THE TRANSITION PROBABILITY MATRIX OF A MARKOV CHAIN MODEL IN AN ATM NETWORK

In this paper we consider a Markov chain model in an ATM network, which has been studied by Dag and Stavrakakis. On the basis of the iterative formulas obtained by Dag and Stavrakakis, we obtain the explicit analytical expression of the transition probability matrix. It is very simple to calculate the transition probabilities of the Markov chain by these expressions. In addition, we obtain some results about the structure of the transition probability matrix, which are helpful in numerical calculation and theoretical analysis.

Using Markov chains to predict the natural progression of diabetic retinopathy

AIM: To study the natural progression of diabetic retinopathy in patients with type 2 diabetes.METHODS: This was an observational study of 153 cases with type 2 diabetes from 2010 to 2013. The state of patient was noted at end of each year and transition matrices were developed to model movement between years. Patients who progressed to severe non-proliferative diabetic retinopathy(NPDR) were treated.Markov Chains and Chi-square test were used for statistical analysis.RESULTS: We modelled the transition of 153 patients from NPDR to blindness on an annual basis. At the end of year 3, we compared results from the Markov model versus actual data. The results from Chi-square test confirmed that there was statistically no significant difference(P =0.70) which provided assurance that the model was robust to estimate mean sojourn times. The key finding was that a patient entering the system in mild NPDR state is expected to stay in that state for 5y followed by 1.07 y in moderate NPDR, be in the severe NPDR state for 1.33 y before moving into PDR for roughly8 y. It is therefore expected that such a patient entering the model in a state of mild NPDR will enter blindness after 15.29 y.CONCLUSION: Patients stay for long time periods in mild NPDR before transitioning into moderate NPDR.However, they move rapidly from moderate NPDR to proliferative diabetic retinopathy(PDR) and stay in that state for long periods before transitioning into blindness.

Characterization of geological uncertainties from limited boreholes using copula-based coupled Markov chains for underground construction

This paper proposes an efficient method for quantifying the stratigraphic uncertainties and modeling the geological formations based on boreholes.Two Markov chains are used to describe the soil transitions along different directions,and the transition probability matrices(TPMs)of the Markov chains are analytically expressed by copulas.This copula expression is efficient since it can represent a large TPM by a few unknown parameters.Due to the analytical expression of the TPMs,the likelihood function of the Markov chain model is given in an explicit form.The estimation of the TPMs is then re-casted as a multi-objective constrained optimization problem that aims to maximize the likelihoods of two independent Markov chains subject to a set of parameter constraints.Unlike the method which determines the TPMs by counting the number of transitions between soil types,the proposed method is more statistically sound.Moreover,a random path sampling method is presented to avoid the directional effect problem in simulations.The soil type at a location is inferred from its nearest known neighbors along the cardinal directions.A general form of the conditional probability,based on Pickard’s theorem and Bayes rule,is presented for the soil type generation.The proposed stratigraphic characterization and simulation method is applied to real borehole data collected from a construction site in Wuhan,China.It is illustrated that the proposed method is accurate in prediction and does not show an inclination during simulation.

Confidence intervals for Markov chain transition probabilities based on next generation sequencing reads data

Background:Markov chains(MC)have been widely used to model molecular sequences.The estimations of MC transition matrix and confidence intervals of the transition probabilities from long sequence data have been intensively studied in the past decades.In next generation sequencing(NGS),a large amount of short reads are generated.These short reads can overlap and some regions of the genome may not be sequenced resulting in a new type of data.Based on NGS data,the transition probabilities of MC can be estimated by moment estimators.However,the classical asymptotic distribution theory for MC transition probability estimators based on long sequences is no longer valid.Methods:In this study,we present the asymptotic distributions of several statistics related to MC based on NGS data.We show that,after scaling by the effective coverage d defined in a previous study by the authors,these statistics based on NGS data approximate to the same distributions as the corresponding statistics for long sequences.Results:We apply the asymptotic properties of these statistics for finding the theoretical confidence regions for MC transition probabilities based on NGS short reads data.We validate our theoretical confidence intervals using both simulated data and real data sets,and compare the results with those by the parametric bootstrap method.Conclusions:We find that the asymptotic distributions of these statistics and the theoretical confidence intervals of transition probabilities based on NGS data given in this study are highly accurate,providing a powerful tool for NGS data analysis.

On Multivariate Markov Chains for Common and Non-Common Objects in Multiple Networks

Node importance or centrality evaluation is an important methodology for network analysis.In this paper,we are interested in the study of objects appearing in several networks.Such common objects are important in network-network interactions via object-object interactions.The main contribution of this paper is to model multiple networks where there are some common objects in a multivariate Markov chain framework,and to develop a method for solving common and non-common objects’stationary probability distributions in the networks.The stationary probability distributions can be used to evaluate the importance of common and non-common objects via network-network interactions.Our experimental results based on examples of co-authorship of researchers in different conferences and paper citations in different categories have shown that the proposed model can provide useful information for researcher-researcher interactions in networks of different conferences and for paperpaper interactions in networks of different categories.

A Markov chain model of mixing kinetics for ternary mixture of dissimilar particulate solids

This paper presents a simple but informative mathematical model to describe the mixing of three dissimilar components of particulate solids that have the tendency to segregate within one another. A nonlinear Markov chain model is proposed to describe the process. At each time step, the exchange of particulate solids between the cells of the chain is divided into two virtual stages. The first is pure stochastic mixing accompanied by downward segregation. Upon the completion of this stage, some of the cells appear to be overfilled with the mixture, while others appear to have a void space. The second stage is related to upward segregation. Components from the overfilled cells fill the upper cells （those with the void space） according to the proposed algorithm. The degree of non-homogeneity in the mixture （the standard deviation） is calculated at each time step, which allows the mixing kinetics to be described. The optimum mixing time is found to provide the maximum homogeneity in the ternary mixture. However, this “common” time differs from the optimum mixing times for individual components. The model is verified using a lab-scale vibration vessel, and a reasonable correlation between the calculated and experimental data is obtained

Generation of rainfall data series by using the Markov Chain model in three selected sites in the Kurdistan Region,Iraq

Rainfall forecasting can play a significant role in the planning and management of water resource systems.This study employs a Markov chain model to examine the patterns,distributions and forecast of annual maximum rainfall(AMR)data collected at three selected stations in the Kurdistan Region of Iraq using 32 years of 1990 to 2021 rainfall data.A stochastic process is used to formulate three states(i.e.,decrease-"d";stability-"s";and increase-"i")in a given year for estimating quantitatively the probability of making a transition to any other one of the three states in the following year(s)and in the long run.In addition,the Markov model is also used to forecast the AMR data for the upcoming five years(i.e.,2022-2026).The results indicate that in the upcoming 5 years,the probability of the annual maximum rainfall becoming decreased is 44%,that becoming stable is 16%,and that becoming increased is 40%.Furthermore,it is shown that for the AMR data series,the probabilities will drop slowly from 0.433 to 0.409 in about 11 years,as indi-cated by the average data of the three stations.This study reveals that the Markov model can be used as an appropri-ate tool to forecast future rainfalls in such semi-arid areas as the Kurdistan Region of Iraq.

用Markov链模型随机模拟储层岩相空间展布

将Markov链模型应用于储层建模来获取二维或三维的储层岩相展布图像是近年来储层随机建模领域的一个研究热点。在应用该模型时,须将一维的Markov链模型拓展到二维、三维,同时要对侧向转移概率矩阵的估算和顶层初始化这2个关键问题给出有效的解决方案。笔者对二维和三维的空间Markov链模型进行了阐述,提出了依据Walther相序定律从岩相状态之间的转移记数矩阵入手,来估算侧向转移概率矩阵的方法,并对顶层初始化的解决办法进行了探索。在相关原理及技术细节问题得到解决的基础上,提出了应用Markov链模型进行二维储层岩相空间分布随机模拟的具体算法。二维空间Markov链模型的实际算例表明,对相关问题的解决方案是有效的。

基于MARKOV理论的扬州市土地利用结构预测

根据扬州市1996—2004年间的土地利用平衡表数据,求得这期间全市土地利用结构的平均转移概率矩阵,运用Markov理论,模拟并检验2004年全市的土地利用结构,发现模拟值与实际值基本吻合,说明运用Markov理论预测扬州的土地利用结构是可行的。由此对全市2010年和2020年土地利用结构变化作预测,结果表明:在今后十几年的土地利用中,耕地、园地会继续减少,工矿用地和交通用地将大幅上升,但是各自的年变化幅度都会降低,同时农村居民点用地会不断减少,表明土地集约化利用将会是扬州市未来土地利用的侧重点。建议扬州市今后能改善农用地结构、引进先进技术以提高农用地效益;同时积极挖掘存量建设用地,提高建设用地的利用率。

蝙蝠算法的Markov链模型分析

针对当前蝙蝠算法的性能改进缺少严谨的收敛性证明,导致算法的改进不具备明确的理论意义的问题,从数学概率以及蝙蝠算法状态转移满足Markov过程的角度为出发点,通过建立合理的Markov链模型研究蝙蝠个体状态的转移行为,论证蝙蝠群体状态空间具有可约性和齐次性,从理论上证明蝙蝠算法满足随机算法的收敛准则,保证算法能100%收敛到全局最优解。

利率服从Markov链的倒按揭模型

建立了利率服从Markov链的倒按揭一般模型,得到了倒按揭模型的定价方程式;在几种特殊情形下,给出了定价的精确公式.

Markov过程在预拌混凝土市场预测中应用

随机运动的系统通常具有无后效性 ,或称具有 Markov链的特性 ,这一特性广泛应用于各个领域 ,包括在管理科学的决策中 .将 Markov链应用于预拌混凝土市场预测中 ,并阐述论证了稳定状态下 Markov链转移概率的性质和平衡方程的数学意义 .

Markov双曲迭代函数系参数与吸引子关系的研究

1 引言迭代函数系IFS(Iterated Function Systems),是混沌分形理论研究的一个重要部分,其理论与方法是分形自然景观模拟及分形图像压缩的理论基础。1985年,Williams和Hutchinson开创了分形几何中IFS的研究,建立了IFS的一般基础理论;M.F.Barnsley和S.Demko的进一步工作使得这一方法成为构造任意维数分形集方便、有效的方法,并将之应用到图像的压缩与处理,使得该方法引起人们的关注。

基于Markov链的最优化模型在长江水质预测中的应用

马尔科夫预测方法在预测领域有着广泛的应用。该方法应用的一个重要问题就是如何估计一步状态转移概率矩阵。在历史资料没有给出系统处于个状态次数的情况下,给出一步状态转移概率矩阵估计的最优化方法。最后讨论了基于Markov链的最优化预测模型在长江水质预测中的应用,表明该模型有效、可行。

基于Markov过程的银行不良资产风险分析

把具有吸收态的马尔可夫链理论 ,运用于银行不良资产风险管理的可行性 ,从而建立相应的分析模型 ,设计出评价指标 。

用Markov链状预测方法估价岩质边坡变形发展的趋势

介绍了有关Markov链的基本概念及一重链状相天预测法的基本原理,根据磐石镍矿七采场二号滑体的位移观测资料,用Markov链状预测方法估价了边坡变形发展趋势。结果表明:用此预测方法估价边坡变形发展的趋势与实际边坡变形发展的趋势相吻合,可见该预测方法是有效且可行的。

高阶Markov链转移概率规律一种新表示法

在许多应用领域,高阶Markov模型正成为研究长期相关性的重要工具之一.为了克服现有高阶Markov模型不能从事动态规律研究这一缺点,讨论从初始状态到任意给定期状态转移概率的表示方法,方便动态数据分析.另外,也给出平稳分布的表达式,为稳态研究提供工具.由于研究结果与一阶Markov链的情况类似,极大地降低了高阶Markov链应用于各领域时的难度.

Markov链的一种随机时间替换

研究Markov链的一种随机时间替换,这种替换有别于通常的Markov过程理论中的随机时间替换。通常的Markov过程的随机时间替换,是通过Markov过程的可加泛函来实现的。而现在,被随机时间替换的过程X={X(n),n=0,1,2…}是一个时间离散的、状态空间可数的、时间齐次的Markov链,用于随机时间替换的过程ι={ιt,t≥0}是一个时间连续的、状态空间为非负整数集的、不降的、空间齐次的Markov链,而且X与独立。证明了随机时间替换后的过程,Y={Y(t),t≥0},Y(t)=X(ιt)是一个Markov链;并求出了Y的转移概率;当是时间齐次时,Y也是时间齐次的。

客户发展关系的Markov链一般模型

根据客户发展关系的Markov链的转移概率矩阵,建立了客户发展关系的一般模型,它包含许多特殊模型,前人研究的一些模型正是该一般模型的特例。一般模型的建立不但刻画了各种客户发展关系,而且为企业对客户发展进行定量分析和管理奠定了基础。

应用Markov理论定量描述土壤剖面中菲的垂向运移分布特征

马尔可夫(Markov)链理论是用来描述随时间(空间)变化的一个离散状态序列的状态转移特性的,本文将其引入来研究土壤剖面多环芳烃有机污染物的垂向变化规律,建立了基于空间离散化的马尔可夫(Markov)链理论数学模型及垂向归趋极限.研究结果表明,基于空间离散化的马尔可夫(Markov)链理论数学模型及垂向归趋极限能够很好地模拟区域土壤剖面多环芳烃(菲)的垂向变化特征,并有迁移规律的识别功能.