  The detrended cross-correlation coefficient $\rho_{\rm DCCA}$ has recently
been proposed to quantify the strength of cross-correlations on different
temporal scales in bivariate, non-stationary time series. It is based on the
detrended cross-correlation and detrended fluctuation analyses (DCCA and DFA,
respectively) and can be viewed as an analogue of the Pearson coefficient in
the case of the fluctuation analysis. The coefficient $\rho_{\rm DCCA}$ works
well in many practical situations but by construction its applicability is
limited to detection of whether two signals are generally cross-correlated,
without possibility to obtain information on the amplitude of fluctuations that
are responsible for those cross-correlations. In order to introduce some
related flexibility, here we propose an extension of $\rho_{\rm DCCA}$ that
exploits the multifractal versions of DFA and DCCA: MFDFA and MFCCA,
respectively. The resulting new coefficient $\rho_q$ not only is able to
quantify the strength of correlations, but also it allows one to identify the
range of detrended fluctuation amplitudes that are correlated in two signals
under study. We show how the coefficient $\rho_q$ works in practical situations
by applying it to stochastic time series representing processes with long
memory: autoregressive and multiplicative ones. Such processes are often used
to model signals recorded from complex systems and complex physical phenomena
like turbulence, so we are convinced that this new measure can successfully be
applied in time series analysis. In particular, we present an example of such
application to highly complex empirical data from financial markets. The
present formulation can straightforwardly be extended to multivariate data in
terms of the $q$-dependent counterpart of the correlation matrices and then to
the network representation.
