第八讲-隐马尔柯夫模型

N α t +1 ( j ) = ∑ α t (i ) aij b j (Ot +1 ), i =1 1 ≤ t ≤ T − 1, 1 ≤ j ≤ N.
– Termination
P (O | λ ) = ∑ α T (i )
i =1 N
– Complexity: O(TN2)
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P(Q λ ) = π q1 aq1q2 aq2q3 ⋯ aqT −1qT
Markov chain of states
– Complexity: O(2TNT)!
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• Evaluation: Forward Procedure
– Define forward variable α t (i ) = P(O1O2 ⋯ Ot , qt = Si | λ ) – Initialization α1 (i ) = π i bi (O1 ), 1 ≤ i ≤ N – Induction
• Sequential (Temporal) Pattern
– Variable length – Distortion – Ambiguous boundary between primitives (symbols)
• Dynamic Time Warping (DTW)
– – – – – Dynamic programming (DP) Tolerant to distortion Computation considerable Model learning non-trivial Suitable for small sample
e.g. 1245526462146146136136661664661636616366163616515615115146123562344 F L F
• Training problem: How “loaded” works? How “fair” works? How often does transitions fair ↔ loaded?
• Evaluation problem: How likely is this sequence, given our model of how the casino works? e.g. P(O|λ)= 1.3 x 10-35 • Decoding problem: What portion of O was generated with the fair die, and what portion with the loaded die?
P(q1 = F ) e.g. 1 π= = P(q1 = L) 1 2 2
F
0.05
L
Given: O=1234561626364656 Infer: Is the loaded die used? Further, where did it roll?
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• Three Basic Problems of HMM
• Observations: temperature, humidity • Hidden states: weather
– Hidden Markov Model (HMM): Doubly embedded stochastic process O O
1 2
OT
P(O1,O2,…,OT) P(q1,q2,…,qT) • Infer states from observations
• Evaluation: Backward Procedure
– Define backward variable βt (i ) = P(Ot +1 ,⋯ , OT | qt = Si , λ ) – Initialization βT (i ) = 1, 1 ≤ i ≤ N – Induction
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• Probabilistic: Hidden Markov Model (HMM)
Markov Chain
• Sequence of States
P (q1q2 ⋯ qT ) = P (q1 ) P (q2 | q1 ) P(q3 | q1q2 )⋯ P (qT | q1 ⋯ qT −1 )
qt ∈ {S1 ,… S N }
0.95 P(1): 1/6 P(2): 1/6 P(3): 1/6 P(4): 1/6 P(5): 1/6 P(6): 1/6 0.05 0.95 P(1): 1/10 P(2): 1/10 P(3): 1/10 P(4): 1/10 P(5): 1/10 P(6): 1/2
0.05
Fair die
• Example: Transition of Weather
State 1: rain or (snow) State 2: cloudy State 3: sunny
0.4 0.3 0.3 A = {aij } = 0.2 0.6 0.2 0.1 0.1 0.8
– N=2, S={ S1=F, S2=L } – M=6, V={1, 2, 3, 4, 5, 6} – Transition probability distribution
P( F | F ) P( F | L) 0.95 0.05 A= = P L F P L L ( | ) ( | ) 0.05 0.95
e.g. 1245526462146146136136661664661636616366163616515615115146123562344
0.95 F
0.05 L 0.05
0.95
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Evaluation Problem
• Given model λ = ( A, B, π) and observation sequence O=O1O2…OT, compute P (O | λ )
– Direct computation
P(O λ ) = =
∑ P(O | Q, λ )P(Q | λ )
all Q
∑
q1 ,q2 , ..., qT
T
π q bq (O1 )aq q bq (O2 )⋯ aq
1 1 1 2 2
T −1qT
bqT (OT )
P(O Q, λ ) = ∏ P(Ot | qt , λ ) Conditional t =1 independence = bq1 (O1 )bq2 (O2 )⋯bqT (OT )
π i = P (q1 = Si ), 1 ≤ i ≤ N
• Concrete Example
– Occasionally dishonest casino
• One die is fair • Another is loaded (unfair): biased to six
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• HMM for Dishonest Casino λ = ( A, B, π)
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Expected number of days for sunny and cloudy?
Hidden Markov Model (HMM)
• Markov Chain: States are Observable • Hidden States: An Example
– Imagine you are in a un-windowed room, cannot see the weather outside. Instead, you can guess the weather from the temperature and humidity in room
中科院自动化所博士生模式识别课程讲义
隐马尔柯夫模型(HMM)
2010年5月18日 刘成林
Outline
• • • • • • • Sequential Pattern Recognition Markov Chain and HMM Three Basic Problems of HMM Types of HMMs Implementation Issues Application to Speech Recognition Extensions
q1
q2
qT
6
qi∈{S1,S2,…,SN}
• Elements of an HMM
λ = ( A , B, π )
– N: number of states in the model, S={S1,S2,…,SN} – M: number of observation symbols, V={v1,v2,…,vM} – State transition probability distribution A={aij}
– Problem 1 (Evaluation):
How to efficiently compute the probability of observation sequence P(O|λ) （训练中要用到） 模式识别的 关键步骤
– Problem 2 (Decoding):
How to choose the best state sequence responding to an observation sequence
1 2 3 d d+1
P(O | Model, q1 = Si ) = (aii ) d −1 (1 − aii ) = pi (d )
– Expected duration of specific state ∞
di = ∑ d pi (d )
d =1
= ∑ d (aii )
d =1
∞
d −1
1 (1 - aii ) = 1 − aii
– Problem 3 (Training):
How to estimate the model parameters