Extreme Values, Time Series and Prediction

本项目主要研究时间序列预测和极值的理论和应用问题。理论研究包括：(1)残差形成可能具有长程依赖形式的未知谱密度的二元时间序列的回归模型拟合的谱方法；(2)将方法扩展到时空序列；(3)贝叶斯方法和频率方法在极值预测中的决策理论比较；(4)将(3)推广到多元过程，包括层次模型和多元极值理论的思想。激发这些主题的应用来自两个不同的领域，即(a)气候学和(b)保险和金融风险评估。在(a)方面，P.I.先前曾与两位主要的气候学家（T. Wigley和B. Santer）合作，研究一种基于自相关的方法，从一系列半球平均温度中探测气候信号。这项新研究继续了这项合作，包括更广泛和严格的技术，用于在背景噪声的统计结构未知的情况下从背景噪声中分离信号。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值，而且通过空间网格框上更详细的总结。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有关。第二个主题(b)涉及“风险价值”概念的扩展，这已经成为保险和金融领域广泛研究的主题。风险价值是评估巨大财务损失可能性的一系列工具，但目前大多数可用的系统都没有使用可用的最先进的统计技术。这项新的研究将使用从极值概率数学理论中衍生出来的统计方法来评估风险的替代措施。

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.