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.******

有限马尔可夫链嵌入 (FMCI) 技术是一种非常规、简单、灵活且计算高效的概率工具，用于评估运行和感兴趣模式的概率和分布。它已成功应用于解决健康科学、基因组分析、可靠性、质量控制、物理学、统计学、应用概率、计算机科学和离散数学等各个领域的复杂且未解决的问题。在本提案中，FMCI 技术将扩展到四个非常重要的应用领域，（i）高维布朗运动的边界交叉概率（BCP），（ii）两个 DNA 序列的匹配概率，最多允许 d 个突变，（iii）在 [S] 指定的随机排列中避免的模式分布，以及（iv）用于比较基因组之间基因表达的全基因组关联研究的凹凸分布。 正常和疾病染色体。以下是预期结果：***A。短期预期结果（未来五年）***(i)   该提案的第一部分将建立一种分析有效的数值方法，用于近似 d 维布朗运动非线性凸边界的边界交叉概率。它将显示收敛速度为 O(1/n1/2) 并且与维数 d 无关。结果将扩展到相关的随机过程，例如 Ornstein-Uhlenbeck 过程和布朗桥。***(ii) 统计量 Ln(d)，即允许最多 d 个突变/插入的两个 DNA 序列的最长匹配长度，建议作为相似性的度量。将导出 Ln(d) 的精确分布，并表明 Ln(d) 的分布可以用扫描统计分布来表示。***(iii) 将获得 [S] 指定的随机排列中要避免的模式的精确分布。为了实现这一目标，从带有[s]指定符号的瓮中进行一一采样，无需放回，形成随机排列。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) “碰撞狩猎”对于正常染色体和疾病染色体之间的全基因组关联研究至关重要。在提案的最后一部分中，凸块由其两个组件建模，即凸块的长度和大小以及凸块数量和长度的接头和边缘分布。****B。长期目标***(i) 长期目标是使用 FMCI 技术来解决尽可能多的复杂且未解决的问题、猜想以及与应用概率和统计中的游程和模式分布相关的新出现的实际问题，尤其是连续情况。例如，跳跃过程、扩散过程和马尔可夫过程的边界交叉概率以及一组 DNA 序列之间最多允许 d 个突变/删除的匹配概率。******