Nonlinear and Nonstationary Time Series

这个项目关注于分析随着时间的推移收集的复杂数据的两个问题，例如每日股市回报。这种不规则的时间序列数据出现在许多不同的领域，如生物和医学、生态学、遗传学、地学、语音识别、计量经济学和金融以及计算机视觉等。由于数据是不规则的，因此很难预测高度波动的时期等问题。一般来说，人们必须依靠数值或基于计算机的优化，而不是使用显式的数学公式。然而，对于这样的数据，现有的方法具有很差的属性。这个项目的目标是极大地改进现有的计算方法。第二个项目专注于快速检测长DNA序列中的基因。虽然已经开发了许多方法来对短序列进行彻底的微观分析，但对于长DNA序列的宏观分析还缺乏功能强大的程序。首先，通过数值方法对非线性和非高斯时间序列模型的分析一直是人们关注的焦点。粒子采样器是经典和贝叶斯估计的一种很有前途的方法，但它们存在粒子退化和混合不良的问题。然而，没有必要放弃粒子方法；它们可以改进，这就是本项目的目标。例如，粒子Gibbs方法可以被塑造成快速和高效的，同时改善取样器的混合性能。其基本思想是构建一个类似粒子过滤器的过程，通过对粒子进行条件处理来避免路径退化。这种条件意味着不变性，这是它作为粒子采样器适用性的关键。不变性也是提供采样器渐近精度的关键。仅仅渐近准确是不够的，因为维度的诅咒是我们试图避免的。此外，虽然该技术并不完美，但该方法可以作为探索更快方法的基础，同时避免糟糕的混合。当似然估计只能用数值计算时，该方法也可用于经典推理中进行无导数的最大似然估计(如EM算法)。第二个项目的主要兴趣在于检测超长DNA序列中的编码(基因)和其他有趣的特征。特别是，重点是基于使用小波基的光谱包络的概念来快速检测长DNA序列中的变化点。基因组序列的快速积累增加了对破译数据库中收集的遗传信息的方法的需求。将统计分析与现代计算机能力相结合，可以高速搜索长序列中的诊断模式。这种组合提供了一种自动方法来评估超长序列中模式之间的异同，并有助于发现隐藏在这些有机分子中的生化信息。

This project focuses on two problems in analyzing complex data collected over time such as daily stock market returns. Such irregular time series data occur in many diverse fields such as biology and medicine, ecology, genetics, geoscience, speech recognition, econometrics and finance, and computer vision to mention a few. Because the data are irregular, problems such as predicting highly volatile periods are difficult. In general, rather than using explicit mathematical formulas, one must rely on numerical or computer-based optimization. However, for such data, existing methods have poor properties. The goal of this project is to vastly improve on the existing computational methods. The second project focuses on the fast detection of genes in long DNA sequences. While many methods have been developed for a thorough micro-analysis of short sequences, there is a shortage of powerful procedures for the macro-analysis of long DNA sequences.The project focuses on two problems in nonlinear and nonstationary time series analysis. First, there has been an intense focus on the analysis of nonlinear and non-Gaussian time series models via numerical methods. Particle samplers are a promising approach for classical and Bayesian estimation, but they are plagued by particle degeneration and by poor mixing. However, there is no need to abandon particle methods; they can be improved, and this is the goal of this project. For example, particle Gibbs methods can be fashioned to be fast and efficient while improving the mixing property of the sampler. The basic idea is to build a particle-filter-like procedure that avoids path degeneracy by conditioning on particles. This conditioning implies an invariance property, which is key to its applicability as a particle sampler. The invariance property is also key to providing the asymptotic accuracy of the sampler. It is not enough to be asymptotically accurate because of the curse of dimensionality, which we try to avoid. Moreover, while the technique is not perfect, the methodology can be used as a basis from which to explore faster methods while avoiding poor mixing. The method can also be used in classical inference to perform derivative free maximum likelihood estimation (e.g., EM algorithm) when the likelihood can only be evaluated numerically. The main interest of the second project is on the detection of coding (genes) and other interesting features in very long DNA sequences. In particular, the focus is on fast detection of change points in long DNA sequences based on the concept of spectral envelope using a wavelet basis. Rapid accumulation of genomic sequences has increased demand for methods to decipher the genetic information gathered in data banks. Combining statistical analysis with modern computer power makes it feasible to search, at high speeds, for diagnostic patterns within long sequences. This combination provides an automated approach to evaluating similarities and differences among patterns in very long sequences and aids in the discovery of the biochemical information hidden in these organic molecules.