Study On The Prediction Method Based On Higher Order Period Markov Chain Model

Stochastic prediction is to reveal the statistical regularities of future developmentof random system through investigation and analysis of the history development.Markov process is a vital tool to characterize the random dynamic phenomenon andwhich has achieved greater prosperity in random prediction theory and application.Aftereffect, the characteristic of Markov process, means the future time variable relyonly on the current close state but does not matter on the past. However, except theideal circumstance, things cannot be totally determined by the closest known state. Thetraditional Markov processes abandon the older useful information in describingrandom phenomena which course the process development roughly. Therefore, the mainpurpose of this article is to establish a higher order Markov chain model, which includesthe more preliminary information into forecasts of future variables and the more precisein random prediction. The study of period of higher order Markov chain mode, as wellas the stationary distribution, improves the theoretical results of parameter estimation.And this article also focus on the combination of higher order multivariate Markovchain model, it proposes a newly research method for the multidimensional sequenceprediction problem of discrete category variables.The main research work includes, based on the theoretical background of thehigher order Markov chain model, put forward the concept of period for higherorder chain and argument the existence of stationary distribution. And it alsoincludes the limit distribution characteristics, parameter estimation and predictionmethod of the higher order Markov chain model. Then the expand study formulti-sequence modeling to explore the multidimensional and higher order Markovchain model with order selection, parameter estimation and forecasting application.Firstly, this article explores the means of the order of the higher order Markovchain model. The order is the number of lag dependencies steps between the internalsequence variables. Higher order Markov chain model take advantage of the multiplehistorical status information when it associate with the development of a dynamicvariable high-level variables. It reconstruct the state space corresponding with the orderon the state space of the traditional Markov chain order and export a class ofreduced-order Markov chain model. Vector in reduced-order model is actually adynamic variable dependency links into for the reconstruction of the state space, and the order of the high-order model performance as the reconstruction of the space dimension.The empirical research shows that the reduced order model of higher order Markovchain model is a promotion of traditional high order mixture transformation model, andit can not only describe the behavior of the traditional model, but also more subtlestructure than mixture transformation model. The disadvantages of the reduced ordermodel include difficulty of parameters estimation and of high capacity of samplesequence requirement.Secondly, this article proposes the period definition of higher order Markov chain.After the investigation of the limit distribution of the higher order Markov chain model,chain transfer distribution can be two features. One is converges to the limit distributionand the other is changes in a period distribution cycle. Then it investigates the existenceconditions for the period of the higher order Markov chain model. This article examinesthe characteristics of the limit distribution with periodic or non-periodic; It gives highorder sub-matrix of the higher order Markov chain connectivity and the equivalentconditions of existence; It derive the period number with each of the higher ordersub-matrix of the chain, as well as the number of chain state space dimensionmonotonic relationship. Finally, this article analysis of the relationship between the highorder model in the multi-step transition probability matrix connectivity with thestationary distribution of the chain, and it prove the existence and uniqueness conditionsof higher order Markov chain stationary distribution, all of that improve the high-orderMarkov chain parameter estimation theory.Further more, this article expanse the high-order methods to multiple sequencemodeling and establish a multivariate higher order Markov chain model. MultivariateMarkov model is applied to the interaction analysis of multiple random sequences in thestochastic system. It builds a formation of cross random sequences inferred mode,which is a useful expand from traditional Markov model. The article presents theconcept of cross-correlation function between the column series, and analysis the orderselection method from the multi-sequence model. For the problem of large numberparameters estimated difficulty, this paper solves it by proposing a minimum predictionerror optimization objective function and pointing to the weighted parameter betweenthe columns batch the optimization.Finally, this article practices an empirical application for the high ordermultivariate Markov chain. Because the higher order Markov chain model involvesmore upfront multivariate information in stochastic modeling, analysis of the model iscloser to the real situation. It has a greater advantage in dynamic description and prediction analysis. The multivariate higher order Markov chain model is applied onfive industry sub-index discrete sequences to find the inherent characteristicdependencies between of each industry sub-index of Shanghai. Multivariate higherorder Markov chain model is applied to the interaction of multiple random sequencedata to predict can not only reflect the chain of higher order information, but also takeadvantage of the mutual information of the random sequence of multiple stochasticsystems. All of those to form a cross between random sequences inferred mode and toimprove forecast accuracy. Multiple interaction information show the relationshipbetween application of stock industry categorical index series data and the model fulfillthe forecastes of the stock categorical index sequences.