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The detrended cross-correlation coefficient $\rho_{\rm DCCA}$ has recentlybeen proposed to quantify the strength of cross-correlations on differenttemporal scales in bivariate, non-stationary time series. It is based on thedetrended cross-correlation and detrended fluctuation analyses (DCCA and DFA,respectively) and can be viewed as an analogue of the Pearson coefficient inthe case of the fluctuation analysis. The coefficient $\rho_{\rm DCCA}$ workswell in many practical situations but by construction its applicability islimited to detection of whether two signals are generally cross-correlated,without possibility to obtain information on the amplitude of fluctuations thatare responsible for those cross-correlations. In order to introduce somerelated flexibility, here we propose an extension of $\rho_{\rm DCCA}$ thatexploits the multifractal versions of DFA and DCCA: MFDFA and MFCCA,respectively. The resulting new coefficient $\rho_q$ not only is able toquantify the strength of correlations, but also it allows one to identify therange of detrended fluctuation amplitudes that are correlated in two signalsunder study. We show how the coefficient $\rho_q$ works in practical situationsby applying it to stochastic time series representing processes with longmemory: autoregressive and multiplicative ones. Such processes are often usedto model signals recorded from complex systems and complex physical phenomenalike turbulence, so we are convinced that this new measure can successfully beapplied in time series analysis. In particular, we present an example of suchapplication to highly complex empirical data from financial markets. Thepresent formulation can straightforwardly be extended to multivariate data interms of the $q$-dependent counterpart of the correlation matrices and then tothe network representation.

Detrended fluctuation analysis made flexible to detect range of cross-correlated fluctuations