HSMM深度解析：人工智能半马尔可夫模型的实战应用与算法
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收藏 558KB PDF 举报 HSMM人工智能文档深入探讨了一种扩展自传统隐马尔可夫模型（Hidden Markov Model, HMM）的高级算法——隐半马尔可夫模型（Hidden Semi-Markov Model, HSMM）。HSMM在人工智能领域具有广泛的应用价值，尤其在处理那些状态具有随机持续时间且在每个状态下可能产生多个观测值的问题上。相比于HMM，HSMM的每一个状态不再假设其停留时间固定，而是允许这些时间是变化的，这使得它在诸如语音识别、生物信息学、自然语言处理和机器学习等复杂任务中表现优异。
文档中，作者Shun-Zheng Yu来自中山大学电子与通信工程系，他介绍了HSMM的基本概念和优势。HSMM的核心特性在于它的灵活性，能够更好地捕捉现实世界中事件的发生和持续时间的不确定性。与HMM中的Viterbi算法类似，HSMM也有自己的前向后向算法（Forward-Backward Algorithm），这一算法对于模型参数的估计、预测概率的计算、观测序列的适应性评估以及潜在状态序列的最佳查找都至关重要。
此外，HSMM还包括了两个扩展类型：显式持续时间HMM和可变持续时间HMM。显式持续时间HMM中，每个状态的持续时间是预先设定的，而可变持续时间HMM则允许更自由的时间动态。这两种变体针对不同的应用场景提供了不同的精确度和效率平衡。
HSMM的使用场景包括但不限于：
1. 语音识别：通过考虑音素的持续时间和发音模式，HSMM能更准确地识别连续的语音信号。
2. 生物信息学：在基因序列分析中，HSMM可以捕捉蛋白质家族成员之间进化过程中的动态行为。
3. 自然语言处理：用于建模文本序列中的词性、语法结构，其中词语出现的非均匀分布和词组长度的变化。
4. 时间序列预测：在经济或市场数据中，HSMM能够识别趋势和异常，预测未来的状态转换。
HSMM人工智能文档为理解和应用这种强大的统计模型提供了详尽的理论基础和实用方法，是深入理解隐半马尔可夫模型及其在复杂系统建模中的关键资源。无论是理论研究者还是实际应用工程师，学习和掌握HSMM都是提高人工智能系统性能的重要一步。
220 S.-Z. Yu / Artificial Intelligence 174 (2010) 215–243
P [o
1:T
|λ]=
j∈S
P [S
t
= j,o
1:T
|λ]=
j∈S
γ
t
( j),
for
d
, d
∈D,
j
∈S ,
i
∈S \{j} and
t
=1,...,T
, where
η
t
( j, d)/P [o
1:T
|λ] represents the probability of being in state
j
having
duration
d
by time
t
given the model and the observation sequence;
ξ
t
(i, d
; j,d)/P [o
1:T
|λ] the probability of transition at
time
t
from state
i
occurred with duration
d
to state
j
having duration
d
given the model and the observation sequence;
ξ
t
(i, j)/P [o
1:T
|λ] the probability of transition at time
t
from state
i
to state
j
given the model and the observation se-
quence;
γ
t
( j)/P [o
1:T
|λ] the probability of state
j
at time
t
given the model and the observation sequence; and
P
[o
1:T
|λ]
the probability that the observed sequence
o
1:T
is generated by the model λ. Obviously, the conditional factor
P
[o
1:T
|λ] is
common for all the posterior probabilities, which will be eliminated when the posterior probabilities are used in parameter
estimation. Therefore, it is often omitted for simplicity in the literature. Similarly, in the rest of this paper, we sometimes
will not explicitly mention this conditional factor in calculating the posterior probabilities by
η
t
( j, d), ξ
t
(i, d
; j,d), ξ
t
(i, j),
and
γ
t
( j).
In considering the following identity
P [S
t:t+1
= j,o
1:T
|λ]=P [S
t
= j,o
1:T
|λ]−P [S
t]
= j,o
1:T
|λ],
P [S
t:t+1
= j,o
1:T
|λ]=P [S
t+1
= j,o
1:T
|λ]−P [S
[t+1
= j,o
1:T
|λ]
we have a recursive formula for calculating γ
t
( j):
γ
t
( j) = γ
t+1
( j) + P [S
t]
= j,o
1:T
|λ]−P [S
[t+1
= j,o
1:T
|λ]=γ
t+1
( j) +
i∈S\{j}
ξ
t
( j, i) −ξ
t
(i, j)
. (9)
Denote
P
[o
1:T
|λ] by
L
in the following expressions. Then using the forward and backward variables, one can compute
various expectations [60]:
(a) The expected number of times state
i
ends before
t
:
1
L
t
t
j∈S\{i}
ξ
t
(i, j); The expected number of times state
i
starts at
t
or before:
1
L
t
t−1
j∈S\{i}
ξ
t
( j, i).
(b) Expected total duration spent in state
i
:
1
L
t
γ
t
(i).
(c) Expected number of times that state
i
occurred with observation
o
t
= v
k
:
1
L
t
γ
t
(i)I(o
t
= v
k
), where the indicator
function
I(x) =1ifx is true and zero otherwise.
(d) Estimated average observable values of state i:
t
γ
t
(i)o
t
t
γ
t
(i)
.
(e) Probability that state i was the first state:
1
L
γ
1
(i).
(f) Expected total number of times state i commenced:
1
L
t
j∈S\{i}
ξ
t
( j, i) or terminated:
1
L
t
j∈S\{i}
ξ
t
(i, j).Forthe
simplifying assumption for the boundary conditions described in the last subsection, we have
T −1
t
=0
j∈S\{i}
ξ
t
( j, i) =
T
t
=1
j∈S\{i}
ξ
t
(i, j).
(g) Estimated average duration of state i:
t
d
η
t
(i,d)d
t
d
η
t
(i,d)
.
2.2.3. MAP and MLE estimate of states
The maximum a posteriori (MAP) estimate of state S
t
given a specific observation sequence o
1:T
can be obtained [60] by
maximizing
γ
t
( j) given by (8), i.e.,
ˆ
s
t
=arg max
i∈S
γ
t
(i)
.
If we choose η
t
(i, d) of (6), instead of γ
t
(i), as the MAP criterion, we obtain the joint MAP estimate of the state that ends
at time t and the duration of this state, when a specific sequence o
1:T
is observed:
(
ˆ
s
t
,
ˆ
d
t
) =arg max
(i,d)
η
t
(i, d). (10)
Viterbi algorithms are the most popular dynamic programming algorithms for the maximum likelihood estimate (MLE)
of state sequence of HMMs. There exist the similar algorithms for the HSMM [115,151,35,26]. Define the forward variable
for the extended Viterbi algorithm by
δ
t
( j,d) max
s
1:t−d
P [s
1:t−d
, S
[t−d+1:t]
= j,o
1:t
|λ]= max
i∈S\{j}, d
∈D
δ
t−d
i, d
a
(i,d
)( j,d)
b
j,d
(o
t−d+1:t
)
, (11)
for 1 t T , j ∈ S, d ∈ D. δ
t
( j, d) represents the maximum likelihood that the partial state sequence ends at t in state
j of duration d. Record the previous state that
δ
t
( j, d) selects by Ψ(t, j,d) (t −d, i
∗
, d
∗
), where i
∗
is the previous state
survived, d
∗
its duration, and (t −d) its ending time. Ψ(t, j, d) is determined by letting
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