Finite Markov Chain Imbedding and Its Applications in Stochastic Processes, biological Sequences, and Discrete Mathematics

有限马尔可夫链嵌入及其在随机过程、生物序列和离散数学中的应用

• 批准号: RGPIN-2015-06698
• 负责人: Fu, James
• 金额: $ 0.8万
• 依托单位: University of Manitoba
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=651245
• 关键词: Finite; Markov; Chain; Imbedding; Its

The finite Markov chain imbedding (FMCI) technique is an unconventional, simple, flexible and computation efficient probabilistic tool to evaluate probabilities and distributions of runs and patterns of interest. It has been successfully applied for solving complex and unsolved problems in various areas such as health science, genomic analysis, reliability, quality control, physics, statistics, applied probability, computer science, and discrete mathematics. In this proposal, the FMCI technique is going to be extended into four very important applied areas, (i) boundary crossing probability (BCP) for high dimensional Brownian motion, (ii) matching probability of two DNA sequences with allowing at most d mutations, (iii) distributions of patterns to avoid in [S]-specified random permutation and (iv) distributions of bumps of genome-wide association studies for comparing gene expressions between normal and disease chromosomes. The following are expected results:  ***A. Short term expected results (next five years)***(i)   The first part of the proposal will establish an analytical and efficient numerical method for approximating the boundary crossing probabilities for non-linear convex boundaries of d-dimensional Brownian motion. It will show the rate of convergence is O(1/n1/2) and independent of the dimensionality d. The results will be extended to related stochastic processes such as Ornstein-Uhlenbeck process and Brownian Bridge.***(ii)  The statistic Ln(d), the length of the longest matching of two DNA sequences with allowing at most d mutations/insertions, is proposed as a measure for similarity. The exact distributions of Ln(d) will be derived and show that the distribution of Ln(d) can be expressed in terms of the distribution of scan statistics.***(iii)  The exact distribution of patterns to avoid in [S]-specified random permutation will be obtained. To achieve the goal, sampling one-by-one from an urn with [s]-specified symbols without replacement to forming random permutation. The results will cover many classical results for example the conditional runs tests and conditional scan statistics.***(iv) "Bump hunting" is vital important in genome-wide association studies between normal and disease chromosomes. In the last part of the proposal, the bump is modeled by its two components, the length and size of the bump and the joint and marginal distributions for the number and length of bumps will be derived.****B. Long term goal***(i)  The long term goal is to use FMCI technique to solve as many complex and unsolved problems, conjectures and newly arise practical problems associated with distributions of runs and patterns in applied probability and statistics, especially the continuous case. For example boundary crossing probabilities for jump processes, diffusion processes and Markov processes and matching probabilities among a set of DNA sequences allowing at most d mutations/deletions.******

有限的马尔可夫链嵌入式（FMCI）技术是一种非常规，简单，灵活和计算有效的有问题的有问题的工具，可评估运行和感兴趣的模式的可能性和分布。它已成功地用于解决各个领域的复杂和未解决的问题，例如健康科学，基因组分析，可靠性，质量控制，物理，统计，应用概率，计算机科学和离散数学。该提议，FMCI技术将扩展到四个非常重要的应用领域，（i）高维的布朗运动的边界越过概率（BCP），（ii）匹配两个DNA序列的概率，允许在[s]上表达的随机置换和（iv）中避免的图案分布的大多数DNA序列与（ii IV）的惯例分布（iv）的传统分布（iv），并匹配。和疾病染色体。以下是预期的结果：*** a。短期预期结果（接下来的五年）***（i）该提案的第一部分将建立一种分析和有效的数值方法，用于近似于d二维布朗尼运动的非线性凸边界的边界交叉可能性。它将显示收敛速率为O（1/N1/2），并且与维度无关。结果将扩展到相关的随机过程，例如Ornstein-Uhlenbeck工艺和Brownian Bridge。***（ii）统计LN（d），两个DNA序列最长匹配的长度与最多允许D突变/插入的最长匹配，被认为是相似性的措施。 LN（d）的确切分布将得出，并表明可以根据扫描统计的分布表示LN（d）的分布。***（iii）将获得[S]指定的随机置换中要避免的模式的确切分布。为了实现该目标，请从具有[S]指定符号的urn一对一取样，而无需替换为随机排列。结果将涵盖许多经典结果，例如条件运行测试和条件扫描统计。***（iv）“凸起狩猎”在正常染色体和疾病染色体之间的全基因组关联研究中至关重要。在提案的最后一部分中，凸起由其两个组件建模，即凸起的长度和大小以及凸起的数量和长度的关节和边缘分布。**** b。长期目标****（i）长期目标是使用FMCI技术来解决与应用概率和统计数据中的运行和模式分布相关的许多复杂和未解决的问题，猜想以及新出现的实际问题，尤其是连续情况。例如，边界跨越跳跃过程，差异过程和马尔可夫过程的可能性，以及一组DNA序列之间的可能性，最多可以在D突变/删除。******