Mathematical Sciences: Time Series and Point Processes: Networks and Wavelets

本项目分为两部分。首先是开发用于处理顶点变量为时间序列或点过程的图形模型的统计技术。部分动机来自于神经科学和系统科学中的问题，在这些问题中，人们设置方框和箭头图，旨在提供相互关联单元之间的信息流。在神经科学的例子中，单位是神经元，数据是放电时间，模型将被开发并适合于单个单元或单元组的相互连接。将对实验数据进行分析，包括从分布在猫的听觉系统中的神经元收集的一些数据。在各种其他实质性领域，人们也对检测连接的存在和估计这些连接的强度感兴趣。该项目的第二部分涉及到一些现有的普通统计技术的随机过程的小波变量的发展。例如，小波可用于从时间序列和图像中提取非平稳信号。该项目的两个部分都需要开发复杂的随机模型及其特性，并考虑到实际实施。时间序列是一条弯曲的线，用来表示一种随时间波动的现象，如地震图、心电图或脑电图。点过程是发生感兴趣的事件的时间集合，例如地震或神经细胞放电或平均水平的变化（例如河流）。网络是相互作用的实体（如神经细胞、城市或计算机）的流程图或接线图。研究的一个重点是在网络的每个单元记录的时间序列或点过程的情况。时间序列和点过程是相互作用的，我们希望了解这些相互作用的优势和方向。小波分析是一种从时间序列或图像记录中提取可能发生变化的现象信息的新方法。检测和估计变化的问题经常出现在环境问题上，特别是全球变暖问题上。它旨在将小波技术扩展到其他类型的演化和分布式数据，如点过程或分类值过程。

DMS 9625774 Brillinger This project has two parts. A first is to develop statistical techniques for dealing with graphical models whose vertex variates are time series or point processes. Part of the motivation comes from problems in the neurosciences and in system sciences where one sets down box and arrow diagrams meant to provide the flow of information between interconnected units. In the neuroscience case the units will be neurons, the data will be the firing times and models will developed and fit for the interconnections of single units or of groups of units. Analyses will be carried out for experimental data including some collected from neurons distributed through the auditory system of the cat. In various other substantive fields one is also interested in detecting the presence of connections and estimating the strength of those connections. The second part of the project concerns the development of wavelet variants, for random processes, of some existing ordinary statistical techniques. Wavelets are of use in eliciting signals of nonstationary character from time series and images for example. Both parts of the project will require development of complex stochastic models and their properties, having in mind practical implementation. A time series is a wiggly line meant to represent a phenomenon fluctuating in time, such as a seismogram, an electrocardiogram or an electroencephalogram. A point process is a collection of times at which an event of interest occurs, such as an earthquake or a nerve cell firing or changes in mean level (for example of a river). A network is a flow or wiring diagram of interacting entities such as nerve cells or cities or computers. A focus of the research is with the case of time series or point processes recorded at each of the units of a network. The time series and point processes are interacting and it is desired to learn the strengths and directions of those interactions. Wavelet analysis is a novel method for elicitilg information concerning phenomena that may be changing, from time series or image records. The problem of detecting and estimating change arises often in problems of the environment and in particular of global warming. It is intended to extend the wavelet technique to other types of evolving and distributed data such as point processes or categorical-valued processes.