Abstract
Based on a recently proposed -dependent detrended cross-correlation coefficient, [J. Kwapień, P. Oświęcimka, and S. Drożdż, Phys. Rev. E 92, 052815 (2015)], we generalize the concept of the minimum spanning tree (MST) by introducing a family of -dependent minimum spanning trees () 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 , 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 . In order to illustrate their performance, we apply the to sample empirical data from the American stock market and discuss the results. We show that the graphs can complement in disentangling “hidden” correlations that cannot be observed in the MST graphs based on , 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 for a spectrum of values we obtain more information about the correlation structure of the data than by using only. More specifically, we show that two sets of signals that differ from each other statistically can give comparable trees for , while only by using the trees for do we become able to distinguish between these sets. We also show that a family of for a range of 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 '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.
- Received 19 October 2016
- Revised 31 March 2017
DOI:https://doi.org/10.1103/PhysRevE.95.052313
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