Minimum spanning tree filtering of correlations for varying time scales and size of fluctuations

Based on a recently proposed $q$-dependent detrended cross-correlation coefficient $\rho_q$, we generalize the concept of minimum spanning tree (MST) by introducing a family of $q$-dependent minimum spanning trees ($q$MST) that are selective to cross-correlations between different fluctuation amplitudes and different time scales. They inherit this ability directly from the coefficients $\rho_q$ that are processed here to construct a distance matrix. Conventional MST with detrending corresponds in this context to $q=2$. We apply the $q$MSTs to sample empirical data from the stock market and discuss the results. We show that the $q$MST graphs can complement $\rho_q$ in disentangling correlations that cannot be observed by the MST graphs based on $\rho_{\rm DCCA}$ and, therefore, they can be useful in many areas where the multivariate cross-correlations are of interest. We apply our method to data from the stock market and obtain more information about correlation structure of the data than by using $q=2$ only. We show that two sets of signals that differ from each other statistically can give comparable trees for $q=2$, while only by using the trees for $q \ne 2$ we become able to distinguish between these sets. We also show that a family of $q$MSTs for a range of $q$ express the diversity of correlations in a manner resembling the multifractal analysis, where one computes a spectrum of the generalized fractal dimensions, the generalized Hurst exponents, or the multifractal singularity spectra: the more diverse the correlations are, the more variable the tree topology is for different $q$s. Our analysis exhibits that the stocks belonging to the same or similar industrial sectors are correlated via the fluctuations of moderate amplitudes, while the largest fluctuations often happen to synchronize in those stocks that do not necessarily belong to the same industry.