Estimating Parameters in Spike-convolution Models and Mixture Models

9971698本研究将尖峰卷积模型中的参数反卷积问题与有限混合模型中的估计问题联系起来。它旨在将双方在算法和渐近方面的良好结果结合起来，并开发新的方法，这些方法在计算上是可实现的，在理论上是有效的。本研究的第一个对象，尖刺-卷积模型，是PI和他的合作者提出的DNA测序模型的一部分。目前以Sanger命名的测序方案结合了三种技术：酶促反应、凝胶或毛细管电泳和基于荧光的检测。这个生化程序为每个DNA片段产生一个四组分矢量时间序列。DNA碱基调用的任务是从上述时间序列中恢复潜在的DNA序列。大部分碱基召唤误差是由电泳的扩散效应引起的。发现这种扩散效应可以用所谓的尖峰-卷积模型很好地描述。当稀疏狄拉克尖峰序列与不动点扩散函数进行卷积时，会产生叠加噪声或测量误差。在这个模型中，反褶积只不过是一个标准参数估计问题，其中参数包括潜在峰值的数量、位置和高度、基线和测量误差方差。本研究的第二个对象是有限混合模型，它是许多统计分析的框架，如鲁棒性检查、聚类、估计密度函数等。然而，混合模型中参数的估计是非常麻烦的，特别是当涉及到许多成分时。认为一类广泛的有限混合模型与尖峰卷积模型密切相关。由于这两种模型的复杂性，它们的估计问题没有简单的解决方案。本研究提出将三角矩法与两阶段模型选择程序、高斯-牛顿算法或EM算法结合起来，视情况而定。研究了新方法及其变体的数值和统计方面，并与现有方法进行了比较。由三角矩构造的Toeplitz形式及其统计性质在所提出的方法中起着关键作用，并对其进行了详细的研究。本文从统一的角度对新提出的尖峰卷积模型和长期存在的有限混合模型进行了研究。前者受到大规模和高通量DNA测序的推动，这是正在进行的人类基因组计划和其他基因组计划的最重要方面之一。缺乏令人满意的反褶积技术和统计模型使得DNA碱基调用（测序的数据分析部分）成为这些项目的瓶颈。有效的反褶积技术是实现快速、可靠的DNA碱基调用的基本前提。事实上，类似的反褶积问题出现在许多其他科学学科中，如地球物理学、光谱学和色谱学。研究结果也有望丰富有限混合模型的理解和方法，这些模型在物理、医学和生物学等多个领域都有应用。

9971698This research links the parametric deconvolution problem in the spike-convolution model with the estimation problem in finite mixture models. It aims to weave together good results on algorithms and asymptotics from both sides, and develop new methodologies, which are implementable in computation and efficient in theory. The first object of this research, the spike-convolution model, is introduced as part of the models proposed for DNA sequencing by the PI and his collaborators. The current sequencing scheme named after Sanger combines three techniques: enzymatic reactions, gel or capillary electrophoresis and fluorescence-based detection. This biochemical procedure produces a four-component vector time series for each DNA fragment. The task of DNA base-calling is to recover the underlying DNA sequence from the above time series. Most of the base-calling errors are caused by the diffusion effect of electrophoresis. It is found that this diffusion effect can be well described by the so-called spike-convolution model. It arises when a sparse Dirac spike train is convolved with a fixed point spread function, and additive noise or measurement error is superimposed. In this model, deconvolution is nothing but a standard parameter estimation problem, where the parameters include the number, locations and heights of the underlying spikes, the baseline and the measurement error variance. The second object of this research, the finite mixture model, is the framework of many statistical analyses like robustness checking, clustering, estimating density functions, etc. However, the estimation of the parameters in mixture models can be very troublesome, especially when many components are involved. It is believed that a broad class of finite mixture models is closely related to the spike-convolution model. No simple solution exists to the estimation problems in these two models because of the complexity. This research proposes to combine the method of trigonometric moments with a two-stage model selection procedure, Gauss-Newton algorithm, or EM algorithm depending on the situations. The numerical and statistical aspects of the new methods and their variants are examined and compared with those of existing methods. The Toeplitz forms constructed from trigonometric moments and their statistical properties play a key role in the proposed methods, and are investigated in full detail.This research studies the newly proposed spike-convolution model and the long-standing finite mixture models from a unified perspective. The former is motivated by the large scale and high throughput DNA sequencing, which is one of the most important aspects of the ongoing Human Genome Project and other genome projects. The lack of satisfactory deconvolution techniques and statistical models has made DNA base-calling---the data analysis part of sequencing---a bottle neck of these projects. An effective deconvolution technique, a target of this research, is a fundamental prerequisite for rapid and reliable DNA base-calling. In fact, similar deconvolution problems arise in many other scientific disciplines like geophysics, spectroscopy, and chromatography. The research results are also expected to enrich the understanding and methodologies of finite mixture models, which have applications to a diversity of fields such as physics, medicine, and biology.