The Strict Definition Of Higher-order Discrete Hidden Markov Modeland Its Equivalent Nature

Hidden Markov model is used as a statistical model,which has a double random process. It has been widely used in speech recognition, biological sequence analysis,image processing, computer vision character recognition and so on. In a first order hidden Markov model, such an approach is effective for some practical application in a way. It is appropriate and simplified calculation, but this model is inability to express the farther distance dependencies between the state. It might not be able to make an accurate estimate to the actual situation.They proposed high-order hidden Markov model which overcomes the shortcomings and deficiencies of first order hidden Markov models in a way, giving a better description in practical processing.Although high-order hidden Markov model has been widely used in many aspects,it has developed many branches to analysis the specific issues efficiently.We provide the specific meaning of the various conditions in the description of the model,but these conditions are not strict.So far, the strict definition of the model has not been given by the researchers, which can not lay a solid foundation at the theoretical level.In this paper, we give the strict definition of second-order discrete Hidden Markov model and high-order discrete Hidden Markov model according to the strict definition of the first-order hidden Markov model, and to study the two relevant equivalent properties.In this paper, we introduced the origin, development process,the advantages,and main application directions of the hidden Markov model. The current situation and purpose of the research is also given in the paper. Then,the part of concept and nature involved of this paper is described in detail.A strict definition and equivalent nature of first order Hidden Markov model is given in the field of mathematics.In the third chapter,strict definition of a second-order hidden Markov model is given. The model includes two processes. One is the state process, describing the transition probability between the states with the two order Markov chain.The other is the observation process.The the issued probability of a symbol of the next step relies only on the previous two states,being independent of the previous status and issuedsymbols under the premise of a given current state.At last,two sufficient and necessary conditions of the model definition are showed on this basis.In the fourth chapter,extending it to more general cases, the strict definition of high-order discrete hidden Markov models is given.The model includes two processes,too. One is the state process, describing the transition probability between the states with the high order Markov chain.The transition probability of the next step relies only on the m previous states,being independent of the previous status and issued symbols under the premise of a given current state.The other is the observation process,describing the statistical relationship between the state and the observed value.The issued probability of a symbol at the next step relies only on the n previous states,being independent of the previous status and issued symbols under the premise of a given current state.At last, two sufficient and necessary the conditions of the model definition are showed in this paper.The second sufficient and necessary condition is researched at m=2,n=3 special circumstance.