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, ρ_{q} [J. Kwapień, P. Oświęcimka, and S. Drożdż, Phys. Rev. E 92, 052815 (2015)PLEEE81539-375510.1103/PhysRevE.92.052815], we generalize the concept of the minimum spanning tree (MST) by introducing a family of q-dependent minimum spanning trees (qMSTs) that are selective to cross-correlations between different fluctuation amplitudes and different time scales of multivariate data. They inherit this ability directly from the coefficients ρ_{q}, which are processed here to construct a distance matrix being the input to the MST-constructing Kruskal's algorithm. The conventional MST with detrending corresponds in this context to q=2. In order to illustrate their performance, we apply the qMSTs to sample empirical data from the American stock market and discuss the results. We show that the qMST graphs can complement ρ_{q} in disentangling "hidden" correlations that cannot be observed in the MST graphs based on ρ_{DCCA}, and therefore, they can be useful in many areas where the multivariate cross-correlations are of interest. As an example, we apply this method to empirical data from the stock market and show that by constructing the qMSTs for a spectrum of q values we obtain more information about the correlation structure of the data than by using q=2 only. More specifically, 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≠2 do we become able to distinguish between these sets. We also show that a family of qMSTs for a range of q expresses 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. As regards the correlation structure of the stock market, 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.