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.

我提出的研究计划属于与定量风险管理（QRM）的应用有关的Copula建模和计算统计领域。 Copula建模与具有连续边缘分布的随机向量的组件之间的依赖性建模。应用程序通常需要从底层copula模型中进行采样，例如，用于计算QRM中的风险度量，金融和保险中的定价应用程序，或者在计算统计应用中稀有事实概率时。尺寸越大，找到给定数据的足够的副群模型就越困难。从实际的角度来看，需要柔性且潜在的高维模型模型，可以轻松地将其拟合到数据中，并且可以快速采样。为了从估计上述数量（例如上述数量）时受益，也希望从各个模型中具有准随机数发生器（QRNG），即，从随机的准蒙特卡洛点设置为相应的copula样品。但是，有许多Copula模型没有或没有有效的QRNG。为此，我的研究计划着重于生成神经网络（GNN），特别是生成力矩匹配网络（GMMN）。我们朝这个新方向的开拓性工作利用GMMN来构建QRNG，从任意的Copula模型中构造QRNG。这种方法的普遍性和可计算性特别吸引人。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。此方式的目标如下：在我们的初始工作中，我们确定了三种情况，这些方案可能会从GMMN正确学习基础分布的能力方面引起问题，目标是解决这些情况。我们还发现，较低的差异特性似乎会恶化，以增加维度，我们计划进一步研究。对于企业来说，一个重要的目标是拥有有意义的统计和图形工具来总结和比较GNN。另一个目标是研究是否可以利用GMMN来构建合适的测试。我们还旨在开发算法和功能，用于建模R中涉及GNNS的任务，该任务使任何人都可以复制和应用我们的研究。我们的最终目标是将我们的发现应用于QRM中挑战性问题，例如（系统性的）风险措施和资本分配，GMMN为各种模型提供了一种有希望的新方法。研究生将成为该研究计划不可或缺的一部分。他们将了解统计和概率的知识，对高维依赖模型的构建和挑战的理解以及计算技能，包括应用神经网络来解决实际相关的依赖性问题。