Extreme Values, Time Series and Prediction

本项目的研究集中在时间序列的预测和极值方面的一些理论和应用课题。理论研究包括：(1)用于拟合回归模型的谱方法，其中残差形成具有可能长期相关形式的未知谱密度的二元时间序列；(2)将该方法扩展到时空序列；(3)用于极值预测的贝叶斯方法和频率方法的决策理论比较；以及(4)将(3)扩展到多变量过程，包括来自分层模型和多变量极值理论的思想。推动这些主题的应用来自两个不同的领域，即(A)气候学，和(B)保险和金融领域的风险评估。在(A)方面，P.I.之前曾与两位领先的气候学家(T.Wigley和B.Santer)合作，研究一种基于自相关的方法，以探测一系列半球平均气温中的气候信号。这项新的研究延续了这种合作，包括更广泛和更严格的技术，用于在背景噪声的统计结构未知的情况下将信号与背景噪声分离。将扩展到描述地球表面温度模式的情况，不仅通过北半球和南半球的平均，而且通过对空间网格框的更详细的总结。这项研究的好处是改进了检测人类活动对地球气候影响的方法，这与联邦环境和全球变化战略利益(EGCH)有关。第二个专题(B)涉及“风险价值”概念的外延，该概念已成为保险和金融领域的一个广泛研究的专题。风险价值是一组工具，用于评估非常大的财务损失的可能性，但目前大多数可用的系统没有使用最先进的统计技术。这项新的研究将使用源自极值概率数学理论的统计方法来评估替代的风险衡量标准。

The research in this project focuses on a number of theoretical and applied topics concerned with prediction and extreme values in time series. Theoretical research includes: (1) spectral methods for fitting regression models where the residuals form a bivariate time series with unknown spectral density of possibly long-range dependent form; (2) extensions of the methodology to spatial-temporal series; (3) decision-theoretic comparisons of Bayesian and frequentist methods for prediction of extreme values, and (4) extensions of (3) to multivariate processes, including ideas both from hierarchical models and multivariate extreme value theory. The applications which have motivated these topics come from two distinct areas, namely (a) climatology, and (b) the evaluation of risk in insurance and finance. On (a), the P.I. has previously collaborated with two leading climatologists (T. Wigley and B. Santer) on an autocorrelation-based method of detecting climate signals in series of hemispheric temperature averages. The new research, which continues this collaboration, includes more extensive and rigorous techniques for separating signals from background noise in cases where the statistical structure of the background noise is unknown. Extensions will be developed to cases in which the patterns of temperature over the earth's surface are described, not just through northern and Southern Hemisphere averages, but through much more detailed summaries over spatial grid boxes. The benefit of this research is improved methods for detecting anthropogenic influence on the earth's climate, which related to the Federal strategic interest in Environment and Global Change (EGCH). The second topic (b) is concerned with extensions of the idea of "value at risk" which has become a widely studied topic in the fields of insurance and finance. Value at risk is a collection of tools for assessing the probability of very large financial losses, but most currently available systems do not use the most advanced statistical techniques available. The new research will assess alternative measures of risk using statistical methods derived from the mathematical theory of extreme value probabilities.