Detrended cross-correlations and their random matrix limit: an example from the cryptocurrency market
Abstract
Correlations in complex systems are often obscured by nonstationarity, long-range memory, and heavy-tailed fluctuations, which limit the usefulness of traditional covariance-based analyses. To address these challenges, we construct scale and fluctuation-dependent correlation matrices using the multifractal detrended cross-correlation coefficient $ρ_r$ that selectively emphasizes fluctuations of different amplitudes. We examine the spectral properties of these detrended correlation matrices and compare them to the spectral properties of the matrices calculated in the same way from synthetic Gaussian and $q$Gaussian signals. Our results show that detrending, heavy tails, and the fluctuation-order parameter $r$ jointly produce spectra, which substantially depart from the random case even under absence of cross-correlations in time series. Applying this framework to one-minute returns of 140 major cryptocurrencies from 2021-2024 reveals robust collective modes, including a dominant market factor and several sectoral components whose strength depends on the analyzed scale and fluctuation order. After filtering out the market mode, the empirical eigenvalue bulk aligns closely with the limit of random detrended cross-correlations, enabling clear identification of structurally significant outliers. Overall, the study provides a refined spectral baseline for detrended cross-correlations and offers a promising tool for distinguishing genuine interdependencies from noise in complex, nonstationary, heavy-tailed systems.
Summary
This paper addresses the challenge of analyzing correlations in complex systems where non-stationarity, long-range memory, and heavy-tailed fluctuations obscure traditional covariance-based analyses. The authors introduce a framework based on multifractal detrended cross-correlation analysis (DCCA), using the coefficient ρ_r to construct scale and fluctuation-dependent correlation matrices. They then analyze the spectral properties of these matrices, comparing them against synthetic Gaussian and q-Gaussian signals to establish a baseline for distinguishing genuine interdependencies from noise. The methodology involves applying DCCA to one-minute returns of 140 major cryptocurrencies from 2021-2024. By examining the eigenvalue spectra of the resulting correlation matrices, the authors identify robust collective modes, including a dominant market factor and sectoral components. A key step is filtering out the market mode, which allows the empirical eigenvalue bulk to align with the limit of random detrended cross-correlations, thereby enabling clearer identification of structurally significant outliers. The research provides a refined spectral baseline for detrended cross-correlations and demonstrates its effectiveness in the cryptocurrency market, offering a promising tool for analyzing complex, non-stationary systems.
Key Insights
- •Detrending, heavy tails (specifically q-Gaussian distributions), and the fluctuation-order parameter *r* jointly produce eigenvalue spectra that deviate significantly from the standard Marchenko-Pastur distribution, even in the absence of cross-correlations in the time series.
- •Empirical analysis of cryptocurrency data reveals a dominant market factor and several sectoral components, whose strength varies depending on the analyzed scale (*s*) and fluctuation order (*r*).
- •Filtering out the market mode allows the empirical eigenvalue bulk to closely align with the random matrix limit of detrended cross-correlations, enabling the identification of structurally significant outliers.
- •The distributions of off-diagonal elements in the correlation matrices deviate significantly from a normal distribution, especially as the parameters *q*, *r*, and scale *s* increase, signaling a reduction in the effective rank of the matrix.
- •The largest eigenvalue (λ_1) of the detrended correlation matrices is often separated from the rest of the spectrum, with the gap increasing with the scale *s*, consistent with the collective movement of prices (market factor) and information transfer delays.
- •The choice of the fluctuation-order parameter *r* influences the identified correlations: *r* = 4 tends to highlight cross-correlations among large-amplitude price movements, while *r* = 2 captures correlations among more medium amplitude flucutations.
- •The study uses both Gaussian (q=1) and q-Gaussian (q=3/2, q=2) distributions to model the synthetic data, with q=3/2 providing a good fit to the empirical cryptocurrency returns data, aligning with the inverse-cubic power law.
Practical Implications
- •The framework can be used for risk management and portfolio optimization in cryptocurrency markets by identifying and filtering out the dominant market mode and sector-specific correlations.
- •Traders and analysts can use the refined spectral baseline to distinguish genuine interdependencies from noise, potentially improving trading strategies and asset allocation.
- •The methodology is applicable to other complex systems characterized by non-stationarity, long-range dependencies, and heavy-tailed fluctuations, such as climate science, neuroscience, and molecular biology.
- •Future research can focus on developing analytical approximations for the random matrix limits of detrended correlation spectra, which would formalize the empirical baselines established in this study.
- •The approach can be extended to incorporate other financial markets and asset classes, providing a more comprehensive understanding of global market dynamics and interdependencies.