stem/AI/Pattern Matching/Markov

Hidden Markov Models - JWMI Github Rabiner - A Tutorial on Hidden Markov Models and Selected Applications in Speech Recognition

• Stochastic sequences of discrete states
  • Transitions have probabilities
• Desired output not always produced the same
  • Same pronunciation

P(X|M)=\left(\prod_{t=1}^Ta_{x_{t-1}x_t}\right)\eta_{x_T}

a_{x_0x_1}=\pi_{x_1}

1st Order

• Depends only on previous state
  • Markov assumption

P(x_t=j|x_{t-1}=i,x_{t-2}=h,...)\approx P(x_t=j|x_{t-1}=i)

• Described by state-transition probabilities

a_{ij}=P(x_t=j|x_{t-1}=i), 1\leq i,j\leq N

• \alpha
  • State transition
• For N states
  • N  by N matrix of state transition probabilities

Weather

A=\left\{a_{ij}\right\}=\begin{bmatrix} 0.4 & 0.3 & 0.3\\ 0.2 & 0.6 & 0.2 \\ 0.1 & 0.1 & 0.8 \end{bmatrix}

rain, cloud, sun across columns and down rows

A=\{\pi_j,a_{ij},\eta_i\}=\{P(x_t=j|x_{t-1}=i)\}

Start/End

• Null states
  • Entry/exit states
  • Don't generate observations

\pi_j=P(x_1=j) \space 1 \leq j \leq N

• Sub j because probability of kicking off into that state

\eta_i=P(x_T=i) \space 1 \leq i \leq N

• Sub i because probability of finishing from that state

State Duration

• Probability of staying in state decays exponentially

p(X|x_1=i,M)=(a_{ii})^{\tau-1}(1-a_{ii})

• Given, a_{33}=0.8
• $\times0.8$ repeatedly
  • Stay in state