Estimating Parameters in Spike-convolution Models and Mixture Models

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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.