Bayesian Models and Methods for Dynamic and Spatio-Dynamic Systems

本文主要研究结构化动态和空间动态多元过程的统计理论、模型和方法。范围包括在多变量和矩阵变量时间序列中动态协方差结构的新型随机过程模型的理论发展，包括用于多变量波动建模的新型平稳马尔可夫过程。理论和应用方面的发展包括对日益高维、时变参数随机系统的稀疏性建模的新方法，应用于动态回归、时变向量自回归、动态因素模型和协方差波动模型。额外的研究集中在新类型的空间晶格和空间变化随机场模型上，再加上时间序列过程来定义时空模型的灵活模型，用于越来越高分辨率的晶格数据随时间观察。研究者开发了基于贝叶斯模拟的统计计算——包括基于gpu的并行算法——用于模型实现，以及金融时间序列中的跨学科应用，以及大气和生物医学科学的研究。面对时空系统研究中产生的日益高维的数据集，统计科学研究的目标是大幅提高表示、分析和使用维度、现实性和复杂性不断增加的数学模型的能力。研究者开发数学和统计建模理论以及相关的基于模拟的计算方法，在一定程度上受到金融、大气科学和神经科学领域跨学科合作应用的激励。统计研究方面的创新包括：(i)新的和改进的模型，用于描述和预测几个或多个时间序列之间复杂关系模式的时间变化——例如财务指标，或基于纳米技术的脑成像神经信号记录；（ii）用于诱导应用随机模型的稀疏性（即控制复杂性）的新理论和方法，以便能够按比例处理日益高维的问题，例如大气研究中的高分辨率卫星成像以及大规模金融时间序列中出现的问题；（iii）统计计算模拟技术的创新，包括并行桌面计算，以提高适应、探索和使用规模和复杂性日益增加的模型以及数据集日益庞大的模型的能力。与跨学科的合作者和学生一起，研究推进了核心的数学和统计建模理论和技术，并在几个特定的应用环境中为建模和数据分析提供了新的，精炼的和相关的方法，并产生了更广泛使用的方法。

The research concerns novel statistical theory, models and methods for structured dynamic and spatio-dynamic multivariate processes. The scope includes theoretical developments of new classes of stochastic process models for dynamic covariancestructures in multi- and matrix-variate time series, including novel classes of stationary Markov processes for multivariate volatility modeling. Theoretical and applied developments include new approaches to sparsity modeling for increasingly high-dimensional, time-varying parameter stochastic systems, applying to dynamic regression, time-varying vector auto-regression, dynamic factor models and covariance volatility models. Additional research focuses on new classes of spatial lattice and spatially-varying random field models, coupled with time series processes to define flexible models of spatio-temporal models for increasingly high-resolution lattice data observed through time. The investigator develops Bayesian simulation-based statistical computation-- including GPU-based parallelized algorithms-- for model implementations, and cross-disciplinary applications in financial time series as well as studies in atmospheric and biomedical sciences.Faced with increasingly high-dimensional data sets generated in studies of temporal and spatial systems, statistical science research aims to substantially advance the ability to represent, analyze and use mathematical models of increasing dimension, realism and complexity. The investigator develops mathematical and statistical modelling theory and associated simulation-based computational methods for a range of contexts, motivated in part by collaborative cross-disciplinary applications in areas of finance, atmospheric science and the neurosciences. Innovations in statistical research include: (i) new and improved models for describing and predicting change in time of the complex patterns of relationships among several or many time series-- such as financial indicators, or nano-technology based recordings of neural signals in brain imaging; (ii) new theory and methods for inducing sparsity-- i.e., controlling complexity-- of applied stochastic models, to enable scaling to increasingly high-dimensional problems, such as arise in high-resolution satellite imaging in atmospheric studies as well as large-scale financial time series;(iii) innovations in simulation techniques for statistical computing, including parallel desktop computing, to advance the ability to fit, explore and use models of increasing scale and complexity and with increasingly large data sets. With cross-disciplinary collaborators and students, the research advances core mathematical and statistical modeling theory and technology, and contributes new, refined and relevant approaches to modeling and dataanalysis in several specific applied contexts as well as generating methods for broader use.