Formal statistical tools for the dependence modeling of environmental data

在环境问题和气候变化问题上做出明智的决定对未来几年的世界决策者来说至关重要。因此，重要的是，基于对自然现象的透彻理解而提出的确凿论据可供公众使用。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，因此对它们之间的关系的研究显然是有兴趣的。为此，一个好的战略是依靠成熟的Copula方法。然而，尽管在过去15年中，在适应Copulas的统计推断方面取得了许多进展，但必须改进现有的工具，以处理现在可获得的环境数据日益复杂的问题。从这一前提出发，本研究计划旨在利用Copulas在依赖关系建模的三个一般方面取得重大进展，这将对复杂环境数据的分析产生影响： (A)由于形式上选择适当的相关结构对于理解自然现象至关重要，基于Copula特征函数的全新统计推断程序将允许测试涉及Copula的许多类型的假设； (B)由于当变量数量变大时，现有Copula模型的灵活性有限，将考虑为高维情况创建模型和统计工具，重点是在空间统计、多元回归和新兴的大数据领域中的应用； (C)由于气候变化探测/建模工具一般假定突然变化的不切实际情景，因此将开发基于新的渐变相关性模型的新的统计检验和建模工具，用于气候时间序列的分析。 此外，为了促进新方法的重现性和传播性，编码程序将通过我的MatLab网页提供给整个研究界。 根据这一建议开发的方法工具通常是非参数和半参数的，这是因为从样本中无法观察到感兴趣总体的Copula，而只能间接地从等级中观察到。然后，新统计方法的形式证明将需要非参数统计和大样本经验过程技术的知识和新发展，以推导秩统计量的一致性和弱收敛，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.