Detrended cross-correlations and their random matrix limit: an example from the cryptocurrency market

TL;DR

This work develops a scale- and fluctuation-dependent framework for detrended cross-correlations using the multifractal coefficient $\rho_r$ to study interdependencies in nonstationary, heavy-tailed systems. By analyzing the eigenvalue spectra of $\boldsymbol{\rho}_r(s)$ and comparing them to random-matrix baselines (Gaussian and $q$-Gaussian), the authors show how detrending, tail heaviness, and the fluctuation-order parameter alter the spectral structure. Applied to 1-minute returns of 140 cryptocurrencies, the approach reveals robust market-wide and sectoral modes, with a market factor dominating the spectrum that, when removed, leaves a bulk consistent with the detrended random limit and highlights genuine outliers. This provides a refined spectral baseline for distinguishing true interdependencies from noise in complex, nonstationary financial systems and beyond.

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.

Figures (10)

• Figure S1: (Main) Evolution of the cumulative log-returns $\hat{R}_i(t_k)$ of the 140 cryptocurrencies over the whole time period from Jan 1, 2021 to Sep 30, 2024. The bulk of the cryptocurrencies is shown in the background (grey lines), while BTC and ETH are distinguished by red and blue lines, respectively. Four specific periods are distinguished by narrow vertical rectangles (a)-(d). (Inset) Evolution of the same data during a selected shorter period (b): Aug 15, 2023 to Aug 18, 2023.
• Figure S2: Complementary cumulative distribution function (CCDF) of the log-returns $R_i(t_k)$ for all 140 cryptocurrencies together with $q$Gaussian ($q=3/2$) and power law ($\gamma=3$) CCDFs.
• Figure S3: Time evolution of the largest eigenvalue $\lambda_1$ of the matrix $\boldsymbol{\rho}_r(s)$ with $r=2$ (red) and $r=4$ (blue) for $s=10$ (top), $s=60$ (middle), and $s=360$ (bottom). A rolling window of length 7 days, shifted by 1 day, was applied. Four specific periods are marked by vertical dashed lines (a)-(d).
• Figure S4: Probability density function (PDF) of the off-diagonal elements of the matrix $\boldsymbol{\rho}_r(s)$ obtained from random uncorrelated time series with $q$Gaussian distribution defined by $q=1$ (i.e., a Gaussian distribution, top), $q=3/2$ (middle), and $q=2$ (bottom). Results for a few even values of the index $r$ in a range $2 \leqslant r \leqslant 10$ and two scales $s=10$ (left) and $s=360$ (right) are shown. Each PDF was created from 100 independent realizations of the random data set. The dashed line represents a Gaussian PDF corresponding to random Wishart matrices.
• Figure S5: Probability density function (PDF) of the off-diagonal elements of the matrix $\boldsymbol{\rho}_r(s)$ obtained from random uncorrelated time series with $q$Gaussian distribution defined by $q=1$ (i.e., a Gaussian distribution, top), $q=3/2$ (middle), and $q=2$ (bottom). Results for different scales $10 \leqslant s \leqslant 1000$ and for two even values of the index $r=2$ (left) and $r=4$ (right) are shown. Each PDF was created from 100 independent realizations of the random data set. The dashed line represents a Gaussian PDF corresponding to random Wishart matrices.