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

有限马尔可夫链嵌入（FMCI）技术是一种非常规的、简单、灵活和计算效率高的概率工具，用于评估运行和感兴趣模式的概率和分布。它已经成功地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和未解决的问题。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穿越概率（BCP）， （ii）允许最多d个突变的两个DNA序列的匹配概率，（iii）在[S]指定的随机排列中避免的模式分布，以及（iv）用于比较正常和疾病染色体之间基因表达的全基因组关联研究的凸起分布。预期结果如下：***A；短期预期结果（未来五年）***(i)提案的第一部分将建立一种解析和有效的数值方法来近似d维布朗运动非线性凸边界的边界穿越概率。它将显示收敛速度为O(1/n1/2)，并且与维数d无关。结果将推广到相关的随机过程，如Ornstein-Uhlenbeck过程和Brownian Bridge。***(ii)提出了两个DNA序列最多允许d个突变/插入的最长匹配长度Ln(d)作为相似性度量。将推导出Ln(d)的确切分布，并表明Ln(d)的分布可以用扫描统计量的分布来表示。***(iii)将得到[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，从一个带有[s]指定符号的瓮中逐个采样，不进行替换，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信息。***(iv)“Bump hunting”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，将凸起由其两个组成部分建模，即凸起的长度和大小，并推导凸起数量和长度的联合分布和边缘分布。****B。长期目标***(i)长期目标是利用FMCI技术解决与应用概率和统计中的运行分布和模式有关的尽可能多的复杂和未解决的问题、猜想和新出现的实际问题，特别是连续情况。例如跳跃过程、扩散过程和马尔可夫过程的边界跨越概率，以及允许最多d个突变/缺失的一组DNA序列之间的匹配概率。******

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.******