T,
  given that at time t, we are at state si.

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HMM Evaluation Backward Algorithm
Initialization
Induction
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Calculating Observation probability

• Traced in a similar way except the direction is
  from tT to 1(backward)

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aß trellis
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Remarks

• The forward algorithm and backward algorithm both
  have complexity of O(TN2)

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HMM Learning Baum-Welch algorithm

• HMM Learning problem
• Given a HMM F and a set of observations, how can
  we adjust the HMM parameter F to maximize the
  joint probability (likehood) ?P(YF)
• Baum-Welch algorithm
• Similar to general EM algorithm
• The updated parameter can be obtained with
  maximizing the auxiliary function Q

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HMM Learning Baum-Welch algorithm

• The probability of taking the transition from
  state i to sate j at time t, given the model and
  observation sequence Y1,T

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HMM Learning Baum-Welch algorithm

• The equation for re-estimating the parameter

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Remarks

• Recursive algorithm
• Unsupervised learning

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HMM Decoding Viterbi Algorithm

• The Decoding Problem (decode the hidden states,
  state estimation) Given a HMM F and a sequence
  of observations, what is the most likely state
  sequence that produces the observation ?

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HMM Decoding Viterbi Algorithm

• Viterbi Algorithm
• A best path finding algorithm in dynamic
  programing
• Recursive algorithm
• The probability of the most likely state sequence
  at time t 1, which has generated the observation
  Y1,t and ends in state j

The best path
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HMM Decoding Viterbi Algorithm

• Modified forward algorithm
• example

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Particle Filter for Solving HMM Decoding

• Suitable for HMM with continuous probability as
  Relations
• Goal estimate the posterior density
• p( x(t) z(t), z(t-1),.z(0))

P(x(t)x(t-1))
P(y(t)x(t))
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Particle Filter

• Particle filter
• Sampling-Importance-Resampling Filter (SIRF)
• Approx. of posterior density
• Estimation with SIRF

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Particle Filter

• Application
• Target tracking
• Navigation
• Blind deconvolution of digital communication
  channels
• Joint channel estimation
• Detection in Rayleigh fading channels
• Digital enhancement of speech and audio signals
• Time-varying spectrum estimation
• Computer vision
• Portfolio allocation
• Sequential estimation of signals under model
  uncertainty
• ..

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Sampling

• Particle generation with importance function
  sampling
• Many choices for importance functions
• Optimal choice
• Transition density p(xn xn-1)
• Local linearizations and Gaussian approximation
  of the optimal choice

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Importance

• The weight updating EQ can be shown to be

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Resampling

• To solve the degeneracy problem
• Many existing resampling method
• Systematic resampling
• Residual resampling
• Residual systematic resampling (proposed)

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Resampling Algorithm Systematic Resampling
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Particle Filter (SIRF) Procedure
Initialization (sampling and assign uniform
weights)
Importance calculation
Resampling
Sampling
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Conclusion and Comments

• HMM
• Can be used to employ temporal relationship
• HMM Evaluation
• Forward, Backward algorithm
• HMM Learning
• Baum-Welch algorithms
• HMM Decoding
• Viterbi
• Particle filter Relations are non-deterministic
  with continuous probability.

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Conclusion and Comments

• Different problems gt different machine learning
  tool that are suitable
• Good features is also important
• Combination of N kinds of machine learning
  algorithms
• Ex. Behavior analysis
• Raw image gt feature extraction gt SVM or
  Adaboost gt HMM