英文标题：
《Topological Data Analysis of Financial Time Series: Landscapes of
  Crashes》
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作者：
Marian Gidea and Yuri Katz
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最新提交年份：
2017
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英文摘要：
  We explore the evolution of daily returns of four major US stock market indices during the technology crash of 2000, and the financial crisis of 2007-2009. Our methodology is based on topological data analysis (TDA). We use persistence homology to detect and quantify topological patterns that appear in multidimensional time series. Using a sliding window, we extract time-dependent point cloud data sets, to which we associate a topological space. We detect transient loops that appear in this space, and we measure their persistence. This is encoded in real-valued functions referred to as a \'persistence landscapes\'. We quantify the temporal changes in persistence landscapes via their $L^p$-norms. We test this procedure on multidimensional time series generated by various non-linear and non-equilibrium models. We find that, in the vicinity of financial meltdowns, the $L^p$-norms exhibit strong growth prior to the primary peak, which ascends during a crash. Remarkably, the average spectral density at low frequencies of the time series of $L^p$-norms of the persistence landscapes demonstrates a strong rising trend for 250 trading days prior to either dotcom crash on 03/10/2000, or to the Lehman bankruptcy on 09/15/2008. Our study suggests that TDA provides a new type of econometric analysis, which goes beyond the standard statistical measures. The method can be used to detect early warning signals of imminent market crashes. We believe that this approach can be used beyond the analysis of financial time series presented here.
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中文摘要：
我们探讨了2000年技术崩溃和2007-2009年金融危机期间四大美国股市指数日收益率的演变。我们的方法基于拓扑数据分析（TDA）。我们使用持久性同源性来检测和量化多维时间序列中出现的拓扑模式。使用滑动窗口，我们提取与时间相关的点云数据集，并将其与拓扑空间相关联。我们检测出现在这个空间中的瞬态循环，并测量它们的持久性。这是在实值函数中编码的，称为“持久性景观”。我们通过其$L^p$标准量化了持久性景观的时间变化。我们在由各种非线性和非平衡模型生成的多维时间序列上测试了这个过程。我们发现，在金融危机附近，美元/便士的标准值在主峰之前表现出强劲的增长，主峰在崩盘期间上升。值得注意的是，在2000年10月3日网络崩溃或2008年9月15日雷曼兄弟破产之前的250个交易日内，持久性景观的美元/便士-标准时间序列低频平均频谱密度呈现出强劲的上升趋势。我们的研究表明，TDA提供了一种新的计量经济分析，它超越了标准的统计指标。该方法可用于检测即将发生的市场崩溃的预警信号。我们认为，这种方法可以用于本文所述的金融时间序列分析之外的其他领域。
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分类信息：

一级分类：Quantitative Finance        数量金融学
二级分类：Mathematical Finance        数学金融学
分类描述：Mathematical and analytical methods of finance, including stochastic, probabilistic and functional analysis, algebraic, geometric and other methods
金融的数学和分析方法，包括随机、概率和泛函分析、代数、几何和其他方法
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一级分类：Mathematics        数学
二级分类：Dynamical Systems        动力系统
分类描述：Dynamics of differential equations and flows, mechanics, classical few-body problems, iterations, complex dynamics, delayed differential equations
微分方程和流动的动力学，力学，经典的少体问题，迭代，复杂动力学，延迟微分方程
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一级分类：Physics        物理学
二级分类：Physics and Society        物理学与社会
分类描述：Structure, dynamics and collective behavior of societies and groups (human or otherwise). Quantitative analysis of social networks and other complex networks. Physics and engineering of infrastructure and systems of broad societal impact (e.g., energy grids, transportation networks).
社会和团体（人类或其他）的结构、动态和集体行为。社会网络和其他复杂网络的定量分析。具有广泛社会影响的基础设施和系统（如能源网、运输网络）的物理和工程。
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