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

古气候重建的目的是重建过去的气候，对于评估现代气温(和其他气候变量)在千禧年的背景下是如何异常的至关重要。文献中提供的大多数方法都是对过去千年及以后的气候变量进行综合估计。人们普遍认识到，气候的变化因空间位置不同而不同；例如，热带地区和极地地区之间估计的温度变化及其变异性是不同的。要对这种现象有详细的了解，就需要制定在空间上分类的古气候重建方法，并对估计的过去气候进行不确定性量化。空间分解的古气候重建基本上是高维统计问题，具有独特的地学背景带来的挑战。特别是，古气候重建依赖于将气候变量与替代变量(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。作为一个具体应用，该项目将在空间上按分类重建过去千年的全球气候，并对这些重建附上信心声明。拟议的工作需要数学科学和地球科学之间的合作，以解决这两个领域交界处的科学问题。该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，这些领域经常研究高维背景下基因之间的关系。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)，依赖于噪声输入数据，并且往往高维数据维度接近或超过样本大小，导致病态或秩亏估计问题。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，对这类问题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。最近，对不确定性进行严格量化的需要引起了人们对贝叶斯气候重建方法的极大兴趣，这些方法通常基于马尔可夫链蒙特卡罗(MCMC)抽样技术。后者提出了两个主要问题：(1)不能保证后验样本是根据所需的分布生成的(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，该方法可以得到后验量的闭合形式的解，从而减轻了大部分计算负担。收敛问题将首先解决，随后将通过利用马尔可夫随机场的丰富理论来实现降维。该项目理论部分的成果将是构建为气候场重建问题量身定做的新颖、健全、高效的高维贝叶斯算法。然后，这些新的统计工具将用于根据共同时代的不同地质指标(树轮、冰芯、洞穴、珊瑚、沉积物)重建全球和区域温度场，并对仪器表面温度和海平面压力进行新的分析。这些解决方案将伴随着可信的间隔，并有望对全球范围内的自然气候变异性产生新的见解。

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.