Advances in sampling methods with a dependence structure

具有依赖结构的采样方法的进展

• 批准号: RGPIN-2020-04019
• 负责人: Lemieux, Christiane
• 金额: $ 1.75万
• 依托单位: University of Waterloo
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=765768
• 关键词: Advances; sampling; methods; dependence; structure

To remain competitive in today's world, the Canadian economy rests on advances in science and industry, which increasingly depend on the availability of efficient computational tools. These tools help scientists and analysts evaluate quantities of interest for a system under study. Many of these tools rely on some form of random sampling to approximate quantities for which no explicit formula exists. Random sampling is typically used to "simulate" scenarios of the system. For each scenario, the corresponding value of the quantity of interest is evaluated. By repeating this process several times, a sample of possible values for this quantity is created, which can then be used for inference. This approach is often referred to as the "Monte Carlo method". A drawback of this method is that by nature, random sampling can produce irregularities. Indeed, since scenarios are sampled independently from one another, we may get too many that are similar and/or not enough of a certain type. Quasi-Monte Carlo methods aim at addressing this issue by replacing random sampling by more structured sampling. More precisely, new scenarios are sampled by implicitly taking into account the scenarios sampled so far. This is achieved through the use of low-discrepancy sequences, which are constructions that attempt to place points in a very uniform way in the space over which they are defined. Sophisticated techniques are then used to transform each point into a scenario of the system. These methods have gained considerable attention over the last 20 to 25 years, as they have proven to be useful for solving high-dimensional problems in finance, e.g., involving the simulation of several financial assets over long periods of time. That is, with the same computational effort, they provide estimators with a smaller error than Monte Carlo-based ones. The main goal of this research program is to advance our understanding of quasi-Monte Carlo methods by focusing on the dependence being induced in their underlying sampling schemes. This new approach has the potential to improve the effectiveness of these methods. In addition, we aim to make significant progress in the design and analysis of algorithms that use low-discrepancy sequences to construct approximations adaptively, i.e., learning along the way some of the features of the system to further direct sampling into important regions. Finally, when using low-discrepancy sequences it is more difficult to apply the techniques by which "points" are transformed into "scenarios". This has limited the kinds of models that can be tackled by quasi-Monte Carlo methods. Our research will attempt to address these limitations. This research program will involve at least 10 students from all levels, who will gain valuable expertise on Monte Carlo and quasi-Monte Carlo methods. This research blends theoretical and practical work, so students will be well equipped to transfer the acquired knowledge to either industry or academia.

为了在当今世界保持竞争力，加拿大经济取决于科学和工业的进步，这越来越依赖于有效的计算工具的可用性。这些工具可帮助科学家和分析师评估研究系统的兴趣数量。这些工具中的许多工具都依靠某种形式的随机抽样来近似不存在明确公式的数量。随机采样通常用于“模拟”系统的情况。对于每种情况，都评估了利益量的相应值。通过重复此过程多次，创建了该数量的可能值的样本，然后可以将其用于推断。这种方法通常称为“蒙特卡洛方法”。 这种方法的缺点是，从本质上讲，随机抽样会产生不规则性。确实，由于场景是从彼此独立进行采样的，因此我们可能会得到太多相似和/或不够某种类型的情况。 准蒙特卡罗方法旨在通过更结构化的采样来替换随机抽样来解决此问题。更确切地说，通过隐式考虑到目前为止采样的情况，可以采样新的情况。这是通过使用低分配序列来实现的，这些序列是试图以非常统一的方式将点放置在其定义的空间中的结构。然后使用复杂的技术将每个点转换为系统的场景。在过去的20到25年中，这些方法引起了很大的关注，因为事实证明，这些方法可用于解决金融方面的高维问题，例如，涉及长时间内几个金融资产的模拟。也就是说，通过相同的计算工作，它们与基于蒙特卡洛的估计量相比，估计器的误差较小。该研究计划的主要目的是通过关注其基础抽样方案中引起的依赖性来提高我们对准蒙特卡洛方法的理解。这种新方法有可能提高这些方法的有效性。此外，我们旨在在使用低静止序列的算法的设计和分析中取得重大进展，以适应构建近似值，即，在系统的某些特征沿途学习，以进一步将采样直接引入重要区域。最后，当使用低分配序列时，更难应用“点”转换为“方案”的技术。这限制了准蒙特卡洛方法可以解决的类型。我们的研究将尝试解决这些局限性。 该研究计划将涉及至少10名来自各个级别的学生，他们将获得有关蒙特卡洛和准蒙特卡洛方法的宝贵专业知识。这项研究融合了理论和实践工作，因此学生将有能力将获得的知识转移到行业或学术界。