Nonlinear and Nonstationary Time Series

该项目侧重于分析随着时间的推移收集的复杂数据，如每日股票市场回报的两个问题。 这种不规则的时间序列数据出现在许多不同的领域，如生物学和医学，生态学，遗传学，地球科学，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，预测高度波动的时期等问题是困难的。 一般来说，不是使用明确的数学公式，而是必须依赖于数值或基于计算机的优化。 然而，对于这样的数据，现有的方法具有较差的性能。该项目的目标是大大改善现有的计算方法。第二个项目的重点是快速检测长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，但缺乏对长DNA序列进行宏观分析的强大程序。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，但它们受到粒子退化和混合不良的困扰。 然而，没有必要放弃粒子方法;它们可以得到改进，这就是本项目的目标。 例如，粒子吉布斯方法可以被塑造成快速和有效的，同时改善采样器的混合特性。 其基本思想是建立一个类似粒子过滤器的过程，通过对粒子进行条件化来避免路径退化。这种调节意味着不变性，这是其作为粒子采样器的适用性的关键。 不变性也是提供采样器的渐近精度的关键。 由于我们试图避免的维数灾难，它是不够的。 此外，虽然该技术并不完美，但该方法可以用作探索更快方法的基础，同时避免混合不良。该方法还可以用于经典推断以执行无导数最大似然估计（例如，EM算法）时，可能性只能进行数值评估。第二个项目的主要兴趣是在非常长的DNA序列中检测编码（基因）和其他有趣的特征。 特别是，重点是快速检测的变化点在长DNA序列的概念的基础上，使用小波基的频谱包络。 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，可以在长序列中高速搜索诊断模式。 这种组合提供了一种自动化的方法来评估非常长的序列中模式之间的相似性和差异，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.