Kun Hu1, Plamen Ch. Ivanov1, 2, Zhi Chen1, Pedro Carpena3, and H. Eugene Stanley1
1 Center for Polymer Studies and Department of Physics,
Boston University, Boston, Massachusetts 02215
2 Harvard Medical School, Beth Israel Deaconess Medical Center, Boston, Massachusetts 02215
3 Department of de Física Applicada II, Universidad de Málaga E-29071, Málaga, Spain
This article originally appeared in Physical Review E, vol. 64, 011114, 2001 (©2001 The American Physical Society, http://link.aps.org/abstract/PRE/v64/e011114). Please cite this publication when referencing this material.
Detrended fluctuation analysis (DFA) is a scaling analysis method used to estimate long-range power-law correlation exponents in noisy signals. Many noisy signals in real systems display trends, so that the scaling results obtained from the DFA method become difficult to analyze. We systematically study the effects of three types of trends -- linear, periodic, and power-law trends, and offer examples where these trends are likely to occur in real data. We compare the difference between the scaling results for artificially generated correlated noise and correlated noise with a trend, and study how trends lead to the appearance of crossovers in the scaling behavior. We find that crossovers result from the competition between the scaling of the noise and the "apparent" scaling of the trend. We study how the characteristics of these crossovers depend on (i) the slope of the linear trend; (ii) the amplitude and period of the periodic trend; (iii) the amplitude and power of the power-law trend, and (iv) the length as well as the correlation properties of the noise. Surprisingly, we find that the crossovers in the scaling of noisy signals with trends also follow scaling laws--i.e., long-range power-law dependence of the position of the crossover on the parameters of the trends. We show that the DFA result of noise with a trend can be exactly determined by the superposition of the separate results of the DFA on the noise and on the trend, assuming that the noise and the trend are not correlated. If this superposition rule is not followed, this is an indication that the noise and the superposed trend are not independent, so that removing the trend could lead to changes in the correlation properties of the noise. In addition, we show how to use DFA appropriately to minimize the effects of trends, how to recognize if a crossover indicates indeed a transition from one type to a different type of underlying correlation, or if the crossover is due to a trend without any transition in the dynamical properties of the noise.
Address for correspondence:
Plamen Ch. Ivanov, Ph.D.
Room 247, Dept. of Physics
590 Commonwealth Avenue
Boston, MA 02215, USA