Equilibrium in Multivariate Nonstationary Time Series

非平稳时间序列系统经常出现在经济学、地震学、神经科学和物理学中，其中平稳性通常与平衡同义。这些系统通常是多维的，它们的建模、预测和控制具有巨大的社会和科学影响。分离和识别平衡或稳态特征在预测和控制此类系统中具有根本的重要性。本研究计划旨在发展方法学，以撷取多元非平稳过程中，显示出平衡或平稳感的方面。它本质上是跨学科的，并直接应用于经济学，地震学和神经科学数据的分析。一名研究生将参与这项研究。本研究旨在将协整的概念和理论从经济学理论中的多元整合时间序列提升到概率统计学中更一般的多元非平稳时间序列。尽管协整理论在过去40年中在计量经济学中发挥了核心作用，在经济学中也有充分的动机，但它仍然需要对序列进行整合（单位根非平稳），并满足向量自回归和移动平均模型。这个项目的目标是避免这样的限制，并专注于一般的多元非平稳时间序列。三种不同的方法计算类似的协整向量和协整秩将开发。第一个是一个时域的方法，在经典的（约翰森的）方法，依赖于降低秩回归和似然比测试。 第二种方法是在谱域中，依赖于投影寻踪的思想。它通过最小化测量时变和恒定谱密度函数之间的差异的投影指数来搜索候选线性组合的系数。第三种方法是关于一个时变协整设置的系数是分段常数随时间的推移。它的成功实现依赖于一个很好的解决问题的变点检测的非平稳过程，和一个新的解决方案，在这项研究中进行了探索。这些结果将对收集多变量时间序列数据的环境产生直接影响，例如金融市场，流行病学，环境监测和全球变化。

Nonstationary time series systems appear routinely in economics, seismology, neuroscience, and physics, where stationarity is usually synonymous with equilibrium. Such systems are usually multidimensional, and their modeling, prediction, and control have tremendous social and scientific impacts. Isolating and identifying equilibrium or stationary features are of fundamental importance in prediction and control of such systems. This research project aims to develop methodologies for extracting aspects of multivariate nonstationary processes that display a sense of equilibrium or stationarity. It is interdisciplinary in nature and has immediate applications to the analysis of economics, seismology, and neuroscience data. A graduate student will be involved in the research. This research project aims to elevate the concept and theory of cointegration from multivariate integrated time series rooted in economics theory to the more general multivariate nonstationary time series setup in probability and statistics. In spite of its central role in econometrics in the last four decades and well-founded motivations in economics, the cointegration theory suffers from the requirements that the series be integrated (unit-root nonstationary) and satisfy a vector autoregressive and moving average model. The goal of this project is to avoid such restrictions and focus on general multivariate nonstationary time series. Three distinct methods for computing analogues of cointegrating vectors and the cointegrating rank will be developed. The first is a time-domain method in line with the classical (Johansen's) approach that relies on the reduced rank regression and likelihood ratio tests. The second method is in the spectral domain and relies on the idea of projection pursuit. It searches for coefficients of candidate linear combinations by minimizing a projection index measuring the discrepancy between time-varying and constant spectral density functions. The third method is concerned with a time-varying cointegration setup where the coefficients are piecewise constant over time. Its successful implementation rests on a good solution of the problem of change-point detection for nonstationary processes, and a novel solution is explored in this research. The results will have immediate impact in settings where multivariate time series data are collected, such as in financial markets, epidemiology, environmental monitoring, and global change.