## 2 Materials and Methods
A special case of \(F_{q}^{\rm XY}\) is \(\text{X}=\text{Y}\), when the detrended cross-correlations become the detrended autocorrelations. In this case, we can omit both the sign term and the modulus in Equation (3), as the detrended variance \(f_{\rm X}^{2}\) is always positive. The MFCCA then reduces to the standard MFDFA with univariate fluctuation functions \(F_{q}^{\rm XX}(s)\) and \(F_{q}^{\rm YY}(s)\). The detrended cross-correlation function plays a role of a mean covariance, while \(F_{q}^{\rm XX}\) and \(F_{q}^{\rm YY}\) play a role of mean variances. If the univariate fluctuation functions are power-law dependent on \(s\), such that \(F_{q}^{\rm XX}\sim s^{h_{\rm XY}(q)}\) and \(F_{q}^{\rm YY}\sim s^{h_{\rm Y}(q)}\), the exponents \(h_{\rm X}(q)\) and \(h_{\rm Y}(q)\) are the generalised Hurst exponents, which for \(q=2\) reduce to the standard definition of the Hurst exponent \(H\). Having calculated all fluctuation functions, we can then introduce the \(q\)-dependent detrended cross-correlation coefficient \(\rho_{q}(s)\)[42], defined as  
\[\rho_{q}(s)=\frac{F_{q}^{\rm XY}(s)}{\sqrt{F_{q}^{\rm XX}(s)F_{q}^{\rm YY}(s )}}. \tag{5}\]  
Formula (5) was proposed in such a form in order to resemble the formula for the Pearson cross-correlation coefficient if \(q=2\). The coefficient \(\rho_{q}(s)\) can be considered, then, as a counterpart of the Pearson coefficient for non-stationary signals. Both coefficients assume values in the range [\(-1\),\(1\)], with \(\rho_{q}(s)=1\) for perfectly correlated time series, \(\rho_{q}(s)=0\) for independent time series, and \(\rho_{q}(s)=-1\) for perfectly anticorrelated time series. It should be noted that, in order to calculate \(\rho_{q}(s)\) the time series do not have to be fractal [42]. If compared with the Pearson coefficient, the coefficient \(\rho_{q}\) offers a few main advantages. The first one is its flexibility of the trend removal. There is no a priori best polynomial to use in Equation (1). The order \(m\) of this polynomial has to be optimised by considering the stability of the final results if we vary \(m\). The lowest order that gives stable results is preferred. Typically, the optimal value of \(m\) is larger than one, so the removed trends can be nonlinear, which is important, especially for the large-scale \(s\). This contrasts with a more standard approach to detrending of the financial data, where the linear trends are considered (e.g., by transforming the original time series to its increments). In general, it is also possible to use \(m\) that is selected individually for each segment, but we shall not consider such a detrending variant here. Another advantage of \(\rho_{q}\) over the Pearson coefficient is that it is not sensitive to the linear correlations only--we can look at its values for different \(q\)s and catch the nonlinear correlations as well. Moreover, with \(\rho_{q}\), we can have some insight into the amplitude of the fluctuations that carry the correlations (bytuning \(q\), we amplify the segments of specific variance/covariance). The fourth advantage is that \(\rho_{q}\) is inherently multiscale oriented. In order to achieve a comparable feature by using the Pearson coefficient, one needs to consider the same observable recorded with different sampling frequencies, while for \(\rho_{q}\), it is a built-in property. The fifth advantage is that the DFA procedure and its multifractal generalisations are able to detect the fractal and multifractal scaling in the data fluctuations, something that is beyond the scope of the Pearson-coefficient-based correlation measures. Detection of the multifractality of data allows one to gain some insight into the nature of the processes that govern the observed time series evolution. It can be of importance if one's goal is to model the empirical data in order to make prediction, for example.