MaxEnt-Fin: computational maximum entropy approach to high-dimensional modeling and analysis in finance

对金融时间序列中的波动(波动性)进行建模和预测是经济学和金融学的核心挑战之一。在过去的几年里，随着大量高维金融数据的积累，用于分析这些数据的计量经济学和机器学习(ML)方法取得了令人印象深刻的发展。除了提供新的令人兴奋的机会外，最近新兴工具在金融数据中的应用揭示了与算法的可扩展性、敏感性和可比性有关的一些新的方法学挑战--以及关于所获得结果的可解释性以及对数学和统计性质的研究。例如，股票回报的时间序列的特征是相对较少的连续观测T(从使用月度数据的几百个观测到使用每日数据的几千个观测)和多个维度n(高达数万或数十万，其中n对应于不同的公司，但也对应于这些公司的特征)。将计量经济学和机器学习等流行的计算数值工具应用于这种“小T，大n”数据，通常旨在找到越来越精细的模型，这些模型具有许多参数，必须根据少数可用的高维观测进行调整。这会导致所谓的“过拟合”问题，即训练数据的良好拟合质量与调整后的模型的预测性能差相结合。另一个限制是常用数值工具的计算成本，随着数据维度n的增加呈多项式增长。在本研究方案中，我们将开发基于最近引入的自适应数据离散化的可伸缩概率近似(SPA)方法与物理学和信息论中的最大熵原理相结合的数值工具。最大熵原理(MaxEnt)的目的是找到尽可能简单(但不是比必要更简单)的模型来拟合数据，在基本假设方面是最小偏差的，在可调参数总数方面是最小的。我们将开发一个数值离散化驱动的MaxEnt框架，用于时间序列分析和不同金融时间序列(如资产回报和信用评级)之间潜在的横截面相互依赖关系的推断，允许计算不确定性量化和基于经济和金融的时间序列的风险预测。

Modeling and prediction of fluctuations (volatilities) in financial time series are among the core challenges in Economics and Finance. In the last several years, the accumulation of massive amounts of high-dimensional financial data was accompanied by an impressive development of econometric and machine learning (ML) approaches for the analysis of these data. Besides offering new exciting opportunities, recent applications of the emerging tools to the financial data has revealed some new methodological challenges related to the scalability, sensitivity, and comparability of the algorithms - as well as with respect to the interpretability of the obtained results and the study of the mathematical and statistical properties. For example, time series of stock returns are characterized by relatively few serial observations T (ranging from a few hundred observations with monthly data to a few thousands with daily data) and by many dimensions n (up to tens or hundreds of thousands, where n corresponds to different companies, but also to characteristics of those companies). Application of popular computational numerical tools from econometrics and machine learning to such “small T, large n” data generally aims at finding increasingly-elaborate models with many parameters that have to be tuned to few high-dimensional observations available. This can lead to a problem known as “overfitting”, i.e. the good quality of fit on the training data is combined with the poor predictive performance of the tuned models. Another limitation is imposed by the computational cost of the common numerical tools, increasing polynomially with the data dimension n. In this research proposal, we will develop numerical tools based on combining the recently-introduced Scalable Probabilistic Approximation (SPA) methods for adaptive data discretization with the Maximum Entropy principle from physics and information theory. Maximum Entropy principle (MaxEnt) aims at finding as simple as possible (but not simpler than necessary) models fitting the data, being least biased in terms of the underlying assumptions and minimal in terms of the total number of tunable parameters. We will develop a numerical discretization-driven MaxEnt framework for time series analysis and inference of latent cross-sectional interdependencies between different financial time series (like asset returns and credit ratings), allowing for computational uncertainty quantification and risk predictions based on time series from Economics and Finance.