高阶隐马氏模型算法理论若干问题的研究
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摘要
隐马氏模型作为一种具有双重随机过程的统计模型,具有可靠的概率统计理论基础和强有力的数学结构,目前已经被广泛应用于语音识别、字符识别、生物序列分析、金融数据分析、图像处理和计算机视觉等领域,但这些应用主要是基于一阶隐马氏模型的算法和理论.在一阶隐马氏模型中,假定状态转移概率和符号发出概率都只取决于当前的状态,而与以前的状态和发出的符号无关.尽管这样的假定在一定程度上对一些实际应用是有效的,并简化了相关的算法理论,但同时这样的假定也使得一阶隐马氏模型存在一定的不足和缺陷,不能对一些实际过程提供更加准确的描述,例如语音过程和DNA碱基排序过程等.
为了克服一阶隐马氏模型的不足和缺陷,一些学者从不同角度对一阶隐马氏模型提出了一些改进措施.其中之一是通过考虑状态转移概率或符号发出概率与更多远程状态之间的依赖关系,从而提出了高阶隐马氏模型.与一阶隐马氏模型相比,高阶隐马氏模型具有更好的时序结构,能够纳入更多的统计特征,在一定程度上克服了一阶隐马氏模型的不足和缺陷,从而能够为一些实际应用提供更好的模型,对一些实际过程给予更加准确的描述.尽管高阶隐马氏模型具有更好的应用潜力,但相关的应用研究还是很少的.究其原因,其中之一是高阶隐马氏模型的算法理论尚不完善,缺少有效的算法.为了让高阶隐马氏模型得以广泛的应用,必须进一步发展和完善高阶隐马氏模型的算法理论.
对于高阶隐马氏模型算法理论的研究有两种基本方法：一种方法是拓展法,就是根据一阶隐马氏模型算法的原理和方法,直接推导出高阶隐马氏模型的算法；另一种方法是模型降阶法,就是通过某种方法将高阶隐马氏模型转换为与之等价的一阶隐马氏模型,然后利用一阶隐马氏模型的标准技术建立高阶隐马氏模型的算法.本文分别使用拓展法和模型降阶法研究了高阶隐马氏模型算法理论中的若干重要问题,进一步发展和完善了高阶隐马氏模型的算法理论,为高阶隐马氏模型的实际应用奠定了一定的理论基础.本文主要获得了如下研究结果：
(1)提出了一类三阶隐马氏模型,其中状态转移概率和符号发出概率都同时取决于当前状态和前面两个状态.通过借鉴一阶隐马氏模型算法的原理和方法,使用拓展法为三阶隐马氏模型定义了相应的向前变量、向后变量和Viterbi变量,研究和推导了三阶隐马氏模型中三个基本问题的算法,即估值问题的向前-向后算法、解码问题的Viterbi算法、单观测序列下学习问题的Baum-Welch算法以及多观测序列下学习问题的Baum-Welch算法.其中,多观测序列可以是相互独立的也可以是统计相关的,并使用组合权重来刻划多观测序列的独立相关性,从而使得三阶隐马氏模型在多观测序列下学习问题的Baum-Welch算法适用于更一般的训练集.为了说明这种一般性,给出了多观测序列下学习问题Baum-Welch算法的两个特例,即多观测序列分别在相互独立时和一致相关时的Baum-Welch算法.另外,还研究了三阶隐马氏模型与一阶隐马氏模型之间的关系,构造了一个与三阶隐马氏模型等价的一阶隐马氏模型,给出并证明了它们之间的等价性定理.
(2)针对一类任意高阶的隐马氏模型,基于Hadar的等价变换方法和Li的组合方法,使用模型降阶法建立了多观测序列下高阶隐马氏模型的Baum-Welch算法,获得了多观测序列下高阶隐马氏模型的参数重估公式,并使用一阶隐马氏模型的标准技术来计算这些参数重估公式.其中,多观测序列可以是相互独立的也可以是统计相关的,并使用组合权重来刻划多观测序列的独立相关性,从而使得高阶隐马氏模型在多观测序列下学习问题的Baum-Welch算法适用于更一般的训练集.为了说明这种一般性,给出了参数重估公式的两个特例,即多观测序列分别在相互独立时和一致相关时的参数重估公式.
(3)参照混合模型的概念,提出了一类混合高阶隐马氏模型.基于模型降阶法,通过Hadar的等价变换方法将混合高阶隐马氏模型转换为与之等价的混合
一阶隐马氏模型,然后利用混合一阶隐马氏模型的Baum-Welch算法建立了混合高阶隐马氏模型的Baum-Welch算法,给出了混合高阶隐马氏模型的参数重估公式,并使用一阶隐马氏模型的标准技术来计算这些参数重估公式.
(4)针对一类任意高阶的隐马氏模型,分别使用拓展法和模型降阶法分析了高阶隐马氏模型的解码问题.至于拓展法,就是通过定义高阶隐马氏模型的Viterbi变量,根据动态规划原理建立了高阶隐马氏模型推广的Viterbi算法,利用推广的Viterbi算法可直接求得高阶隐马氏模型的最佳路径.至于模型降阶法,就是通过Hadar的等价变换方法将高阶隐马氏模型转换为与之等价的一阶隐马氏模型,然后利用一阶隐马氏模型的Viterbi算法求得这个等价的一阶隐马氏模型的最佳路径,最后根据这个等价的一阶隐马氏模型的最佳路径与原高阶隐马氏模型最佳路径之间的关系间接地求出原高阶隐马氏模型的最佳路径.
Hidden Markov model as a statistical model with a doubly stochastic process hasbeen widely applied to speech recognition, character recognition, biological sequenceanalysis, financial data analysis, image processing and computer vision, etc. not onlybecause of its reliable theoretical basis of probability and statistics but also because ofits powerful mathematical structure. These applications, however, are mainly based onthe first-order hidden Markov model in which there exists an assumption that the statetransition probability and output observation probability depend only on the current stateand have nothing with the previous states and output observations. While it can simplifythe algorithms of the first-order hidden Markov model and is also valid to some extentin some applications, this assumption causes some shortages for the first-order hiddenMarkov model and can not provide more accurate description for some real processessuch as speech process and DNA bases sorting process etc.
To overcome the shortages of the first-order hidden Markov model, some re-searchers have proposed several refinements from diferent perspectives. One of these re-finements is to extend the first-order hidden Markov model to high-order hidden Markovmodel that takes into consideration the dependency of state transition probability or out-put observation probability on more long-distance states. By contrast with the first-orderhidden Markov model, high-order hidden Markov model has better temporal structureand can contain more statistical characteristics, so that it can partly compensate for theshortages of the first-order hidden Markov model and more exactly match some real pro-cesses. In spite of its potential, high-order hidden Markov model has been found littleapplication in the existing literatures. One of the reasons for little application is the lackof efective algorithms of high-order hidden Markov model due to its imperfect theory.To obtain more wide applications of high-order hidden Markov model, the algorithms ofhigh-order hidden Markov model must be further developed and improved.
There are two basic approaches to study the algorithms of high-order hiddenMarkov model. The first one is called the extended approach, which is to extend directlythe existing algorithms of the first-order hidden Markov model to high-order hiddenMarkov model by using the principles and techniques of the algorithms of the first-orderhidden Markov models. The second one is called the model reduction method, which isto transform high-order hidden Markov to an equivalent first-order hidden Markov modelby some means and then to establish the algorithms of high-order hidden Markov modelby using standard techniques applicable to the first-order hidden Markov model. Thisdissertation uses the extended approach and the model reduction method respectively to discuss several important problems on the algorithms of high-order hidden Markovmodel, and then further develops and improves the algorithms of high-order hiddenMarkov model. This dissertation lays a theory foundation for the application of high-order hidden Markov model. The main findings of this dissertation are listed as follows:
(1) A class of third-order hidden Markov model is proposed. In this proposedmodel, both state transition probability and output observation probability depend onthe current state and on the two preceding states as well. With reference of the prin-ciples and techniques of the algorithms of the first-order hidden Markov models andby using the extended method, three variables for third-order hidden Markov model aredefined, including the forward variable, the backward variable and the Viterbi variable,and then three algorithms of third-order hidden Markov model are developed and de-rived, including the forward-backward algorithm for observation sequence evaluation,the Viterbi algorithm for determining the optimal state sequence, the Baum-Welch al-gorithm for training third-order hidden Markov model with single observation sequenceand the Baum-Welch algorithm for training third-order hidden Markov model with mul-tiple observation sequences. In this treatment, multiple observation sequences are eitherindependent of each other or statistically correlated, and the dependence-independenceproperty of multiple observation sequences is characterized by combinational weights,so that the Baum-Welch algorithm for training third-order hidden Markov model withmultiple observation sequences can be suitable for a more general training set. To showthis generality, two special cases of the Baum-Welch algorithm are given when multipleobservation sequences are independent of each other and uniformly dependent on oneanother respectively. In addition, a first-order hidden Markov model equivalent to thethird-order hidden Markov model is constructed, and then a theorem of their equivalenceis proposed and proved.
(2) Aiming at a class of any high-order hidden Markov model, based on Hadar’sequivalent transformation method and Li’s combinatorial method, the Baum-Welch algo-rithm for training high-order hidden Markov model with multiple observation sequencesis developed and derived by using the model reduction method, and then the correspond-ing parameter reestimation formula can be expressed by using standard techniques ap-plicable to the first-order hidden Markov model. In this treatment, multiple observa-tion sequences are either independent of each other or statistically correlated, and thedependence-independence property of multiple observation sequences is characterizedby combinational weights, so that the Baum-Welch algorithm for training high-orderhidden Markov model with multiple observation sequences can be suitable for a moregeneral training set. To show this generality, two special cases of the parameter reesti- mation formula are given when multiple observation sequences are independent of eachother and uniformly dependent on one another respectively.
(3) With reference of the definition of mixture model, a class of mixture high-orderhidden Markov model is proposed. Based on the model reduction method, mixture high-order hidden Markov model is transformed into two equivalent mixture first-order hid-den Markov models respectively by using Hadar’s equivalent transformation method,and then the Baum-Welch algorithm of mixture high-order hidden Markov model is de-veloped and derived by means of the Baum-Welch algorithm of an equivalent mixturefirst-order hidden Markov model. Also, the parameter reestimation formula of mixturehigh-order hidden Markov model can be expressed by using standard techniques appli-cable to the first-order hidden Markov model.
(4) Aiming at a class of any high-order hidden Markov model, the extended ap-proach and the model reduction method are used respectively to decode high-order hid-den Markov model. As for the extended approach, the Viterbi variable of high-orderhidden Markov model is firstly defined, and then the extended Viterbi algorithm of high-order hidden Markov model is established according to the dynamic programming princi-ple. The extended Viterbi algorithm can determine directly the optimal state sequence ofhigh-order hidden Markov model. As for the model reduction method, high-order hid-den Markov model is transformed into an equivalent first-order hidden Markov modelby Hadar’s equivalent transformation method, and then the optimal state sequence of theequivalent first-order hidden Markov model is determined by the existing Viterbi algo-rithm of the first-order hidden Markov model. Based on their equivalence, the optimalstate sequence of high-order hidden Markov model is inferred from the optimal statesequence of the equivalent first-order hidden Markov model.
为了克服一阶隐马氏模型的不足和缺陷,一些学者从不同角度对一阶隐马氏模型提出了一些改进措施.其中之一是通过考虑状态转移概率或符号发出概率与更多远程状态之间的依赖关系,从而提出了高阶隐马氏模型.与一阶隐马氏模型相比,高阶隐马氏模型具有更好的时序结构,能够纳入更多的统计特征,在一定程度上克服了一阶隐马氏模型的不足和缺陷,从而能够为一些实际应用提供更好的模型,对一些实际过程给予更加准确的描述.尽管高阶隐马氏模型具有更好的应用潜力,但相关的应用研究还是很少的.究其原因,其中之一是高阶隐马氏模型的算法理论尚不完善,缺少有效的算法.为了让高阶隐马氏模型得以广泛的应用,必须进一步发展和完善高阶隐马氏模型的算法理论.
对于高阶隐马氏模型算法理论的研究有两种基本方法：一种方法是拓展法,就是根据一阶隐马氏模型算法的原理和方法,直接推导出高阶隐马氏模型的算法；另一种方法是模型降阶法,就是通过某种方法将高阶隐马氏模型转换为与之等价的一阶隐马氏模型,然后利用一阶隐马氏模型的标准技术建立高阶隐马氏模型的算法.本文分别使用拓展法和模型降阶法研究了高阶隐马氏模型算法理论中的若干重要问题,进一步发展和完善了高阶隐马氏模型的算法理论,为高阶隐马氏模型的实际应用奠定了一定的理论基础.本文主要获得了如下研究结果：
(1)提出了一类三阶隐马氏模型,其中状态转移概率和符号发出概率都同时取决于当前状态和前面两个状态.通过借鉴一阶隐马氏模型算法的原理和方法,使用拓展法为三阶隐马氏模型定义了相应的向前变量、向后变量和Viterbi变量,研究和推导了三阶隐马氏模型中三个基本问题的算法,即估值问题的向前-向后算法、解码问题的Viterbi算法、单观测序列下学习问题的Baum-Welch算法以及多观测序列下学习问题的Baum-Welch算法.其中,多观测序列可以是相互独立的也可以是统计相关的,并使用组合权重来刻划多观测序列的独立相关性,从而使得三阶隐马氏模型在多观测序列下学习问题的Baum-Welch算法适用于更一般的训练集.为了说明这种一般性,给出了多观测序列下学习问题Baum-Welch算法的两个特例,即多观测序列分别在相互独立时和一致相关时的Baum-Welch算法.另外,还研究了三阶隐马氏模型与一阶隐马氏模型之间的关系,构造了一个与三阶隐马氏模型等价的一阶隐马氏模型,给出并证明了它们之间的等价性定理.
(2)针对一类任意高阶的隐马氏模型,基于Hadar的等价变换方法和Li的组合方法,使用模型降阶法建立了多观测序列下高阶隐马氏模型的Baum-Welch算法,获得了多观测序列下高阶隐马氏模型的参数重估公式,并使用一阶隐马氏模型的标准技术来计算这些参数重估公式.其中,多观测序列可以是相互独立的也可以是统计相关的,并使用组合权重来刻划多观测序列的独立相关性,从而使得高阶隐马氏模型在多观测序列下学习问题的Baum-Welch算法适用于更一般的训练集.为了说明这种一般性,给出了参数重估公式的两个特例,即多观测序列分别在相互独立时和一致相关时的参数重估公式.
(3)参照混合模型的概念,提出了一类混合高阶隐马氏模型.基于模型降阶法,通过Hadar的等价变换方法将混合高阶隐马氏模型转换为与之等价的混合
一阶隐马氏模型,然后利用混合一阶隐马氏模型的Baum-Welch算法建立了混合高阶隐马氏模型的Baum-Welch算法,给出了混合高阶隐马氏模型的参数重估公式,并使用一阶隐马氏模型的标准技术来计算这些参数重估公式.
(4)针对一类任意高阶的隐马氏模型,分别使用拓展法和模型降阶法分析了高阶隐马氏模型的解码问题.至于拓展法,就是通过定义高阶隐马氏模型的Viterbi变量,根据动态规划原理建立了高阶隐马氏模型推广的Viterbi算法,利用推广的Viterbi算法可直接求得高阶隐马氏模型的最佳路径.至于模型降阶法,就是通过Hadar的等价变换方法将高阶隐马氏模型转换为与之等价的一阶隐马氏模型,然后利用一阶隐马氏模型的Viterbi算法求得这个等价的一阶隐马氏模型的最佳路径,最后根据这个等价的一阶隐马氏模型的最佳路径与原高阶隐马氏模型最佳路径之间的关系间接地求出原高阶隐马氏模型的最佳路径.
Hidden Markov model as a statistical model with a doubly stochastic process hasbeen widely applied to speech recognition, character recognition, biological sequenceanalysis, financial data analysis, image processing and computer vision, etc. not onlybecause of its reliable theoretical basis of probability and statistics but also because ofits powerful mathematical structure. These applications, however, are mainly based onthe first-order hidden Markov model in which there exists an assumption that the statetransition probability and output observation probability depend only on the current stateand have nothing with the previous states and output observations. While it can simplifythe algorithms of the first-order hidden Markov model and is also valid to some extentin some applications, this assumption causes some shortages for the first-order hiddenMarkov model and can not provide more accurate description for some real processessuch as speech process and DNA bases sorting process etc.
To overcome the shortages of the first-order hidden Markov model, some re-searchers have proposed several refinements from diferent perspectives. One of these re-finements is to extend the first-order hidden Markov model to high-order hidden Markovmodel that takes into consideration the dependency of state transition probability or out-put observation probability on more long-distance states. By contrast with the first-orderhidden Markov model, high-order hidden Markov model has better temporal structureand can contain more statistical characteristics, so that it can partly compensate for theshortages of the first-order hidden Markov model and more exactly match some real pro-cesses. In spite of its potential, high-order hidden Markov model has been found littleapplication in the existing literatures. One of the reasons for little application is the lackof efective algorithms of high-order hidden Markov model due to its imperfect theory.To obtain more wide applications of high-order hidden Markov model, the algorithms ofhigh-order hidden Markov model must be further developed and improved.
There are two basic approaches to study the algorithms of high-order hiddenMarkov model. The first one is called the extended approach, which is to extend directlythe existing algorithms of the first-order hidden Markov model to high-order hiddenMarkov model by using the principles and techniques of the algorithms of the first-orderhidden Markov models. The second one is called the model reduction method, which isto transform high-order hidden Markov to an equivalent first-order hidden Markov modelby some means and then to establish the algorithms of high-order hidden Markov modelby using standard techniques applicable to the first-order hidden Markov model. Thisdissertation uses the extended approach and the model reduction method respectively to discuss several important problems on the algorithms of high-order hidden Markovmodel, and then further develops and improves the algorithms of high-order hiddenMarkov model. This dissertation lays a theory foundation for the application of high-order hidden Markov model. The main findings of this dissertation are listed as follows:
(1) A class of third-order hidden Markov model is proposed. In this proposedmodel, both state transition probability and output observation probability depend onthe current state and on the two preceding states as well. With reference of the prin-ciples and techniques of the algorithms of the first-order hidden Markov models andby using the extended method, three variables for third-order hidden Markov model aredefined, including the forward variable, the backward variable and the Viterbi variable,and then three algorithms of third-order hidden Markov model are developed and de-rived, including the forward-backward algorithm for observation sequence evaluation,the Viterbi algorithm for determining the optimal state sequence, the Baum-Welch al-gorithm for training third-order hidden Markov model with single observation sequenceand the Baum-Welch algorithm for training third-order hidden Markov model with mul-tiple observation sequences. In this treatment, multiple observation sequences are eitherindependent of each other or statistically correlated, and the dependence-independenceproperty of multiple observation sequences is characterized by combinational weights,so that the Baum-Welch algorithm for training third-order hidden Markov model withmultiple observation sequences can be suitable for a more general training set. To showthis generality, two special cases of the Baum-Welch algorithm are given when multipleobservation sequences are independent of each other and uniformly dependent on oneanother respectively. In addition, a first-order hidden Markov model equivalent to thethird-order hidden Markov model is constructed, and then a theorem of their equivalenceis proposed and proved.
(2) Aiming at a class of any high-order hidden Markov model, based on Hadar’sequivalent transformation method and Li’s combinatorial method, the Baum-Welch algo-rithm for training high-order hidden Markov model with multiple observation sequencesis developed and derived by using the model reduction method, and then the correspond-ing parameter reestimation formula can be expressed by using standard techniques ap-plicable to the first-order hidden Markov model. In this treatment, multiple observa-tion sequences are either independent of each other or statistically correlated, and thedependence-independence property of multiple observation sequences is characterizedby combinational weights, so that the Baum-Welch algorithm for training high-orderhidden Markov model with multiple observation sequences can be suitable for a moregeneral training set. To show this generality, two special cases of the parameter reesti- mation formula are given when multiple observation sequences are independent of eachother and uniformly dependent on one another respectively.
(3) With reference of the definition of mixture model, a class of mixture high-orderhidden Markov model is proposed. Based on the model reduction method, mixture high-order hidden Markov model is transformed into two equivalent mixture first-order hid-den Markov models respectively by using Hadar’s equivalent transformation method,and then the Baum-Welch algorithm of mixture high-order hidden Markov model is de-veloped and derived by means of the Baum-Welch algorithm of an equivalent mixturefirst-order hidden Markov model. Also, the parameter reestimation formula of mixturehigh-order hidden Markov model can be expressed by using standard techniques appli-cable to the first-order hidden Markov model.
(4) Aiming at a class of any high-order hidden Markov model, the extended ap-proach and the model reduction method are used respectively to decode high-order hid-den Markov model. As for the extended approach, the Viterbi variable of high-orderhidden Markov model is firstly defined, and then the extended Viterbi algorithm of high-order hidden Markov model is established according to the dynamic programming princi-ple. The extended Viterbi algorithm can determine directly the optimal state sequence ofhigh-order hidden Markov model. As for the model reduction method, high-order hid-den Markov model is transformed into an equivalent first-order hidden Markov modelby Hadar’s equivalent transformation method, and then the optimal state sequence of theequivalent first-order hidden Markov model is determined by the existing Viterbi algo-rithm of the first-order hidden Markov model. Based on their equivalence, the optimalstate sequence of high-order hidden Markov model is inferred from the optimal statesequence of the equivalent first-order hidden Markov model.
引文
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