基于高阶周期Markov链模型的预测方法研究
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摘要
随机预测是通过对随机系统的历史和现状进行科学的调查和分析，揭示其未来发展的统计规律。Markov过程是刻画动态随机现象的一个重要工具，在随机预测理论和应用领域已取得较大的繁荣。Markov过程的特征是无后效性，这些研究仅考虑时间变量的状态仅与相邻状态有关，而与过去没有关系。然而除理想状态以外，事物的发展并不是以最近已知的状态就可以完全决定。传统的Markov过程在描述随机现象时摒弃了这类远期相关的有用信息，在刻画系统的发生变化过程上较为粗糙。因此，本文研究的主要目的是建立高阶Markov链模型，将更多的前期信息融入未来变量的预测中，提高随机预测的精度。通过高阶Markov链模型周期以及平稳分布的研究，丰富模型参数估计的理论成果。高阶与多元Markov链模型的结合，为离散属性变量多维序列的预测问题研究提供了一个新的研究方法。
本论文的主要研究工作是基于高阶Markov链模型的理论背景，提出链周期的概念，论证链平稳分布的存在性，对高阶Markov链模型的极限分布特征、参数估计、预测方法进行研究。并将高阶方法拓展到多元序列的建模，探讨了多元高阶Markov链模型的阶数判定、参数估计和预测应用。
首先，本文探究了高阶Markov链模型阶数的内涵，阶数是序列内部变量间相依滞后的步数。高阶Markov链模型在解决动态变量的发展时，利用了前当变量与多个历史状态相联系的高阶信息。通过对传统高阶Markov链模型的状态空间按对应的阶数进行向量重构，导出一类在重构状态空间上的降阶Markov链模型。降阶模型实际上是将动态变量的相依联系转化为重构的状态空间，高阶模型的阶数表现为重构空间的维数。经数据算例研究表明，高阶Markov链模型降阶模型是对传统高阶混合转移模型的一个推广，不但可以描述传统模型的全部性态，更能表达较混合转移模型更加细微的随机结构。降阶模型的缺点是参数估计较困难，对样本序列容量要求很高。
其次，本文给出了高阶Markov链周期的定义。通过对高阶Markov链模型的极限分布的研究，发现链的转移分布可能呈两种特征，即收敛于极限分布或以一组循环分布周期变化，进而探讨了高阶Markov链模型中周期的存在性条件和性质。论文研究了周期模型与非周期模型对链的转移分布的极限特征的影响；给出了高阶Markov链中高阶子矩阵的连通性与周期的存在性的等价条件；推导出了高阶Markov链的周期数与各高阶子矩阵的周期，以及链的状态空间维数的数量单调关系。最后，分析了高阶模型中多步转移概率矩阵的连通性与链的平稳分布的关系，证明了高阶Markov链平稳分布的存在性与唯一性条件，完善了高阶Markov链的参数估计理论。
再次，本文将高阶方法拓展到多元序列的建模，建立了多元高阶Markov链模型。多元Markov模型将随机系统中多个随机序列的交互信息应用于建模分析，形成随机序列间的交叉推断模式，是传统Markov模型的良好拓展。论文提出了列间互相关函数的概念，分析给出多元序列模型阶数选择方法。针对多元Markov链模型分析中，待估参数数量大而导致估计困难的问题，本文提出以一元预测误差最小为优化目标，对列间权重参数进行了分批次优化求解的改进方法。
最后，论文对高阶多元Markov链作了应用算例研究。由于高阶Markov链模型采用更多的前期多元信息应用于随机建模，使模型分析更贴近于真实的情形，在动态描述和预测分析上有较大的优势。将多元高阶Markov链模型应用于我国上证五大行业分类指数离散化序列，研究了各行业分类指数相互间的内在相依特征。多元高阶Markov链模型应用于相互影响的多元随机序列数据预测中，既反映了链的高阶信息，又能利用随机系统中多个随机序列的交互信息，形成随机序列间的交叉推断的模式，较好地提高预测精确度。应用股票行业分类指数序列间的多元交互信息，实现了分类股指序列的预测。
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.
本论文的主要研究工作是基于高阶Markov链模型的理论背景，提出链周期的概念，论证链平稳分布的存在性，对高阶Markov链模型的极限分布特征、参数估计、预测方法进行研究。并将高阶方法拓展到多元序列的建模，探讨了多元高阶Markov链模型的阶数判定、参数估计和预测应用。
首先，本文探究了高阶Markov链模型阶数的内涵，阶数是序列内部变量间相依滞后的步数。高阶Markov链模型在解决动态变量的发展时，利用了前当变量与多个历史状态相联系的高阶信息。通过对传统高阶Markov链模型的状态空间按对应的阶数进行向量重构，导出一类在重构状态空间上的降阶Markov链模型。降阶模型实际上是将动态变量的相依联系转化为重构的状态空间，高阶模型的阶数表现为重构空间的维数。经数据算例研究表明，高阶Markov链模型降阶模型是对传统高阶混合转移模型的一个推广，不但可以描述传统模型的全部性态，更能表达较混合转移模型更加细微的随机结构。降阶模型的缺点是参数估计较困难，对样本序列容量要求很高。
其次，本文给出了高阶Markov链周期的定义。通过对高阶Markov链模型的极限分布的研究，发现链的转移分布可能呈两种特征，即收敛于极限分布或以一组循环分布周期变化，进而探讨了高阶Markov链模型中周期的存在性条件和性质。论文研究了周期模型与非周期模型对链的转移分布的极限特征的影响；给出了高阶Markov链中高阶子矩阵的连通性与周期的存在性的等价条件；推导出了高阶Markov链的周期数与各高阶子矩阵的周期，以及链的状态空间维数的数量单调关系。最后，分析了高阶模型中多步转移概率矩阵的连通性与链的平稳分布的关系，证明了高阶Markov链平稳分布的存在性与唯一性条件，完善了高阶Markov链的参数估计理论。
再次，本文将高阶方法拓展到多元序列的建模，建立了多元高阶Markov链模型。多元Markov模型将随机系统中多个随机序列的交互信息应用于建模分析，形成随机序列间的交叉推断模式，是传统Markov模型的良好拓展。论文提出了列间互相关函数的概念，分析给出多元序列模型阶数选择方法。针对多元Markov链模型分析中，待估参数数量大而导致估计困难的问题，本文提出以一元预测误差最小为优化目标，对列间权重参数进行了分批次优化求解的改进方法。
最后，论文对高阶多元Markov链作了应用算例研究。由于高阶Markov链模型采用更多的前期多元信息应用于随机建模，使模型分析更贴近于真实的情形，在动态描述和预测分析上有较大的优势。将多元高阶Markov链模型应用于我国上证五大行业分类指数离散化序列，研究了各行业分类指数相互间的内在相依特征。多元高阶Markov链模型应用于相互影响的多元随机序列数据预测中，既反映了链的高阶信息，又能利用随机系统中多个随机序列的交互信息，形成随机序列间的交叉推断的模式，较好地提高预测精确度。应用股票行业分类指数序列间的多元交互信息，实现了分类股指序列的预测。
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.
引文
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