Formal statistical tools for the dependence modeling of environmental data

未来几年，在环境问题和气候变化问题上做出明智的决策对世界各国的决策者来说至关重要。因此，为公众提供基于对自然现象的透彻理解的可靠论据是很重要的。从这个角度来看，用正式的统计方法分析复杂的环境数据是必须迅速发展的一个关键因素。由于给定的环境现象通常涉及几个随机变量，因此研究它们之间的关系显然是有兴趣的。为此，一个好的策略是依靠已建立的copula方法。然而，尽管在过去的15年里，统计推断在适应copula方面取得了许多进展，但现有的工具必须得到改进，以处理目前可用的环境数据日益复杂的情况。从这个前提出发，本研究计划旨在利用copulas在依赖性建模的三个一般方面取得重大进展，这些方面将对复杂环境数据的分析产生影响：(A)因为正式选择适当的依赖结构对于理解自然现象至关重要，基于联结特征函数的全新统计推断程序将允许测试涉及联结的许多类型的假设；(B)由于当变量数量变大时，可用的联结模型的灵活性受到限制，因此将考虑为高维环境创建模型和统计工具，重点关注空间统计、多元回归和新兴的大数据领域的应用；(C)由于检测/模拟气候变化的工具通常假设一个不现实的突变情景，因此将开发新的统计试验和基于新的渐变依赖模式的模拟工具，用于分析气候时间序列。此外，为了促进新方法的可重复性和传播，编码程序将通过我的Matlab网页提供给整个研究界。根据这一建议将开发的方法工具通常是非参数和半参数的，这是由于不能从样本中观察到感兴趣的总体的联结，而只能间接地从秩中观察到。因此，新统计方法的正式证明将需要非参数统计和大样本经验过程技术方面的知识和新发展，以得出秩统计的一致性和弱收敛性，以及适当适应的重新抽样技术的有效性等。

Smart decisions on environmental issues and climate changes are crucial for the decision makers of the world in the coming years. It is thus important that solid arguments based on a thorough understanding of natural phenomena are available for public use. From this perspective, the analysis of complex environmental data with formal statistical methods is a key element that must be rapidly developed. Since a given environmental phenomenon typically involve several random variables, there is a clear interest for the study of their relationships. To this end, a good strategy is to rely on the well-established copula approach. However, although many advances in statistical inference adapted to copulas took place in the last fifteen years, the available tools must be improved to deal with the growing complexity of the environmental data that are now available. Setting out from this premise, this research program aims at making significant advances in three general aspects of dependence modeling with copulas that will have an impact on the analysis of complex environmental data: (A) Because the formal choice of an appropriate dependence structure is crucial for the understanding of a natural phenomenon, brand new statistical inference procedures based on copula characteristic functions will allow for the testing of many types of hypotheses involving copulas; (B) Since the flexibility of the available copula models is limited when the number of variables gets large, the creation of models and statistical tools for high-dimensional contexts will be considered, with a focus on applications in spatial statistics, multivariate regression, and the emerging field of Big Data; (C) Because the tools for the detection/modeling of climatic changes generally assume an unrealistic scenario of an abrupt change, new statistical tests and modeling tools based on a novel gradual-change dependence model will be developed for the analysis of climatic time series. Moreover, in order to promote the reproducibility and dissemination of the new methodologies, the coded procedures will be made available to the entire research community via My Matlab webpage. The methodological tools that will be developed with this proposal are typically nonparametric and semi-parametric, a consequence of the fact that the copula of a population of interest is not observable from the sample, but only indirectly from the ranks. The formal justification of the new statistical methods will then require knowledge and new developments in nonparametric statistics and large-sample empirical processes techniques to derive the consistency and weak convergence of rank statistics, as well as for the validity of suitably adapted resampling techniques, among other things.