[Paper Review] Topological recognition of critical transitions in time series of cryptocurrencies

Marian Gidea, Daniel Goldsmith+3 more|arXiv (Cornell University)|Sep 3, 2018

Topological and Geometric Data AnalysisComputer Science56 references62 citations

TL;DR

The paper presents a topological data analysis (TDA) pipeline combined with k-means clustering to detect early warning signals of critical transitions in cryptocurrency time series, demonstrated on four major coins before the 2017-2018 crash.

ABSTRACT

We analyze the time series of four major cryptocurrencies (Bitcoin, Ethereum, Litecoin, and Ripple) before the digital market crash at the end of 2017 - beginning 2018. We introduce a methodology that combines topological data analysis with a machine learning technique -- $k$-means clustering -- in order to automatically recognize the emerging chaotic regime in a complex system approaching a critical transition. We first test our methodology on the complex system dynamics of a Lorenz-type attractor, and then we apply it to the four major cryptocurrencies. We find early warning signals for critical transitions in the cryptocurrency markets, even though the relevant time series exhibit a highly erratic behavior.

Motivation & Objective

• Investigate whether TDA can identify approaching critical transitions in noisy, non-stationary cryptocurrency time series.
• Develop a pipeline that transforms time-series data into topological summaries via delay-coordinate embedding and persistence landscapes.
• Use unsupervised learning to automatically recognize topologically distinct regimes preceding crashes.
• Validate the approach on chaotic reference systems and real cryptocurrency data from 2016–2018.

Proposed method

• Embed time series using time-delay coordinates to reconstruct phase space.
• Apply sliding windows to generate time-varying point clouds from the embedded data.
• Construct Vietoris-Rips filtrations and compute persistent homology, focusing on 1-dimensional features.
• Convert persistence diagrams into persistence landscapes and measure their L1 norms as a topological feature time series.
• Integrate L1-norm time series with log-prices and log-returns using k-means clustering to identify topologically distinct regimes.
• Use short sliding windows to detect topology changes and relate them to potential transitions.

Experimental results

Research questions

• RQ1 Can TDA-derived features (persistence landscapes) provide early indicators of regime changes in financial time series?
• RQ2 Do k-means clusters on combined financial and topological features reveal topologically distinct regimes preceding cryptocurrency crashes?
• RQ3 Are the L1-norms of persistence landscapes robust enough to signal transitions in highly noisy, non-stationary data?
• RQ4 How does the methodology perform on both chaotic reference systems and real cryptocurrency data?
• RQ5 What is the relationship between topology changes and the observed log-returns during critical periods?

Key findings

• The pipeline detects significant changes in the L1-norms of persistence landscapes prior to critical transitions in simulated noisy chaotic systems.
• No-noise Lorenz-type system shows little pre-transition signal in the L1-norms, while the noisy system exhibits a sharp increase before bifurcation.
• In cryptocurrency data, the L1-norms tend to peak near crashes and show increased dynamics in their first difference, indicating potential early signals.
• k-means clustering applied to log-prices, log-returns, and L1-norms yields clusters corresponding to topologically distinct regimes preceding crashes.
• The method demonstrates that topological signals can be extracted from short windows and non-stationary data, offering a potential early-warning framework for financial markets.