Estimating Parameters in Spike-convolution Models and Mixture Models

9971698这项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。 它的目的是将两者在算法和渐近性方面的好结果编织在一起，发展新的方法，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，尖峰卷积模型，介绍了PI和他的合作者提出的DNA测序模型的一部分。 目前以桑格命名的测序方案结合了三种技术：酶反应、凝胶或毛细管电泳和基于荧光的检测。 这个生化过程为每个DNA片段产生一个四组分向量时间序列。 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。 大多数碱基识别错误是由电泳的扩散效应引起的。 研究发现，这种扩散效应可以很好地描述所谓的尖峰卷积模型。 当稀疏狄拉克尖峰序列与固定点扩展函数卷积时，会出现加性噪声或测量误差叠加。 在该模型中，反卷积只是一个标准的参数估计问题，其中参数包括潜在尖峰的数量、位置和高度、基线和测量误差方差。 本研究的第二个对象，有限混合模型，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，混合模型中的参数估计可能是非常麻烦的，特别是当涉及到许多组件。 据信，一类广泛的有限混合模型与尖峰卷积模型密切相关。 由于这两个模型的复杂性，对于这两个模型中的估计问题不存在简单的解决方案。 本研究提出将联合收割机的三角矩方法与两阶段的模型选择过程，高斯-牛顿算法，或EM算法根据情况相结合。 的数值和统计方面的新方法和它们的变种进行检查，并与现有的方法进行比较。 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，并进行了详细的研究。本研究从统一的角度研究了新提出的尖峰卷积模型和长期存在的有限混合模型。 大规模、高通量的DNA测序是人类基因组计划和其他基因组计划的重要内容之一。 由于缺乏令人满意的去卷积技术和统计模型，DNA碱基识别（测序的数据分析部分）成为这些项目的瓶颈。 一个有效的去卷积技术，这项研究的目标，是快速和可靠的DNA碱基调用的基本先决条件。 事实上，类似的反卷积问题出现在许多其他科学学科中，如天体物理学，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，该模型可应用于物理，医学和生物学等多种领域。

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.