It was shown for variables across different complex systems that their fluctuation functions calculated with detrending methods of scaling analysis are rarely, as in theory, ideal linear functions on log-log graphs of scale dependence. Instead, they frequently exhibit existence of transient crossovers in behavior, signs of trends that arise as effects of periodic or aperiodic cycles (Hu et al., 2001). The use of global and local wavelet transform spectral analysis (WTS) and their detrended fluctuation analysis (DFA) variants provides a possibility to detect these cyclic trends and to investigate their timing, nature and effects on the analyzed time series.
We recently developed a technique to cluster or differentiate records from an arbitrary complex system dataset based on the presence and influence of these cycles in data (Stratimirović et al., 2018), which we dubbed the Hurst Space Analysis (HSA). We defined a space of p-vectors h^{ts} that represent record ts in the dataset, which we called the Hurts space. Vectors h^{ts} are populated by scaling exponents \alpha calculated on subsets of time scale windows of time series ts that bound cyclic peaks in WTS, by way of use of time dependent detrended moving average analysis (tdDMA; Carbone, Castelli and Stanley, 2004). The length p of h^{ts} depends on the number of WTS peaks in that complex system. This number is, as was shown across complex systems, universal. In order to be able to quantify any time series ts with a single number, we projected its relative Hurst space unit vectors s^{ts}=\frac{h^{ts}-m}{\sqrt{\sum_{i=1}^{n}\left(h_i^{ts}-m_i\right)^2}} (with m_i=\frac{1}{n}\sum_{ts=1}^{n}h_i^{ts}) onto a unit vector e of an assigned preferred direction in the Hurst space. The definition of the ’preferred’ direction depends on the characteristic behavior one wants to investigate with HSA - projection of unit vectors s^{ts} of any record ts with a ’preferred’ behavior onto the unit vector e will then always be positive.
The HSA procedure can serve to examine and differentiate records within datasets of randomly selected time series of any complex variable. We used HSA to differentiate complex time series of stock market data, based on the preferred characteristic of marked development, and to cluster datasets of observed temperature records from land stations from different climatically and topologically homogeneous regions, based on the ‘belonging to a continent’ preference.
References:
Hu, K. et al. (2001) ‘Effect of trends on detrended fluctuation analysis’, Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics, 64(1), p. 19. doi: 10.1103/PhysRevE.64.011114.
Stratimirović, Dj. et al. (2018) ‘Analysis of cyclical behavior in time series of stock market returns’, Communications in Nonlinear Science and Numerical Simulation, 54. doi: 10.1016/j.cnsns.2017.05.009.
Carbone, A., Castelli, G. and Stanley, H. E. (2004) ‘Time-dependent Hurst exponent in financial time series’, in Physica A: Statistical Mechanics and its Applications. doi: 10.1016/j.physa.2004.06.130.