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Topological Data Analysis of Financial Time Series: Landscapes of Crashes
Marian Gidea, Yuri Katz
TL;DR
The paper asks whether TDA can detect growing systemic risk and early warning signals in financial markets, where crashes are difficult to predict. It applies persistence homology to sliding-window point clouds from four market indices, representing loop persistence with persistence landscapes and their L^p-norms. These norms and their low-frequency spectral density rise before the dotcom crash and Lehman bankruptcy, supporting TDA as a novel econometric signal for imminent crashes.
Problem
The paper investigates whether TDA can detect growing systemic risk in financial markets, where catastrophic meltdowns are difficult to predict because the system is complex and non-stationary.
Method
The authors apply 1D persistence homology to sliding-window 4D point clouds of daily log-returns from four US stock indices, then track persistence-landscape L^p-norms.
Results
Low-frequency spectral density of persistence-landscape L^p-norms rises strongly for 250 trading days before the dotcom crash and Lehman bankruptcy.
Takeaways & Limitations
TDA offers a novel econometric method and a new category of early warning signals for imminent market crashes.
Takeaways & Limitations
The simulation models are not claimed to represent how financial crises occur in actual markets.
Abstract
from arXiv · showhide
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.
1. Introduction
The introduction motivates TDA as a way to detect growing systemic risk in financial markets, where crashes are difficult to predict because financial systems are complex and non-stationary. The paper applies persistence landscapes to multidimensional market data and reports rising signals before two major crashes.
- Motivation: TDA combines statistical, computational, and topological methods to identify shape-like structures in noisy multidimensional data through persistent homology.It studies holes across scales, including connected components and loops, retaining their persistence as a measure of significance.
- Motivation: Financial-crash prediction is difficult because markets are complex and non-stationary, motivating searches for early warning signals of systemic risk.Prior observations associate crashes with changes such as increasing variance and shifting behavior in market indices.
- Approach: The study analyzes daily log-returns from four US indices and converts sliding-window observations into 4D point clouds.The indices are S&P 500, DJIA, NASDAQ, and Russell 2000; each window advances by one day.
- Approach: For each point cloud, the authors compute 1D persistence landscapes of loops and track their L^p-norms over time.They use p = 1 and p = 2 to quantify temporal changes in the market’s topological state.
- Findings: The L^p-norms grow strongly before their primary crash-related peak, while low-frequency spectral density rises for 250 trading days before either crash.The reference events are the dotcom crash on 03/10/2000 and Lehman bankruptcy on 09/15/2008.
2. Background
The background introduces persistent homology, Rips filtrations, persistence diagrams, and persistence landscapes as tools for representing multiscale topological features. Persistence landscapes provide a Banach-space representation that is more suitable for statistical analysis than persistence diagrams alone.
- TDA foundations: TDA extracts robust topological information from noisy data by examining features across multiple scales and ranking them by persistence.This is useful when the underlying stochastic process is unknown.
- Rips filtration: A Rips complex represents a point cloud at resolution ε, including a simplex when every pair of its vertices is separated by less than ε.As ε increases, the complexes form a filtration, allowing homology to track features across resolutions.
- Homology: The generators of 0-dimensional homology represent connected components, while 1-dimensional homology generators represent independent loops.The paper focuses on 1-dimensional homology and gives a four-point loop as an example.
- Persistent homology: Persistent homology assigns each topological feature birth and death values, with their difference measuring persistence across the filtration.Longer-lived features are treated as more significant, while shorter-lived features may be noisy; no artificial signal-noise cutoff is required.
- Persistence diagrams: Persistence diagrams encode each k-dimensional homology class as a point whose coordinates are its birth and death values.The diagram also contains infinitely many diagonal points representing trivial generators.
- Persistence landscapes: Persistence landscapes embed persistence diagrams as sequences of functions in L^p(N × R), enabling vector-space operations and statistical treatment.The paper uses L1 and L2 norms, although averages of landscapes need not correspond to persistence diagrams.
3. Description of the method and testing on synthetic time series
The paper applies persistence landscapes and their L1- and L2-norms to sliding-window point clouds, testing whether they track structural changes in synthetic multidimensional time series. The experiments show norm growth near transitions to chaos and with increasing noise variance, while the authors restrict the simulations to methodological tests rather than representative crisis models.
- 3.1. Method: The method converts time-ordered multidimensional observations into sliding-window point clouds, then computes Rips-filtration persistence diagrams, persistence landscapes, and Lp-norms.For financial data, the point clouds contain w observations in Rd, with d equal to the number of time series.
- 3.1. Method: Compared with time-delay embedding, the approach represents separate stochastic time series directly in a low-dimensional Rd space rather than seeking an attractor embedding.Its main tuning parameter is the sliding-window size w.
- 3.2. Chaotic time series with noise: The synthetic experiments indicate that persistence-landscape norms detect transitions from regular to chaotic dynamics as slowly evolving parameters change.The authors relate this signal to significant changes in the topology of the attractor.
- 3.4. White noises with Gamma-distributed inverse variance: With Gamma-distributed inverse variance, decreasing α produces a sharp L1-norm increase corresponding to growing variance, while the L2 behavior is nearly indistinguishable.The experiment uses 100 point clouds and averages results over 100 realizations.
4. Empirical analysis of financial data
The study applies sliding-window persistence landscapes to four-index financial return data, finding stronger topological signals and rising low-frequency variability before the 2000 and 2008 crises.
- Data and point clouds: 7301 trading days of daily log-returns from four US indices are organized into 4D point clouds using 50- and 100-day windows.The indices are S&P 500, DJIA, NASDAQ, and Russel 2000; windows advance one trading day.
- Landscape norms: The L1- and L2-norms summarize persistence landscapes for each rolling window and form daily time series for tracking changing topological structure.Figure 9 displays normalized L1 and L2 norm series.
- Topological features: The 2D return plots illustrate how loops arise in point clouds, while persistence diagrams and landscapes distinguish connected components from loops.Figure 8 uses black dots for connected components and red triangles for loops.
- Crash-period behavior: Strong L1 fore-shocks precede the primary peak around both crashes, while changing the window from 50 to 100 days flattens spikes and shifts the maximum later.The L2 behavior closely resembles the L1 pattern.
- Statistical indicators: 0.89 and 1.00 are the Kendall-tau rank correlations for rising low-frequency spectral density during the 250 trading days before the dotcom crash and Lehman bankruptcy, respectively.Variance also grows substantially, whereas the lag-1 ACF shows no trend; doubling the point-cloud size does not significantly change the picture.
5. Conclusions
The paper presents TDA as an econometric approach for tracking changing topological features in multidimensional financial data. Its results associate increased loop persistence and rising landscape variability with the 250 trading days preceding two major market crashes.
- Method: TDA uses one sliding window across multiple financial series and Lp-norms of persistence landscapes to quantify topological-feature stability.The approach differs from two-parameter time-delay embedding by using only the window size w.
- Findings: The empirical study finds increased persistence of loops as markets transition from an ordinary to a “heated” state.This behavior is observed in point clouds formed from four stock-market return series.
- Implications: The study suggests that TDA supplies a novel category of early warning signals for imminent market crashes and may extend beyond financial time-series analysis.This is presented as a proposed use of the approach rather than a universal guarantee of prediction.