Copula modeling with generative neural networks

使用生成神经网络进行 Copula 建模

• 批准号: RGPIN-2020-04897
• 负责人: Hofert, JanMarius
• 金额: $ 3.13万
• 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: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=740279
• 关键词: Copula; modeling; generative; neural; networks

My proposed research program falls in the area of copula modeling and computational statistics with applications to quantitative risk management (QRM). Copula modeling is concerned with the modeling of the dependence between the components of a random vector with continuous marginal distributions. Applications typically require sampling from the underlying copula model, for example, for computing risk measures in QRM, pricing applications in finance and insurance, or when computing rare-event probabilities in statistical applications. The larger the dimension, the more difficult it is to find an adequate copula model for given data. From a practical point of view, flexible and potentially high-dimensional copula models are needed that can easily be fitted to data and that are fast to sample from. To benefit from variance reduction when estimating quantities such as those above, one would also like to have a quasi-random number generator (QRNG) from the respective model, that is, a discrepancy-preserving transformation from a randomized quasi-Monte Carlo point set to the respective copula sample. However, there are many copula models for which there is no or no efficient QRNG. To this end, my research program focuses on generative neural networks (GNNs), in particular, generative moment matching networks (GMMNs). Our pioneering work in this new direction utilized GMMNs to construct QRNGs from an arbitrary copula model. Particularly appealing are the universality and the computability of this approach. The overarching goal of my proposed research program is to determine to what extent GNNs such as GMMNs can address the limitations of classical parametric models. Objectives along this way are the following: In our initial work we identified three scenarios which can cause problems in terms of the ability of GMMNs to properly learn the underlying distributions and a goal is to address these scenarios. We also found that the low discrepancy property seems to deteriorate for increasing dimension which we plan to investigate further. A goal important for businesses is to have meaningful statistics and graphical tools to summarize and compare GNNs. Another goal is to investigate whether GMMNs can be utilized to construct goodness-of-fit tests. We also aim at developing algorithms and functions for modeling tasks involving GNNs in R which allows anyone to reproduce and apply our research. Our final goal is to apply our findings to challenging problems in QRM such as the estimation of (systemic) risk measures and capital allocations, where GMMNs provide a promising new approach for a wide variety of models. Graduate students will be an integral part of this research program. They will gain knowledge of statistics and probability, an understanding of the construction and challenges of high-dimensional dependence models, as well as computational skills including the ability to apply neural networks to solve practically relevant dependence problems.

我提出的研究项目属于联结建模和计算统计领域，并应用于定量风险管理（QRM）。 Copula 建模涉及对具有连续边缘分布的随机向量的分量之间的依赖性进行建模。应用程序通常需要从底层 copula 模型中进行采样，例如，计算 QRM 中的风险度量、金融和保险中的定价应用程序，或者计算统计应用程序中的罕见事件概率时。维度越大，为给定数据找到合适的 copula 模型就越困难。从实践的角度来看，需要灵活且潜在高维的 copula 模型，该模型可以轻松地拟合数据并且可以快速采样。为了在估计上述数量时受益于方差减少，人们还希望拥有来自相应模型的准随机数生成器（QRNG），即来自随机准蒙特卡洛点集的差异保留变换到相应的 copula 样本。然而，有许多 copula 模型没有或没有有效的 QRNG。为此，我的研究项目重点关注生成神经网络（GNN），特别是生成矩匹配网络（GMMN）。我们在这个新方向上的开创性工作利用 GMMN 从任意 copula 模型构建 QRNG。特别有吸引力的是这种方法的普遍性和可计算性。我提出的研究计划的总体目标是确定 GMMN 等 GNN 在多大程度上可以解决经典参数模型的局限性。这种方式的目标如下：在我们最初的工作中，我们确定了三种可能导致 GMMN 正确学习底层分布的能力出现问题的场景，目标是解决这些场景。我们还发现，随着维度的增加，低差异特性似乎会恶化，我们计划进一步研究这一点。对于企业来说，一个重要的目标是拥有有意义的统计数据和图形工具来总结和比较 GNN。另一个目标是研究 GMMN 是否可用于构建拟合优度检验。我们还致力于开发涉及 R 中 GNN 的建模任务的算法和函数，让任何人都可以重现和应用我们的研究。我们的最终目标是将我们的发现应用于 QRM 中具有挑战性的问题，例如（系统性）风险度量和资本分配的估计，其中 GMMN 为各种模型提供了一种有前景的新方法。研究生将成为该研究计划不可或缺的一部分。他们将获得统计和概率知识，了解高维依赖模型的构造和挑战，以及计算技能，包括应用神经网络解决实际相关依赖问题的能力。