CMG Collaborative Research: Efficient high dimensional Bayesian methods for climate field reconstruction

古气候重建旨在重建过去的气候，对于评估现代气温（和其他气候变量）在千年背景下的异常程度至关重要。文献中可用的大多数方法对过去一千年及以后的气候变量给出了综合估计。人们普遍认识到，气候变化因空间位置而异；例如，热带地区和极地地区之间估计的温度变化及其变率是不同的。要对这些现象有详细的了解，就需要发展空间分解的古气候重建方法，并对估计的过去气候进行不确定性量化。空间分解古气候重建基本上是高维统计问题，具有地球科学背景带来的独特挑战。特别是，古气候重建依赖于将气候变量与代理变量（如树木年轮、冰芯等）联系起来。该项目将为利用现代统计方法重建高维古气候奠定理论基础。作为一项具体应用，该项目将在空间分解水平上重建过去一千年的全球气候，并为这些重建附上信心声明。拟议的工作需要数学和地球科学之间的合作，以解决这两个领域交界的科学问题。项目期间开发的方法在其他领域也有广泛的应用，如基因组学，在高维环境中经常研究基因之间的关系。气候场重建本质上是多变量推理问题（例如，需要大网格上的空间数据），依赖于有噪声的输入数据，并且通常是如此高维以至于数据维数接近或超过样本量，导致不适定或秩不足估计问题。在这种情况下，高维均值和协方差估计往往是推理问题的中心。此外，这类问题的有意义的解决方案需要对估计模型参数的不确定性有可靠的了解。对不确定性进行严格量化的需求最近激发了人们对气候重建贝叶斯方法的极大兴趣，这种方法通常基于马尔可夫链蒙特卡罗（MCMC）采样技术。后者提出了两个主要问题，即：(1)不能保证从所需的分布生成后验样本（收敛问题）；(2)它们依赖于计算量大的算法，这限制了它们的适用性（效率、可扩展性和适用性）。我们建议通过使用灵活但高维的贝叶斯方法来克服这两个限制，该方法导致后验量的封闭形式解，从而减轻了许多计算负担。首先将解决收敛问题，然后通过利用马尔可夫随机场的丰富理论实现降维。该项目的理论组成部分的结果将是为气候场重建问题量身定制的新颖、健全、高效的高维贝叶斯算法的构建。然后，这些新的统计工具将被应用于重建全球和区域温度场，这些温度场是基于非均匀的地质代用物（树木年轮、冰芯、洞穴、珊瑚、沉积物），以及对仪器表面温度和海平面压力的新分析。这些解决方案将伴随着可信的时间间隔，并有望对全球尺度上的自然气候变化产生新的见解。

Paleoclimate reconstructions aim to recreate past climates and are critical in assessing how modern day temperatures (and other climate variables) are anomalous in a millennial context. Most of the methods available in the literature give consolidated estimates of climate variables over the past millennium and beyond. It is widely recognized that changes in climate differ by spatial location; for example, estimated changes in temperature, and variability thereof, between the tropics and the polar regions are different. Obtaining a detailed understanding of such phenomena requires developing methodology for paleoclimate reconstructions that are spatially disaggregated, with uncertainty quantifications of estimated past climate. Spatially disaggregate paleoclimate reconstructions are fundamentally high dimensional statistical problems, with unique challenges imposed by the geosciences context. In particular, paleoclimate reconstruction rely on relating climate variables to proxy variables (such as tree rings, ice cores etc..). The project will lay the theoretical foundation for high dimensional paleoclimate reconstructions using modern day statistical methods. As a concrete application the project will reconstruct global past climates at a spatially disaggregated levels for the past millennium, and also attach confidence statements to these reconstructions. The proposed work entails a collaboration between the mathematical and geo-sciences to solve a scientific problem at the interface of both fields. The methods developed during the project also have broad applications in other fields, such as genomics, where relationships between genes in a high dimensional context are often studied.Climate field reconstructions are inherently multivariate inference problems (e.g., spatial data on a large grid are required), rely on noisy input data, and are often so high-dimensional that the data dimension is close to or exceeds the sample size, resulting in ill-posed or rank-deficient estimation problems. In this context high dimensional mean and covariance estimation is often at the center of the inferential problem. Furthermore, meaningful solutions to such problems require a reliable knowledge of the uncertainty in estimated model parameters. The need for a rigorous quantification of uncertainties has recently spurred much interest in Bayesian methods for climate reconstruction, usually based on Markov Chain Monte Carlo (MCMC) sampling techniques. The latter pose two major issues, as: (1) there is no guarantee that posterior samples are generated from the required distribution (convergence issue); (2) they rely on computationally-heavy algorithms which limit their applicability (efficiency, scalability and applicability). We propose to overcome both limitations by using a flexible but high dimensional Bayesian approach that leads to closed-form solutions for posterior quantities, hence alleviating much of the computational burden. Convergence issues will be addressed first, and dimensionality reduction will subsequently be implemented by exploiting the rich theory of Markov Random fields. The outcome of the theoretical component of the project will be the construction of novel and sound, efficient high-dimensional Bayesian algorithms tailor-made for climate field reconstruction problems. These new statistical tools will then be applied to the reconstruction of global and regional temperature fields from heterogeneous geological proxies (tree rings, ice cores, speleothems, corals, sediments) over the Common Era, and new analysis of instrumental surface temperature and sea-level pressure. The solutions will be accompanied by credible intervals, and promises to yield new insights into natural climate variability at global scales.